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, 9th to 12th, 3rd to 5th
Mathematical Practices
Algebraic Thinking
Expression/Equation
Make sense of problems and persevere in solving them.
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.
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.2c, 3.OA.A.3, 3.OA.D.8, 4.OA.A.2, 4.OA.A.3, CCSS.Math.Practice.MP1
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, 9th to 12th
The Number System
Num & Ops Base Ten
Mathematical Practices
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.
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, 5.NBT.A.3b, 5.NBT.B.6, 6.NS.B.2
Imagine that you bought a Beanie Baby® for $6, sold it for $7, bought it back for $8, then sold it for $9. How much profit did you make?
Problems
Grades: 6th to 8th, 3rd to 5th, 9th to 12th
Mathematical Practices
Measurement & Data
Algebraic Thinking
Make sense of problems and persevere in solving them.
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Use the four operations with whole numbers to solve problems.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.8, 4.OA.A.3, 4.MD.A.2, CCSS.Math.Practice.MP1
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, 9th to 12th
The Number System
Ratio & Proportion
Num & Ops Base Ten
Mathematical Practices
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.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
Attend to precision.
Make sense of problems and persevere in solving them.
Compute fluently with multi-digit numbers and find common factors and multiples.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.2, 6.RP.A.3b, 6.NS.B.3, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6, 5.NBT.B.7, 7.RP.A.1, 7.NS.A.3, 7.NS.A.2a, 7.NS.A.2c
If each side of the triangle in Figure 1 is 1 inch long, this means the triangle has a perimeter of 3 inches. Suppose you continued the pattern in the diagram until you reached Figure 5. What is the sum of the perimeters of all the white triangles in Figure 5?
Problems
Grades: 6th to 8th, 3rd to 5th, 9th to 12th
Mathematical Practices
Algebraic Thinking
Measurement & Data
Functions
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Look for and make use of structure.
Attend to precision.
Generate and analyze patterns.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Interpreting Functions
HSF-IF.A.3, 3.OA.D.9, 3.MD.D.8, 4.OA.C.5, CCSS.Math.Practice.MP6, CCSS.Math.Practice.MP7, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
Suppose you love chocolate. The top of each cookie is covered with the same thickness of chocolate. If you wanted to choose the cookie with more chocolate, which would you pick?
Problems
Grades: 6th to 8th, 3rd to 5th, 9th to 12th
Mathematical Practices
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1
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
Mathematical Practices
Algebraic Thinking
Num & Ops Base Ten
Functions
Expression/Equation
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
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.
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.
Interpreting Functions
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.1, HSF-IF.A.3, 4.NBT.B.5, 5.NBT.B.5, 3.OA.A.3, 3.OA.C.7, 3.OA.D.8, 3.OA.D.9, 4.OA.A.2, 4.OA.A.3, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
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: 6th to 8th, 3rd to 5th, 9th to 12th
Geometry
Measurement & Data
Mathematical Practices
Solve real-world and mathematical problems involving area, surface area, and volume.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Make sense of problems and persevere in solving them.
Geometric Measurement and Dimension
HSG-GMD.A.3, 5.MD.C.5a, 5.MD.C.5b, CCSS.Math.Practice.MP1, 5.MD.C.4, 8.G.C.9, 6.G.A.2
Two players each roll an ordinary six-sided die. Of the two numbers showing, the smaller is subtracted from the larger. If the difference is 0, 1, or 2, player A gets 1 points. If the difference is 3, 4, or 5, Player B gets 1 point. The game ends after 12 rounds. The player with the most points wins the game. Is the game fair?
Problems
Grades: 6th to 8th, 9th to 12th, 3rd to 5th
Stats & Probability
Mathematical Practices
Investigate chance processes and develop, use, and evaluate probability models.
Using Probability to Make Decisions
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Look for and make use of structure.
Attend to precision.
