Generated by All in One SEO v4.9.9, this is an llms.txt file, used by LLMs to index the site. # Open Middle® ## Sitemaps - [XML Sitemap](https://www.openmiddle.com/sitemap.xml): Contains all public & indexable URLs for this website. ## Posts - [Markup & Discount 2](https://www.openmiddle.com/markup-discount-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create the least expensive item after discount. Source: Robert Kaplinsky - [Mixed Number and Fraction Greater Than One](https://www.openmiddle.com/mixed-number-and-fraction-greater-than-one/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equality true. Source: Graeme Lachance - [A New Way to Use Open Middle: Interactive Problems](https://www.openmiddle.com/a-new-way-to-use-open-middle-interactive-problems/) - Learn how interactive Open Middle problems let students drag, try, and revise ideas directly on screen. - [Systems of Equations 2](https://www.openmiddle.com/systems-of-equations-2/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to create a system of equations with a solution in that’s as close to the origin as possible. Source: Robert Kaplinsky - [Write a Linear Function](https://www.openmiddle.com/write-a-linear-function/) - Directions: Using the digits 1 to 8 [You will use each number only once, except for one number that will be used twice in the same coordinate point. i.e.(1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7) or (8,8)], find three coordinate points that lie on the same line. Write the equation of the line represented by - [Using 1/2 as a Benchmark](https://www.openmiddle.com/using-1-2-as-a-benchmark/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make true statements. For the fraction less than 1/2, try to make the greatest number possible. For the fraction greater than 1/2, try to make the least number possible. Source: Alyson Eaglen - [Two-Step Equations 3](https://www.openmiddle.com/two-step-equations-3/) - Challenge students to use the digits 1-9 to find the values of x and y in this DOK 3 problem. Great for 7th and 8th grade students practicing expressions & equations. - [Trinomial Function Features 2](https://www.openmiddle.com/trinomial-function-features-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a function with the corresponding range and roots that are as close together as possible. The closest the two roots can be to each other that has been found so far is 2 from the - [The Triangle Inequality](https://www.openmiddle.com/the-triangle-inequality/) - Directions: Using the integers 1 to 10, at most one time each (though 7 and 9 can still be used), place an integer in each box to complete the scenarios below: Source: Shaun Errichiello - [The Largest Fraction That Is Less Than One Half](https://www.openmiddle.com/the-largest-fraction-that-is-less-than-one-half/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create the largest fraction possible that is less than 1/2 and has a single digit in both the numerator and denominator. Source: Dr. Brian Lack - [System of Equations, Special Case Infinitely Many Solutions](https://www.openmiddle.com/system-of-equations-special-case-infinitely-many-solutions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that there are infinitely many solutions to the system of equations. Source: Nanette Johnson - [Subtraction without Regrouping](https://www.openmiddle.com/subtraction-without-regrouping/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that you would not need to regroup when you subtract. Make sure your number is less than 63. Extension: Explain why you do NOT need to regroup using your number. Source: Chase Orton - [Subtracting Two-Digit Numbers (Middle School)](https://www.openmiddle.com/subtracting-two-digit-numbers-middle-school/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) difference. Note: This problem's difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both. Source: Robert Kaplinsky - [Subtracting Multi-Digit Numbers](https://www.openmiddle.com/subtracting-multi-digit-numbers/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Smarter Balance - [Subtracting Multi-Decimals](https://www.openmiddle.com/subtracting-multi-decimals/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the difference is as close to 50 as possible. NOTE: The digits used in the difference can be repeated. Source: Giselle Garica - [Subtracting Decimals to Make Them As Close to One as Possible](https://www.openmiddle.com/subtracting-decimals-to-make-them-as-close-to-one-as-possible/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to get the difference that is as close to 1 as possible. Source: Giselle Garcia - [Solving Quadratic Equations](https://www.openmiddle.com/solving-quadratic-equations/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a problem that has the solutions x = 4 and x = -1/2. Source: Amy Herzog - [Solving Linear Inequalities](https://www.openmiddle.com/solving-linear-inequalities/) - Directions: Using digits -9 to 9, at most one time each, place a digit in each box to create an inequality that has a solution of x > 2. Source: Sarah Furman - [Smallest Possible LCM](https://www.openmiddle.com/smallest-possible-lcm/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the smallest possible least common multiple. Source: Howie Hua - [Similar Shapes](https://www.openmiddle.com/similar-shapes/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that one rectangle is a scaled drawing of the other. Source: Gian Cavaliere - [Scientific Notation 2](https://www.openmiddle.com/scientific-notation-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a product that equals 800,000,000. Source: Robert Kaplinsky - [Rational Exponents 4](https://www.openmiddle.com/rational-exponents-4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the inequality true. Source: Bryan Anderson - [Rational Exponents 3](https://www.openmiddle.com/rational-exponents-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Bryan Anderson - [Rational Exponents 2](https://www.openmiddle.com/rational-exponents-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Bryan Anderson - [Rational and Irrational Roots 7](https://www.openmiddle.com/rational-and-irrational-roots-7/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create expressions that produce one rational root and three irrational roots. Source: Norma Gordon - [Rational and Irrational Roots 6](https://www.openmiddle.com/rational-and-irrational-roots-6/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create expressions that produce two rational roots and two irrational roots. Source: Norma Gordon - [Rational and Irrational Roots 4](https://www.openmiddle.com/rational-and-irrational-roots-4/) - Directions: Using the digits 2 to 9, at most one time each, place a digit in each box to create expressions that produce two rational roots and two irrational roots. Source: Norma Gordon - [Rational and Irrational Roots 3](https://www.openmiddle.com/rational-and-irrational-roots-3/) - Directions: Using the digits 2 to 9, at most one time each, place a digit in each box to create expressions that produce one rational root and three irrational roost. Source: Norma Gordon - [Rational and Irrational Roots 2](https://www.openmiddle.com/rational-and-irrational-roots-2/) - Directions: Using the digits 2 to 9, at most one time each, place a digit in each box to create expressions that produce three rational roots and one irrational root. Source: Norma Gordon - [Rational and Irrational Roots](https://www.openmiddle.com/rational-and-irrational-roots/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create the following number types. Source: Shaun Errichiello - [Rational and Irrational Numbers 2](https://www.openmiddle.com/rational-and-irrational-numbers-part-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create the following number types: Source: Bryan Anderson - [Properties of Exponents 3](https://www.openmiddle.com/properties-of-exponents-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Robert Kaplinsky - [Properties Of Exponents 2](https://www.openmiddle.com/properties-of-exponents-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make a product that is as close to zero as possible without being exactly zero. Source: Robert Kaplinsky - [Properties Of Exponents 1](https://www.openmiddle.com/properties-of-exponents-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box twice to make a positive product and a negative product. You may reuse all the integers each product. Source: Robert Kaplinsky - [Product with Scientific Notation](https://www.openmiddle.com/product-with-scientific-notation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Luke Cole - [Polar and Cartesian form of complex numbers](https://www.openmiddle.com/polar-and-cartesian-form-of-complex-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the result is as close as possible to the number i. Source: David K Butler - [Perpendicular Lines and Slope](https://www.openmiddle.com/perpendicular-lines-and-slope/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the lines through each pair of points are perpendicular. Source: Nanette Johnson - [Perimeter & Circumference](https://www.openmiddle.com/perimeter-circumference/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to create the largest and smallest combined perimeter/circumference for the rectangle and circle. Source: Christin Smith - [Percent of a Quantity 2](https://www.openmiddle.com/percent-of-a-quantity-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true and have the greatest possible whole without rounding. Source: Robert Kaplinsky - [Ordering Fractions Greater Than One](https://www.openmiddle.com/ordering-fractions-greater-than-one/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the inequality true. Try to find solutions where the fractions are in their lowest terms. Source: Charlotte Hawthorne - [Operations with Time](https://www.openmiddle.com/operations-with-time/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a time that is 4:37 pm. Source: Robert Kaplinsky - [Open Number Line - Integers](https://www.openmiddle.com/open-number-line-integers/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make the number line true. Note: The number line is not necessarily drawn to scale. Source: Amy Bloom - [Open Number Line](https://www.openmiddle.com/open-number-line/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box on the number line to make the number line true. NOTE: number line not drawn to scale Source: Amanda Dey - [One Solution, No Solutions, Infinite Solutions](https://www.openmiddle.com/one-solution-no-solutions-infinite-solutions/) - Directions: Using integers, at most one time each, place an integer in each box to create the following types of Linear Equations Source: Bryan Anderson - [Multiplying Two Fractions to Get a Mixed Number](https://www.openmiddle.com/multiplying-two-fractions-to-get-a-mixed-number/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Joseph Nguyen - [Multiplying Mixed Numbers by Whole Numbers](https://www.openmiddle.com/multiplying-mixed-numbers-by-whole-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) product. Source: Robert Kaplinsky - [Multiplying Complex Numbers 1](https://www.openmiddle.com/multiplying-complex-numbers-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box: once to make a positive real number product and once to make a negative real number product. You may reuse all the integers for each product. Source: Robert Kaplinsky in Open Middle Math - [Matrix Multiplication](https://www.openmiddle.com/matrix-multiplication/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each blank to create the smallest possible value for a. Source: Bryan Anderson - [Marble Madness 1](https://www.openmiddle.com/marble-madness-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each blank to make the statements true. Barbara has ___ ___ ___ marbles. She gives her sister ___ ___ ___ marbles. She now has ___ ___ ___ marbles left. Source: Chase Orton - [Logs 2](https://www.openmiddle.com/logs-2/) - Directions: Using the integers 1 to 9, at most one time each, place an integer in each box to create a log that satisfies the follow constraints. Source: Bryan Anderson - [Logarithms with Fractions](https://www.openmiddle.com/logarithms-with-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a problem that has the smallest possible positive answer. Source: Noel Chang - [Logarithmic Function Features 2](https://www.openmiddle.com/logarithmic-function-features-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a logarithmic function with the greatest possible y-intercept. Source: Robert Kaplinsky - [Logarithmic Function Features 1](https://www.openmiddle.com/logarithmic-function-features-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box and create a logarithmic function with its corresponding y-intercept. Source: Robert Kaplinsky - [Logarithm Laws 2](https://www.openmiddle.com/logarithm-laws-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the values of each expression increases from least to greatest. Each number may only be used once. Source: John Rowe - [Linear Equation with One Solution](https://www.openmiddle.com/linear-equation-with-one-solution/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a linear equation with one solution. Source: Bryan Anderson - [Line Tangent to a Parabola](https://www.openmiddle.com/line-tangent-to-a-parabola/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the line is tangent to the parabola. Source: Erin Stenger - [Limits](https://www.openmiddle.com/limits/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. Source: Julia Anker - [Least Common Multiple](https://www.openmiddle.com/least-common-multiple/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. Source: Wendy Taylor - [Laws of Exponents](https://www.openmiddle.com/laws-of-exponents/) - Directions: Using the integers 1 to 20, at most one time each, place an integer in each box to create equivalent expressions. Source: Shaun Errichiello - [Graphing Linear Equations 2](https://www.openmiddle.com/graphing-linear-equations-2/) - Directions: Using the integers -9 to 9, at most one time each, place a digit in each box to make a linear equation which goes through (5, 4) and has a slope that’s as close to 0 as possible without being horizontal. Source: Robert Kaplinsky - [Graphing Linear Equations 1](https://www.openmiddle.com/graphing-linear-equations-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make two linear equations which go through (5, 4): one with a negative slope and one with a positive slope. You may reuse all the integers for each equation. Source: Robert Kaplinsky - [Exponential Function Features 2](https://www.openmiddle.com/exponential-function-features-2/) - Directions: Using the integers -9 to 9, at most two times each, place an integer in each box to create an exponential growth function with the greatest possible y-intercept. Source: Robert Kaplinsky - [Exponent Exploration](https://www.openmiddle.com/exponent-exploration/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Robert Kaplinsky - [Exponent (Maximum Value)](https://www.openmiddle.com/exponent-maximum-value/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a result that has the greatest value possible. Source: Robert Kaplinsky - [Exploring Equations](https://www.openmiddle.com/exploring-equations/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the the greatest possible value for x. Source: Chase Orton and Mark Goldstein - [Evaluating a Decimal Expression](https://www.openmiddle.com/evaluating-a-decimal-expression/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the value of this expression is as large as possible. Challenge: Try to make this expression as close to 30 as possible (for an added challenge, try to make this expression as close to 30 - [Equations of Perpendicular Lines](https://www.openmiddle.com/equations-of-perpendicular-lines-2/) - Directions: Using the integers -9 to 9 (excluding 0) at most one time each, place an integer in each box to create two distinct perpendicular lines. Source: Louise Pepper with answers from the students of Kings College Alicante, Spain - [Equations of Circles 2](https://www.openmiddle.com/equations-of-circles-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a circle and a point on the circle with the point being as close to the origin as possible. Source: Robert Kaplinsky - [Equation of the Biggest Circle](https://www.openmiddle.com/equation-of-the-biggest-circle/) - Directions: Using any integers, place an integer in each box so that: The equation’s graph is a circle. The circle has the biggest area The circle is completely inside the first quadrant The circle’s radius is a whole number 1 through 9. Source: Nanette Johnson - [Equality 2](https://www.openmiddle.com/equality-2-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation have the greatest possible value. Source: Robert Kaplinsky - [Equality](https://www.openmiddle.com/equality/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Graham Fletcher - [Dividing Two-Digit Numbers (Elementary)](https://www.openmiddle.com/dividing-two-digit-numbers-elementary/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) quotient. Source: Robert Kaplinsky - [Dividing Monomials](https://www.openmiddle.com/dividing-monomials/) - Directions: Using the integers 1 to 10, at most one time each, place an integer in each box to make the equation true. Source: Richard Hung - [Dividing by 1-digit numbers](https://www.openmiddle.com/dividing-by-1-digit-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create the smallest (or largest) whole number quotient. Source: Ellen Metzger - [Distributive Property 4](https://www.openmiddle.com/distributive-property-4/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make a true equation. Source: Linda Cochran - [Derivative Power Rule](https://www.openmiddle.com/derivative-power-rule/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to fill in the boxes to make the equation true. Source: Melissa Flynn - [Create A Pattern](https://www.openmiddle.com/create-a-pattern/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make a pattern that changes by the same amount each time. Source: Brian Errey - [Area of Three Triangles](https://www.openmiddle.com/area-of-three-triangles/) - Directions: Use the integers 2 to 10, at most one time each, as lengths of individual sides to form three triangles. What is the smallest total area of the three triangles you can create? What is the largest? Source: Dan Wulf - [Approximating Irrationals 2](https://www.openmiddle.com/approximating-irrationals-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the greatest possible irrational number. Source: Robert Kaplinsky - [Approximating Irrationals 1](https://www.openmiddle.com/approximating-irrationals-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. Source: Robert Kaplinsky - [Adding Vectors](https://www.openmiddle.com/adding-vectors/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a true statement. Source: Bryan Anderson - [Adding Multiple Decimals](https://www.openmiddle.com/addting-multi-decimals/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the sum is as close to 10 as possible. Source: Giselle Garcia - [Add Some, Subtract Some](https://www.openmiddle.com/add-some-subtract-some/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to complete the equation. What is the greatest result you can make? What is the least result you can make? Source: Molly Rawding - [Absolute Deviation](https://www.openmiddle.com/absolute-deviation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a set of data with the largest possible absolute deviation. Source: Mark Alvaro - [Factoring Quadratics](https://www.openmiddle.com/factoring-quadratics/) - Directions: Fill in the blanks with integers so that the quadratic expression is factorable. Source: Nanette Johnson - [Exponential Function Features 1](https://www.openmiddle.com/exponential-function-features-1/) - Directions: Using the integers -9 to 9, at most two times each, place an integer in each box to create an exponential growth function with its y-intercept. Source: Robert Kaplinsky - [Fraction Quotient Closest to 4/11](https://www.openmiddle.com/fraction-quotient-closest-to-411/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make two fractions that have a quotient that is as close to 4/11 as possible. Source: Robert Kaplinsky - [Linear and Quadratic System](https://www.openmiddle.com/linear-and-quadratic-system/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a system of equations as well as two solutions that make the system true. Source: Cody Pritchard - [Supplementary Angles or Adding Angles](https://www.openmiddle.com/supplementary-angles-or-adding-angles/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the angles add up to be a straight angle. *Note: drawing not to scale. Source: Catherine Castillo - [Three Points on a Line](https://www.openmiddle.com/three-points-on-a-line/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to find three points on the same line and the slope. Source: Maggie Lee McHugh - [Maximize Slope](https://www.openmiddle.com/maximize-slope/) - Directions: Given the point (3,5), use the digits 1 to 9, at most one time each, to find a point (__, __) that maximizes the slope of the line that passes through the two points. The slope cannot be undefined. Source: Andrew Constantinescu - [Greatest Difference of Two Decimal Numbers](https://www.openmiddle.com/greatest-difference-of-two-decimal-numbers/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two numbers that both round to 5 and have the greatest (or least) possible difference with 5. Source: Mike Wiernicki - [Fraction Exponents](https://www.openmiddle.com/fraction-exponents/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Shaun Errichiello - [Exponential Powers 2](https://www.openmiddle.com/exponential-powers-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Pedro Suber - [Finding Intercepts](https://www.openmiddle.com/finding-intercepts/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a linear equation that has an x- and y-intercept with integer values. Source: Jeffrey Mashbitz - [Interior Angles](https://www.openmiddle.com/interior-angles/) - Challenge students to use the digits 0-9 to add interior angles of a triangle in this DOK 2 problem. Great for 8th grade students practicing geometry. - [Largest Possible GCF #2](https://www.openmiddle.com/largest-possible-gcf-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the largest possible greatest common factor. Source: Howie Hua - [Largest Possible GCF](https://www.openmiddle.com/largest-possible-gcf/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the largest possible greatest common factor. Source: Howie Hua - [Intercept Form Equations](https://www.openmiddle.com/intercept-form-equations/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to write an equation of a line in standard form with given x- and y-intercepts. Each number can only be used at most once. Source: Andy Schwen - [Derivative of Trig Functions 2](https://www.openmiddle.com/derivative-of-trig-functions-2/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to make the largest value for D (the derivative). Source: Chris Luzniak - [Derivative of Trig Functions 1](https://www.openmiddle.com/derivative-of-trig-functions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make as many possible solutions as you can. Source: Chris Luzniak - [Definite Integral 2](https://www.openmiddle.com/definite-integral-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box and make a positive and a negative solution. Source: Robert Kaplinsky - [Decimals on a Number Line](https://www.openmiddle.com/decimals-on-a-number-line/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create numbers on the number line. Source: Anne Oliveira - [Comparing Fractions](https://www.openmiddle.com/comparing-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two different fractions: one that is less than one half and one that is more than one half. Source: Robert Kaplinsky - [Comparing Fractions 2](https://www.openmiddle.com/comparing-fractions-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a fraction that is as close to 5/11 as possible. Source: Robert Kaplinsky - [Create an Inequality, Given a Solution Set](https://www.openmiddle.com/create-an-inequality-given-a-solution-set/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create an inequality whose solution set is x < -1/2. Source: Daniel Luevanos - [Combinations and Permutations](https://www.openmiddle.com/combinations-and-permutations/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the statement is true. Source: Mark Alvaro and Kerri Swails - [Combinations](https://www.openmiddle.com/combinations/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the statement is true. Can you find more than one? Source: Kerri Swails, Mark Alvaro - [Similar Triangles And Slope](https://www.openmiddle.com/similar-triangles-and-slope/) - Directions: The three triangles on the line are similar. Using the digits 0 to 9, at most one time each, place a digit in each box for the legs of the right triangles. Source: Jay Sydow - [Solving Equations In Two Variables](https://www.openmiddle.com/solving-equations-in-two-variables/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that x equals y. Source: Arnav Gulati and Daniel Luevanos - [Subtracting Decimals (Middle School)](https://www.openmiddle.com/subtracting-decimals-middle-school/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) difference. Note: This problem’s difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both. Source: Robert Kaplinsky - [Subtracting Decimals (Elementary)](https://www.openmiddle.com/subtracting-decimals-elementary/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) difference. Source: Robert Kaplinsky - [Solving One-Step Equations 2](https://www.openmiddle.com/solving-one-step-equations-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that x has the greatest possible value. Source: Robert Kaplinsky - [Rational and Irrational Numbers](https://www.openmiddle.com/rational-and-irrational-numbers/) - Directions: Using the digits 1 to 8, at most one time each, place a digit in each box to create the following number types: Source: Bryan Anderson - [Pythagorean Theorem](https://www.openmiddle.com/pythagorean-theorem/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to find two pairs of possible lengths for the missing sides. Source: Robert Kaplinsky - [Properties of Exponents 4](https://www.openmiddle.com/properties-of-exponents-4/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make an equation where the product’s exponent has the greatest possible value. Source: Robert Kaplinsky - [Pythagorean Theorem 2](https://www.openmiddle.com/pythagorean-theorem-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to find the lengths of the missing sides such that the missing leg's length is as long as possible. Source: Robert Kaplinsky - [Perfect Squares](https://www.openmiddle.com/perfect-squares/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make each expression evaluate to a perfect square number. Extension/Challenge: What is the largest/smallest square number you can make? How many different perfect square numbers could be made? Source: Erick Lee - [Polynomial Function Features 1](https://www.openmiddle.com/polynomial-function-features-1/) - Directions: Using the integers -9 to 9, at most two times each, place an integer in each box to create a polynomial function with matching roots. Source: Robert Kaplinsky - [Parallel Lines and Slope](https://www.openmiddle.com/parallel-lines-and-slope/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the lines through each pair of points are parallel. Source: Nanette Johnson - [Linear Inequalities in Two Variables](https://www.openmiddle.com/linear-inequalities-in-two-variables/) - Directions: Using the digits 0 to 9, exactly one time each, create 5 ordered pairs. Then, create a linear inequality such that: 1. Two of the ordered pairs are solutions to the linear inequality. 