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    If x2 + y2 = 36, xy = 32, what is the positive value of x + y?
    Problems
    Grades: 9th to 12th
    Functions
    Algebra
    Interpreting Functions
    Reasoning with Equations and Inequalities
    HSA-REI.B.4b, HSF-IF.C.8a

    How do I love thee?  Let me graph the ways! 

    Can you come up with one or more equations to graph a heart on the coordinate plane? The equations can be rectangular, polar, or parametric.

    Bonus: Can you shift your heart so the graph or its interior includes the point (2, 14)?

    Problems
    Grades: 9th to 12th
    Functions
    Algebra
    Building Functions
    Interpreting Functions
    Reasoning with Equations and Inequalities
    Creating Equations
    HSA-CED.A.2, HSA-REI.D.10, HSF-IF.B.4, HSF-IF.C.7b, HSF-BF.B.3

    In the chart, color each square according to the clues below.

    • Two positive odd numbers that have a sum of 40 and the largest possible product.
    • The smallest square number that is the sum of two non‑zero square numbers.
    • The next five numbers in the arithmetic sequence 8, 19, 30, __, __, __, __, __.
    • The maximum possible number of givens in a standard 9 × 9 Sudoku grid that does not render a unique solution.
    • Two different odd numbers, one of whose digits are the reverse of the other, whose sum is 154.
    • The two prime numbers whose product is 4 less than 5 2 .
    • In a normal distribution, the percent of values within one standard deviation of the mean.
    • The 43 rd positive even number.
    • The first four positive multiples of 4.
    • The integer lengths of three sides of a right triangle whose area is 600 square units.
    • The value of the sum 2 0 + 2 1 + 2 2 + 2 3 .
    • The value of the sum 2 0 + 2 1 + 2 2 + 2 3 + 2 4 .
    Problems
    Grades: 6th to 8th, 9th to 12th, 3rd to 5th
    Expression/Equation
    Functions
    Stats & Probability
    Num & Ops Base Ten
    Algebraic Thinking
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Interpreting Functions
    Summarize and describe distributions.
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    Generate and analyze patterns.
    Gain familiarity with factors and multiples.
    Solve problems involving the four operations, and identify and explain patterns in arithmetic.
    Multiply and divide within 100.
    3.OA.C.7, 3.OA.D.9, 3.NBT.A.2, 4.OA.B.4, 4.OA.C.5, 4.NBT.B.4, 4.NBT.B.5, 6.SP.B.5c, HSF-IF.A.3, 6.EE.A.1

    The Fibonacci sequence is shown below, with each term equal to the sum of the previous two terms. If you take the ratios of successive terms, you get 1, 2, \frac{3}{2} , \frac{5}{3} , \frac{8}{5} , \frac{{13}}{8} , and so on. But as you proceed through the sequence, these ratios get closer and closer to a fixed number, known as the Golden Ratio.

    1, 1, 2, 3, 5, 8, 13, …  

    Using the rule that defines the Fibonacci sequence, can you determine the value of the Golden Ratio?

    Problems
    Grades: 6th to 8th, 9th to 12th
    Ratio & Proportion
    Functions
    Stats & Probability
    Analyze proportional relationships and use them to solve real-world and mathematical problems.
    Interpreting Functions
    Investigate patterns of association in bivariate data.
    Understand ratio concepts and use ratio reasoning to solve problems.
    6.RP.A.1, 8.SP.A.1, HSF-IF.A.3, 7.RP.A.2a
    Mark McGwire became baseball's home run king in 1998 with 70 home runs. His 70th home run ball sold for slightly more than $3 million in 1999. Babe Ruth, an earlier home-run king, hit 60 in 1927. His home-run ball was donated to the Hall of Fame. Suppose that Ruth's ball was valued at $3000 in 1927 and, like many good investments, doubled its value every seven years. Would you rather have the value of Ruth's ball or McGwire's?
    Problems
    Grades: 6th to 8th, 9th to 12th, 3rd to 5th
    Expression/Equation
    Functions
    Num & Ops Base Ten
    Algebraic Thinking
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Interpreting Functions
    Perform operations with multi-digit whole numbers and with decimals to hundredths.
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    Use the four operations with whole numbers to solve problems.
    Solve problems involving the four operations, and identify and explain patterns in arithmetic.
    Multiply and divide within 100.
    Represent and solve problems involving multiplication and division.
    3.OA.A.3, 3.OA.C.7, 3.OA.D.8, 3.OA.D.9, 4.OA.A.2, 4.OA.A.3, 4.NBT.B.5, 5.NBT.B.5, HSF-IF.A.3, 6.EE.A.1

