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, HSSMD.B.5a
A rectangular wooden block (not necessarily a cube) is painted on the
outside and then divided into oneunit 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 realworld 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, or ?
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 realworld 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 reallife and mathematical problems using numerical and algebraic expressions and equations.
Represent and analyze quantitative relationships between dependent and independent variables.
Reason about and solve onevariable 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, HSACED.A.2, HSACED.A.3, HSAREI.A.1, HSFBF.A.1a, HSFLE.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 reallife and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.4, HSGGMD.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 ,
where C is the circumference, but
please do not make a mistake. The diameter of the log is not .
Problems
Grades: 9th to 12th, 6th to 8th
Geometry
Geometric Measurement and Dimension
Circles
Solve reallife and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.4, HSGC.A.3, HSGGMD.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,
,
,
,
, 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 realworld 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, HSFIF.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, HSGSRT.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, HSAREI.C.6
A
rectangular wooden block (not necessarily a cube) is painted on the outside and
then divided into oneunit 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 realworld 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 homerun king, hit 60 in 1927. His homerun 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 multidigit whole numbers and with decimals to hundredths.
Use place value understanding and properties of operations to perform multidigit 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, HSFIF.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 AJ 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 realworld 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 realworld and mathematical problems.
5.G.A.1, 6.NS.C.8, 6.G.A.3, 4.G.A.1, 8.G.B.8, HSGGPE.B.6, HSGGPE.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 reallife 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 multidigit 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 345 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, HSGSRT.C.8, HSGGPE.B.7