2. Two of the ordered pairs are not solutions to the linear inequality. 3. One of the ordered pairs is on - [Equivalent Ratios](https://www.openmiddle.com/equivalent-ratios/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make as many ratios as you can that are equivalent to 2:3. Source: Robert Kaplinsky - [Dividing Fractions](https://www.openmiddle.com/dividing-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) quotient. Source: Robert Kaplinsky - [Creating Right Triangles 2](https://www.openmiddle.com/creating-right-triangles-2/) - Directions: Using the digits 1 to 8, at most one time each, place a digit in each blank to create the vertices of a right triangle: A(__, __), B(__, __), C(__, __) Extension: Try to do this using only the digits 1 to 6. Source: Erick Lee - [Benchmark Fractions](https://www.openmiddle.com/benchmark-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create three fractions that are as close to zero, one half and one as possible. NOTE: Close as possible is measured by adding up all the differences and making it the least possible value. Source: Darbie - [Rolling with the Same Probability](https://www.openmiddle.com/rolling-with-the-same-probability/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each blank to complete this sentence: Rolling a sum of ___ on two ___-sided dice is the same probability as rolling a sum of ___ on two ___-sided dice. Source: Audrey Mendivil, Daniel Luevanos, and Robert Kaplinsky - [Related Percentages](https://www.openmiddle.com/related-percentages/) - Directions: Using the digits 0 to 9, as many times as you want, place a digit in each box to create a correct number sentence. Source: Erick Lee - [Rational, Irrational or Whole Roots](https://www.openmiddle.com/rational-irrational-or-whole-roots/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create the following number types: Source: Bryan Anderson - [Sine and Cosine Waves](https://www.openmiddle.com/sine-and-cosine-waves/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two functions that do not intersect and have the following values. Source: Tim Brzezinski - [Systems of Equations 1](https://www.openmiddle.com/systems-of-equations-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a system of equations with a solution in Quadrant 2. Source: Robert Kaplinsky - [Unit Rates with Fractions 1](https://www.openmiddle.com/unit-rates-with-fractions-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two unit rates. You may reuse all the digits each equation. Source: Robert Kaplinsky - [Undefined Quotient with Fraction Division](https://www.openmiddle.com/undefined-products-with-fraction-division/) - Challenge students to use the digits 0-9 to find unidentified quotents in this DOK 2 problem. Great for 6th and 7th grade students working on the number system. - [Scatter Plots 2](https://www.openmiddle.com/scatter-plots-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create the strongest possible linear association. Source: Robert Kaplinsky - [Scatter Plots 1](https://www.openmiddle.com/scatter-plots-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create two sets of six points: one that has a positive association and one that has a negative association. You may reuse all the integers for each equation. Source: Robert Kaplinsky - [Planting Carrots 1](https://www.openmiddle.com/planting-carrots-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each blank to make the statement true. Sarah planted __ __ carrots in her garden. She planted them in __ rows. Each row had __ carrots. Source: Chase Orton - [Operations with Time](https://www.openmiddle.com/operations-with-time-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the latest possible time. Source: Robert Kaplinsky - [Multiplying And Adding Fractions](https://www.openmiddle.com/multiplying-and-adding-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Joseph Nguyen - [Highest Degree Polynomials](https://www.openmiddle.com/highest-degree-polynomials/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a polynomial of the highest degree. Source: Robert Kaplinsky - [Fractions Less Than One Half](https://www.openmiddle.com/fractions-less-than-one-half/) - Challenge students to use the digits 1-9 to make fractions smaller than 1/2 in this DOK 2 problem. Great for 4th grade students working with fractions. - [Equivalent Expressions 3](https://www.openmiddle.com/equivalent-expressions-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two equivalent expressions. Source: Will Case - [Equidistant Points 2](https://www.openmiddle.com/equidistant-points-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two points that are equidistant from (4,-1). Source: Bryan Anderson - [Equal Tips](https://www.openmiddle.com/equal-tips/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the statement true. Leaving a ____ dollar tip for a bill of _____ is the same as leaving a ____ dollar tip for a bill of _____ Source: Bryan Anderson - [Divisibility](https://www.openmiddle.com/divisibility/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a three-digit number. Try to create a three-digit number divisible by the greatest (or fewest) amount of the following factors: 2, 3, 4, 5, 6, 8, 9 or 10. Source: Kelly Zinck - [Dividing Two-Digit Numbers (Middle School)](https://www.openmiddle.com/dividing-two-digit-numbers-middle-school/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) quotient. Note: This problem's difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both. Source: Robert Kaplinsky - [Dividing Mixed Numbers](https://www.openmiddle.com/dividing-mixed-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) quotient. Source: Robert Kaplinsky - [Dividing Decimals (Middle School)](https://www.openmiddle.com/dividing-decimals-middle-school/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) quotient. Note: This problem’s difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both. Source: Robert Kaplinsky - [Dividing Decimals (Elementary)](https://www.openmiddle.com/dividing-decimals-elementary/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) quotient. Source: Robert Kaplinsky - [Distributive Property 5](https://www.openmiddle.com/distributive-property-5/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make an equation where both sides have the greatest possible value. Source: Adrianne Burns and Robert Kaplinsky - [Discriminant](https://www.openmiddle.com/discriminant/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make one function have no real roots, another function have one real root, and the last function have two real roots. Source: Lynda Chung - [Derivatives Power Rule 2](https://www.openmiddle.com/derivatives-power-rule-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a function such that at x = 2, the derivative (at that point) is closest to the value of 449. Source: Gregory L. Taylor, Ed.D. - [Derivatives - Power Rule](https://www.openmiddle.com/derivatives-power-rule/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a function such that at x = 2, the derivative (at that point) would fall in the interval of {0, 48} Source: Gregory L. Taylor, Ed.D. - [Create Squares](https://www.openmiddle.com/create-squares/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a square with one of the vertices at (2,3). Source: John Mahlstedt - [Converting Between Fractions and Decimals](https://www.openmiddle.com/converting-between-fractions-and-decimals/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the fraction equals the decimal. Source: Robert Kaplinsky - [Complimentary and Supplementary Angles 2](https://www.openmiddle.com/complimentary-and-supplementary-angles-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create supplementary and complementary angles where the measures of each pair of angles are as close together as possible. Source: Brian Anderson with Robert Kaplinsky - [Circle Radius and Area 2](https://www.openmiddle.com/circle-radius-and-area-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a circle with the smallest difference between the area estimates. Source: Robert Kaplinsky - [Circle Radius and Area 1](https://www.openmiddle.com/circle-radius-and-area-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two possible circles. You may reuse all the digits for each statement. Source: Robert Kaplinsky - [Adding Two-Digit Numbers (Middle School)](https://www.openmiddle.com/adding-two-digit-numbers-middle-school/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) sum. Note: This problem's difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both. Source: Robert Kaplinsky - [Adding Decimals (Middle School)](https://www.openmiddle.com/adding-decimals-middle-school/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) sum. Note: This problem’s difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both. Source: Robert Kaplinsky - [Adding Decimals (Elementary)](https://www.openmiddle.com/adding-decimals-elementary/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) sum. Source: Robert Kaplinsky - [Law of Cosines Triangle](https://www.openmiddle.com/law-of-cosines-triangle/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to fill in the circles of a triangle. The sum of the numbers on each side of the triangle is equal to the length of that side. What is the triangle with the largest (or smallest) angle - [Linear Function from Table of Values](https://www.openmiddle.com/linear-function-from-table-of-values/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a table of values that represent a linear function. Source: Robert Kaplinsky - [Multiplying Two-Digit Numbers (Middle School)](https://www.openmiddle.com/multiplying-two-digit-numbers-middle-school/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) product. Note: This problem's difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both. Source: Robert Kaplinsky - [Multiplying Mixed Numbers](https://www.openmiddle.com/multiplying-mixed-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) product. Source: Robert Kaplinsky - [Multiplying Decimals (Middle School)](https://www.openmiddle.com/multiplying-decimals-middle-school/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) product. Note: This problem’s difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both. Source: Robert Kaplinsky - [Multiplying Decimals (Elementary)](https://www.openmiddle.com/multiplying-decimals-elementary/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) product. Source: Robert Kaplinsky - [Max Intercept](https://www.openmiddle.com/max-intercept/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to write the equation of a line that passes through the point with the largest possible y-intercept. How many solutions can you find? Source: Andy Schwen - [Markup & Discount 1](https://www.openmiddle.com/markup-discount-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two true statements without rounding. You may reuse all the digits for each statement. Source: Robert Kaplinsky - [Marble Madness 2](https://www.openmiddle.com/marble-madness-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the following problem true. Jenny has ? ? ? marbles. Her brother has ? ? ? marbles. Together they have ? ? ? marbles. Source:Chase Orton - [Properties of Integer Exponents 3](https://www.openmiddle.com/properties-of-integer-exponents-3/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to generate equivalent numerical expressions: Source: Bryan Anderson - [Properties of Integer Exponents 2](https://www.openmiddle.com/properties-of-integer-exponents-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to generate equivalent numerical expressions: Source: Bryan Anderson - [Properties of Integer Exponents](https://www.openmiddle.com/properties-of-integer-exponents/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to generate equivalent numerical expressions: Source: Bryan Anderson - [Perpendicular lines through a given point](https://www.openmiddle.com/perpendicular-lines-through-a-given-point/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. Source: Andy Schwen - [Perpendicular Lines 2](https://www.openmiddle.com/perpendicular-lines-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two perpendicular lines whose solution is as close to the origin as possible. Source: Robert Kaplinsky - [Perpendicular Lines 1](https://www.openmiddle.com/perpendicular-lines-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two perpendicular lines. Source: Robert Kaplinsky - [Percents on a Linear Model 4](https://www.openmiddle.com/percents-on-a-linear-model-4/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create an accurate number line. How many solutions can you find? Source: Adrianne Burns - [Percent of a Quantity 1](https://www.openmiddle.com/percent-of-a-quantity-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make two true statements without rounding. You may reuse all the digits for your second statement. Source: Robert Kaplinsky - [Partitioning a Line Segment](https://www.openmiddle.com/partitioning-a-line-segment/) - Directions: Using the digits 1 to 8, exactly one time each, place a digit in each box to create a line segment AB, where between point A and point B, there exists a point P so that it partitions line segment AB into a ratio. Source: Jon Henderson - [Parallel Lines and Transversals](https://www.openmiddle.com/parallel-lines-and-transversals/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that 2 of the lines are parallel and the third line is a transversal. Source: Shelli Foust - [Parallel Lines and Perpendicular Transversals](https://www.openmiddle.com/parallel-lines-and-perpendicular-transversals/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that 2 of the lines are parallel and the third line is a transversal that is as close to perpendicular to the parallel lines as possible. Source: Shelli Foust and Robert Kaplinsky - [Percents on a Linear Model 3](https://www.openmiddle.com/percents-on-a-linear-model-3/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create an accurate number line. Source: Adrianne Burns - [Polynomial Function Features 2](https://www.openmiddle.com/polynomial-function-features-2/) - Directions: Using the integers -9 to 9, at most two times each, place an integer in each box to create a polynomial function with matching roots that have the least range possible. Source: Robert Kaplinsky - [Period of Trig Function 3](https://www.openmiddle.com/period-of-trig-function-3/) - Directions: Using the digits 0 to 9, at most one time, place a digit in each box to create sine and cosine functions with the given periods in degrees. Source: Kate Nerdypoo - [Equilateral Triangle](https://www.openmiddle.com/equilateral-triangle-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to fill in the circles of the triangle. The sum of the numbers on each side of the triangle is equal to the length of that side. Arrange the numbers so that the triangle is an equilateral - [Square Root Function Features 2](https://www.openmiddle.com/square-root-function-features-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a square root function, its domain, and the greatest possible x-intercept. Source: Robert Kaplinsky - [Square Root Function Features 1](https://www.openmiddle.com/square-root-function-features-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a square root function, its domain, and the x-intercept. Source: Robert Kaplinsky - [Standard Form](https://www.openmiddle.com/standard-form/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find: the steepest slope the flattest slope the greatest y-intercept the least y-intercept Source: Dane Ehlert - [Subtracting Mixed Numbers](https://www.openmiddle.com/subtracting-mixed-numbers-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create three different mixed numbers that will make the equation true. You may reuse the same numbers for each of the three mixed numbers. Source: Robert Kaplinsky - [Multiplying And Dividing Rational Numbers 2](https://www.openmiddle.com/multiplying-and-dividing-rational-numbers-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a quotient with the greatest possible value. Source: Robert Kaplinsky - [Multiplying And Dividing Rational Numbers 1](https://www.openmiddle.com/multiplying-and-dividing-rational-numbers-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create two equations. You may reuse all the integers for each equation. Source: Robert Kaplinsky - [Multiplying Complex Numbers 2](https://www.openmiddle.com/multiplying-complex-numbers-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make a real number product with the greatest possible value. Source: Robert Kaplinsky in Open Middle Math - [Midpoint Of A Line Segment: positive And Negative Slopes](https://www.openmiddle.com/midpoint-of-a-line-segment-positive-and-negative-slopes/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create endpoints for two different line segments whose midpoint is (1, 3). One line segment should have a positive slope and the other should have a negative slope. You may reuse all the integers for each - [Midpoint Of A Line Segment: Longest Line Segment](https://www.openmiddle.com/midpoint-of-a-line-segment-longest-line-segment/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create endpoints for the longest possible line segment whose midpoint is (1, 3). Source: Robert Kaplinsky in Open Middle Math - [Maximum Value of a Quadratic in Standard Form](https://www.openmiddle.com/maximum-value-of-a-quadratic-in-standard-form/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a quadratic equation with the greatest possible maximum value. Source: Robert Kaplinsky - [Maximizing Rectangular Prism Volume](https://www.openmiddle.com/maximizing-rectangular-prism-volume/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to list the dimensions of a rectangular prism with the greatest volume. Source: Robert Kaplinsky - [Maximizing Rectangular Prism Surface Area](https://www.openmiddle.com/maximizing-rectangular-prism-surface-area/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to list the dimensions of a rectangular prism with the greatest possible surface area. Source: Robert Kaplinsky - [Area of a Triangle in the Coordinate Plane](https://www.openmiddle.com/area-of-a-triangle-in-the-coordinate-plane/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create ordered pairs for all three points, such that the area of Triangle ABC is closest to 6 square units. A ( ___, ___ ) B ( ___, ___ ) C ( ___, ___ ) Source: - [Area on a Coordinate Plane 1](https://www.openmiddle.com/area-on-a-coordinate-plane-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create coordinates that represent the vertices of two triangles: one with an area of less than 55 units2 and one with an area of more than 55 units2. You may reuse all the integers each time. - [Area on a Coordinate Plane 2](https://www.openmiddle.com/area-on-a-coordinate-plane-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create coordinates that represent the vertices of the triangle with the smallest possible area. Source: Robert Kaplinsky - [Identical Quadratics](https://www.openmiddle.com/identical-quadratics/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create three equations that produce the exact same parabola. Source: Zack Miller - [Factoring Quadratics - Fraction Solution](https://www.openmiddle.com/factoring-quadratics-fraction-solution/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that at least one of the solutions is a fraction. Source: Daniel Luevanos - [Factoring Quadratics - Integer Solutions](https://www.openmiddle.com/factoring-quadratics-integer-solutions/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the solutions are integers. Source: Daniel Luevanos - [Factoring Quadratics - One Solution](https://www.openmiddle.com/factoring-quadratics-one-solution/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that there is only one solution. Source: Daniel Luevanos - [Factoring Quadratics 2 - Fraction Solution](https://www.openmiddle.com/factoring-quadratics-2-fraction-solution/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that at least one of the solutions is a fraction. Source: Daniel Luevanos - [Factoring Quadratics 2 - Integer Solutions](https://www.openmiddle.com/factoring-quadratics-2-integer-solutions/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the solutions are integers. Note that you can use a 05 to make a 5. Source: Daniel Luevanos - [Factoring Quadratics 2 - One Solution](https://www.openmiddle.com/factoring-quadratics-2-one-solution/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that there is only one integer solution. Note that you can use a 05 to make a 5. Source: Daniel Luevanos - [Equality](https://www.openmiddle.com/equality-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two true number sentences. You may reuse all the digits for each number sentence. Source: Robert Kaplinsky - [Dividing Fractions](https://www.openmiddle.com/dividing-fractions-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make a true fraction division sentence. Source: Shanelia Rhome-Shannon and Aiden (Student) - [Distributive Property with Four](https://www.openmiddle.com/distributive-property-with-four/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a true statement. Source: Nova Katz - [Comparing and Identifying Fractions on a Number Line](https://www.openmiddle.com/comparing-and-identifying-fractions-on-a-number-line/) - Challenge students to use the digits 1-9 to put fractions on a number line in this DOK 3 problem. Great for 4th grade students working with fractions. - [Adding One-Digit Numbers (< 5)](https://www.openmiddle.com/adding-one-digit-numbers-5/) - Directions: Using the digits 1 to 5, at most one time each per number sentence, place a digit in each box to create two or more true number sentences. Source: Robert Kaplinsky - [Integer Sums and Differences](https://www.openmiddle.com/integer-sums-and-differences/) - Directions: Using the integers -3 to 3, at most one time each, place an integer in each box to make both equations true. Source: Jeanmarie Mullen - [Creating Inequalities](https://www.openmiddle.com/creating-inequalities/) - Directions: Using the integers -4 to 4, at most one time each, place an integer in each box create an inequality with solutions of x > 2/3. Source: Robert Kaplinsky - [Absolute Value 3](https://www.openmiddle.com/absolute-value-3/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make the equation true. Source: Owen Kaplinsky - [Multi-Digit Division 2](https://www.openmiddle.com/multi-digit-division-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create the smallest whole number quotient possible. Source: Robert Kaplinsky - [Multi-Digit Division 1](https://www.openmiddle.com/multi-digit-division-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two different whole number quotients: one that is greater than 300 and one that is less than 300. You may reuse all the digits for each quotient. Source: Robert Kaplinsky - [Solution of Two Linear Equations](https://www.openmiddle.com/solution-of-two-linear-equations/) - Directions: Using the integers 0 to 9, at most one time each, provide four sets of points that represent two distinct lines. These lines can be written as two linear equations. Then provide a fifth point that represents the intersection (or solution) of those equations. Line 1: (__, __) and (__, __) Line 2: (__, - [Solving Equations with Variables on Both Sides (Fractions)](https://www.openmiddle.com/solving-equations-with-variables-on-both-sides-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create an equation such that the solution is the largest integer possible. Source: Kristen Amaral - [Tangent to a Cubic Graph](https://www.openmiddle.com/tangent-to-a-cubic-graph/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box make the statement true. Source: Catriona Shearer - [Writing Equivalent Polynomial Expressions](https://www.openmiddle.com/writing-equivalent-polynomial-expressions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a true statement. Source: Andrew Stadel - [Radical