    Equations to solve in your head:

    \begin{array}{l}
 6,751x + 3,249y = 26,751 \\ 
 3,249x + 6,751y = 23,249 \\ 
 \end{array}

    Is this a joke? Not if you can multiply the first equation by 6,751 and the second by 3,249 in your head, and not if you use a second, simpler method.

    Problems
    Grades: 9th to 12th, 6th to 8th
    Algebra
    Expression/Equation
    Reasoning with Equations and Inequalities
    Analyze and solve linear equations and pairs of simultaneous linear equations.
    8.EE.C.8b, HSA-REI.C.6, HSA-REI.C.5
    A bowl contains 75 candies, identical except for color. Twenty are red, 25 are green, and 30 are brown. Without looking, what is the least number of candies you must pick in order to be absolutely certain that three of them are brown?
    Problems
    Grades: 9th to 12th, 6th to 8th
    Stats & Probability
    Using Probability to Make Decisions
    Investigate chance processes and develop, use, and evaluate probability models.
    7.SP.C.5, 7.SP.C.7a, HSS-MD.B.5a

    A 10 × 10 grid is painted with three primary colors (red, yellow, and blue) and three secondary colors (green, purple, and orange). The secondary colors are made by mixing equal parts of the appropriate primary colors — that is, red and yellow are mixed to make orange, red and blue to make purple, and yellow and blue to make green.

    The figure at left shows squares that were painted red and blue. No other squares were painted either red or blue. 

    Suppose that each small square requires a quart of paint. Altogether, 31 quarts of red paint, 40 quarts of blue paint, and 29 quarts of yellow paint were used to paint the entire 10 × 10 grid.

    Given this information, can you determine if there were more yellow or purple squares? And how many more?

    Problems
    Grades: 9th to 12th, 6th to 8th
    Algebra
    Expression/Equation
    Reasoning with Equations and Inequalities
    Analyze and solve linear equations and pairs of simultaneous linear equations.
    8.EE.C.8b, HSA-REI.C.6

    To the left is a circle with an inscribed square. Obviously, there isn’t room for another nonoverlapping square of the same size within the circle. But suppose that you divided the square into n2 smaller squares, each with side length 1/n. Would one of those smaller squares fit in the space between the large square and the circle? As shown to the left, this works if n = 16 and the large square were divided into 256 smaller squares. But it would work for smaller values of n, too.

    What is the smallest value of n such that one of the smaller squares would fit between the larger square and the circle?

    Problems
    Grades: 9th to 12th, 6th to 8th
    Geometry
    Circles
    Similarity, Right Triangles, and Trigonometry
    Understand and apply the Pythagorean Theorem.
    8.G.B.7, HSG-SRT.C.8, HSG-C.A.3

    Label the ten points in the grid shown with the letters A-J so that

    AB < BC < CD < … < HI < IJ.

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Geometry
    The Number System
    Expressing Geometric Properties with Equations
    Understand and apply the Pythagorean Theorem.
    Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
    Solve real-world and mathematical problems involving area, surface area, and volume.
    Apply and extend previous understandings of numbers to the system of rational numbers.
    Graph points on the coordinate plane to solve real-world and mathematical problems.
    5.G.A.1, 6.NS.C.8, 6.G.A.3, 4.G.A.1, 8.G.B.8, HSG-GPE.B.6, HSG-GPE.B.7

    A plywood sheet is 45 by 45 inches. What is the approximate diameter of the log the sheet was made from?

