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    A Friedman number is a number that can be represented with an expression that uses only the digits in the number. In addition, the expression can include +, –, ×, ÷, exponents and parentheses, but nothing else. For instance, 25 is a Friedman number because it can be represented as 52. A nice Friedman number is a Friedman number for which the digits occur in the same order in the expression as they do in the original number.
    So, 343 is a nice Friedman number, because it can be represented by an expression with the digits 3, 4, and 3 in the same order:

    343 = (3 + 4)3

    The first seven nice Friedman numbers are 127, 343, 736, 1285, 2187, 2502, 2592. Can you find an expression for each of them?

    Problems
    Grades: 6th to 8th, 3rd to 5th
    Expression/Equation
    Algebraic Thinking
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Write and interpret numerical expressions.
    Multiply and divide within 100.
    3.OA.C.7, 5.OA.A.1, 6.EE.A.1

    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
    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
    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
    If you rearrange the letters S, T, O, and P, what is the probability that you’ll end up with a common English word?
    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
    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

    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

    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
    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

     7772 means 777 × 777,
    7773 means 777 × 777 × 777,
    and so on.

    Suppose 7777 is completely multiplied out. What is the units digit of
    the resulting product?

    Problems
    Grades: 6th to 8th, 3rd to 5th
    Expression/Equation
    Num & Ops Fractions
    Algebraic Thinking
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
    Multiply and divide within 100.
    3.OA.C.7, 5.NF.B.5a, 6.EE.A.1

    The odometer of the family car shows 15,951 miles. The driver noticed that this number is palindromic: it reads the same backward as forward.

    “Curious,” the driver said to himself. “It will be a long time before that happens again.”

    But 2 hours later, the odometer showed a new palindromic number.

    How fast was the car traveling in those 2 hours

    Problems
    Grades: 6th to 8th
    Ratio & Proportion
    Analyze proportional relationships and use them to solve real-world and mathematical problems.
    Understand ratio concepts and use ratio reasoning to solve problems.
    6.RP.A.2, 6.RP.A.3b, 7.RP.A.1
    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
    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

    Create two nine-digit numbers, using the digits 1-9 in some order, so that one can be used as the numerator of a fraction and the other as the denominator to yield an extremely good approximation of \pi . Each digit 1-9 will be used exactly twice, once in the numerator and once in the denominator.

    How close can you get to the exact value of \pi ?

    Problems
    Grades: 3rd to 5th, 6th to 8th, 9th to 12th
    Num & Ops Fractions
    Algebraic Thinking
    The Number System
    Num & Ops Base Ten
    Number & Quantity
    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.
    Develop understanding of fractions as numbers.
    Multiply and divide within 100.
    Know that there are numbers that are not rational, and approximate them by rational numbers.
    Compute fluently with multi-digit numbers and find common factors and multiples.
    Perform operations with multi-digit whole numbers and with decimals to hundredths.
    Understand the place value system.
    Understand decimal notation for fractions, and compare decimal fractions.
    Quantities
    HSN-Q.A.3, 4.NF.C.7, 5.NBT.A.3b, 5.NBT.B.6, 6.NS.B.2, 8.NS.A.1, 3.OA.C.7, 3.NF.A.1, 3.NF.A.3c, 4.NF.B.4a, 5.NF.B.3
    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
    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
    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
    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

    In May 1999, two National League baseball players, Joe McEwing of the St.Louis Cardinals and Mike Lieberthal of the Philadelphia Phillies, each had the batting averages, as show below.

    Lieberthal: Bats- 132; Hits- 45; Batting Average- 0.341

    McEwing: Bats- 132; Hits- 45; Batting Average- 0.341

    Suppose McEwing then batted 0.800 (4 hits in 5 at bats), and Lieberthal was perfect (3 hits in 3 at bats). Which player now has the higher batting average? Are you surprised?

    Problems
    Grades: 6th to 8th, 3rd to 5th
    The Number System
    Num & Ops Base Ten
    Num & Ops Fractions
    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.
    Understand the place value system.
    Understand decimal notation for fractions, and compare decimal fractions.
    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, 4.OA.A.2, 4.OA.A.3, 4.NF.C.7, 5.NBT.A.3b, 5.NBT.B.6, 6.NS.B.2
    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
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