And Linear Function Intersection](https://www.openmiddle.com/radical-and-linear-function-intersection/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make one set of functions intersect exactly twice, one set of functions intersect exactly once, and one set of functions never intersect. Source: Mike Fouchet - [Properties of Logarithms](https://www.openmiddle.com/properties-of-logarithms/) - Directions: Using the integers 1 to 9, at most one time each, find the value of x that is closest to 0. Extension: Find more than one set of numbers that would make x = 0. Source: Claire Verti - [Pythagorean Inequality](https://www.openmiddle.com/pythagorean-inequality/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to find three side lengths that are two-digits each and form an acute triangle. Source: Samantha Cruz - [Logs](https://www.openmiddle.com/logs/) - Directions: Using the integers 1 to 9, fill in the red and blue boxes so that the chart is accurate. You can only use a number once per red box and once per blue box. Source: Bryan Anderson - [Line of Best Fit](https://www.openmiddle.com/line-of-best-fit/) - Directions: Using the integers 0 to 9, at most one time each, create 4 points that could generate a line of best fit with the equation y=-x+8. (___,___) (___,___) (___,___) (___,___) Source: Bryan Anderson - [Fraction and Decimal](https://www.openmiddle.com/fraction-and-decimal/) - Directions: Using the digits 0 to 9, at most one each time, create an an equivalent fraction and decimal number. Source: Giselle Garcia - [Factoring Quadratics](https://www.openmiddle.com/factoring-quadratics-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the quadratic is factorable. Source: Mark Baethke - [Dividing Fractions 3](https://www.openmiddle.com/dividing-fractions-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two true equations: one where the quotient is greater than 40 and one where it’s less than 40. You may reuse the same digits for each of the equations. Source: Robert Kaplinsky - [Definite Integral](https://www.openmiddle.com/definite-integral/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the equation is a true statement. Source: Daniel Torres-Rangel - [Central, Inscribed, & Circumscribed Angles 1](https://www.openmiddle.com/central-inscribed-circumscribed-angles-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box two times: once where the central angle is greater than 130° and once where it is less than 130°. You may reuse all the digits each time. Source: Robert Kaplinsky - [Comparing Functions](https://www.openmiddle.com/comparing-functions/) - Directions: Using the digits 0 to 9, at most one time each, create five ordered pair that represent a linear function that has a greater rate of change than the following: (___,___) (___,___) (___,___) (___,___) (___,___) How many different ones are there? Source: Bryan Anderson - [Comparing and Ordering Radicals](https://www.openmiddle.com/comparing-and-ordering-radicals/) - Directions: Using the digits 1 to 9, at most one time each, create a sequence that is in numerical order and cannot be simplified anymore. Source: Phillip Haislip-Hansberry - [Closest Difference to 200](https://www.openmiddle.com/closest-difference-to-200/) - Directions: Using the digits 1 to 9, at most one time each, find the closest difference to 200. Source: Jessica Goree - [Decimal Division](https://www.openmiddle.com/decimal-division/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to find the quotient closest to 1. Source: Michael Dennis - [Creating Sequences](https://www.openmiddle.com/creating-sequences/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to complete the first three terms of the arithmetic and geometric sequences. What sequences result in the greatest sum of their second terms? (e.g. 3, 5, 7 and 2, 6, 18 would result in a sum of - [Equations of Parallel Lines](https://www.openmiddle.com/equations-of-parallel-lines/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two distinct parallel lines. Source: Bryan Anderson - [Equations of Circles 1](https://www.openmiddle.com/equations-of-circles-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a circle and a point on the circle. Source: Robert Kaplinsky - [Evaluating Expressions 1](https://www.openmiddle.com/evaluating-expressions-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two true statements: one where the value on each side of the equal sign is greater than 30 and one where it’s less than 30. You may reuse all the digits for each equation. Source: - [Greatest Difference of Two Rounded Numbers](https://www.openmiddle.com/greatest-difference-of-two-rounded-numbers/) - Directions: Using the digits 0 to 9, at most one time each, find two numbers that round to 500, and have the greatest possible difference. Source: Michael Wiernicki, Graham Fletcher, and Rachel Nelli. - [Infinitely Many Solutions](https://www.openmiddle.com/infinitely-many-solutions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that there are infinitely many solutions. Source: Jordan Dodge - [Rational Exponents](https://www.openmiddle.com/rational-exponents/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to make the greatest or least value. Extension: How close to 1 can you get? Source: Erick Lee - [Rational Function Features 2](https://www.openmiddle.com/rational-function-features-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a rational function, its vertical asymptote, and the greatest possible solution. Source: Robert Kaplinsky - [Rational Function Features 1](https://www.openmiddle.com/rational-function-features-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a rational function, its vertical asymptote, and its solution. Source: Robert Kaplinsky - [Scientific Notation 2](https://www.openmiddle.com/scientific-notation-2-2/) - Directions: Using the digits 1 to 9, at most two times each, place a digit in each box to make the sum of the four expressions the greatest possible value. Source: Catriona Shearer - [Subtracting Fractions](https://www.openmiddle.com/subtracting-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) difference. Source: Robert Kaplinsky - [Subtracting Mixed Numbers](https://www.openmiddle.com/subtracting-mixed-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the least possible difference. Source: Robert Kaplinsky - [Window Sum](https://www.openmiddle.com/window-sum/) - Directions: Using the digits 0 to 9, at most one time each, complete the puzzle so that the sum of each side is equivalent. Source: Joshua Nelson and Renee Owen - [Systems of Three Equations - No Solution](https://www.openmiddle.com/systems-of-three-equations-no-solution/) - Directions: Using the digits 1 to 9, at most one time each, create a system of equations that has no solutions. Source: Nanette Johnson and Robert Kaplinsky - [Systems of Three Equations - Multiple Solutions](https://www.openmiddle.com/systems-of-three-equations-multiple-solutions/) - Directions: Using the digits 1 to 9, at most one time each, create a system of equations that has as many solutions as possible. Source: Audrey Mendivil, Daniel Luevanos, and Robert Kaplinsky - [Decomposing tenths & hundredths](https://www.openmiddle.com/decomposing-tenths-hundredths/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Christine Jenkins - [Decimal Approximations of Roots](https://www.openmiddle.com/decimal-approximations-of-roots/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to create a true statement with the smallest possible interval: Source: Bryan Anderson - [Creating Right Triangles](https://www.openmiddle.com/creating-right-triangles/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a right triangle from the vertex (2,3): (__,__) and (__,__) Source: Bryan Anderson - [Creating Rectangles](https://www.openmiddle.com/creating-parallelograms/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a rectangle from the vertex (2,3): (__,__), (__,__), and (__,__) Source: Bryan Anderson - [Creating Parallelograms](https://www.openmiddle.com/creating-parallelograms-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a parallelogram from the vertex (2,3): (__,__), (__,__), and (__,__) Source: Bryan Anderson - [Creating 3 Lines to form Right Triangle](https://www.openmiddle.com/creating-3-lines-to-form-right-triangle/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a right triangle on the coordinate plane. Source: Tracy Conte - [Converting Fractions to Repeating Decimals](https://www.openmiddle.com/converting-fractions-to-repeating-decimals/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Daniel Luevanos - [Complementary and Supplementary Angles](https://www.openmiddle.com/complementary-and-supplementary-angles/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statements true. Source: Bryan Anderson - [Building Shelves 1](https://www.openmiddle.com/building-shelves-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the following problem true. Ricky is building ? sets of shelves for the office. It takes him ? ? minutes to do each set of shelves. He’ll be done building the sets in ? hours. - [Biggest Product 3](https://www.openmiddle.com/biggest-product-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to and make the greatest/least product. Source: Nanette Johnson - [Biggest Product 2](https://www.openmiddle.com/biggest-product-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the biggest/smallest product. Source: Nanette Johnson - [Biggest Product](https://www.openmiddle.com/biggest-product/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the biggest/smallest product. Source: Nanette Johnson - [Solution to a Linear and Quadratic System of Equations](https://www.openmiddle.com/solution-to-a-linear-and-quadratic-system-of-equations/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the solution(s) to the system are integers. Source: Ashley Taplin - [Sums With Scientific Notation](https://www.openmiddle.com/sums-with-scientific-notation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create the largest (or smallest) sum possible. Source: Annie DeAngelo - [Three Digit Integer Sums](https://www.openmiddle.com/three-digit-integer-sums/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the second greatest solution. Source: Neil Hamilton - [Trig Functions](https://www.openmiddle.com/trig-functions/) - Directions: Using the digits 1 to 9, at most one time each, fill in the empty blanks so that you create a triangle whose Cos Θ = √2/2: (5, 4), (__,__) and (__,__). Source: Bryan Anderson - [Trigonometric Ratios](https://www.openmiddle.com/trigonometric-ratios/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a right triangle where 𝜃 is as close to 10° as possible. Source: Thomas Derstein - [Two Step Inequality with Fractional Coeffcient](https://www.openmiddle.com/two-step-inequality-with-fractional-coeffcient/) - Directions: Using the digits 1 to 7, at most one time each, place a digit in each box to create an inequality that match the solution of the inequality on the number line. Source: Jay Sydow - [Two-Step Equations: Positive And Negative Values](https://www.openmiddle.com/two-step-equations-positive-and-negative-values/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two equations: one where x has a positive value and one where x has a negative value. You may reuse all the digits for each equation. Source: Robert Kaplinsky in Open Middle Math - [Unit Rate](https://www.openmiddle.com/unit-rate/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make two equivalent ratios where one of the ratios is a unit rate. Source: Scott Houston - [Volume of a Rectangular Prism and Rectangular Pyramid](https://www.openmiddle.com/volume-of-a-rectangular-prism-and-rectangular-pyramid/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to list the dimensions of a rectangular prism and rectangular pyramid so that both shapes have equal volumes. NOTE: Images are not drawn to scale. Source: Aaron Arispe - [Volume of Three Rectangular Prisms](https://www.openmiddle.com/volume-of-three-rectangular-prisms/) - Directions: Using the digits 1 to 9, at most one time each, find the dimensions of three rectangular prisms so that their volumes are as close as possible. Note: diagram may not be drawn to scale. Source: Daniel Walker - [The Modulus Of A Complex Number](https://www.openmiddle.com/the-modulus-of-a-complex-number/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to find an odd modulus, an even modulus, and the smallest possible modulus. Source: Mark Ward - [Subtracting 3-Digit Numbers 1](https://www.openmiddle.com/subtracting-3-digit-numbers-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make two different pairs of three-digit numbers that form a true number sentence. You may reuse all the digits each difference. Source: Robert Kaplinsky - [Simplifying Rational Expressions](https://www.openmiddle.com/simplifying-rational-expressions/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a true statement Source: Dwight Stephenson - [Simplifying Complex Roots](https://www.openmiddle.com/simplifying-complex-roots/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to to create a true statement. Source: Paige Sheehan - [Simplify Rational Expressions 2](https://www.openmiddle.com/simplify-rational-expressions-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a true equation. Source: Eduardo Bernal - [Simplify Rational Expressions 1](https://www.openmiddle.com/simplify-rational-expressions-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a true equation. Source: Eduardo Bernal - [Similar Triangles 2](https://www.openmiddle.com/similar-triangles-2/) - Directions: Using the digits 0 to 9, at most one time each, create two similar triangles. You may have as many leading zeros as you like. Source: Drew Ross - [Similar Triangles](https://www.openmiddle.com/similar-triangles/) - Directions: Using the digits 0 to 9, at most one time each and as many leading zeros as you like, place a digit in each box to create two similar triangles. Source: Drew Ross - [Sector Area 2](https://www.openmiddle.com/sector-area-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the radius and angle measure result in the sector area is as close to 60 units2 as possible. Source: Robert Kaplinsky - [Sector Area 1](https://www.openmiddle.com/sector-area-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the radius and angle measure result in the sector area. Source: Robert Kaplinsky - [Scientific Notation](https://www.openmiddle.com/scientific-notation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the greatest (or least) product. Source: Bryan Meyer - [Rational and Irrational Roots 8](https://www.openmiddle.com/rational-and-irrational-roots-8/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create 3 rational and 2 irrational numbers. Source: Jennifer Kolis - [Quadratic In Vertex Form With Given Root And Maximum Value](https://www.openmiddle.com/quadratic-in-vertex-form-with-given-root-and-maximum-value/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two different quadratic equations that have a root at 4 and a maximum value of 4. You may reuse all the digits for each equation. Source: Robert Kaplinsky in Open Middle Math - [Quadratic Equations: Distance Between Solutions](https://www.openmiddle.com/quadratic-equations-distance-between-solutions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a quadratic equation such that the distance between the solutions is greater than 1. Source: Mong Kon Mo - [Product Close to 1,000](https://www.openmiddle.com/product-close-to-1000/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the product as close to 1,000 as possible. Source: Ellen Metzger - [Polynomial Division Given a Remainder](https://www.openmiddle.com/polynomial-division-given-a-remainder/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to so that the remainder when dividing the two is 14. Source: Kyle Prince - [Polynomial Division](https://www.openmiddle.com/polynomial-division/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a division problem with a solution of 2x + 5. Source: Andrew King - [Placing Fractions on A Number Line](https://www.openmiddle.com/placing-fractions-on-a-number-line/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create five fractions and place them all on a number line with the correct order and spacing. Source: Robert Kaplinsky - [Period of Trig Function 2](https://www.openmiddle.com/period-of-trig-function-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a sine function with the given period in degrees. Source: Kate Nerdypoo - [Percent Of A Quantity 3](https://www.openmiddle.com/percent-of-a-quantity-3/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a true statement. Source: Matthew Miller - [Parabola's Vertex](https://www.openmiddle.com/parabolas-vertex/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a correct sentence: The vertex of the parabola, y = ▢ x² + ▢ x + ▢, lies on the horizontal axis Source: Cecilia Calvo - [Multiplying Two-Digit Numbers - Closest to 7,000](https://www.openmiddle.com/multiplying-two-digit-numbers-closest-to-7000/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the product as close to 7,000 as possible. Source: Paolo Tolomeo - [Multiplying Multiples Of Ten 1](https://www.openmiddle.com/multiplying-multiples-of-ten-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make two different true number sentences: one with a product that’s less than 500 and one with a product that’s greater than 500. You may reuse all the digits each product. Source: Robert Kaplinsky - [Multiplying Monomials](https://www.openmiddle.com/multiplying-monomials/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Anthony Meli - [Multiply and Divide Within A Hundred 1](https://www.openmiddle.com/multiply-and-divide-within-a-hundred-1/) - Directions: Using the digits 2 to 9, at most one time each, place a digit in each box to make two correct equations: one where the value is greater than 30 and one less than 30. You may reuse all the digits each equation. Source: Robert Kaplinsky - [Multi-Digit Division 3](https://www.openmiddle.com/multi-digit-division-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a true statement. Source: Michael Minas - [Mean of Frequency Table](https://www.openmiddle.com/mean-of-frequency-table/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a frequency table that has the mean in the box at the top. Source: Phillip Haislip-Hansberry - [Maximum Value of a Quadratic in Vertex Form](https://www.openmiddle.com/maximum-value-of-a-quadratic-in-vertex-form/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a quadratic equation with the greatest possible maximum value. Source: Robert Kaplinsky - [Logarithms 1-B](https://www.openmiddle.com/logarithms-1-b/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. Source: Cassie Winchell - [Logarithms 1-A](https://www.openmiddle.com/logarithms-1-a/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. Source: Cassie Winchell - [Linear Equations In One Variable 2](https://www.openmiddle.com/linear-equations-in-one-variable-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create an equation with a solution that’s as close to zero as possible. Source: Robert Kaplinsky - [Linear Equations In One Variable 1](https://www.openmiddle.com/linear-equations-in-one-variable-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two equations: one where x has a positive value and one where x has a negative value. Source: Robert Kaplinsky - [It's About Time 1](https://www.openmiddle.com/its-about-time-1/) - Directions: Using the digits 0 to 9, at most one time each, fill in the question marks to make the following problem true. Suzie leaves work at ? : ? ?. She get’s home at ? : ? ?. Therefore, her commute is ? ? minutes long. Source: Chase Orton - [Integer Y-Intercepts](https://www.openmiddle.com/integer-y-intercepts/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create three different linear equations whose y-intercepts have integer values. Source: Michelle Kovesi and Harold Jacobs - [Infinitely Many Solutions System of Equations](https://www.openmiddle.com/infinitely-many-solutions-system-of-equations/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a a system of equations with infinitely many solutions. Source: Mike - [Imaginary Solutions to a Quadratic Equation](https://www.openmiddle.com/imaginary-solutions-to-a-quadratic-equation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a quadratic equation with an imaginary solution of the form ±𝒃𝒊 where 𝒃 is a whole number. Source: Bradley Springer - [Identical Quadratics 2](https://www.openmiddle.com/identical-quadratics-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that each quadratic is the same. Source: John Rowe - [Geometric Series](https://www.openmiddle.com/geometric-series/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the largest/smallest possible sum of the three terms in this finite geometric series. Source: Dana Harrington - [Fundamental Theorem Of Calculus](https://www.openmiddle.com/fundamental-theorem-of-calculus/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make a derivative as close to 100 as possible. Source: Stephen Spinelli - [Factoring Quadratics (a=4)](https://www.openmiddle.com/factoring-quadratics-a4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to construct four different quadratic expressions that can be factored as two binomials with integer coefficients and terms. Source: Giles Fox - [Factoring Quadratics (a=3)](https://www.openmiddle.com/factoring-quadratics-a3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to construct four different quadratic expressions that can be factored as two binomials with integer coefficients and terms. Source: Giles Fox - [Factoring Quadratics (a=2)](https://www.openmiddle.com/factoring-quadratics-a2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to construct four different quadratic expressions that can be factored as two binomials with integer coefficients and terms. Source: Giles Fox - [Factoring Quadratics (a=1)](https://www.openmiddle.com/factoring-quadratics-a1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create four different quadratic expressions that can be factored as two binomials with integer coefficients and terms. Source: Giles Fox - [Exponential Powers](https://www.openmiddle.com/exponential-powers/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a result with the greatest exponent. Source: Kjersti Oliver - [Evaluating Trigonometric Functions 2](https://www.openmiddle.com/evaluating-trigonometric-functions-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the function has the greatest possible value. Source: Robert Kaplinsky in Open Middle Math - [Evaluating Trigonometric Functions 1](https://www.openmiddle.com/evaluating-trigonometric-functions-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make five true number sentences. You may reuse all the digits for each number sentence. Source: Robert Kaplinsky in Open Middle Math - [Equation of a Circle 2](https://www.openmiddle.com/equation-of-a-circle-2/) - Directions: Using the digits 1 to 9, at most two times each, place a digit in each box to make a circle with the least possible area. Source: Robert Kaplinsky - [Equation of a Circle 1](https://www.openmiddle.com/equation-of-a-circle-1/) - Directions: Using the digits 1 to 9, at most two times each, place a digit in each box to make two circles: one with an area of less than 10 units2 and one with more than 10 units2. Source: Robert Kaplinsky - [Dividing Fractions To Make 2/3](https://www.openmiddle.com/dividing-fractions-to-make-2-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make two different pairs of fractions that have a quotient of 2/3. Source: Robert Kaplinsky in Open Middle Math - [Difference of Squares and Sum of Cubes](https://www.openmiddle.com/difference-of-squares-and-sum-of-cubes/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make both expressions factorable. Source: Jack Assaf - [Definite Integral 3](https://www.openmiddle.com/definite-integral-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a solution that is as close to 100 as possible. Source: Robert Kaplinsky - [Closest Quotient to 250](https://www.openmiddle.com/closest-quotient-to-250/) - Directions: Using the digits 2 to 9, at most one time each, place a digit in each box to find the closest quotient to 250. Source: Jessica Goree - [Central, Inscribed, & Circumscribed Angles 2](https://www.openmiddle.com/central-inscribed-circumscribed-angles-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the central angle has the greatest possible value. Source: Robert Kaplinsky - [Building Shelves 2](https://www.openmiddle.com/building-shelves-2/) - Directions: Using the digits 1 to 9, at most one time each, fill in the question marks to make the following problem true. Ricky is building ? sets of shelves for the office. It takes him ? ? minutes to do each set of shelves. He’ll be done building the sets in ? hours and - [Binomial Expansion](https://www.openmiddle.com/binomial-expansion/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the largest or smallest possible coefficient of the third term in the expansion. Source: Dana Harrington - [Area of a Quadrilateral on a Coordinate Plane](https://www.openmiddle.com/area-of-a-quadrilateral-on-a-coordinate-plane/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a quadrilateral with an area of 16 square units. Source: Daniel Luevanos - [Creating Rectangles 2](https://www.openmiddle.com/creating-rectangles-2/) - Directions: Using the digits 1 to 8, at most one time each, fill in the coordinates to create the vertices of a rectangle: A(__, __), B(__, __), C(__, __), D(__, __). Extension: What is the rectangle with the largest/smallest area/perimeter that you can find? Source: Erick Lee - [Constant of Proportionality](https://www.openmiddle.com/constant-of-proportionality/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to complete the ordered pairs for a proportional relationship with the greatest constant of proportionality possible. Source: Jenny Wilcox - [Comparing Hundredths and Tenths 