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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

    Consider three six-sided dice A, B, and C, with the following numbers on their sides:

    A: 2, 2, 4, 4, 9, 9

    B: 1, 1, 6, 6, 8, 8

    C: 3, 3, 5, 5, 7, 7

    What is the probability that:

    ·     A produces a higher number than B?

    ·     B produces a higher number than C?

    ·     C produces a higher number than A?

    Can you find another set of face values for A, B, and C that yield the same properties? (Does such a set even exist?)

    Problems
    Grades: 6th to 8th
    Stats & Probability
    Investigate chance processes and develop, use, and evaluate probability models.
    7.SP.C.5, 7.SP.C.7a

    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
    One invention saves 30% on fuel; a second, 45%; and a third, 25%. If you use all three inventions at once can you save 100%? If not, how much?
    Problems
    Grades: 6th to 8th
    The Number System
    Ratio & Proportion
    Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
    Understand ratio concepts and use ratio reasoning to solve problems.
    6.RP.A.3c, 7.NS.A.3
    A rectangular wooden block (not necessarily a cube) is painted on the outside and then divided into one-unit cubes. As it happens, the total number of painted faces equals the total number of unpainted faces. What were the dimensions of the block before it was painted?
    Problems
    Grades: 3rd to 5th, 6th to 8th
    Measurement & Data
    Geometry
    Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
    Solve real-world and mathematical problems involving area, surface area, and volume.
    5.MD.C.3a, 5.MD.C.3b, 6.G.A.2, 6.G.A.4, 5.MD.C.4, 5.MD.C.5a

    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 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
    If 18 students occupy  \frac{3}{5} of the seats in the classroom, how many students would occupy  \frac{2}{3} of the seats in the room?
    Problems
    Grades: 3rd to 5th, 6th to 8th
    Num & Ops Fractions
    Ratio & Proportion
    Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
    Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
    Understand ratio concepts and use ratio reasoning to solve problems.
    5.NF.B.4a, 6.RP.A.1, 6.RP.A.3a, 4.NF.B.4b, 4.NF.B.4c, 5.NF.B.7a, 5.NF.B.7b, 5.NF.B.7c

    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
    What is the smallest integer n > 1 for which 3n > n9?
    Problems
    Grades: 6th to 8th, 3rd to 5th
    Expression/Equation
    Num & Ops Base Ten
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Generalize place value understanding for multi-digit whole numbers.
    4.NBT.A.2, 6.EE.A.1
    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
    Suppose you found an old roll of 15¢ stamps. Can you use a combination of 33¢ stamps and 15¢ stamps to mail a package for exactly $1.77?
    Problems
    Grades: 6th to 8th, 3rd to 5th
    Ratio & Proportion
    Num & Ops Base Ten
    Algebraic Thinking
    Understand ratio concepts and use ratio reasoning to solve problems.
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    Multiply and divide within 100.
    3.OA.C.7, 4.NBT.B.5, 6.RP.A.3a
    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
    If the sum of three numbers equals zero, and the sum of their cubes equals 90, what is their product?
    Problems
    Grades: 6th to 8th
    Expression/Equation
    The Number System
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
    Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
    7.NS.A.1d, 7.NS.A.3, 7.EE.B.3, 6.EE.A.1
    A grocery store sells brown rice in 3-pound bags and white rice in 5-pound bags. Katrina bought a total of 22 pounds of rice. How many bags of rice 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
    Every year, Arctic terns fly from the Arctic to the Antarctic and back, a distance of about 9000 miles each way. Suppose the birds fly at an average speed of 25 miles per hour for 12 hours a day. How many days of flying would be necessary to make the roundtrip?
    Problems
    Grades: 6th to 8th, 3rd to 5th
    The Number System
    Num & Ops Base Ten
    Algebraic Thinking
    Compute fluently with multi-digit numbers and find common factors and multiples.
    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.
    Multiply and divide within 100.
    3.OA.C.7, 4.NBT.B.5, 5.NBT.B.5, 5.NBT.B.6, 6.NS.B.2
    If all grape juice concentrates are the same strength, which recipe would you expect to have the strongest grape taste?
    Problems
    Grades: 6th to 8th, 3rd to 5th
    Ratio & Proportion
    Num & Ops Fractions
    Analyze proportional relationships and use them to solve real-world and mathematical problems.
    Understand ratio concepts and use ratio reasoning to solve problems.
    Extend understanding of fraction equivalence and ordering.
    Develop understanding of fractions as numbers.
    3.NF.A.3d, 4.NF.A.2, 6.RP.A.1, 6.RP.A.3a, 7.RP.A.3
    The Environmental Protection Agency (EPA) estimates for gas mileage on 1999 cars vary widely. Which of these cars should go the farthest on one tank of gas?
    Problems
    Grades: 6th to 8th, 3rd to 5th
    The Number System
    Ratio & Proportion
    Num & Ops Base Ten
    Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
    Analyze proportional relationships and use them to solve real-world and mathematical problems.
    Compute fluently with multi-digit numbers and find common factors and multiples.
    Understand ratio concepts and use ratio reasoning to solve problems.
    Perform operations with multi-digit whole numbers and with decimals to hundredths.
    7.NS.A.3, 5.NBT.B.7, 6.RP.A.2, 6.RP.A.3b, 6.NS.B.3, 7.RP.A.1, 7.NS.A.2a, 7.NS.A.2c
    Some doctors use body-mass index (BMI) as a health risk indicator. According to The Old Farmer's Almanac 2000, BMI can be found using the formula: BMI = [(W×705)÷H]/H, where H is height in inches and W is weight is pounds. An index greater than 27 or less than 19 indicates an increased risk for health problems. Helix is 5 feet, 2 inches tall and weighs 110 pounds. Is his health at risk?
    Problems
    Grades: 6th to 8th, 3rd to 5th
    Expression/Equation
    Algebraic Thinking
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Use the four operations with whole numbers to solve problems.
    Solve problems involving the four operations, and identify and explain patterns in arithmetic.
    Represent and solve problems involving multiplication and division.
    3.OA.A.3, 3.OA.D.8, 4.OA.A.2, 4.OA.A.3, 6.EE.A.2c
    Find four distinct integers a, b, c, and d such that ab = c + d and cd = a + b.
    Problems
    Grades: 6th to 8th
    The Number System
    Expression/Equation
    Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
    Use properties of operations to generate equivalent expressions.
    Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
    7.EE.B.3, 7.NS.A.1d, 7.NS.A.3, 7.NS.A.2a, 7.EE.A.2, 7.NS.A.2c
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