     

    The diameter d of a circle equals \frac{C}{\pi }, where C is the circumference, but please do not make a mistake. The diameter of the log is not \frac{{45}}{\pi }.

    Problems
    Grades: 9th to 12th, 6th to 8th
    Geometry
    Geometric Measurement and Dimension
    Circles
    Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
    7.G.B.4, HSG-C.A.3, HSG-GMD.A.1
    What is the smallest integer that can be the hypotenuse of two different right triangles, each of which has legs whose lengths are also integers?
    Problems
    Grades: 9th to 12th, 6th to 8th
    Geometry
    Similarity, Right Triangles, and Trigonometry
    Understand and apply the Pythagorean Theorem.
    8.G.B.7, HSG-SRT.C.8
    Take two sheets of 8.5 by 11 inch paper. Roll one into a short cylinder and the other into a tall cylinder. Does one hold more than the other?
    Problems
    Grades: 3rd to 5th, 9th to 12th, 6th to 8th
    Measurement & Data
    Geometry
    Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
    Geometric Measurement and Dimension
    Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
    Solve real-world and mathematical problems involving area, surface area, and volume.
    6.G.A.2, 8.G.C.9, HSG-GMD.A.3, 5.MD.C.4, 5.MD.C.5a, 5.MD.C.5b
    Solve puzzles to strengthen understanding of expanding and factoring polynomials.
    Lesson Plan
    Grades: 9th to 12th, High School
    Algebra
    Arithmetic with Polynomials and Rational Functions
    Seeing Structure in Expressions
    HSA-SSE.A.2, HSA-SSE.B.3a, HSA-APR.A.1
    Use proportions and similar figures to adjust the size of the New York City Subway Map so that it is drawn to scale.
    Lesson Plan
    Grades: 9th to 12th, 6th to 8th, High School
    Geometry
    Number & Quantity
    Ratio & Proportion
    Similarity, Right Triangles, and Trigonometry
    Congruence
    Understand congruence and similarity using physical models, transparencies, or geometry software.
    Quantities
    Draw construct, and describe geometrical figures and describe the relationships between them.
    Understand ratio concepts and use ratio reasoning to solve problems.
    6.RP.A.1, 6.RP.A.3a, 7.G.A.1, HSN-Q.A.3, 8.G.A.1a, 8.G.A.1b, 8.G.A.1c, 8.G.A.5, HSG-CO.A.1, HSG-SRT.A.2, HSG-SRT.A.3, HSG-SRT.B.4, HSG-SRT.B.5
    Model linear functions using Barbie dolls and rubber bands.
    Lesson Plan
    Grades: High School, 6th to 8th, 9th to 12th
    Stats & Probability
    Interpreting Categorical and Quantitative Data
    Investigate patterns of association in bivariate data.
    8.SP.A.1, 8.SP.A.2, 8.SP.A.3, HSS-ID.B.6a, HSS-ID.B.6c, HSS-ID.C.7
    Manipulate and fold three paper circles to explore four circle theorems.
    Lesson Plan
    Grades: 9th to 12th
    Geometry
    Circles
    HSG-C.A.2
    Use polydrons to build nets and create the most appealing fish tank.
    Lesson Plan
    Grades: 9th to 12th, High School, 6th to 8th
    Geometry
    Congruence
    Geometric Measurement and Dimension
    Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
    Solve real-world and mathematical problems involving area, surface area, and volume.
    6.G.A.2, 6.G.A.4, 8.G.C.9, HSG-GMD.A.3, HSG-CO.A.1
    Participate in a modeling activity where they will learn the rules for translations, rotations, and reflections.
    Lesson Plan
    Grades: 9th to 12th, 6th to 8th
    Geometry
    Congruence
    Understand congruence and similarity using physical models, transparencies, or geometry software.
    8.G.A.3, HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.B.6
    Use problem-solving skills to find the solution to a multi-variable problem that is solved by manipulating linear equations.
    Lesson Plan
    Grades: High School, 9th to 12th
    Algebra
    Reasoning with Equations and Inequalities
    Creating Equations
    HSA-CED.A.3, HSA-REI.C.6, HSA-REI.D.11
    1 - 20 of 76 results