2 Open Middle](https://www.openmiddle.com/comparing-hundredths-and-tenths-2-open-middle/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make each statement true. Source: The Open Middle Elementary Team and Dean Johnstone - [Comparing Hundredths and Tenths 1 Open Middle](https://www.openmiddle.com/comparing-hundredths-and-tenths-1-open-middle/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make each statement true. Source: The Open Middle Elementary Team and Dean Johnstone - [Comparing Fractions 3](https://www.openmiddle.com/comparing-fractions-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a true statement. Source: Robert Kaplinsky - [Comparing Decimals 2](https://www.openmiddle.com/comparing-decimals-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two decimals that are close to 5 as possible but also equally far away from 5. Source: Robert Kaplinsky - [Comparing Decimals 1](https://www.openmiddle.com/comparing-decimals-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two different decimals: one that is greater than 5 and one that is less than 5. Source: Robert Kaplinsky - [Closest product to 500 - Two-digit times two-digit](https://www.openmiddle.com/closest-product-to-500-two-digit-times-two-digit/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the closest product to 500 Source: Jessica Goree - [Closest product to 500 - Two-digit times one-digit](https://www.openmiddle.com/closest-product-to-500-two-digit-times-one-digit/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the closest product to 500 Source: Jessica Goree - [Closest Product to 500 - Three-digit times one-digit 2](https://www.openmiddle.com/closest-product-to-500-three-digit-times-one-digit-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to find the closest product to 500 Source: Jessica Goree - [Closest Product to 500 - Three-digit times one-digit](https://www.openmiddle.com/closest-product-to-500-three-digit-times-one-digit/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the closest product to 500. Source: Jessica Goree - [Closest Difference to 200 - Problem 2](https://www.openmiddle.com/closest-difference-to-200-problem-2/) - Directions: Using the digits 1 to 9, exactly one time each, place a digit in each box to make the difference as close to 200 as possible. Source: Tara Trifiletti and Jessica Goree - [Adding And Subtracting Rational Numbers 2](https://www.openmiddle.com/adding-and-subtracting-rational-numbers-2/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to create an equation where each side has the greatest possible value. Source: Robert Kaplinsky - [Adding And Subtracting Rational Numbers 1](https://www.openmiddle.com/adding-and-subtracting-rational-numbers-1/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to create two equations. You may reuse all the integers for each equation. Source: Robert Kaplinsky - [Adding and Subtracting Integers](https://www.openmiddle.com/adding-and-subtracting-integers/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box so that top two equations are equal and the bottom equation has a greatest value than the other two. Source: Kate Nerdypoo - [Adding 3-Digit Numbers](https://www.openmiddle.com/adding-3-digit-numbers/) - Directions: Using the digits 1 to 9, exactly one time each, place a digit in each box two times: once to make a sum that is greater than 700 and once to make a sum that is less than 700. You may reuse all the digits for each sum. Source: Robert Kaplinsky - [Add Fractions with Decimal Sums](https://www.openmiddle.com/add-fractions-with-decimal-sums/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest possible decimal sum. Source: Kari Frazier - [Solving One-Step Equations](https://www.openmiddle.com/solving-one-step-equations/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make each equation true. Source: Robert Kaplinsky - [Percentages](https://www.openmiddle.com/percentages/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. Source: Cecilia Calvo - [Two-Digit Multiplication - Equivalent Expressions](https://www.openmiddle.com/two-digit-multiplication-equivalent-expressions/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Travis Drake - [Summation of Product Values](https://www.openmiddle.com/summation-of-product-values/) - Directions: Using the digits 0 to 9, at most three times each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Subtracting Vectors](https://www.openmiddle.com/subtracting-vectors/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to make the equations true. Source: Owen Kaplinsky - [Solving Two-Step Equations](https://www.openmiddle.com/solving-two-step-equations/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equations true. Source: Michele Dijkstra - [Equivalent Expressions with Fractions](https://www.openmiddle.com/equivalent-expressions-with-fractions/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box and choose either multiplication/division or addition/subtraction to make the equation true. Source: Brian Errey - [Interpreting Graphs](https://www.openmiddle.com/interpreting-graphs/) - Directions: Using the digits 1 to 6 at most one time each, place a digit in each blank and create a graph to make the statements true. Source: Bryan Anderson - [Negative Exponents - Closest to Zero](https://www.openmiddle.com/negative-exponents-closest-to-zero/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make a result that is as close to zero as possible. Source: Daniel Luevanos - [Negative Exponents](https://www.openmiddle.com/negative-exponents/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Daniel Luevanos - [Multiplying Fraction by a 2-Digit Whole Number](https://www.openmiddle.com/multiplying-fraction-by-a-2-digit-whole-number/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Ellen Metzger - [Multiplying Differences](https://www.openmiddle.com/multiplying-differences/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Multiplying 3 Fractions to Get 1](https://www.openmiddle.com/multiplying-3-fractions-to-get-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky, answer by Joseph Nguyen - [Composite Volume](https://www.openmiddle.com/composite-volume/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. The composite figure is made of a rectangular prism and a rectangular pyramid; the base for each is congruent. All dimensions must be greater than zero. Source: Jedidiah Butler - [Area of a House](https://www.openmiddle.com/area-of-a-house/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. (Note: Image is not to scale) Source: Owen Kaplinsky - [Equivalent Ratios 2](https://www.openmiddle.com/equivalent-ratios-2/) - Directions: Using the digits 0 to 6, exactly one time each, place a digit in each box to make two equivalent ratios (also known as a proportion). Source: AnneMarie Untalan - [Box Plots](https://www.openmiddle.com/box-plots/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to represent a data set with: a. The smallest possible interquartile range, largest possible range, and that is skewed right. b. An interquartile range greater than 5, range that is greater than 7, and that is skewed - [Magnitude of Vectors](https://www.openmiddle.com/magnitude-of-vectors/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make the statement true. Source: Owen Kaplinsky - [Unit Fraction Proportion](https://www.openmiddle.com/unit-fraction-proportion/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Bryan Anderson - [Implicit Differentiation at a Point](https://www.openmiddle.com/implicit-differentiation-at-a-point/) - Directions: Using the integers -9 through 9, at most two times each, place an integer in each box to make the equations true. Source: Owen Kaplinsky - [Average Rate of Change](https://www.openmiddle.com/average-rate-of-change/) - Directions: Using the digits 0 to 9, at most two times each, place a digit in each box to make the equations true. Source: Owen Kaplinsky - [Adding Multiples 2](https://www.openmiddle.com/adding-multiples-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the greatest possible total. Source: Owen Kaplinsky and Robert Kaplinsky - [Adding Decimals to Make Them As Close to One as Possible](https://www.openmiddle.com/adding-decimals-to-make-them-as-close-to-one-as-possible/) - Challenge students to use the digits 1-9 to add 3 decimals to make 1. Great for 5th and 6th grade students working with the base ten number system. - [Cubed Roots](https://www.openmiddle.com/cubed-roots/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Ryun Deckert and Brock Montgomery - [Dimensions of Rectangles - Area and Perimeter](https://www.openmiddle.com/dimensions-of-rectangles-area-and-perimeter/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create dimensions so that rectangle B has a perimeter double & an area quadruple that of rectangle A. Source: Jessica Goree - [Adding Mixed Numbers 5](https://www.openmiddle.com/adding-mixed-numbers-5/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest possible sum. Source: Robert Kaplinsky - [Adding Mixed Numbers 4](https://www.openmiddle.com/adding-mixed-numbers-4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Robert Kaplinsky - ["Given" With Probabilities](https://www.openmiddle.com/given-with-probabilities/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equations true. Assume A and B are events in the same sample space. Source: Owen Kaplinsky - [Multiplying A Decimal By A Fraction to Get a Whole Number](https://www.openmiddle.com/multiplying-a-decimal-by-a-fraction-to-get-a-whole-number/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each to make the equation true. Source: Owen Kaplinsky - [Equivalent Expressions 1](https://www.openmiddle.com/equivalent-expressions-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two expressions that are equivalent to one another. Source: Will Case - [Polynomial Expansion](https://www.openmiddle.com/polynomial-expansion/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Solving One-Step Inequalities with Addition](https://www.openmiddle.com/solving-one-step-inequalities-with-addition/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create an inequality with a solution of x - [Multiplying Normal Distributions](https://www.openmiddle.com/multiplying-normal-distributions/) - Directions: Using the digits 0 to 9, at most two times each, place a digit in each box to make a valid transformation from the original normal distribution. Source: Owen Kaplinsky - [Multiplying Fractions 4](https://www.openmiddle.com/multiplying-fractions-4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the greatest possible product. Source: Marc DeArmond - [Rounding Decimals](https://www.openmiddle.com/rounding-decimals/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to make the largest (or smallest) number that rounds to 5. Source: Annie Forest - [Adding Fractions 7](https://www.openmiddle.com/adding-fractions-7/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Decimal Addition 2](https://www.openmiddle.com/decimal-addition-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Adding Two-Digit Numbers (Elementary)](https://www.openmiddle.com/adding-two-digit-numbers-elementary/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) sum. Source: Robert Kaplinsky Challenge students to use the digits 1-9 to make a large or small sum in this DOK 3 problem. Great for 1st and 2nd grade students practicing operations in base ten. - ["And" With Probabilities](https://www.openmiddle.com/and-with-probabilities/) - Directions: Using the digits 0 to 9, at most two times each, place a digit in each box to make the equations true. Assume A and B are events in the same sample space. Source: Owen Kaplinsky - [Maximum Volume of a Cylinder](https://www.openmiddle.com/maximum-volume-of-a-cylinder/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to give the cylinder the maximum volume possible. Source: Kyle Leinweber - [Adding Multiples](https://www.openmiddle.com/adding-multiples/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the statement true. Source: Owen Kaplinsky - [Adding One-Digit Numbers (< 5) 2](https://www.openmiddle.com/adding-one-digit-numbers-5-2/) - Directions: Using the digits 1 to 5, at most one time each, place a digit in each box to create a true number sentence with the greatest possible sum. Source: Robert Kaplinsky - [Adding Fractions](https://www.openmiddle.com/adding-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) possible sum. Source: Robert Kaplinsky - [Adding 3 Fractions to Get 1](https://www.openmiddle.com/adding-3-fractions-to-get-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Whole Number Division](https://www.openmiddle.com/whole-number-division/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Create a System of Two Equations](https://www.openmiddle.com/create-a-system-of-two-equations/) - Directions: Using the integers 1 to 30, at most one time each, place an integer in each box to create a system of two linear equations where (3, 2) is the solution to the system. Source: Daniel Luevanos - [Adding Mixed Numbers 2](https://www.openmiddle.com/adding-mixed-numbers-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Ellen Metzger - [Adding Fractions 6](https://www.openmiddle.com/adding-fractions-6/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Comparing Radicals](https://www.openmiddle.com/comparing-radicals/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make both equations true. Source: Kate Nerdypoo - [Derivative of e](https://www.openmiddle.com/derivative-of-e/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to create an exponential function of base e whose derivative at x = 3 is 2. Source: Christine Relleva - [Absolute Value 2](https://www.openmiddle.com/absolute-value-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Bryan Anderson - [Adding Mixed Numbers 3](https://www.openmiddle.com/adding-mixed-numbers-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box make the largest possible sum. Source: Robert Kaplinsky and Ellen Metzger - [Adding Fractions 5](https://www.openmiddle.com/adding-fractions-5/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Giselle Garcia - [Adding Fractions 3](https://www.openmiddle.com/sum-of-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the sum is equal to 1/2. Source: Daniel Luevanos - [Which Circle Is Bigger? (High School)](https://www.openmiddle.com/which-circle-is-bigger-high-school/) - Directions: Which circle is bigger? Circle A with an area of 25 square units or Circle B with the equation x^2 + y^2 = 25 Source: Robert Kaplinsky - [Writing Linear Equations](https://www.openmiddle.com/writing-linear-equations/) - Directions: Make a table with three points in the same line with 1) a slope not equal to zero 2) and the y-intercept is not a whole number Write the equation for the line. Source: Lane H. Walker - [Trig Ratios](https://www.openmiddle.com/trig-ratios/) - Directions: Using the following trig ratios, complete the following table: cos 30º, sin 30º, cos 45º, sin 45º, tan 30º, cot 30º, sec 30º, csc 30º Source: Bryan Anderson - [Transformations](https://www.openmiddle.com/transformations/) - Directions: Given triangle ABC with vertices (-8,2), (-2,2), and (-2, 8), create triangle DEF in quadrant one that uses a translation, rotation, and reflection (in any order) to take that triangle to triangle ABC and show congruence. Source: Jon Henderson - [Three Triangles And A Wannabe](https://www.openmiddle.com/three-triangles-and-a-wannabe/) - Directions: Using the values 8, 10, 12, 14, 16, 18, and 20, determine lengths for an acute triangle, a right triangle, an obtuse triangle, and a non-triangle. Source: Jonathan Lees - [Square Circumscribed about a Circle](https://www.openmiddle.com/square-circumscribed-about-a-circle/) - Directions: What is the area of the smallest square that could fit around (circumscribed about) a circle with area 100π square units? Source: Nanette Johnson - [Quadrilateral Inscribed in a Circle](https://www.openmiddle.com/quadrilateral-inscribed-in-a-circle/) - Directions: What is the biggest area of a quadrilateral you can fit in a circle that has a circumference of 20π units? Source: Nanette Johnson - [Quadratics with Defined Roots in Vertex Form](https://www.openmiddle.com/quadratics-with-defined-roots-in-vertex-form/) - Directions: Create three equations for quadratics in vertex form that have roots at 3 and 5 but have different maximum and/or minimum values. Source: Robert Kaplinsky - [Quadratics with Defined Roots in Standard Form](https://www.openmiddle.com/quadratics-with-defined-roots-in-standard-form/) - Directions: Create three equations for quadratics in standard form that have roots at 3 and 5. Source: Robert Kaplinsky - [Quadratic Formula](https://www.openmiddle.com/quadratic-formula-2/) - Directions: What are the maximum and minimum values for c if x^2 + 12x + 32 = (x+a) (x+b) + c? Source: Jedidiah Butler - [Multiply Complex Numbers](https://www.openmiddle.com/multiply-complex-numbers/) - Directions: Create two complex numbers (a + bi) such that the product of your numbers is 67. Each value of a, b must be non-zero. Source: Chris Duran - [Longest Chord in a Circle](https://www.openmiddle.com/longest-chord-in-a-circle/) - Directions: What is the longest chord in a circle that has an area of 25π square units? Source: Nanette Johnson - [Line of Reflections on Isosceles Triangles](https://www.openmiddle.com/line-of-reflections-on-isosceles-triangles/) - Directions: How many ways can you determine the location of the line of reflection for isosceles triangle XYZ that maps Point X to Point Z? Source: Irvine Math Project, Nanette Johnson, and Robert Kaplinsky. - [Line Builders](https://www.openmiddle.com/line-builders/) - Directions: Complete the table & graph below or here on Desmos to create a linear relation. Find the equation of the linear relation. Fill in the table again and again to create as many different linear relations as you can. What do the graphs have in common? What do the equations have in common? Source: Jon - [Is the Quadrilateral a Square?](https://www.openmiddle.com/is-the-quadrilateral-a-square/) - Directions: What is the least number of geometric markings needed to demonstrate that a quadrilateral is a square? Source: A collaborative effort of Jose De La Torre and Nanette Johnson answer by Ricardo Navarro with help from Robert Kaplinsky - [Is the Quadrilateral a Kite?](https://www.openmiddle.com/is-the-quadrilateral-a-kite/) - Directions: What is the least number of geometric markings to show that a quadrilateral is a kite? Source: Nanette Johnson - [Geometric Proofs](https://www.openmiddle.com/geometric-proofs/) - Directions: Using exactly five geometric markings to show that a quadrilateral is a square. Source: Robert Kaplinsky - [Finding the Length of a Right Triangle's Altitude](https://www.openmiddle.com/finding-the-length-of-a-right-triangles-altitude/) - Directions: The black triangle is a right triangle with legs 8 and 6. The vertices are at the points (0,0), (0,8), and (6,0). The red line segment is perpendicular to hypotenuse. Find the length of the red line segment. Source: Kate Nerdypoo - [Factoring Polynomials](https://www.openmiddle.com/factoring-polynomials/) - Directions: What numbers go in the blanks to make the equation true? Source: Robert Kaplinsky - [Factoring Complex Numbers](https://www.openmiddle.com/factoring-complex-numbers/) - Directions: Find Integers a,b,c,d such that: Source: Bryan Anderson - [Equilateral Triangle Side Length](https://www.openmiddle.com/equilateral-triangle-side-length/) - Directions: What is the side length of an equilateral triangle that has an area of 5 square units? Source: Robert Kaplinsky - [Domain and Range](https://www.openmiddle.com/domain-and-range/) - Directions: Create 3 lines that have the same domain. Then, create 3 more lines that have the same range. Source: Dane Ehlert - [Create a Quadratic Equation, Given Constraints](https://www.openmiddle.com/create-a-quadratic-equation-given-constraints/) - Directions: Write a quadratic equation that has a y-intercept of 24 and the distance between the x-intercepts is 10. Bonus: find more than 2 quadratic equations. Source: Daniel Luevanos - [Arithmetic vs Geometric](https://www.openmiddle.com/arithmetic-vs-geometric/) - Directions: Which is bigger? The common ratio, r, in a geometric sequence with OR the common difference, d, in an arithmetic sequence with Source: Nanette Johnson - [Angles of a Polygon](https://www.openmiddle.com/angles-of-a-polygon/) - Directions: The measures of the angles of a convex polygon form an arithmetic sequence. The smallest angle has a measurement of 129 degrees. The largest angle has a measurement of 159 degrees. Find the number of sides in this polygon. Source: Ricardo Navarro - [Absolute Value Equation](https://www.openmiddle.com/absolute-value-equation/) - Directions: Create an absolute value equation such that x = - 2 is an extraneous solution. Source: Daniel Luevanos - [Trapezoids: Maximizing Area](https://www.openmiddle.com/trapezoids-maximizing-area/) - Directions: What is the greatest area you can make with a right trapezoid that has a perimeter of 46 units? Source: Patrick McGowan - [Order of Operations 3](https://www.openmiddle.com/order-of-operations-3/) - Directions: Write an expression that is equivalent to 64 using each of the following numbers and symbols once in the expression: 7, 7, 7, 2, + , ÷, ( ) Source: Smarter Balanced Assessment Consortium - [Order of Operations](https://www.openmiddle.com/order-of-operations/) - Directions: Use the Order of Operations with the numbers shown on the card below (in any order) so that when you simplify the expression, the answer is 24. Source: This problem was adapted from the 24 Math Game. - [Median with Constraints](https://www.openmiddle.com/median-with-constraints/) - Directions: Create a statistical data set of at least 10 numbers such that: 1. All of the numbers in the data set are whole numbers. 2. The median is not a whole number. 3. The median is not part of the data set. Source: Daniel Luevanos - [Mean, Median, and Range 2](https://www.openmiddle.com/mean-median-and-range-2/) - Directions: Create a set of five positive integers from 1 to 20 so that the values of their mean, median, and range are the same and have the greatest possible value. Source: Eric Berchtold, Melissa Minnix, and Robert Kaplinsky - [Mean, Median, and Range](https://www.openmiddle.com/mean-median-and-range/) - Directions: Create a set of five positive integers from 1 to 20 that have the same mean, median, and range. Source: Eric Berchtold and Melissa Minnix - [Mean Absolute Deviation](https://www.openmiddle.com/mean-absolute-deviation/) - Directions: Give an example of two sets of numbers that form identical box plots (also called box-and-whisker plots) but have different mean absolute deviation values. Source: Robert Kaplinsky with help from Pamela Franklin - [Lower and Upper Quartiles with Constraints](https://www.openmiddle.com/lower-and-upper-quartiles-with-constraints/) - Directions: Create a statistical data set of at least 10 numbers such that: 1. All of the numbers in the data set are whole numbers. 2. The lower and upper quartiles are not whole numbers. 3. The lower and upper quartiles are not part of the data set. Source: Daniel Luevanos - [Least / Greatest Inequality Values](https://www.openmiddle.com/least-greatest-inequality-values/) - Directions: What's the solution that has the least / greatest value to the inequality 4x > 12? Source: Robert Kaplinsky - [Inequality with No Solution](https://www.openmiddle.com/inequality-with-no-solution/) - Directions: Create an inequality that has no solution. Source: Kara Colley - [Inequalities with Same Number of Solutions](https://www.openmiddle.com/inequalities-with-same-number-of-solutions/) - Directions: Create two inequalities that have the same number of solutions. Source: Robert Kaplinsky - [Get MAD!](https://www.openmiddle.com/get-mad/) - Directions: Construct two sets of 9 numbers that have a mean of 6 and and MAD (mean absolute deviation) of 2. Source: Patrick Sullivan - [Equations](https://www.openmiddle.com/equations/) - Directions: Write three equations whose solution is x = 3. Source: Dan Meyer - [Dividing Fractions 2](https://www.openmiddle.com/fraction-division/) - Directions: Find two fractions whose quotient is 1/20. Source: Kara Colley - [Converting Between Fahrenheit And Celsius](https://www.openmiddle.com/converting-between-fahrenheit-and-celsius/) - Directions: When is the value of the temperature in Fahrenheit the same as the value of the temperature in Celsius? Source: Robert Kaplinsky - [Divisibility 2](https://www.openmiddle.com/divisibility-kitchen-sink/) - Directions: What is the smallest number, greater than zero, that is divisible by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10? Source: Brian Lack - [Pocket Change](https://www.openmiddle.com/pocket-change/) - Directions: You have $1.00 in your pocket. You only have pennies, nickels, and dimes. You don't have any quarters or other coins. What coins are in your pocket? Source: Andrew Gael - [Pocket Change 2](https://www.openmiddle.com/pocket-change-2/) - Directions: You have the same number of pennies, nickels, and dimes in your pocket. You have $1.44. You don't have any other coins or bills. How many of each coin do you have? Source: Andrew Gael - [Pocket Change 3](https://www.openmiddle.com/pocket-change-3/) - Directions: You have $1.00 in change in your pocket. You have 15 coins. What coins do you have? Source: Andrew Gael - [Sum to 10,000](https://www.openmiddle.com/sum-to-10000/) - Directions: Using the digits 1-9 at most one time each, find the closest sum to 10,000 using two 4-digit addends. Source: Jessica Goree - [Which Quadrilateral Has A Greater Area?](https://www.openmiddle.com/which-quadrilateral-has-a-greater-area/) - Directions: Which quadrilateral has a greater area? Quadrilateral A has its perimeter equal to 44 units. Quadrilateral B has the sum of its interior angles equal to 360 degrees. Source: Robert Kaplinsky - [Volume](https://www.openmiddle.com/volume/) - Directions: A rectangular prism has a volume of 144 cubic units and a base of 48 square units. What could the possible dimensions be? Source: Stem Savvy Girls - [Subtracting Fractions 2](https://www.openmiddle.com/fraction-subtraction/) - Directions: Find two fractions whose difference is 1/20. Source: Kara Colley - [Operation Symbols](https://www.openmiddle.com/operation-symbols/) - Directions: Use the operation symbols (+, -, x, and ÷) to make the equation true. Operation symbols may be used more than once. Source: Joshua Nelson - [Multiplying Products to Get as Close to 10000](https://www.openmiddle.com/get-as-close-as-you-can-to-10000/) - Directions: Using the digits 1-9 only once, create two factors that will result in a product as close to 10,000, without going over. Source: Danielle McNichol - [Multiplying Fractions 2](https://www.openmiddle.com/fraction-multiplication/) - Directions: Find two fractions whose product is 1/20. Source: Kara Colley - [Multiplication of large numbers](https://www.openmiddle.com/multiplication-of-large-numbers/) - Directions: Use the digits 1 to 9, at most one time each, to create two numbers that have a product as close to 500,000 as possible. NOTE: You may use any length of factors as you would need. Ex 8 digit by 1 digit. 