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    Assuming that the circumference of each circle below passes through the centers of the other two, and that the radius of each circle is 1, what is the total gray area?
    Problems
    Grades: 9th to 12th, 6th to 8th
    Geometry
    Geometric Measurement and Dimension
    Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
    7.G.B.4, HSG-GMD.A.1
    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

    The triangle at left lies on a flat surface and is pushed at the top vertex. The length of the congruent sides does not change, but the angle between the two congruent sides will increase, and the base will stretch. Initially, the area of the triangle will increase, but eventually the area will decrease, continuing until the triangle collapses.

    What is the maximum area achieved during this process? And, what is the length of the base when this process is used to create a different triangle whose area is the same as the triangle above?

    Problems
    Grades: 9th to 12th, 6th to 8th, 3rd to 5th
    Geometry
    Measurement & Data
    Similarity, Right Triangles, and Trigonometry
    Understand and apply the Pythagorean Theorem.
    Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
    Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
    3.MD.C.5a, 3.MD.C.5b, 3.MD.C.6, 3.MD.D.8, 8.G.B.7, HSG-SRT.C.8

    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
    Use research on the Amazon deforestation to determine whether it is occurring exponentially.
    Lesson Plan
    Grades: 9th to 12th
    Stats & Probability
    Interpreting Categorical and Quantitative Data
    HSS-ID.B.6a, HSS-ID.B.6c, HSS-ID.C.7

    The diagram at left shows the top of a regular pentagon with the top of a square inscribed in it. The shapes share a vertex at the top, and the other two vertices of the square lie on the sides of the pentagon. If the diagram were continued to include the entire pentagon and the entire square, which shape would extend below the other?

    In other words, does the whole square fit inside the pentagon, does the square protrude at the bottom, or do the square and pentagon meet at a single point?

    Problems
    Grades: 9th to 12th
    Geometry
    Circles
    HSG-C.A.3
    Two chess players compete in a best-of-five match. If Chekmatova has a 60% chance of winning any particular game, what is the likelihood that she will win the match?
    Problems
    Grades: 9th to 12th, 6th to 8th
    Stats & Probability
    Conditional Probability and the Rules of Probability
    Investigate chance processes and develop, use, and evaluate probability models.
    7.SP.C.5, 7.SP.C.7a, 7.SP.C.8a, 7.SP.C.8b, HSS-CP.B.8, HSS-CP.B.9

    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
    Juliet bought 10 beads for $18. The beads she bought are red, blue or silver. Red beads are $1 each, blue beads are $2 each and silver beads are $5 each.  If she bought at least one of each, how many red beads did she buy?
    Problems
    Grades: 6th to 8th, 9th to 12th
    Expression/Equation
    Algebra
    Analyze and solve linear equations and pairs of simultaneous linear equations.
    Reasoning with Equations and Inequalities
    Creating Equations
    8.EE.C.8b, HSA-CED.A.3, HSA-REI.C.6, 8.EE.C.8c
    A regular octagon is inscribed inside a square. Another square is inscribed inside the octagon. What is the ratio of the area of the smaller square to the area of the larger square?
    Problems
    Grades: 9th to 12th, 3rd to 5th, 6th to 8th
    Geometry
    Measurement & Data
    Ratio & Proportion
    Circles
    Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
    Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
    Understand ratio concepts and use ratio reasoning to solve problems.
    6.RP.A.1, 3.MD.C.7b, 4.MD.A.3, HSG-C.A.3
    Create and use linear models to make projections about what impact increased portion sizes may have on weight.
    Lesson Plan
    Grades: 9th to 12th
    Functions
    Algebra
    Linear, Quadratic, and Exponential Models
    Building Functions
    Interpreting Functions
    Reasoning with Equations and Inequalities
    Creating Equations
    HSA-CED.A.2, HSA-CED.A.3, HSA-REI.A.1, HSF-IF.C.7a, HSF-BF.A.1a, HSF-LE.A.1a, HSF-LE.A.2, HSF-LE.B.5

    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 circle of radius 1 unit is inscribed inside a right triangle that has height a and base b. If b is an integer, what are the possible values of a?
    Problems
    Grades: 9th to 12th
    Geometry
    Circles
    HSG-C.A.3
    Examine and draw representations of cubes and then learn how to analyze these representations using complex numbers.
    Lesson Plan
    Grades: 9th to 12th
    Number & Quantity
    The Complex Number System
    HSN-CN.A.1, HSN-CN.A.2, HSN-CN.B.5, HSN-CN.B.6
    develop a delicious new drink by mixing various concentrations of a two-fold dilution series.
    Lesson Plan
    Grades: High School, 9th to 12th
    Functions
    Linear, Quadratic, and Exponential Models
    HSF-LE.A.1c
    Explore the concept of genetics and inheritance using probability.
    Lesson Plan
    Grades: 9th to 12th, 6th to 8th
    Stats & Probability
    Making Inferences and Justifying Conclusions
    Investigate patterns of association in bivariate data.
    Investigate chance processes and develop, use, and evaluate probability models.
    Summarize and describe distributions.
    6.SP.B.5b, 7.SP.C.6, 7.SP.C.7a, 7.SP.C.7b, 8.SP.A.1, HSS-IC.A.2
    Discover how to determine the appropriate number of digits that should be reported.
    Lesson Plan
    Grades: High School, 9th to 12th
    Number & Quantity
    Quantities
    HSN-Q.A.3
    Investigate the properties of regression lines and correlation using an online interactive tool.
    Lesson Plan
    Grades: High School, 9th to 12th
    Stats & Probability
    Interpreting Categorical and Quantitative Data
    HSS-ID.B.6a, HSS-ID.B.6b, HSS-ID.B.6c, HSS-ID.C.7, HSS-ID.C.8
    Discover the pattern of Pick’s Theorem using physical or virtual manipulatives.
    Lesson Plan
    Grades: 9th to 12th
    Geometry
    Congruence
    HSG-CO.D.12

    Look at the panel of elevator buttons shown. Can you find a set of three buttons whose centers form the vertices of a right triangle and whose numbers are the side lengths of a right triangle? (The classic 3-4-5 right triangle doesn’t work, because the 3, 4, and 5 buttons don’t form a right triangle on the elevator panel.)

    And after you’ve found one set, can you find another?

    Problems
    Grades: 9th to 12th, 6th to 8th
    Geometry
    Expressing Geometric Properties with Equations
    Similarity, Right Triangles, and Trigonometry
    Understand and apply the Pythagorean Theorem.
    8.G.B.7, HSG-SRT.C.8, HSG-GPE.B.7
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