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    The interactive Paper Pool game provides an opportunity for students to develop their understanding of ratio, proportion, greatest common factor and least common multiple.

    In this investigation, students are asked to play a game called Paper Pool. The game is played on rectangular grids made of congruent squares.

    The Paper Pool unit was adapted with permission and guidance from the Connected Mathematics Project.

    Math Content

    Students will:
    • Develop their understanding of ratios, proportions, and equivalent fractions.
    • Find the greatest common factor and the least common multiple.
    • Investigate similar figures.
    • Gather and organize data.
    • Search for patterns.
    Grades: 6th to 8th, 3rd to 5th
    Stats & Probability
    Algebraic Thinking
    Summarize and describe distributions.
    Generate and analyze patterns.
    4.OA.C.5, 6.SP.B.5b
    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 rectangular wooden block (not necessarily a cube) is painted on the outside and then divided into one-unit cubes. It turns out that exactly half of the cubes have paint on them. 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

    Which is bigger, \sqrt {10}  + \sqrt {29} or \sqrt {73}

    Don’t even think about using a calculator for this one.

    Problems
    Grades: 6th to 8th
    Expression/Equation
    The Number System
    Work with radicals and integer exponents.
    Know that there are numbers that are not rational, and approximate them by rational numbers.
    8.NS.A.2, 8.EE.A.2

    The number groups below are the last five digits of the fifth powers of the numbers 31 through 39. However, the groups aren't in the right order to represent the fifth powers of 31 through 39 sequentially. Using only these digits, and without using a calculator, can you place the groups in the correct order?

    35393       35424       29151
    24199   21875   35168
    54432   43957   66176

     

    Problems
    Grades: 6th to 8th, 3rd to 5th
    Expression/Equation
    Num & Ops Fractions
    Apply and extend previous understandings of arithmetic to algebraic expressions.
    Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
    5.NF.B.5a, 6.EE.A.1
    Mara has 3 times as many dollars as her brother, Timmy. If Mara is given $20 by their mother, she will have 7 times as many dollars as Timmy. How many dollars does Timmy have?
    Problems
    Grades: 6th to 8th, 9th to 12th, 3rd to 5th
    Ratio & Proportion
    Functions
    Algebra
    Expression/Equation
    Algebraic Thinking
    Analyze proportional relationships and use them to solve real-world and mathematical problems.
    Linear, Quadratic, and Exponential Models
    Building Functions
    Reasoning with Equations and Inequalities
    Creating Equations
    Use functions to model relationships between quantities.
    Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
    Represent and analyze quantitative relationships between dependent and independent variables.
    Reason about and solve one-variable equations and inequalities.
    Multiply and divide within 100.
    3.OA.C.7, 6.EE.B.6, 6.EE.C.9, 7.EE.B.4a, 8.F.B.4, HSA-CED.A.2, HSA-CED.A.3, HSA-REI.A.1, HSF-BF.A.1a, HSF-LE.A.2, 7.RP.A.2c
    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

    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
    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

    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
    The following isosceles trapezoid is composed of 7 matches. Modify the position of three matches in order to obtain two equilateral triangles.
    Problems
    Grades: 6th to 8th
    Geometry
    Draw construct, and describe geometrical figures and describe the relationships between them.
    7.G.A.2

    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

    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 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
    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 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

    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
    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
    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

    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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