4 digit by 3 digit. Source: Miles Knight - [Balanced Equations 2](https://www.openmiddle.com/balanced-equations-2/) - Directions: Use the operation symbols (+, -, x, ÷) and equal sign (=) to make a true equation. Operation symbols may be used more than once. What is the least value for each part of the equation? What is the greatest value for each part of the equation? Can you complete the equation with at - [Balanced Equation](https://www.openmiddle.com/balanced-equation/) - Directions: Use the operation symbols (+, -, x, and ÷) to make the equation true. Operations may be used more than once. Source: Joshua Nelson - [Adding Fractions 2](https://www.openmiddle.com/fraction-addition/) - Directions: Find two fractions whose sum is 1/20. Source: Kara Colley - [Which Rectangle is More Like a Square?](https://www.openmiddle.com/which-rectangle-is-more-like-a-square/) - Directions: Which rectangle is more like a square? Rectangle A is 13 units by 15 units. Rectangle B is 2 units by 3 units. Source: Robert Kaplinsky - [Which Circle Is Bigger? (Middle School)](https://www.openmiddle.com/which-circle-is-bigger-middle-school/) - Directions: Which circle is bigger: one with an area of 30 square units or one with a circumference of 30 units? How do you know? Source: Robert Kaplinsky - [What Measurements Make a Circle?](https://www.openmiddle.com/what-measurements-make-a-circle/) - Directions: What is one possible set of values for the area and circumference of the same circle? Source: Nathan Charlton - [Sides of a Triangle](https://www.openmiddle.com/sides-of-a-triangle/) - Directions: The perimeter of a triangle is 20 units. Using whole numbers, how many sets of side lengths can you find for this triangle? Source: Christina Ploeckelman - [Rectangular Prism Surface Area Versus Volume](https://www.openmiddle.com/rectangular-prism-surface-area-versus-volume/) - Directions: What is least amount of surface area possible on a rectangular prism with a volume of 64 cubic inches? Source: Robert Kaplinsky - [Rectangular Prism Surface Area](https://www.openmiddle.com/rectangular-prism-surface-area/) - Directions: List the measurements of three different rectangular prisms that each have a surface area of 72 square units? Source: Robert Kaplinsky - [Product of Distributive Property](https://www.openmiddle.com/product-of-distributive-property/) - Directions: Decide if 30x - 12 could be a result of using the distributive property. If it is, find the possible combinations of factors whose product would be 30x - 12 (using integer coefficients and constants). Source: adapted from Nathan Charlton - [Probability with Spinners](https://www.openmiddle.com/probability-with-spinners/) - Directions: Select three of the spinners from the image below (you may pick more than one of each) such that the total number of sectors in all three spinners totals 10. Select spinners so that the probability of all three spinners landing in the shaded sector is the smallest (or largest). Extension: How would the - [Probability of Rolling Two Six Sided Dice](https://www.openmiddle.com/probability-of-rolling-two-six-sided-dice/) - Directions: What value(s) have a 1/12 chance of being rolled as the sum of two 6-sided dice? Source: Daniel Luevanos - [Multiplying Fractions 3](https://www.openmiddle.com/multiplying-fractions-3/) - Directions: Find three fractions whose product is -5/24. You may use fractions between -8/9 to 8/9 no more than one time each. Find at least 2 possible combinations. Source: Al Oz - [Maximizing Surface Area](https://www.openmiddle.com/reduce-volume-keep-surface-area-constant/) - Directions: The following prism is made up of 27 identical cubes. What is the greatest possible surface area the prism can have after removing 1 or more cubes from the outside? Source: Brian Lack - [Maximizing Rectangular Prism Volume Versus Surface Area](https://www.openmiddle.com/maximizing-rectangular-prism-volume-versus-surface-area/) - Directions: What is the greatest volume you can make with a rectangular prism that has a surface area of 20 square units? Source: Robert Kaplinsky - [Interpreting Percentages](https://www.openmiddle.com/interpretting-percentages/) - Directions: What is the fewest number of people surveyed if exactly 93.6% of people completed a survey? Source: Robert Kaplinsky - [Circles and Square](https://www.openmiddle.com/circles-and-square/) - Directions: The diagram shows a square and four semicircles formed using each side of the square as a diameter. What fraction of the square is shaded? Source: Dylan Kane - [Circle Radius](https://www.openmiddle.com/circle-radius/) - Directions: What is a radius for a circle that has an area of 20 to 25 square feet? Source: Nathan Charlton - [Area/Circumference of Circles](https://www.openmiddle.com/areacircumference-of-circles/) - Directions: If possible, find the radius of a circle where the area of the circle and the circumference of the circle are equal. Is there more than one possible answer? Source: Karen Bloom - [Incorrect Linear Equations](https://www.openmiddle.com/incorrect-linear-equations/) - Directions: Based on the graph of this line, write a linear equation that is incorrect. Write an a linear equation that could be correct. Source: Sam Olderbak - [Transformations - Three Sequences](https://www.openmiddle.com/transformations-three-sequences/) - Directions: List three sequences of transformations that take pre-image ABCD to image A'B'C'D'. Source: Robert Kaplinsky - [Interior and Exterior Angles #1](https://www.openmiddle.com/interior-and-exterior-angles-1/) - Directions: How can you tell what kind of regular polygon has the same measure for its interior and exterior angles? Source: Ricardo Navarro, Gladys Velasquez, Bryan Aguilar, and Eddie Galaviz - [Table of Values: Not a Function](https://www.openmiddle.com/table-of-values-not-a-function/) - Directions: Create a table of values that is not a function. Source: Nanette Johnson and Robert Kaplinsky - [Y-Intercept](https://www.openmiddle.com/y-intercept/) - Directions: Write a linear equation with a y-intercept of (0, -100). Source: Kara Colley - [X-Intercept](https://www.openmiddle.com/x-intercept/) - Directions: Write a linear equation that has an x-intercept of (-100, 0). Source: Kara Colley - [Two-Way Tables](https://www.openmiddle.com/two-way-tables/) - Directions: Using the whole numbers 0 to 20 no more than one time each, fill in the two-way frequency table so that chocolate has a higher relative frequency than vanilla and that 8th grade students has a lower frequency than 7th grade students. Source: Eric Zuercher - [Slope](https://www.openmiddle.com/slope/) - Directions: Write a linear equation that has a slope of -100. Source: Kara Colley - [Transformations - Shortest Sequence](https://www.openmiddle.com/transformations-shortest-sequence/) - Directions: What is the fewest number of transformations needed to take pre-image ABCD to image A'B'C'D'? Source: Robert Kaplinsky - [Table of Values: Function](https://www.openmiddle.com/table-of-values-function/) - Directions: Create a table of values that is a function. Source: Nanette Johnson and Robert Kaplinsky - [Solving Radical Equations](https://www.openmiddle.com/solving-radical-equations/) - Source: Smarter Balanced Practice Question - [Maximizing Volume of a Cylinder, Given Lateral Area](https://www.openmiddle.com/maximizing-volume-of-a-cylinder-given-lateral-area/) - Directions: Find at least 3 possible measures for the height and the radius of a cylinder with a lateral area of 144pi square centimeters. Which of your dimensions will give you the largest volume? Source: Jason Miller - [Pythagorean Theorem](https://www.openmiddle.com/pythagorean-theorem-prob/) - Directions: What could the lengths of the legs be such that the lengths are integers and x is an irrational number between 5 and 7? Source: Daniel Luevanos - [Non-Linear Correlations](https://www.openmiddle.com/non-linear-correlation/) - Directions: Using the integers 0-9 (without repeating any number), create a set of points that have the following characteristics: Non-linear Positive Correlation (__,__) (__,__) (__,__) (__,__) (__,__) Non-linear Negative Correlation (__,__) (__,__) (__,__) (__,__) (__,__) No Correlation (__,__) (__,__) (__,__) (__,__) (__,__) Source: Bryan Anderson - [Perimeter and Pythagorean Theorem](https://www.openmiddle.com/perimeter-and-pythagorean-theorem/) - Directions: What could the lengths of the legs be such that the lengths of the legs are integers and x is an irrational number between 5 and 7? Source: Daniel Luevanos - [Interior and Exterior Angles #2](https://www.openmiddle.com/interior-and-exterior-angles-2/) - Directions: If n is equal to the number of sides on a regular polygon, what is the lowest value of n where each interior angle's measure is less (or greater) than the exterior angle? Source: Ricardo Navarro, Gladys Velasquez, Bryan Aguilar, and Eddie Galaviz - [Interior and Exterior Angles #3](https://www.openmiddle.com/interior-and-exterior-angles-3/) - Directions: If n is equal to the number of sides, when is the sum of the interior angles less than (or greater than or equal to) the sum of the exterior angles? Source: Ricardo Navarro, Gladys Velasquez, Bryan Aguilar, and Eddie Galaviz - [Equidistant Points](https://www.openmiddle.com/equidistant-points/) - Directions: How many points with two integer coordinates are 5 units away from (-2, 3)? Source: Dylan Kane - [Cylinders](https://www.openmiddle.com/cylinders/) - Source: Smarter Balanced Practice Test - [Create a System of Equations, Given 1 Equation and the Solution](https://www.openmiddle.com/create-a-system-of-equations-given-1-equation-and-the-solution/) - Directions: Write at least two linear equations so that the solution of the system of equations of that line and 4x + y = 8 is (3, -4) Source: Nanette Johnson - [Commuting Exponents](https://www.openmiddle.com/commuting-exponents/) - Directions: Place a set of parenthesis on each term to make the inequality below true: 10^10^100 < 10^10^100 Source: Shaun Errichiello - [Pythagorean Shell](https://www.openmiddle.com/pythagorean-shell/) - Directions: Find the length of side x in the diagram below Source: Dylan Kane - [Cone and Cylinder Volumes](https://www.openmiddle.com/cone-and-cylinder-volumes/) - Directions: A cone is twice as tall as a cylinder of the same radius. What is the ratio of their volumes? Source: Dylan Kane - [Got change for a dollar?](https://www.openmiddle.com/got-change-for-a-dollar/) - Directions: What is the largest amount in coins you can have and not be able to make change for a dollar without shorting yourself or cheating the other person? Source: Glenn Waddell - [How Many Numbers Are There?](https://www.openmiddle.com/how-many-numbers-are-there/) - Directions: How many numbers are between 1 and 3? Source: Robert Kaplinsky - [Identify a Fraction on a Number Line](https://www.openmiddle.com/identify-a-fraction-on-a-number-line/) - Directions: Label the point where 3/4 belongs on the number line. Be as precise as possible. Source: Inspired by Illustrative Mathematics - [Perimeter](https://www.openmiddle.com/perimeter/) - Directions: Draw three rectangles with a perimeter of 20 units. Source: Dan Meyer - [Rectangles: Maximizing Area](https://www.openmiddle.com/rectangles-maximizing-area/) - Directions: What is the greatest area you can make with a rectangle that has a perimeter of 24 units? Source: Robert Kaplinsky - [Rectangles: Maximizing Perimeter](https://www.openmiddle.com/rectangles-maximizing-perimeter/) - Directions: What is the greatest perimeter you can make with a rectangle that has an area of 24 square units? Source: Robert Kaplinsky - [Rectangles: Perimeter v. Area](https://www.openmiddle.com/rectangles-perimeter-v-area/) - Directions: How can you tell which rectangle is bigger: a rectangle with a perimeter of 24 units or a rectangle with an area of 24 square units? Source: Robert Kaplinsky - [Squares: Perimeter v. Area](https://www.openmiddle.com/squares-perimeter-v-area/) - Directions: How can you tell which square is bigger: a square with a perimeter of 25 units or a square with an area of 25 square units? Source: Robert Kaplinsky - [Fractions On A Number Line](https://www.openmiddle.com/fractions-on-a-number-line/) - Directions: Using the digits 0 to 5 at most one time each, place a digit to create five fractions and place them all on a number line with the correct order and spacing. Source: Robert Kaplinsky - [Time Twister](https://www.openmiddle.com/time-twister/) - Directions: Using the digits 0 to 9, at most one time each, create three different times on the clocks where the span of the times are between 12 noon and 7 pm. How can you make the difference between the times the greatest? closest times together? Source: Jason Kornoely - [Making Change 2](https://www.openmiddle.com/making-change-2/) - Directions: Make 47¢ using exactly 6 coins with either quarters, dimes, nickels, or pennies. Source: Thad Domina and Robert Kaplinsky - [Making Change](https://www.openmiddle.com/making-change/) - Directions: Make 47¢ in three different ways with either quarters, dimes, nickels, or pennies. Source: Thad Domina and Robert Kaplinsky - [Drawing and Naming Shapes by Angles](https://www.openmiddle.com/drawing-and-naming-shapes-by-angles/) - Directions: Draw and name a shape that has the following characteristics: Has 3 angles Has 4 angles Has 5 angles Has 6 angles Has two equal sides Has five equal sides Source: Bryan Anderson - [Constructing Rectangles](https://www.openmiddle.com/constructing-rectangles/) - Directions: Using the following squares, how many different rectangles can you make? Source: Bryan Anderson - [Shape Partitions](https://www.openmiddle.com/shape-partitions/) - Directions: Using the same cut pattern for each figure, partition each shape into fourths. Using different cut patterns for each figure, partition each shape into fourths Source: Bryan Anderson - [Representing Data](https://www.openmiddle.com/representing-data/) - Directions: Using the counting numbers 1 to 6, each only once, fill in the graph and blanks to make the statements true. There are twice as many dogs as cats. There are twice as many cats as birds. There are ___ dogs, ___ cats and ___ birds in class. Source: Bryan Anderson - [Parts Unknown Problems](https://www.openmiddle.com/math-story/) - Directions: Complete the story problem and answer statement. Version 1 (Difficult) Lucy has _____ apples. She has nine _____ (more/less) than Marcus. How many apples does _____ (Lucy/Marcus) have? _____ (Lucy/Marcus) has _____ apples. Version 2 (Medium Difficulty) Lucy has _____ apples. She has nine less than Marcus. How many apples does _____ (Lucy/Marcus) have? - [Ordering Shapes](https://www.openmiddle.com/ordering-shapes/) - Directions: Order the squares from shortest to tallest. Order the rectangles from tallest to shortest. Order both the squares and rectangles from tallest to shortest. Source: Bryan Anderson - [Interpreting Data 2](https://www.openmiddle.com/interpreting-data-2/) - Directions: Make a graph that shows a possible result of 7 students’ favorite color with red being the most popular color. Source: Robert Kaplinsky - [Interpreting Data](https://www.openmiddle.com/interpreting-data/) - Directions: Make a graph that shows a possible result of 7 students’ favorite color. Source: Robert Kaplinsky - [Drawing and Naming Shapes by Sides](https://www.openmiddle.com/drawing-and-naming-shapes-by-sides/) - Directions: Draw and name a shape that has the following characteristics: Has 3 sides Has 4 sides Has 5 sides Has 6 sides Has no straight sides Source: Bryan Anderson - [Domino Window 1](https://www.openmiddle.com/domino-window-1/) - Directions: Use four of these dominoes to form a square with the same number of dots on each side. Source: Joshua Nelson - [Domino Window](https://www.openmiddle.com/domino-window/) - Directions: Use four of these dominoes to form a square with the same number of dots on each side. Source: Joshua Nelson - [Composite 2D Shapes](https://www.openmiddle.com/composite-2d-shapes/) - Directions: What shapes could be used to create this picture? Make a list of the shapes needed, and how many of each you would need. Source: Bryan Anderson - [Ten Frame Challenge](https://www.openmiddle.com/ten-frame-challenge/) - Directions: I have a horizontal ten-frame that has some counters on it. One row of the frame is full and one is not. What is the largest number I could make? What is the smallest number I could make? Source: Elizabeth Brandenburg - [Teen Number with 10 Frames](https://www.openmiddle.com/teen-number-with-10-frames/) - Directions: I have 2 ten-frames that have counters on them. One is full and one is not. What is the largest number I could make? What is the smallest number I could make? Source: Brian Kelley - [Sum of 5](https://www.openmiddle.com/sum-of-5/) - Directions: I rolled 2 dice and when I counted the pips (dots), there were 5 altogether. What could I have rolled on the dice? I rolled again and got 5 again, but I didn’t get the same numbers as before. What could my new roll be? Source: Brian Kelley - [Identifying Shapes](https://www.openmiddle.com/identifying-shapes/) - Directions: Using the digits 1 to 6, at most once each time, fill in boxes and identify a shape in the blank to make as many of the following statements true as you can. Source: Bryan Anderson - [Dot Card Counting](https://www.openmiddle.com/dot-card-counting/) - Challenge students to count the number of dots in this Open Middle problem. Great for 1st grade students practicing counting and algebraic thinking. - [Domino Friends of Ten](https://www.openmiddle.com/domino-friends-of-ten/) - Directions: I picked 3 dominoes out of a bag and they all had exactly 10 pips, but the same number was not on both sides of any of the dominoes. Which dominoes could I have picked? Is there more than one answer? Source: Brian Kelley - [Describing Shapes](https://www.openmiddle.com/describing-shapes/) - Directions: Using the following picture, complete the following sentences (using the phrases: above, below, beside, in front of, behind, and next to) The cube is ___________ the sphere and ___________ the triangle. The hexagon is __________ the pentagon and __________ the circle. Use the shape names to complete the following statements: The ________ is next - [Decomposing Numbers Less Than Or Equal to 10](https://www.openmiddle.com/decomposing-numbers-less-than-or-equal-to-10/) - Directions: Add spots to each dog to make the total number of spots equal 10. Source: Unknown - [Caterpillar Counting](https://www.openmiddle.com/caterpillar-counting/) - Directions: Complete the following number sequences and create spots on the caterpillar's body that represents the number above it. Source: Bryan Anderson - [Analyzing Shapes](https://www.openmiddle.com/analyzing-shapes/) - Directions: Using the diagram, fill in the blanks with the names of the shapes to make each statement true. __________ has more sides than __________ __________ has the same number of sides as __________ __________ has more vertices than __________ Source: Bryan Anderson - [Multiplying Decimals Given One](https://www.openmiddle.com/multiplying-decimals-given-one/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the statement true. Source: Robert Kaplinsky with help verifying the answer from Marcia and Rick Casterline - [Asymptotes of Rational Functions](https://www.openmiddle.com/asymptotes-of-rational-functions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the statements true. Source: Dana Harrington - [Operations with Rational Numbers](https://www.openmiddle.com/operations-with-rational-numbers/) - Directions: Using the digits 2 to 9, at most one time each, place a digit in each box to make the statements true. Source: Bryan Anderson - [Sums with the Same Digit (Decimal Addition)](https://www.openmiddle.com/sums-with-the-same-digit-decimal-addition/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to find a sum where all the digits are the same. What is the greatest sum? What is the least sum? Source: Elizabeth Trochil - [Reciprocal Fractions](https://www.openmiddle.com/reciprocal-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the maximum possible value of the expression given. Source: Jacob Johanssen - [Square and Circle Area](https://www.openmiddle.com/square-and-circle-area/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to create the largest possible combined area for the rectangle and circle. Source: Mike Chamberlain - [Inequality Expressions 2](https://www.openmiddle.com/inequality-expressions-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a true inequality. Source: Bryan Anderson - [Create an Equation with a Solution Closest to Zero](https://www.openmiddle.com/create-an-equation-with-a-solution-closest-to-zero/) - Challenge students to use the digits 1-9 to find a solution close to zero in this DOK 3 problem. Great for 8th grade students practicing expressions and equations. - [Quadratics and Number of Solutions](https://www.openmiddle.com/quadratics-and-number-of-solutions/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create three quadratic equations: one with two imaginary solutions, one with one real solution, and one with two real (rational or irrational) solutions. Source: Ryan D. Fox - [Circle Tangent to Line](https://www.openmiddle.com/circle-tangent-to-line/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a circle tangent to the line x+y=5. Source: Linnea Reyes-LaMon - [Rational Roots](https://www.openmiddle.com/rational-roots/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create expressions that produce rational roots. Source: Norma Gordon - [Triangle Sum Theorem](https://www.openmiddle.com/triangle-sum-theorem/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that when you solve for x, it is a whole number. Source: Franco D. Adkins - [Radical Equations](https://www.openmiddle.com/radical-equations/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make both of these equations true. Source: Jonathan Newman - [Evaluating Logs](https://www.openmiddle.com/evaluating-logs/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Dana Harrington - [Ordering Rational Numbers](https://www.openmiddle.com/ordering-rational-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the inequality true. Source: Robert Millett - [Equivalent Expressions 4](https://www.openmiddle.com/equivalent-expressions-4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: James Cresswell - [Fraction Multiplication Equal to 1](https://www.openmiddle.com/fraction-multiplication-equal-to-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the three fractions have a product as close to 1 as possible. Source: Patrick Vennebush - [Subtracting Decimals 2](https://www.openmiddle.com/subtracting-decimals-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Adina Rochkind - [Adding Fractions 4](https://www.openmiddle.com/adding-fractions-4/) - Directions: Using the integers 1 to 10, at most one time each, place an integer in each box so that the sum is equal to 1. Source: Joshua Nelson - [Comparing Fractions to Decimals](https://www.openmiddle.com/comparing-fractions-to-decimals/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the inequality true. Source: Owen Kaplinsky - [Exponent Products](https://www.openmiddle.com/exponent-products/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Stephen Cox - [Place Value](https://www.openmiddle.com/place-value/) - Directions: Using the digits 1 to 9, exactly two times each, place a digit in each box to create a nine digit number and its corresponding place values. Source: Owen Kaplinsky - [Four Digit Products](https://www.openmiddle.com/four-digit-products/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Adina R - [Simple Patterns 1](https://www.openmiddle.com/simple-patterns-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a true pattern. Source: Robert Kaplinsky - [Fraction Equivalence](https://www.openmiddle.com/fraction-equivalence/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a fraction that correctly completes each equation. Source: Ian Kerr - [Adding Three Fractions](https://www.openmiddle.com/adding-three-fractions/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Denise White - [Multiplying Multiples Of Ten 2](https://www.openmiddle.com/multiplying-multiples-of-ten-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make a product that’s as close to 500 as possible. Source: Robert Kaplinsky - [Create an Equation](https://www.openmiddle.com/create-an-equation/) - Directions: Using the digits 1 to 7, at most one time each, place a digit in each box to create a true statement. Source: Eric Appleton Challenge students to use the digits 1-7 to in this DOK 2 problem. Great for 2nd grade students practicing operations in base ten. - [Subtraction with Regrouping 2](https://www.openmiddle.com/subtraction-with-regrouping-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the difference equal to 39. Source: Chris Ignaciuk - [Standard Deviation](https://www.openmiddle.com/standard-deviation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create an ordered set of data with the largest possible standard deviation. Source: Mark Alvaro - [Slope From Two Points](https://www.openmiddle.com/slope-from-two-points/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the greatest and least possible slope. Source: Dane Ehlert - [Comparing Fractions to Decimals 2](https://www.openmiddle.com/comparing-fractions-to-decimals-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the inequality true. Source: Owen Kaplinsky - [Distributive Property 2](https://www.openmiddle.com/distributive-property-2-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Adrianne Burns - [Distributive Property 3](https://www.openmiddle.com/distributive-property-3/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Julie Wright - [Equivalent Exponents](https://www.openmiddle.com/equivalent-exponents/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create as many true equations as possible. Source: Annie DeAngelo and Maeve O'Connell - [Prime Factorization 2](https://www.openmiddle.com/prime-factorization-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the greatest possible product. Source: Robert Kaplinsky - [Subtracting 3-Digit Numbers 2](https://www.openmiddle.com/subtracting-3-digit-numbers-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a difference that is as close to 329 as possible. Source: Robert Kaplinsky - [Make The Time](https://www.openmiddle.com/make-the-time/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make a time that is 3:57 pm. Source: Patty Stephens - [Unit Rates with Fractions 2](https://www.openmiddle.com/unit-rates-with-fractions-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a unit rate with the greatest possible value. Source: Robert Kaplinsky - [Multiplying Integers 2](https://www.openmiddle.com/multiplying-integers-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make the greatest possible product. Source: Robert Kaplinsky - [Inequality Expressions 4](https://www.openmiddle.com/inequality-expressions-4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a true inequality. Source: Bryan Anderson - [Inequality Expressions 3](https://www.openmiddle.com/inequality-expressions-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a true inequality. Source: Bryan Anderson - [Volume of Rectangular Prisms](https://www.openmiddle.com/volume-of-rectangular-prisms-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a rectangular prism with a volume that is less than 100 cubic units. What's the least volume? What's the greatest volume? Source: Kari Frazier - [Fractions: Sum of 2](https://www.openmiddle.com/fractions-sum-of-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that the sum is equal to 2 wholes. Source: Joshua Nelson - [Multiplying Fractions 5](https://www.openmiddle.com/multiplying-fractions-5/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Robert Kaplinsky - [Multiplying Fractions 6](https://www.openmiddle.com/multiplying-fractions-6/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a product that's as close to 4/11 as possible. Source: Robert Kaplinsky - [Multiplying Fractions](https://www.openmiddle.com/multiplying-fractions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) possible product. Source: Robert Kaplinsky - [Adding Mixed Numbers](https://www.openmiddle.com/adding-mixed-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the largest sum possible. Source: Robert Kaplinsky - [Make the Biggest Circle](https://www.openmiddle.com/make-the-biggest-circle/) - Directions: Using the digits 1 to 9, as many times as you want, place a digit in each box to make the biggest circle possible. Source: Robert Kaplinsky - [What's Your Sine?](https://www.openmiddle.com/whats-your-sine/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make three true statements. Source: Zack Miller - [Quadratic Formula](https://www.openmiddle.com/quadratic-formula/) - Directions: Using the digits 1 to 9, at most one time each, find the result with the greatest and least value. Source: Dane Ehlert - [Percents on a Linear Model 5](https://www.openmiddle.com/percents-on-a-linear-model-5/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create an accurate number line. Source: Adrianne Burns - [Biggest Rectangle](https://www.openmiddle.com/biggest-rectangle/) - Directions: Using the digits 1 to 9, at most one time each, find the largest possible area for the rectangle. Source: Nanette Johnson, Inspired by Mike Chamberlain's Problem - [Missing Digits](https://www.openmiddle.com/missing-digits/) - Challenge students to use digits to find an answer close to 200 in this DOK 2 problem. Great for 3rd grade students practicing operations in base ten. - [Complex Number Products (Greatest Value)](https://www.openmiddle.com/complex-number-products-greatest-value/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make a real number product with the greatest value. Source: Robert Kaplinsky - [Complex Number Products](https://www.openmiddle.com/complex-number-products/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to make a positive real number product and then repeat the process to make a negative real number product. You may use all the integers each time. Source: Robert Kaplinsky - [Compound Inequalities 2](https://www.openmiddle.com/compound-inequalities-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make two compound inequalities that are equivalent to 2 ≤ x < 4. Source: Robert Kaplinsky - [Compound Inequalities 1](https://www.openmiddle.com/compound-inequalities-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a compound inequality that has the largest interval. Source: Robert Kaplinsky - [Equivalent lines in slope-intercept and standard form](https://www.openmiddle.com/equivalent-lines-in-slope-intercept-and-standard-form-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to complete the statement below. Source: Andy Schwen - [Create an Equation with a Given Solution](https://www.openmiddle.com/create-an-equation-with-a-given-solution/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to write three equations whose solution is -1/2. Source: Daniel Luevanos - [Simplifying Exponential Expressions](https://www.openmiddle.com/simplifying-exponential-expressions/) - Directions: Using the integers 1 to 10, at most one time each, place an integer in each box so that the result is closest to 1. Source: Daniel Luevanos - [Volume of Rectangular Prisms 2](https://www.openmiddle.com/volume-of-rectangular-prisms-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two rectangular prisms where the larger one has the greatest possible volume and is double the volume of the other. Source: Joe Schwartz and Robert Kaplinsky - [Volume of Rectangular Prisms](https://www.openmiddle.com/volume-of-rectangular-prisms/) - Challenge students to use the digits 1-9 to create 2 rectangular prisms in this DOK 2 problem. Great for 5th grade students practicing measurement & data. - [Volume of Rectangular Prisms](https://www.openmiddle.com/volume-of-rectangular-prisms-4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a rectangular prism with a volume that is greater than 100 cubic units. What's the least volume? What's the greatest volume? Source: Kari Frazier - [Sum of Fractions Closest to 10](https://www.openmiddle.com/sum-of-fractions-closest-to-10/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation as close to 9.8 as possible. Source: Nanette Johnson, based on Giselle Garcia's problem - [Decimal Product Close To 50](https://www.openmiddle.com/decimal-product-close-to-50/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the product is as close to 50 as possible. Source: Robert Kaplinsky - [Constructing a Nondifferentiable Function](https://www.openmiddle.com/constructing-a-nondifferentiable-function/) - Directions: Using the digits -9 to 9, at most two times each, place a digit in each box so that the function is nondifferentiable at the given x value. Extension: Can you make the function continuous but nondifferentiable? Source: Owen Kaplinsky - [How to Use Embedded Open Middle Problems](https://www.openmiddle.com/about-embedding/) - Want to add Open Middle problems to your Canvas or website? Learn how to embed problems and customize them for your students in this quick guide. - [Piecewise Continuity](https://www.openmiddle.com/piecewise-continuity/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the function continuous/discontinuous at 3. Source: Owen Kaplinsky - [Adding Fractions to Make a Whole Number](https://www.openmiddle.com/adding-fractions-to-make-a-whole-number/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a whole number sum. Can you make all whole numbers from 1 to 9? Source: Owen Kaplinsky - [Area of an Obtuse Triangle](https://www.openmiddle.com/area-of-an-obtuse-triangle/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a triangle with side lengths that give the corresponding area. Source: Owen Kaplinsky - [Area of a Rectangle](https://www.openmiddle.com/area-of-a-rectangle/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make it so that the value for the area of the rectangle (in square units) is greater that the value for the perimeter (in linear units). What is the greatest difference you can find between the - [Area & Perimeter of a Rectangle](https://www.openmiddle.com/area-perimeter-of-a-rectangle/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a rectangle with an area as close to 500 and a perimeter as close to 100 as possible. Source: Owen Kaplinsky - [Dividing Whole Numbers With A Decimal Quotient](https://www.openmiddle.com/dividing-whole-numbers-with-a-decimal-quotient/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Multiplying Differences 2](https://www.openmiddle.com/multiplying-differences-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a product that's as close to 50 as possible. Source: Owen Kaplinsky - [Decimal Subtraction 2](https://www.openmiddle.com/decimal-subtraction-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a difference with the least possible value. Source: Owen Kaplinsky and Robert Kaplinsky - [Decimal Addition 3](https://www.openmiddle.com/decimal-addition-3/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a sum with the greatest possible value. Source: Owen Kaplinsky and Robert Kaplinsky - [Multiplying Decimals to Make a Whole Number Product](https://www.openmiddle.com/multiplying-decimals-to-make-a-whole-number-product/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a whole number product. Source: Owen Kaplinsky - [Composite Numbers](https://www.openmiddle.com/composite-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make 5 composite numbers. Source: Owen Kaplinsky - [Two-Step Equations 2](https://www.openmiddle.com/two-step-equations-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the largest (or smallest) possible values for x. Source: Chase Orton and Mark Goldstein - [Two-Step Equations](https://www.openmiddle.com/two-step-equations/) - Challenge students to use the digits 1-9 to find values for x in this DOK 3 problem. Great for 7th grade students practicing expressions & equations. - [Greatest Common Factor](https://www.openmiddle.com/greatest-common-factor/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a true statement. Source: Cecilia Calvo Pesce - [Fraction of an Amount](https://www.openmiddle.com/fraction-of-an-amount/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make as many true statements as possible. Source: Rochelle Telfer - [Multiplying a Two-Digit Number by a Single-Digit Number](https://www.openmiddle.com/multiplying-a-two-digit-number-by-a-single-digit-number/) - Challenge students to use the digits 1-4 to make the largest possible product in this DOK 3 problem. Great for 3rd grade students practicing operations in base ten. - [Subtraction with Zeros](https://www.openmiddle.com/subtraction-with-zeros/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the greatest/smallest difference. Source: Ellen Metzger - [Subtracting Two-Digit Numbers (Elementary)](https://www.openmiddle.com/subtracting-two-digit-numbers-elementary/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the smallest (or largest) difference. Source: Robert Kaplinsky - [Adding and Subtracting Two-Digit Whole Numbers](https://www.openmiddle.com/adding-and-subtracting-two-digit-whole-numbers/) - Challenge students to use the digits 0-9 to make a true statement in this DOK 3 problem. Great for 2nd grade students practicing operations in base ten. - [Sum to 1,000 - Two Addends](https://www.openmiddle.com/sum-to-1000-two-addends/) - Directions: Using the digits 1 to 6, at most one time each, place a digit in each box to create two 3-digit numbers. Make the sum as close to 1000 as possible. Source: Ian Kerr - [Multiplying Two-Digit Numbers (Elementary)](https://www.openmiddle.com/multiplying-two-digit-numbers-elementary/) - Challenge students to use the digits 1-9 to make the smallest or largest product in this DOK 3 problem. Great for 4th and 5th grade students practicing operations in base ten. - [Solving Equations with Variables on Both Sides](https://www.openmiddle.com/solving-equations-with-variables-on-both-sides/) - Challenge students to use the digits 1-9 to make an equation with no solutions in this DOK 2 problem. Great for 8th grade students practicing reasoning with expressions. - [Equivalent Exponents](https://www.openmiddle.com/equivalent-exponents-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a true in equality and a true equation. An extension is to do so while leaving the greatest or smallest number unused. Source: John Joseph Vasko jr - [Solving Quadratic Functions using the Square Root Method](https://www.openmiddle.com/solving-quadratic-functions-using-the-square-root-method/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two separate quadratic equations. Source: Avanti Yamamoto - [Factoring using Greatest Common Factor](https://www.openmiddle.com/factoring-using-greatest-common-factor/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the largest possible Greatest Common Factor for the polynomial. Then factor out the GCF. Extension: Repeat the activity with a two-digit GCF. Source: David Groat - [Completing the Square](https://www.openmiddle.com/completing-the-square/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Kate Nerdypoo - [Summation of Multiplication Max/Min Value](https://www.openmiddle.com/summation-of-multiplication-max-min-value/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the largest/smallest possible sum. Source: Owen Kaplinsky - [Closest to One](https://www.openmiddle.com/closest-to-one/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a fraction as close to one as possible. Source: Peter Morris - [Adding Parts of a Whole](https://www.openmiddle.com/adding-parts-of-a-whole/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Miles Knight - [Dividing Fractions 4](https://www.openmiddle.com/dividing-fractions-4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create an equation with the greatest possible quotient. Source: Robert Kaplinsky - [Square Root Expression](https://www.openmiddle.com/square-root-expression/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the following expression as close to 0 as possible. Source: Erick Lee - [Differences in Scientific Notation](https://www.openmiddle.com/differences-in-scientific-notation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the largest (or smallest) absolute difference. Source: Marie Isaac - [Finding Equivalent Ratios](https://www.openmiddle.com/finding-equivalent-ratios/) - Challenge students to use the digits 1-9 to create 3 equivalent ratios in this DOK 2 problem. Great for 6th grade students practicing ratios & proportions. - [Finding Equivalent Ratios 2](https://www.openmiddle.com/finding-equivalent-ratios-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create equivalent ratios. Source: Lorna McClory - [Fraction Division](https://www.openmiddle.com/fraction-division-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Shaun Errichiello - [Equivalent Expressions 2](https://www.openmiddle.com/equivalent-expressions-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Will Case - [Solving One-Step Equations (Greatest Solution)](https://www.openmiddle.com/solving-one-step-equations-greatest-solution/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create an equation where x has the greatest possible value. Source: Robert Kaplinsky - [Sine Functions 2](https://www.openmiddle.com/sine-functions-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to find the function's greatest possible value. Source: Robert Kaplinsky - [Sine Functions](https://www.openmiddle.com/sine-functions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box and make two true number sentences. Source: Robert Kaplinsky - [Second Derivative](https://www.openmiddle.com/second-derivative/) - Directions: Using the digits 0 to 9, at most two times each, place a digit in each box so that each expression is the correct derivative of the one above it. Source: Owen Kaplinsky - [Powers of "i"](https://www.openmiddle.com/powers-of-i/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Jason Curley - [Power Rule with Polynomials](https://www.openmiddle.com/power-rule-with-polynomials/) - Directions: Using the digits 0 to 9, at most two times each, place a digit in each box so that the expression for f'(x) is the correct derivative of f(x). Source: Owen Kaplinsky - [Function Notation](https://www.openmiddle.com/function-notation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the values of the two functions are equivalent. Source: Steven Midzak - [Fractional Power to a Power](https://www.openmiddle.com/fractional-power-to-a-power/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Kate Nerdypoo - [Multiplying Radicals](https://www.openmiddle.com/multiplying-radicals/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Katelyn Devine - [Minimum Value of a Quadratic in Factored Form](https://www.openmiddle.com/minimum-value-of-a-quadratic-in-factored-form/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create a quadratic in factored form with the lowest minimum. Source: Ryan Kimes - [Trigonometric Equation](https://www.openmiddle.com/trigonometric-equation/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the trigonometric equation true. Source: Kevin Rees - [Systems of Inequalities 2](https://www.openmiddle.com/systems-of-inequalities-2/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a system of inequalities as well as an included and excluded point. Make the points as close together as possible. Source: Robert Kaplinsky - [Systems of Inequalities 1](https://www.openmiddle.com/systems-of-inequalities-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a system of inequalities as well as an included and excluded point. Source: Robert Kaplinsky - [Coordinate Parallelograms](https://www.openmiddle.com/coordinate-parallelograms/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that the points make a parallelogram. Source: Daniel Torres-Rangel - [Adding to Normal Distributions](https://www.openmiddle.com/adding-to-normal-distributions/) - Directions: Using the digits 0 to 9, at most two times each, place a digit in each box to make a valid transformation from the original normal distribution. Source: Owen Kaplinsky - [Ratios 2](https://www.openmiddle.com/ratios-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make an equivalent ratio with a unit rate that has greatest possible value. Source: Robert Kaplinsky - [Ratios 1](https://www.openmiddle.com/ratios-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box make an equivalent ratio. Source: Robert Kaplinsky - [Decimal Addition](https://www.openmiddle.com/decimal-addition/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Shaun Errichiello - [Subtracting Decimals To Get Close To 0](https://www.openmiddle.com/subtracting-numbers-to-get-close-to-0/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to get a difference closest to 0. Source: Owen Kaplinsky - [Multiplying Fractions to Make a Whole Number](https://www.openmiddle.com/multiplying-fractions-to-make-a-whole-number/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a whole number product. Source: Owen Kaplinsky - [Multiplying A Fraction By A Whole Number To Make 1](https://www.openmiddle.com/multiplying-a-fraction-by-a-whole-number-to-make-1/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Rounding Decimals 3](https://www.openmiddle.com/rounding-decimals-3/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two different decimals that are equivalent when rounded to the nearest tenth and have the least possible value. Source: Robert Kaplinsky - [Rounding Decimals 2](https://www.openmiddle.com/rounding-decimals-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create two different decimals that are equivalent when rounded to the nearest tenth. Source: Robert Kaplinsky - [Order of Operations 5](https://www.openmiddle.com/order-of-operations-5/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box so that each expression is simplified to a different odd number. Source: Molly Rawding - [Order of Operations 4](https://www.openmiddle.com/order-of-operations-4/) - Challenge students to use the digits 0-9 to make a true inequality in this open-ended order of operations problem. Great for 5th grade students practicing PEMDAS reasoning. - [Multi-Digit Multiplication 2](https://www.openmiddle.com/multi-digit-multiplication-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a true equation with the greatest possible product. Source: Robert Kaplinsky - [Multi-Digit Multiplication 1](https://www.openmiddle.com/multi-digit-multiplication-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a true equation. Source: Robert Kaplinsky - [Evaluating Expressions 2](https://www.openmiddle.com/evaluating-expressions-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create the greatest possible value. Source: Robert Kaplinsky - [Decimal Subtraction](https://www.openmiddle.com/decimal-subtraction/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Exponents 2](https://www.openmiddle.com/exponents-2/) - Directions: Using the integers -8 to 8, at most one time each, place an integer in each box to make the greatest possible value. Source: Robert Kaplinsky - [Square Root Expression 2](https://www.openmiddle.com/square-root-expression-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a true equation. Source: Jos Bertemes - [Creating Zero](https://www.openmiddle.com/creating-zero/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equality true. Source: Bryan Anderson - [Converting a Fraction to a Decimal](https://www.openmiddle.com/converting-a-fraction-to-a-decimal/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equality true. Source: Owen Kaplinsky - [Absolute Value](https://www.openmiddle.com/absolute-value/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the statement true. Source: Bryan Anderson - [Make it Equal](https://www.openmiddle.com/equivalent-statements/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Molly Rawding - [Adding with Parenthesis](https://www.openmiddle.com/adding-with-parenthesis/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Adding Two-Digit Numbers Given One](https://www.openmiddle.com/adding-two-digit-numbers-given-one/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Robert Kaplinsky - [Adding Single Digits](https://www.openmiddle.com/adding-single-digits/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the equation true. Source: Stephen Caviness - [Adding and Subtracting Within 10](https://www.openmiddle.com/adding-and-subtracting-within-10/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true. Source: Owen Kaplinsky - [Add and Subtract Mixed Numbers](https://www.openmiddle.com/add-and-subtract-mixed-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create an equation using addition and subtraction that equals 4. Source: Erik Almer - [Simple Patterns 2](https://www.openmiddle.com/simple-patterns-2/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make a true pattern where the pattern increases by the smallest amount possible. Source: Robert Kaplinsky - [Comparing Fractions](https://www.openmiddle.com/comparing-fractions-4/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make two true statements. Source: Kate Nerdypoo - [Equivalent Fractions](https://www.openmiddle.com/equivalent-fractions/) - Challenge students to use the digits 1-9 to make equivalent fractions in this DOK 2 problem. Great for 4th grade students working with fractions. - [Prime Numbers](https://www.openmiddle.com/prime-numbers/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make 5 prime numbers. Source: Owen Kaplinsky - [Rounding 2](https://www.openmiddle.com/rounding-2/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make the greatest possible three-digit number that still rounds (to the nearest hundred) to 500. Source: Robert Kaplinsky - [Rounding 1](https://www.openmiddle.com/rounding-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make two different three-digit numbers that round (to the nearest hundred) to 500. Source: Robert Kaplinsky - [Multiply and Divide Within A Hundred 2](https://www.openmiddle.com/multiply-and-divide-within-a-hundred-2/) - Directions: Using the digits 2 to 9, at most one time each, place a digit in each box to make a correct equations where the value is as close to 38 as possible. Source: Robert Kaplinsky - [Sums to 100](https://www.openmiddle.com/sums-to-100/) - Challenge students to use the digits 1-9 to make a sum close to 100 in this DOK 2 problem. Great for 2nd grade students practicing base ten operations. - [Subtraction to Get the Smallest Difference](https://www.openmiddle.com/subtraction-to-get-the-smallest-difference/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create the smallest possible difference. Source: Graham Fletcher Challenge students to use the digits 1-9 to create the smallest possible difference in this DOK 3 problem. Great for 2nd and 3rd grade students practicing operations in base ten. - [Close to 1000](https://www.openmiddle.com/close-to-1000/) - Challenge students to use the digits 1-9 to make 1000 in this DOK 3 problem. Great for 2nd and 3rd grade students practicing base ten operations. - [Order Numbers](https://www.openmiddle.com/order-numbers/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to make a two-digit number that has a value between given numbers. Source: Thach-Thao Phan - [Order of Operations 2](https://www.openmiddle.com/order-of-operations-2/) - Challenge students to use the digits 0-9 to create the largest or smallest expression in this open-ended order of operations problem. Great for upper elementary and middle school students practicing PEMDAS reasoning. - [Multiplying Binomials](https://www.openmiddle.com/multiplying-binomials/) - Directions: Using any numbers, fill in the boxes to make the equation true. Source: Dane Ehlert - [Systems of Equations, Special Case No Solution](https://www.openmiddle.com/systems-of-equations-special-case-no-solution/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box so that there is no solution to the system of equations. Source: Nanette Johnson - [Solving One-Step Equations (Negative and Positive Solutions)](https://www.openmiddle.com/solving-one-step-equations-negative-and-positive-solutions/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two equations: one where x has a positive value and one where x has a negative value. Source: Robert Kaplinsky - [Logarithms](https://www.openmiddle.com/logarithms/) - Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to create two different solutions to the problem. Source: Noel Chang - [Supplementary Angles 2](https://www.openmiddle.com/supplementary-angles-2/) - Directions: Using the digits from 1 – 9, at most one time each, find the measures of the two angles that form supplementary angles where their difference is as large as possible. Source: Debra Schneider - [Period of Trig Function 1](https://www.openmiddle.com/period-of-trig-function-1/) - Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create a cosine function with the given period in degrees. Source: Kate Nerdypoo - [Limits on a Graph](https://www.openmiddle.com/limits-on-a-graph/) - Directions: Using the options listed at most once each, create three true limit statements. Source: Dana Harrington - [Baking Cookies](https://www.openmiddle.com/baking-cookies/) - Directions: Daniel was making chocolate cookies. He had _ _ cookies in each row and _ _ many rows. There were a total of 84 cookies. How many cookies were there in each row and how many rows of cookies were there? Draw a model to support your answer. You may use the digits 0-9 Challenge students to fill in the blanks to make a true statement in this DOK 2 problem. Great for 3rd grade students practicing algebraic thinking. - [System of Equations - Table of Value and Slope Intercept Form](https://www.openmiddle.com/system-of-equations-table-of-value-and-slope-intercept-form/) - Directions: Using the integers -4 to 4, at most one time each, create a system of linear equations that has a solution in quadrant 4. Source: Robert Kaplinsky - [Functions](https://www.openmiddle.com/functions/) - Directions: List 2 points form a line that satisfies the following (you can use numbers 0-5, but you can only use a number once). Write the equation of the line represented by the points: a) Its rate of change must be larger than 2 b) Its y-intercept must be smaller than 3 c) It must - [Adding Numbers](https://www.openmiddle.com/adding-numbers/) - Directions: Fill in the blanks with numbers that make the equation true. Source: Nanette Johnson - [Factoring Quadratics With Undefined C](https://www.openmiddle.com/factoring-quadratics-with-undefined-c/) - Directions: Place an integer in the blank to find the largest and smallest value that will make the quadratic expression factorable. Source: Robert Kaplinsky - [What is it Not?](https://www.openmiddle.com/what-is-it-not/) - Directions: Use the terms square, rhombus, kite, parallelogram, trapezoid, rectangle, irregular quadrilateral at most one time each to complete two sentences Source: Alice Keeler and Miguel Ruiz - [Rectas Perpendiculares y Pendiente](https://www.openmiddle.com/es/rectas-perpendiculares-y-pendiente/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera que las rectas que pasan por cada pareja de puntos sean perpendiculares. Origen: Nanette Johnson - [Mean Median Mode](https://www.openmiddle.com/mean-median-mode/) - Directions: Using the digits 1 to 9, find a six number data set that has a Mode of 1, Median of 2 and Mean of 3. Digits can be repeated. Source: Harold Jacobs - [Interior and Exterior Angles of Triangles](https://www.openmiddle.com/interior-and-exterior-angles-of-triangles/) - Directions: In triangle ABC, angle ABC is obtuse. Using the digits 1 to 9 at most one time each, place a digit in each box to make angle ACB the smallest possible acute angle. Source: Jay Sydow - [Trinomial Function Features 1](https://www.openmiddle.com/trinomial-function-features-1/) - Directions: Using the integers -9 to 9, at most one time each, place an integer in each box to create a function with the corresponding range and roots. Source: Robert Kaplinsky - [Systems of Equations 4](https://www.openmiddle.com/systems-of-equations-4/) - Directions: Using the integers −9 to 9 at most one time each, place an integer in each box to create a system of equations whose solution is as close to the origin as possible. Source: Robert Kaplinsky - [Systems of Equations 3](https://www.openmiddle.com/systems-of-equations-3/) - Directions: Using the integers −9 to 9 at most one time each, place an integer in each box to create a system of equations and its solution. Source: Robert Kaplinsky - [Arithmetic Sequences 2](https://www.openmiddle.com/arithmetic-sequences-2/) - Directions: Using the integers −9 to 9 at most one time each, place an integer in each box to create an arithmetic sequence so that the coefficient in the function that represents it is the greatest possible value. Source: Robert Kaplinsky - [Arithmetic Sequences 1](https://www.openmiddle.com/arithmetic-sequences-1/) - Directions: Using the integers −9 to 9 at most one time each, place an integer in each box to create an arithmetic sequence and a function that represents it. Source: Robert Kaplinsky - [Exponents 1](https://www.openmiddle.com/exponents-1/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to make two values: one that is positive and one that is negative. You may reuse all the integers each time. Source: Robert Kaplinsky in Open Middle Math - [Adding Polynomials 1](https://www.openmiddle.com/adding-polynomials-1/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to make two expressions: one that has three or more terms and one that has fewer than three terms. You may reuse all the integers for each expression. Source: Robert Kaplinsky in Open Middle Math - [Adding Polynomials 2](https://www.openmiddle.com/adding-polynomials-2/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to create a polynomial with the least amount of terms. Source: Robert Kaplinsky in Open Middle Math - [Exponents 4](https://www.openmiddle.com/exponents-4/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to make a value that is as close to zero as possible without being exactly 0. Source: Robert Kaplinsky in Open Middle Math - [Exponents 3](https://www.openmiddle.com/exponents-3/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to make the least possible value. Source: Robert Kaplinsky in Open Middle Math - [Multiplying Integers 1](https://www.openmiddle.com/multiplying-integers-1/) - Directions: Using the integers -9 to 9 at most one time each, place an integer in each box to make two products: one where the product is positive and one where the product is negative. You may reuse all the integers for each product. Source: Robert Kaplinsky in Open Middle Math - [Order of Operations 6](https://www.openmiddle.com/order-of-operations-6/) - Directions: Using the digits 1 to 5, at most one time each, place a digit in each box to create an expression with the largest possible value. Source: Matt Donahue - [Adding Products](https://www.openmiddle.com/adding-products/) - Directions: Old Mother Hubbard is baking cookies so her cupboards won't be bare anymore! She bakes 109 cookies in all. She bakes the cookies on 4 cookie sheets. Each cookie sheet is arranged into equal rows and columns, but not every cookie sheet has the same number of rows and columns. Using digits 0-9, at - [Multiplication Decisions](https://www.openmiddle.com/multiplication-decisions/) - Directions: Using the digits 5, 6, 7, and 8 exactly once and picking one of the expressions below, create the greatest product possible out of the two expressions. Source: Howie Hua - [Supplementary Angles](https://www.openmiddle.com/supplementary-angles/) - Directions: Using the digits from 0 – 9, at most one time each, find the measures of the two angles forming supplementary angles as close as possible in size. Source: Debra Schneider - [Desviación Absoluta](https://www.openmiddle.com/es/desviacion-absoluta/) - Usando los dígitos del 1 al 9, sin repetir, rellena los cuadros para crear el conjunto de datos con la mayor desviación absoluta posible. Origen: Mark Alvaro - [Valor Absoluto](https://www.openmiddle.com/es/valor-absoluto/) - Usando los dígitos del 1 al 9 sin repetir, rellena los espacios para que el enunciado sea cierto: "____ y ____ están a ____ unidades del ____." Origen: Bryan Anderson - [Number Pattern](https://www.openmiddle.com/number-pattern/) - Directions: Using the digits 1 to 7, at most one time each, place a digit in each circle so that the sum of the numbers in 3 squares (the middle horizontal line or 2 diagonals) are same. e.g A+B+C or D+B+E or F+B+G Is there more than one solution? Source: Al Oz - [Creando Rectángulos 2](https://www.openmiddle.com/es/creando-rectangulos-2/) - Instrucciones: Usando los dígitos del 1 al 8, como máximo una vez cada uno, completa los recuadros para obtener las coordenadas de los vértices de un rectángulo: A(__, __), B(__, __), C(__, __), D(__, __). Ampliación: ¿Cuál es el rectángulo con la mayor/menor área/perímetro que puedes encontrar? Origen: Erick Lee - [Creando Cuadrados](https://www.openmiddle.com/es/creando-cuadrados/) - Instrucciones: Usando los dígitos del 0 al 9, como máximo una vez cada uno, completa los recuadros para crear un cuadrado junto con el vértice (2,3) Origen: John Mahlstedt - [Multi-Step Equations - Positive (or Negative) Solution](https://www.openmiddle.com/multi-step-equations-positive-or-negative-solution/) - Directions: Using the digits 1 to 9, at most one time each, create an equation where x has a positive (or negative) value. Source: Daniel Luevanos - [Multi-Step Equations - Smallest (or Largest) Solution](https://www.openmiddle.com/multi-step-equations-smallest-or-largest-solution/) - Directions: Using the digits 1 to 9, at most one time each, create an equation where x has the smallest (or greatest) possible value. Source: Daniel Luevanos - [Paralelogramos Coordenados](https://www.openmiddle.com/es/paralelogramos-coordenados/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros para obtener las coordenadas de los vértices de un paralelogramo. Origen: Daniel Torres-Rangel - [Área de un Triángulo en el Plano Coordenado](https://www.openmiddle.com/es/area-de-un-triangulo-en-el-plano-coordenado/) - Instrucciones: Usando los dígitos del 0 al 9, como máximo una vez cada uno, completa los recuadros para obtener las coordenadas de tres puntos que determinen el triángulo ABC de área más cercana a 6 unidades cuadradas A ( ___, ___ ) B ( ___, ___ ) C ( ___, ___ ) Origen: Henry Wadsworth - [Puntos Equidistantes 2](https://www.openmiddle.com/es/puntos-equidistantes-2/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros para crear dos puntos que estén a la misma distancia del (4,-1). Origen: Bryan Anderson - [Ángulos Interiores](https://www.openmiddle.com/es/angulos-interiores/) - Instrucciones: Usando los dígitos del 0 al 9, como máximo una vez cada uno, completa los recuadros para que respeten la suma de los ángulos interiores de un triángulo Origen: Ashley Henderson - [Ángulos Complementarios y Suplementarios](https://www.openmiddle.com/es/angulos-complementarios-y-suplementarios/) - Instrucciones: Usando los dígitos del 0 al 9, como máximo una vez cada uno, completa los recuadros para que las afirmaciones sean ciertas __ __ y __ __ son ángulos complementarios __ __ y __ __ __ son ángulos suplementarios Origen: Bryan Anderson - [Área de un Cuadrilátero en el Plano Coordenado](https://www.openmiddle.com/es/area-de-un-cuadrilatero-en-el-plano-coordenado/) - Instrucciones: Usando los dígitos del 0 al 9, como máximo una vez cada uno, completa los recuadros para crear un cuadrilátero de 16 unidades de área. Origen: Daniel Luevanos - [Perímetro & Circunferencia](https://www.openmiddle.com/es/perimetro-circunferencia/) - Instrucciones: Usando los dígitos del 1 al 6, como máximo una vez cada uno, completa los recuadros para conseguir el mayor y menor perímetro, combinando las dos figuras: el rectángulo y la circunferencia. Origen: Christin Smith - [Maximizar el Área Lateral de un Prisma Rectangular](https://www.openmiddle.com/es/maximizar-el-area-lateral-de-un-prisma-rectangular/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros para indicar las dimensiones de un prisma rectangular con la mayor área lateral posible. Origen: Robert Kaplinsky - [Rational and Irrational Roots 5](https://www.openmiddle.com/rational-and-irrational-roots-5/) - Directions: Using the digits 1 to 9, find one pair of digits to fill in the boxes to create expressions that produce one rational root and three irrational roots. Source: Norma Gordon - [Funciones de Primer Grado a Partir de su Tabla de Valores](https://www.openmiddle.com/es/funciones-de-primer-grado-a-partir-de-su-tabla-de-valores/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa una tabla de valores que represente a una función de primer grado. Origen: Robert Kaplinsky - [Ecuación de Valor Absoluto](https://www.openmiddle.com/es/ecuacion-de-valor-absoluto/) - Crea una ecuación de valor absoluto en la que x = – 2 sea una solución extraña o espuria. Origen: Daniel Luevanos - [Equations de Radicaux](https://www.openmiddle.com/fr/equations-de-radicaux/) - En utilisant les chiffres de 0 à 9 au plus une fois chacun, créer ces 2 expressions telles qu’elles soient vraies. La Source: Jonathan Newman - [Comparer et Ordonner des Radicaux](https://www.openmiddle.com/fr/comparer-et-ordonner-des-radicaux/) - En utilisant les chiffres de 1 à 9 au plus une fois chacun, créer une sequence dans l’ordre croissant et qui ne peut plus être simplifiée. La Source: Phillip Haislip-Hansberry - [Exposants et Ordre des Opérations](https://www.openmiddle.com/fr/exposants-et-ordre-des-operations/) - Trouver 3 entiers positifs dont la somme est 10. Placer chaque nombre dans une des cases pour trouver le résultat le plus grand possible. La Source: Zack Miller (@zmill415) - [Propriétés D’exposants Entiers 2](https://www.openmiddle.com/fr/proprietes-dexposants-entiers-2/) - En utilisant les chiffres 0 à 9 au plus une fois chacun, remplir les cases pour créer des expressions numériques équivalentes La Source: Bryan Anderson - [Propriétés D’exposants Entiers](https://www.openmiddle.com/fr/proprietes-dexposants-entiers/) - En utilisant les chiffres 0 à 9 au plus une fois chacun, remplir les cases pour créer des expressions numériques équivalentes La Source: Bryan Anderson - [Le Plus Grand PGCD Possible](https://www.openmiddle.com/fr/le-plus-grand-pgcd-possible/) - En utilisant les chiffres 0-9 au plus une fois, remplir les cases pour obtenir le plus grand diviseur commun le plus grand possible. La Source: Howie Hua - [Le Plus Grand PGCD Possible 2](https://www.openmiddle.com/fr/le-plus-grand-pgcd-possible-2/) - En utilisant les chiffres 0-9 au plus une fois, remplir les cases pour obtenir le plus grand diviseur commun le plus grand possible. La Source: Howie Hua - [Le Plus Petit PPCM Possible](https://www.openmiddle.com/fr/le-plus-petit-ppcm-possible/) - En utilisant les chiffres 0-9 au plus une fois, remplir les cases pour obtenir le plus petit multiple commun le plus petit possible. La Source: Howie Hua - [Expressions Èquivalentes Avec des Puissances](https://www.openmiddle.com/fr/expressions-equivalentes-avec-des-puissances/) - Trouver des valeurs pour a et b qui rendent cette expression équivalente, en supposant que a n’est pas égal à b. La Source: Owen Kaplinsky - [Ordre des Opérations](https://www.openmiddle.com/fr/ordre-des-operations/) - Créer l’expression la plus grande (ou la plus petite) en utilisant les chiffres de 0-9, pas plus qu’une fois chacun, dans les cases ci-dessous. La Source: Robert Kaplinsky avec réponse de Michael Fenton et ses étudiants. - [Exposant (Valeur Maximale)](https://www.openmiddle.com/fr/exposant-valeur-maximale/) - Utiliser les chiffres 1 à 9, au plus une fois chacun, pour remplir les cases pour obtenir un résultat qui a la plus grande valeur possible. La Source: Robert Kaplinsky - [Suma y Resta de Enteros](https://www.openmiddle.com/es/suma-y-resta-de-enteros/) - Instrucciones: Usando los dígitos del 1 al 6 sin repetir, rellena los recuadros de manera que las primeras dos expresiones den el mismo resultado y la tercera expresión tenga le mayor valor posible. Source: Kate Nerdypoo - [Exploration D’exposants](https://www.openmiddle.com/fr/exploration-dexposants/) - Utiliser les chiffres 1 à 9, au plus une fois chacun, pour remplir les cases pour créer deux phrases numériques vraies. La Source: Robert Kaplinsky - [Suma Hasta 1,000- Dos Adendos](https://www.openmiddle.com/es/suma-hasta-1000-dos-adendos/) - Acomoda los dígitos del 1-6 en dos números enteros de tres dígitos. Haz la suma lo más cercana al número 1,000. Origen: Ian Kerr - [Operaciones con Tiempo II](https://www.openmiddle.com/es/operaciones-con-tiempo-ii/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera de conseguir que la respuesta sean las 4:37 Origen: Robert Kaplinsky - [Operaciones con Tiempo I](https://www.openmiddle.com/es/operaciones-con-tiempo-i/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera de conseguir la hora más tardía posible. Origen: Robert Kaplinsky - [Division Euclidienne](https://www.openmiddle.com/fr/division-euclidienne/) - Complétez: Écrivez dans chaque case un chiffre de 1 à 9. Les égalités devront être vraies. Un chiffre ne peut servir qu’une seule fois. La Source: Owen Kaplinsky - [Addition de Nombres Décimaux.](https://www.openmiddle.com/fr/addition-de-nombres-decimaux-3/) - Complétez: Chaque case avec un chiffre de 1 à 9. L’égalité devra être vraie. Comment obtenir la plus grande somme possible ? Chaque chiffre ne peut servir qu’une seule fois. La Source: Owen Kaplinsky et Robert Kaplinsky - [Propiedad Distributiva 2](https://www.openmiddle.com/es/propiedad-distributiva-2/) - Instrucciones: Usando los dígitos del 1 al 9, sin repetir, rellena los recuadros de manera que el enunciado sea cierto. Origen: Adrianne Burns - [Graphing Points on a Coordinate Plane](https://www.openmiddle.com/graphing-points-on-a-coordinate-plane/) - Directions: Make four points using the integers -4 to 4 at most one time each so that each point is in a different quadrant. Source: Robert Kaplinsky - [Soustraction de Nombres Décimaux.](https://www.openmiddle.com/fr/soustraction-de-nombres-decimaux/) - Complétez chaque case avec un chiffre de 1 à 9. L’égalité devra être vraie. Chaque chiffre ne peut servir qu’une seule fois. Trouvez deux solutions différentes. Origen: Owen Kaplinsky - [Figuras Semejantes](https://www.openmiddle.com/es/4308-2/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera que los rectángulos resultantes sean semejantes entre sí. Origen: Gian Cavaliere - [Propiedad Distributiva 3](https://www.openmiddle.com/es/propiedad-distributiva-3/) - Instrucciones: Usando los dígitos del 0 al 9, como máximo una vez cada uno, completa los recuadros de manera las igualdades sean correctas. Origen: Julie Wright - [Soustraction de Nombres Décimaux 2.](https://www.openmiddle.com/fr/soustraction-de-nombres-decimaux-2-2/) - Complétez chaque case avec un chiffre de 1 à 9. L’égalité devra être vraie. Comment obtenir la plus petite différence possible ? Chaque chiffre ne peut servir qu’une seule fois. La Source: Owen Kaplinsky et Robert Kaplinsky - [Addition de Nombres Décimaux 2](https://www.openmiddle.com/fr/addition-de-nombres-decimaux-2-2/) - Complétez chaque case avec un chiffre de 1 à 9. L’égalité devra être vraie. Chaque chiffre ne peut servir qu’une seule fois. Trouvez deux solutions différentes. La Source: Owen Kaplinsky - [Addition de Nombres Décimaux.](https://www.openmiddle.com/fr/4332-2/) - Complétez chaque case avec un chiffre de 0 à 9. L'addition devra être correcte. Chaque chiffre ne peut servir qu’une seule fois. La Source: Shaun Errichiello - [Soustraction de Nombres Décimaux.](https://www.openmiddle.com/fr/soustraction-de-nombres-decimaux-3/) - Complétez chaque case avec un chiffre de 1 à 9. L’égalité devra être vraie. Chaque chiffre ne peut servir qu’une seule fois. Trouvez deux solutions différentes. La Source: Owen Kaplinsky - [Inecuaciones Compuestas 2](https://www.openmiddle.com/es/inecuaciones-compuestas-2/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera que la inecuación compuesta resultante tenga por solución el intervalo 2 ≤ x < 4. Origen: Robert Kaplinsky - [Desigualdades 2](https://www.openmiddle.com/es/desigualdades-2/) - ¿Qué sucede cuando sumas números racionales? Origen: Bryan Anderson - [Distributive Property](https://www.openmiddle.com/distributive-property-2/) - Directions: Using the digits 1 to 9 at most one time each, fill in the boxes to make a true equation. Source: Adrianne Burns - [Inecuaciones Compuestas](https://www.openmiddle.com/es/inecuaciones-compuestas/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera que la inecuación compuesta resultante tenga por solución el intervalo de mayor longitud posible. Origen: Robert Kaplinsky - [Multiplicación de Decimales con Resultado Entero](https://www.openmiddle.com/es/multiplicacion-de-decimales-con-resultado-entero/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera que el resultado de la multiplicación sea un número entero. Origen: Owen Kaplinsky - [La Distributivité.](https://www.openmiddle.com/fr/la-distributivite-2/) - Complétez chaque case avec un chiffre de 0 à 9. Les égalités seront vraies. Chaque chiffre ne peut servir qu’une seule fois. La Source: Julie Wright - [La Distributivité 2](https://www.openmiddle.com/fr/la-distributivite-2-2/) - Complétez chaque case avec un chiffre de 0 à 9. L'égalité devra être vraie. Chaque chiffre ne peut servir qu'une seule fois. La Source: Adrianne Burns - [La Distributivité.](https://www.openmiddle.com/fr/la-distributivite-3/) - Complétez chaque case avec un chiffre de 1 à 9. L'égalité devra être vraie. Chaque chiffre ne peut servir qu’une seule fois. La Source: Adrianne Burns - [Ecuaciones Lineales de la Forma ax + b = c (3)](https://www.openmiddle.com/es/ecuaciones-lineales-de-la-forma-ax-b-c-3/) - Instrucciones: Escribe los dígitos del 1 al 9 sin repetir dentro de los recuadros, de manera que obtengas el mayor (o menor) valor para la suma de "x" y "y". Origen: Erick Lee - [Propiedades de los Logaritmos 2](https://www.openmiddle.com/es/propiedades-de-los-logaritmos-2-3/) - Utilizando los dígitos del 0 al 9 sin repetir, completa los casilleros de modo que cada expresión sea menor a la siguiente. Origen: John Rowe - [Triángulo con ley de Cosenos](https://www.openmiddle.com/es/triangulo-con-ley-de-cosenos/) - Instrucciones: Escribe dentro de cada círculo uno de los dígitos del 1 al 9, sin repetir, de manera que, siendo la suma de los círculos de cada lado la medida correspondiente a ese lado, se forme el triángulo con el mayor (o menor) ángulo posible. Origen: Erick Lee - [Suma de Fracciones 7](https://www.openmiddle.com/es/suma-de-fracciones-7/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera que la igualdad sea correcta. Origen: Owen Kaplinsky - [Suma de Fracciones 6](https://www.openmiddle.com/es/suma-de-fracciones-6/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera que la igualdad sea correcta. Origen: Owen Kaplinsky - [División de Números Enteros con Cociente Decimal](https://www.openmiddle.com/es/division-de-numeros-enteros-con-cociente-decimal/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los recuadros de manera que la igualdad sea correcta. Origen: Owen Kaplinsky - [Triángulo con ley de Cosenos](https://www.openmiddle.com/es/triangulo-con-ley-de-cosenos-2/) - Instrucciones: Escribe dentro de cada círculo uno de los dígitos del 1 al 9, sin repetir, de manera que, siendo la suma de los círculos de cada lado la medida correspondiente a ese lado, se forme el triángulo con el mayor (o menor) ángulo posible. Origen: Erick Lee - [Minimize Slope](https://www.openmiddle.com/minimize-slope/) - Directions: Given the point (3,5), use digits 1-9, at most one time, to find a point (__, __) that minimizes the slope of the line that passes through the two points. The slope cannot be undefined. Source: Nanette Johnson (Problem Based on Andrew Constantinescu's Problem) and Andrew Constantinescu - [Suma de Fracciones 3](https://www.openmiddle.com/es/suma-de-fracciones-3/) - Instrucciones: Usando los dígitos del 1 al 9, como máximo una vez cada uno, completa los cuadros para conseguir el resultado más más cercano posible a 1/2. Origen: Daniel Luevanos - [Porcentajes](https://www.openmiddle.com/es/porcentajes/) - Instrucciones: Usando los dígitos del 0 al 9, como máximo una vez cada uno, completa los cuadros para crear una igualdad correcta. Origen: Cecilia Calvo - [La Distributivité.](https://www.openmiddle.com/es/la-distributivite/) - "Complétez chaque case avec un chiffre de 0 à 9. Les égalités seront vraies. Chaque chiffre ne peut servir qu’une seule fois." Origen: Julie Wright - [Corte con el eje OX](https://www.openmiddle.com/es/corte-con-el-eje-ox/) - Escribe la ecuación de una recta que corta al eje OX en el punto (-100,0) Origen: Kara Colley - [Rational Number Computation](https://www.openmiddle.com/rational-number-computation/) - Directions: Using the integers -5 to 5, at most one time each, write an expression that will have the greatest (or least) absolute value. Source: Michael Wiernicki - [Dividing Rational Expressions](https://www.openmiddle.com/dividing-rational-expressions/) - Directions: Determine values to place in the missing spots to solve the equation below. You may use integer values: Source: Sandra Crawford - [Division Fill in the Blanks (no remainder)](https://www.openmiddle.com/division-fill-in-the-blanks-no-remainder/) - Directions: Fill in the blanks using all different non-zero digits (except the numbers 1 and 4, which have already been used) to make the greatest possible quotient. Source: Brian Lack - [System of Inequalities](https://www.openmiddle.com/system-of-inequalities-2/) - Directions: Fill each blank with a different integer such that the point (4,4) is within the solution region created by the constraints. Source: Erick Lee - [Midpoint Formula](https://www.openmiddle.com/midpoint-formula/) - Directions: Create two pairs of coordinates on the same line segment that have M (3,4) as their midpoint. Source: Dane Ehlert - [L'Hospital's Rule Exploration](https://www.openmiddle.com/lhospitals-rule-exploration/) - Directions: Using the digits 1 to 9, at most one time each, create 3 different expressions such that their graphs contains any 2 of the 3 following criteria: 1) Horizontal Asymptote @ y = some positive rational number 2) Slant Asymptote with a slope such that: 1 < m ≤ 2 3) Two Vertical Asymptotes - [Equations of Perpendicular Lines](https://www.openmiddle.com/equations-of-perpendicular-lines/) - Directions: Using the digits 1 to 9 at most one time each, fill in the blanks to create two distinct perpendicular lines. Note that the coefficient for the second line's y is negative. __ x + __ y = __ __ x + -__ y = __ Source: Bryan Anderson - [Adding Decimals 2](https://www.openmiddle.com/adding-decimals/) - Directions: Using the digits 1 to 9 at most one time each, fill in the boxes to make the largest (or smallest) sum. Source: Daniel Luevanos - [Probability with Marbles](https://www.openmiddle.com/probability-with-marbles/) - Directions: There are _____ red marbles and _____ blue marbles in Bag A. There are _____ red marbles and _____ green marbles in Bag B. Using the digits 1 to 9 at most one time each, fill in the boxes to make the probability of drawing a red marble from either bag the same. Extension: - [Laws of Logarithms](https://www.openmiddle.com/laws-of-logarithms/) - Directions: Using the integers 0 to 9, fill in the red and blue boxes so that the chart is accurate. You can only use a number once per red box and once per blue box. (Logs are in base 10) Source: Bryan Anderson - [Subtraction with Regrouping](https://www.openmiddle.com/subtraction-with-regrouping/) - Directions: Fill in the boxes so that you would need to regroup when you subtract. Make sure that your number is less than 63. Extension: Explain why you need to regroup using your number. Source: Chase Orton - [Equivalent Expressions with Powers](https://www.openmiddle.com/equivalent-expressions-with-powers/) - Directions: Find values for a and b that will make the expressions equivalent, assuming that a does not equal b. Source: Owen Kaplinsky - [Solving Trigonometric Equations](https://www.openmiddle.com/solving-trigonometric-equations/) - Directions: Using the digits 1 to 9, at most one time each, find the equation whose solution is the largest value of x (from 0 to 360, or 0 to 2π). Source: Mishaal Surti - [Systems of Equations - One Solution](https://www.openmiddle.com/systems-of-equations-one-solution/) - Directions: Using the integers from -9 to 9, at most one time each, create a system of three-equations such that the solution is (1,1). Source: Audrey Mendivil, Daniel Luevanos, and Robert Kaplinsky - [Parallel and Perpendicular Lines](https://www.openmiddle.com/parallel-and-perpendicular-lines/) - Directions: Using the digits 0 to 9, at most once each time, fill in blanks to create a set of 4 points that create either parallel or perpendicular lines, depending on how you connect them. ( ___, ___ ) ( ___, ___ ) ( ___, ___ ) ( ___, ___ ) Source: Bryan Anderson - [Exponents and Order of Operations](https://www.openmiddle.com/exponents-and-order-of-operations/) - Directions: Find 3 positive integers that add up to 10. Place each number into one of the blanks to find the largest possible result. Source: Zack Miller (@zmill415) - [Equivalent Equations](https://www.openmiddle.com/equivalent-equations-2/) - Source: Smarter Balance 7th grade practice test - [Multiply to Make -64](https://www.openmiddle.com/multiply-to-make-64/) - Directions: Find three numbers whose product is -64. You may use integers from -10 to 10. You may not use the same absolute value twice. Find all possible combinations. Source: Nathan Charlton and Daniel Martinez - [Solve Linear Equations with Special Cases](https://www.openmiddle.com/solve-linear-equations-with-special-cases/) - Source: SBAC Practice Test 8th Grade - [How Many Squares?](https://www.openmiddle.com/how-many-squares/) - Directions: How many squares are shown in the diagram below? Source: This problem was found in the Fifth Edition Elementary Geometry for College Students by Alexander and Koeberlein ## Pages - [Home](https://www.openmiddle.com/) - Open Middle offers interactive math problems designed to foster critical thinking. Explore unique challenges that boost problem-solving skills and creativity. - [Casa](https://www.openmiddle.com/es/casa/) - PROBLEMAS MATEMÁTICOS DESAFIANTES QUE VALE LA PENA RESOLVER CONSULTA NUESTRO LIBRO PARA SABER MÁS SOBRE OPEN MIDDLE Consigue nuestros favoritos Realice el taller en línea USAR PROBLEMAS DE MEDIO ABIERTO EN LÍNEA CONOCE NUESTROS NUEVOS PROBLEMAS INTERACTIVOS (INGLÉS) LEA EL ARTÍCULO ¿BUSCAS UN LIBRO DE OPEN MIDDLE? COMPRA EL LIBRO DE OPEN MIDDLE: PROBLEMAS QUE - [Accueil](https://www.openmiddle.com/fr/accueil/) - DÉFIS MATHÉMATIQUES QUI MÉRITENT D’ÊTRE RÉSOLUS CONSULTEZ NOTRE LIVRE POUR EN SAVOIR PLUS SUR OPEN MIDDLE Obtenez nos problèmes préférés Suivre L’atelier En Ligne UTILISER LES PROBLÈMES DU MILIEU OUVERT EN LIGNE DÉCOUVREZ NOS NOUVEAUX PROBLÈMES INTERACTIFS (ANGLAIS) LIRE L'ARTICLE VOUS CHERCHEZ UN LIVRE INTERMÉDIAIRE OUVERT ? OUVREZ LES MATHS DU MILIEU : DES PROBLÈMES - [Recherche avancée](https://www.openmiddle.com/fr/recherche-avancee/) - Niveaux scolaires Mat/J 1ière 2e 3e 4e 5e 6e 7e 8e Secondarie Niveaux DOK DOK 2: Compétence / Concept DOK 3: Réflexion Stratégique Appliquer les filtres - [Búsqueda Avanzada](https://www.openmiddle.com/es/busqueda-avanzada/) - Niveles de grado Jardín 1er Gr 2do Gr 3er Gr 4to Gr 5to Gr 6to Gr 7mo Gr 8vo Gr Secundaria Niveles DOK DOK 2: Habilidad / Concepto DOK 3: Pensamiento Estrategico Aplicar filtros - [Advanced Search](https://www.openmiddle.com/advanced-search/) - Grade Levels Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Grade 6 Grade 7 Grade 8 High School DOK Levels DOK 2 - Skill / Concept DOK 3 - Strategic Thinking Apply Filters - [What's Open Middle?](https://www.openmiddle.com/whats-open-middle/) - "Open Middle" problems refer to a specific type of problem that fosters critical thinking. Find out more in this explanation. - [Open Middle Team](https://www.openmiddle.com/open-middle-team/) - Open Middle is run by an all volunteer team. Here are the people who make things happen. - [Submit a Problem](https://www.openmiddle.com/submit/) - We are always looking for new problems to add to this website. If you have an idea for a problem, please show us! - [Qu'est-ce que l'Open Middle?](https://www.openmiddle.com/fr/quest-ce-que-lopen-middle/) - Le nom «Open Middle» peut sembler étrange pour un site Web traitant de problèmes de mathématiques. Cependant, il fait référence à un type de problème très spécifique que nous essayons d'encourager ici. La plupart des problèmes sur ce site ont: un «début fermé» signifiant qu'ils commencent tous par le même problème initial. une «fin fermée» - [Equipo Open Middle](https://www.openmiddle.com/es/equipo-open-middle/) - Open Middle es manejado por un equipo de voluntarios. Aquí están las personas que hacen esta página posible. Principal Open Middle y Co-Fundador Robert Kaplinsky Principales Contribuyentes Kjersti Oliver Administrador del sitio web Owen Kaplinsky K-5 Equipo de Problemas Ashley Powell Kate Fallon Molly Rawding Tara Trifiletti 6-8 Equipo de Problemas Ashley Weaver Gabriel Despatie - [Équipe Open Middle](https://www.openmiddle.com/fr/equipe-open-middle/) - Open Middle est géré par une équipe entièrement composée de bénévoles. Voici les personnes qui rendent ce site possible. Principaux Open Middle et Co-Fondateur Robert Kaplinsky Principaux Contributeurs Kjersti Oliver Gestionnaire de site Web Owen Kaplinsky Equipe de Problèmes M-5 Ashley Powell Kate Fallon Molly Rawding Tara Trifiletti Equipe de Problèmes 6-8 Ashley Weaver Gabriel - [¿Qué es Open Middle?](https://www.openmiddle.com/es/que-es-open-middle/) - El nombre "Open Middle" puede sonar como un nombre extraño para un sitio web de problemas matemáticos. Sin embargo, hace referencia a un tipo de problema muy particular que buscamos fomentar. La mayoría de los problemas de este sitio tienen: un "comienzo cerrado": lo que significa que todos comienzan con el mismo problema inicial. un - [Ingresar un Problema](https://www.openmiddle.com/es/ingresar-un-problema/) - Estamos siempre abiertos a recibir nuevos problemas que enriquecerán a nuestro sitio. Si quieres compartir tu idea con nosotros, deberás llenar el siguiente formulario. Los campos marcados con asterisco son obligatorios ¡Muchas gracias! Loading... - [Soumettre un Problème](https://www.openmiddle.com/fr/soumettre-un-probleme/) - Nous sommes toujours à la recherche de nouveaux problèmes à ajouter à ce site. Si vous avez une idée pour un problème, veuillez remplir le formulaire ci-dessous. Les champs obligatoires sont suivis d'un astérisque rouge. Merci! 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[Andy Schwen](https://www.openmiddle.com/tag/andy-schwen/) - [Ryan Kimes](https://www.openmiddle.com/tag/ryan-kimes/) - [Will Case](https://www.openmiddle.com/tag/will-case/) - [6.EE.4](https://www.openmiddle.com/tag/6-ee-4/) - [Erick Lee](https://www.openmiddle.com/tag/erick-lee/) - [N-RN.1](https://www.openmiddle.com/tag/n-rn-1/) - [Jason Miller](https://www.openmiddle.com/tag/jason-miller/) - [Karen Bloom](https://www.openmiddle.com/tag/karen-bloom/) - [Chase Orton](https://www.openmiddle.com/tag/chase-orton/) - [7.EE.4a](https://www.openmiddle.com/tag/7-ee-4a/) - [A-APR.7](https://www.openmiddle.com/tag/a-apr-7/) - [6.RP.3c](https://www.openmiddle.com/tag/6-rp-3c/) - [7.RP.3](https://www.openmiddle.com/tag/7-rp-3/) - [6.G.3](https://www.openmiddle.com/tag/6-g-3/) - [6.NS.6b](https://www.openmiddle.com/tag/6-ns-6b/) - [G-SRT.11](https://www.openmiddle.com/tag/g-srt-11/) - [G-SRT.4](https://www.openmiddle.com/tag/g-srt-4/) - [7.G.2](https://www.openmiddle.com/tag/7-g-2/) - [Irvine Math Project](https://www.openmiddle.com/tag/irvine-math-project/) - [Shannon Andrews](https://www.openmiddle.com/tag/shannon-andrews/) - [Ian Kerr](https://www.openmiddle.com/tag/ian-kerr/) - [5.NF.5](https://www.openmiddle.com/tag/5-nf-5/) - [Jon Henderson](https://www.openmiddle.com/tag/jon-henderson/) - [7.EE.1](https://www.openmiddle.com/tag/7-ee-1/) - [Claire Verti](https://www.openmiddle.com/tag/claire-verti/) - [F-LE.4.3 (CA)](https://www.openmiddle.com/tag/f-le-4-3-ca/) - [Patrick McGowan](https://www.openmiddle.com/tag/patrick-mcgowan/) - [Bowen Kerins](https://www.openmiddle.com/tag/bowen-kerins/) - [Kate Nowak](https://www.openmiddle.com/tag/kate-nowak/) - [2.NBT.6](https://www.openmiddle.com/tag/2-nbt-6/) - [3.MD.3](https://www.openmiddle.com/tag/3-md-3/) - [1.MD.4](https://www.openmiddle.com/tag/1-md-4/) - [4.MD.2](https://www.openmiddle.com/tag/4-md-2/) - [Andrew Gael](https://www.openmiddle.com/tag/andrew-gael/) - [Glenn Waddell](https://www.openmiddle.com/tag/glenn-waddell/) - [7.SP.8c](https://www.openmiddle.com/tag/7-sp-8c/) - [John Ulbright](https://www.openmiddle.com/tag/john-ulbright/) - [6.EE.8](https://www.openmiddle.com/tag/6-ee-8/) - [Laura Wagenman](https://www.openmiddle.com/tag/laura-wagenman/) - [Cecilia Calvo](https://www.openmiddle.com/tag/cecilia-calvo/) - [3.MD.1](https://www.openmiddle.com/tag/3-md-1/) - [3.OA.3](https://www.openmiddle.com/tag/3-oa-3/) - [Steven Midzak](https://www.openmiddle.com/tag/steven-midzak/) - [F-IF.2](https://www.openmiddle.com/tag/f-if-2/) - [Mishaal Surti](https://www.openmiddle.com/tag/mishaal-surti/) - [F-TF.3](https://www.openmiddle.com/tag/f-tf-3/) - [Daniel Torres-Rangel](https://www.openmiddle.com/tag/daniel-torres-rangel/) - [Calculus](https://www.openmiddle.com/tag/calculus/) - [John Rowe](https://www.openmiddle.com/tag/john-rowe/) - [A-REI.12](https://www.openmiddle.com/tag/a-rei-12/) - [7.EE.3](https://www.openmiddle.com/tag/7-ee-3/) - [8.EE.6](https://www.openmiddle.com/tag/8-ee-6/) - [F-BF.5](https://www.openmiddle.com/tag/f-bf-5/) - [Peter Morris](https://www.openmiddle.com/tag/peter-morris/) - [Joe Schwartz](https://www.openmiddle.com/tag/joe-schwartz/) - [5.MD.5](https://www.openmiddle.com/tag/5-md-5/) - [Adrianne Burns](https://www.openmiddle.com/tag/adrianne-burns/) - [Mark Alvaro](https://www.openmiddle.com/tag/mark-alvaro/) - [S-ID.4](https://www.openmiddle.com/tag/s-id-4/) - [Marie Isaac](https://www.openmiddle.com/tag/marie-isaac/) - [Ellen Metzger](https://www.openmiddle.com/tag/ellen-metzger/) - [Al Oz](https://www.openmiddle.com/tag/al-oz/) - [Marilyn Burns](https://www.openmiddle.com/tag/marilyn-burns/) - [Kevin Rees](https://www.openmiddle.com/tag/kevin-rees/) - [Annie Forest](https://www.openmiddle.com/tag/annie-forest/) - [5.NBT.4](https://www.openmiddle.com/tag/5-nbt-4/) - [Joshua Nelson](https://www.openmiddle.com/tag/joshua-nelson/) - [Richard Hung](https://www.openmiddle.com/tag/richard-hung/) - [8.EE.7b](https://www.openmiddle.com/tag/8-ee-7b/) - [K.OA.5](https://www.openmiddle.com/tag/k-oa-5/) - [Christine Newell](https://www.openmiddle.com/tag/christine-newell/) - [Jonathan Newman](https://www.openmiddle.com/tag/jonathan-newman/) - [N-RN.2](https://www.openmiddle.com/tag/n-rn-2/) - [6.RP.3](https://www.openmiddle.com/tag/6-rp-3/) - [Kerri Swails](https://www.openmiddle.com/tag/kerri-swails/) - [S-CP.9](https://www.openmiddle.com/tag/s-cp-9/) - [Giselle Garcia](https://www.openmiddle.com/tag/giselle-garcia/) - [4.NF.6](https://www.openmiddle.com/tag/4-nf-6/) - [Lynda Chung](https://www.openmiddle.com/tag/lynda-chung/) - [Ashley Henderson](https://www.openmiddle.com/tag/ashley-henderson/) - [Julie Wright](https://www.openmiddle.com/tag/julie-wright/) - [8.F.5](https://www.openmiddle.com/tag/8-f-5/) - [K.G.4](https://www.openmiddle.com/tag/k-g-4/) - [K.G.1](https://www.openmiddle.com/tag/k-g-1/) - [K.G.2](https://www.openmiddle.com/tag/k-g-2/) - [Wendy Taylor](https://www.openmiddle.com/tag/wendy-taylor/) - [6.NS.4](https://www.openmiddle.com/tag/6-ns-4/) - [Patrick Sullivan](https://www.openmiddle.com/tag/patrick-sullivan/) - [6.SP.3a](https://www.openmiddle.com/tag/6-sp-3a/) - [Molly Rawding](https://www.openmiddle.com/tag/molly-rawding/) - [G-GPE.6](https://www.openmiddle.com/tag/g-gpe-6/) - [Phillip Haislip-Hansberry](https://www.openmiddle.com/tag/phillip-haislip-hansberry/) - [Daniel Rocha](https://www.openmiddle.com/tag/daniel-rocha/) - [3.OA.1](https://www.openmiddle.com/tag/3-oa-1/) - [Owen Kaplinsky](https://www.openmiddle.com/tag/owen-kaplinsky/) - [AnneMarie Untalan](https://www.openmiddle.com/tag/annemarie-untalan/) - [Gregory L. Taylor](https://www.openmiddle.com/tag/gregory-l-taylor/) - [Christine Jenkins](https://www.openmiddle.com/tag/christine-jenkins/) - [4.NF.5](https://www.openmiddle.com/tag/4-nf-5/) - [4.NF.3B](https://www.openmiddle.com/tag/4-nf-3b/) - [G-CO.2](https://www.openmiddle.com/tag/g-co-2/) - [G-CO.6](https://www.openmiddle.com/tag/g-co-6/) - [7.G.5](https://www.openmiddle.com/tag/7-g-5/) - [Norma Gordon](https://www.openmiddle.com/tag/norma-gordon/) - [Darbie Valenti](https://www.openmiddle.com/tag/darbie-valenti/) - [Miles Knight](https://www.openmiddle.com/tag/miles-knight/) - [Patrick Vennebush](https://www.openmiddle.com/tag/patrick-vennebush/) - [Giselle Garica](https://www.openmiddle.com/tag/giselle-garica/) - [Gian Cavaliere](https://www.openmiddle.com/tag/gian-cavaliere/) - [Christin Smith](https://www.openmiddle.com/tag/christin-smith/) - [7.NS.1c](https://www.openmiddle.com/tag/7-ns-1c/) - [Ryun Deckert](https://www.openmiddle.com/tag/ryun-deckert/) - [Brock Montgomery](https://www.openmiddle.com/tag/brock-montgomery/) - [Franco D. Adkins](https://www.openmiddle.com/tag/franco-d-adkins/) - [Andrew Stadel](https://www.openmiddle.com/tag/andrew-stadel/) - [Catriona Shearer](https://www.openmiddle.com/tag/catriona-shearer/) - [Erin Stenger](https://www.openmiddle.com/tag/erin-stenger/) - [Melissa Flynn](https://www.openmiddle.com/tag/melissa-flynn/) - [Chris Luzniak](https://www.openmiddle.com/tag/chris-luzniak/) - [Christine Relleva](https://www.openmiddle.com/tag/christine-relleva/) - [Henry Wadsworth](https://www.openmiddle.com/tag/henry-wadsworth/) - [G-GPE.7](https://www.openmiddle.com/tag/g-gpe-7/) - [Dan Wulf](https://www.openmiddle.com/tag/dan-wulf/) - [Samantha Cruz](https://www.openmiddle.com/tag/samantha-cruz/) - [Elizabeth Brandenburg](https://www.openmiddle.com/tag/elizabeth-brandenburg/) - [K.CC.6](https://www.openmiddle.com/tag/k-cc-6/) - [S-ID.1](https://www.openmiddle.com/tag/s-id-1/) - [S-ID.2](https://www.openmiddle.com/tag/s-id-2/) - [David K Butler](https://www.openmiddle.com/tag/david-k-butler/) - [N-CN.5](https://www.openmiddle.com/tag/n-cn-5/) - [5.NBT.1](https://www.openmiddle.com/tag/5-nbt-1/) - [K.OA.2](https://www.openmiddle.com/tag/k-oa-2/) - [DOK 2: Habilidad / Concepto](https://www.openmiddle.com/es/tag/dok-2-habilidad-concepto/) - [DOK 3: Pensamiento Estrategico](https://www.openmiddle.com/es/tag/dok-3-pensamiento-estrategico/) - [N-VM.8](https://www.openmiddle.com/tag/n-vm-8/) - [N-VM.4](https://www.openmiddle.com/tag/n-vm-4/) - [N-RN.3](https://www.openmiddle.com/tag/n-rn-3/) - [F-TF.4](https://www.openmiddle.com/tag/f-tf-4/) - [F-LE.4](https://www.openmiddle.com/tag/f-le-4/) - [Erick Lee](https://www.openmiddle.com/es/tag/erick-lee-es/) - [6.RP.3c](https://www.openmiddle.com/es/tag/6-rp-3c-es/) - [Cecilia Calvo](https://www.openmiddle.com/es/tag/cecilia-calvo-es/) - [8.F.4](https://www.openmiddle.com/es/tag/8-f-4-es/) - [Kara Colley](https://www.openmiddle.com/es/tag/kara-colley-es/) - [F-BF.5](https://www.openmiddle.com/es/tag/f-bf-5-es/) - [John Rowe](https://www.openmiddle.com/es/tag/john-rowe-es/) - [6.EE.3](https://www.openmiddle.com/es/tag/6-ee-3-es/) - [Julie Wright](https://www.openmiddle.com/es/tag/julie-wright-es/) - [G-SRT.11](https://www.openmiddle.com/es/tag/g-srt-11-es/) - [G-SRT.4](https://www.openmiddle.com/es/tag/g-srt-4-es/) - [5.NF.1](https://www.openmiddle.com/es/tag/5-nf-1-es/) - [Owen Kaplinsky](https://www.openmiddle.com/es/tag/owen-kaplinsky-es/) - [DOK 2: Compétence / Concept](https://www.openmiddle.com/fr/tag/dok-2-competence-concept/) - [6.EE.3](https://www.openmiddle.com/fr/tag/6-ee-3-fr/) - [Julie Wright](https://www.openmiddle.com/fr/tag/julie-wright-fr/) - [5.NBT.7](https://www.openmiddle.com/es/tag/5-nbt-7-es/) - [Adrianne Burns](https://www.openmiddle.com/fr/tag/adrianne-burns-fr/) - [Michael Dennis](https://www.openmiddle.com/tag/michael-dennis/) - [Denise White](https://www.openmiddle.com/tag/denise-white/) - [Andrew Constantinescu](https://www.openmiddle.com/tag/andrew-constantinescu/) - [Howie Hua](https://www.openmiddle.com/tag/howie-hua/) - [A-REI.3](https://www.openmiddle.com/es/tag/a-rei-3-es/) - [Robert Kaplinsky](https://www.openmiddle.com/es/tag/robert-kaplinsky-es/) - [6.RP.3a](https://www.openmiddle.com/tag/6-rp-3a/) - [7.EE.4b](https://www.openmiddle.com/es/tag/7-ee-4b-es/) - [Bryan Anderson](https://www.openmiddle.com/es/tag/bryan-anderson-es/) - [G-GPE.5](https://www.openmiddle.com/es/tag/g-gpe-5-es/) - [Nanette Johnson](https://www.openmiddle.com/es/tag/nanette-johnson-es/) - [8.F.2](https://www.openmiddle.com/es/tag/8-f-2-es/) - [7.G.1](https://www.openmiddle.com/es/tag/7-g-1-es/) - [Gian Cavaliere](https://www.openmiddle.com/es/tag/gian-cavaliere-es/) - [5.NBT.7](https://www.openmiddle.com/fr/tag/5-nbt-7-fr/) - [Owen Kaplinsky](https://www.openmiddle.com/fr/tag/owen-kaplinsky-fr/) - [DOK 3: Réflexion Stratégique](https://www.openmiddle.com/fr/tag/dok-3-reflexion-strategique/) - [Robert Kaplinsky](https://www.openmiddle.com/fr/tag/robert-kaplinsky-fr/) - [6.NS.8](https://www.openmiddle.com/tag/6-ns-8/) - [Adrianne Burns](https://www.openmiddle.com/es/tag/adrianne-burns-es/) - [4.NBT.6](https://www.openmiddle.com/fr/tag/4-nbt-6-fr/) - [3.MD.1](https://www.openmiddle.com/es/tag/3-md-1-es/) - [2.NBT.7](https://www.openmiddle.com/es/tag/2-nbt-7-es/) - [Ian Kerr](https://www.openmiddle.com/es/tag/ian-kerr-es/) - [6.NS.4](https://www.openmiddle.com/fr/tag/6-ns-4-fr/) - [Howie Hua](https://www.openmiddle.com/fr/tag/howie-hua-fr/) - [5.OA.1](https://www.openmiddle.com/fr/tag/5-oa-1-fr/) - [6.EE.2c](https://www.openmiddle.com/fr/tag/6-ee-2c-fr/) - [Michael Fenton](https://www.openmiddle.com/fr/tag/michael-fenton-fr/) - [8.EE.1](https://www.openmiddle.com/fr/tag/8-ee-1-fr/) - [Alyson Eaglen](https://www.openmiddle.com/tag/alyson-eaglen/) - [1.OA.2](https://www.openmiddle.com/tag/1-oa-2/) - [7.NS.1](https://www.openmiddle.com/es/tag/7-ns-1-es/) - [Kate Nerdypoo](https://www.openmiddle.com/es/tag/kate-nerdypoo-es/) - [6.EE.1](https://www.openmiddle.com/fr/tag/6-ee-1-fr/) - [8.EE.1a](https://www.openmiddle.com/fr/tag/8-ee-1a-fr/) - [Bryan Anderson](https://www.openmiddle.com/fr/tag/bryan-anderson-fr/) - [Zack Miller](https://www.openmiddle.com/fr/tag/zack-miller-fr/) - [N-RN.2](https://www.openmiddle.com/fr/tag/n-rn-2-fr/) - [Phillip Haislip-Hansberry](https://www.openmiddle.com/fr/tag/phillip-haislip-hansberry-fr/) - [Jonathan Newman](https://www.openmiddle.com/fr/tag/jonathan-newman-fr/) - [6.NS.1b](https://www.openmiddle.com/es/tag/6-ns-1b-es/) - [Mark Alvaro](https://www.openmiddle.com/es/tag/mark-alvaro-es/) - [A-CED.1](https://www.openmiddle.com/es/tag/a-ced-1-es/) - [Daniel Luevanos](https://www.openmiddle.com/es/tag/daniel-luevanos-es/) - [Christina Ploeckelman](https://www.openmiddle.com/tag/christina-ploeckelman/) - [Matt Donahue](https://www.openmiddle.com/tag/matt-donahue/) - [Mike Wiernicki](https://www.openmiddle.com/tag/mike-wiernicki/) - [Joseph Nguyen](https://www.openmiddle.com/tag/joseph-nguyen/) - [5.OA.3](https://www.openmiddle.com/tag/5-oa-3/) - [5.NF.6](https://www.openmiddle.com/tag/5-nf-6/) - [5.NF.2](https://www.openmiddle.com/tag/5-nf-2/) - [Amanda Dey](https://www.openmiddle.com/tag/amanda-dey/) - [1.NBT.3](https://www.openmiddle.com/tag/1-nbt-3/) - [Eric Appleton](https://www.openmiddle.com/tag/eric-appleton/) - [2.NBT.9](https://www.openmiddle.com/tag/2-nbt-9/) - [6.G.4](https://www.openmiddle.com/es/tag/6-g-4-es/) - [7.G.6](https://www.openmiddle.com/es/tag/7-g-6-es/) - [7.G.4](https://www.openmiddle.com/es/tag/7-g-4-es/) - [Christin Smith](https://www.openmiddle.com/es/tag/christin-smith-es/) - [6.G.1](https://www.openmiddle.com/es/tag/6-g-1-es/) - [6.G.3](https://www.openmiddle.com/es/tag/6-g-3-es/) - [6.NS.6b](https://www.openmiddle.com/es/tag/6-ns-6b-es/) - [7.G.5](https://www.openmiddle.com/es/tag/7-g-5-es/) - [8.G.5](https://www.openmiddle.com/es/tag/8-g-5-es/) - [Ashley Henderson](https://www.openmiddle.com/es/tag/ashley-henderson-es/) - [8.G.8](https://www.openmiddle.com/es/tag/8-g-8-es/) - [G-GPE.1](https://www.openmiddle.com/es/tag/g-gpe-1-es/) - [6.SP.5c](https://www.openmiddle.com/es/tag/6-sp-5c-es/) - [4.NF.3C](https://www.openmiddle.com/tag/4-nf-3c/) - [Renee Owen](https://www.openmiddle.com/tag/renee-owen/) - [Chris Ignaciuk](https://www.openmiddle.com/tag/chris-ignaciuk/) - [Adina R](https://www.openmiddle.com/tag/adina-r/) - [Annie DeAngelo](https://www.openmiddle.com/tag/annie-deangelo/) - [Maeve O'Connell](https://www.openmiddle.com/tag/maeve-oconnell/) - [Rochelle Telfer](https://www.openmiddle.com/tag/rochelle-telfer/) - [Marc DeArmond](https://www.openmiddle.com/tag/marc-dearmond/) - [Louise Pepper](https://www.openmiddle.com/tag/louise-pepper/) - [Linda Hutcheson](https://www.openmiddle.com/tag/linda-hutcheson/) - [Brian Errey](https://www.openmiddle.com/tag/brian-errey/) - [Jason Kornoely](https://www.openmiddle.com/tag/jason-kornoely/) - [Paolo Tolomeo](https://www.openmiddle.com/tag/paolo-tolomeo/) - [2.MD.7](https://www.openmiddle.com/tag/2-md-7/) - [4.OA.3](https://www.openmiddle.com/tag/4-oa-3/) - [Daniel Torres-Rangel](https://www.openmiddle.com/es/tag/daniel-torres-rangel-es/) - [G-GPE.4](https://www.openmiddle.com/es/tag/g-gpe-4-es/) - [G-CO.11](https://www.openmiddle.com/es/tag/g-co-11-es/) - [John Mahlstedt](https://www.openmiddle.com/es/tag/john-mahlstedt-es/) - [Arnav Gulati](https://www.openmiddle.com/tag/arnav-gulati/) - [Robert Millett](https://www.openmiddle.com/tag/robert-millett/) - [6.NS.7a](https://www.openmiddle.com/tag/6-ns-7a/) - [Thach-Thao Phan](https://www.openmiddle.com/tag/thach-thao-phan/) - [The Open Middle Elementary Team](https://www.openmiddle.com/tag/the-open-middle-elementary-team/) - [Dean Johnstone](https://www.openmiddle.com/tag/dean-johnstone/) - [4.NF.7](https://www.openmiddle.com/tag/4-nf-7/) - [Nova Katz](https://www.openmiddle.com/tag/nova-katz/) - [Anne Oliveira](https://www.openmiddle.com/tag/anne-oliveira/) - [4.OA.5](https://www.openmiddle.com/tag/4-oa-5/) - [Adina Rochkind](https://www.openmiddle.com/tag/adina-rochkind/) - [Debra Schneider](https://www.openmiddle.com/tag/debra-schneider/) - [G-CO.9](https://www.openmiddle.com/tag/g-co-9/) - [Linda Cochran](https://www.openmiddle.com/tag/linda-cochran/) - [Sarah Furman](https://www.openmiddle.com/tag/sarah-furman/) - 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