HSS-MD.B.6, CCSS.Math.Practice.MP6, CCSS.Math.Practice.MP7, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, HSS-MD.B.5a, 7.SP.C.7a
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, 9th to 12th, 3rd to 5th
Mathematical Practices
Num & Ops Fractions
Ratio & Proportion
Attend to precision.
Make sense of problems and persevere in solving them.
Develop understanding of fractions as numbers.
Understand ratio concepts and use ratio reasoning to solve problems.
Extend understanding of fraction equivalence and ordering.
Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.RP.A.3, 4.NF.A.2, 6.RP.A.1, 6.RP.A.3a, 3.NF.A.3d, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
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: 3rd to 5th, 6th to 8th, 9th to 12th
Algebraic Thinking
Ratio & Proportion
Num & Ops Base Ten
Mathematical Practices
Multiply and divide within 100.
Understand ratio concepts and use ratio reasoning to solve problems.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Look for and make use of structure.
Attend to precision.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6, CCSS.Math.Practice.MP7, 4.NBT.B.5, 6.RP.A.3a, 3.OA.C.7
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, 9th to 12th
The Number System
Num & Ops Base Ten
Mathematical Practices
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.
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, 5.NBT.B.6, 6.NS.B.2
Would you rather work seven days at $20 per day or be paid $2 the first day and have your salary double every day for a week?
Problems
Grades: 3rd to 5th, 6th to 8th, 9th to 12th
Algebraic Thinking
Mathematical Practices
Num & Ops Base Ten
Functions
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Look for and make use of structure.
Generalize place value understanding for multi-digit whole numbers.
Interpreting Functions
HSF-IF.A.3, 4.NBT.A.2, CCSS.Math.Practice.MP7, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, 3.OA.D.9
Your team is down by one point. Your teammate, who makes free throws about three-fourths of the time, is at the free throw line. She gets a second shot if she makes the first one. Each free throw she makes is worth 1 point. If there is no time left, what are the chances you win the game without overtime?
Problems
Grades: 6th to 8th, 9th to 12th, 3rd to 5th
Stats & Probability
Mathematical Practices
Investigate chance processes and develop, use, and evaluate probability models.
Using Probability to Make Decisions
Make sense of problems and persevere in solving them.
Conditional Probability and the Rules of Probability
HSS-CP.A.2, HSS-CP.B.8, HSS-CP.B.9, CCSS.Math.Practice.MP1, HSS-MD.B.5a, 7.SP.C.7a, 7.SP.C.8a, 7.SP.C.8b
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, 3rd to 5th, 9th to 12th
Ratio & Proportion
Expression/Equation
Functions
Algebraic Thinking
Mathematical Practices
Algebra
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Represent and analyze quantitative relationships between dependent and independent variables.
Use functions to model relationships between quantities.
Multiply and divide within 100.
Make sense of problems and persevere in solving them.
Reason about and solve one-variable equations and inequalities.
Linear, Quadratic, and Exponential Models
Building Functions
Reasoning with Equations and Inequalities
Creating Equations
HSA-CED.A.2, HSA-CED.A.3, HSA-REI.A.1, HSF-BF.A.1a, HSF-LE.A.2, 6.EE.B.6, CCSS.Math.Practice.MP1, 3.OA.C.7, 8.F.B.4, 6.EE.C.9, 7.EE.B.4a, 7.RP.A.2c
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, 3rd to 5th
Expression/Equation
Mathematical Practices
Algebra
Analyze and solve linear equations and pairs of simultaneous linear equations.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Reasoning with Equations and Inequalities
Creating Equations
HSA-CED.A.3, HSA-REI.C.6, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, 8.EE.C.8c, 8.EE.C.8b
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: 6th to 8th, 3rd to 5th, 9th to 12th
Geometry
Measurement & Data
Mathematical Practices
Solve real-world and mathematical problems involving area, surface area, and volume.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Make sense of problems and persevere in solving them.
5.MD.C.5a, CCSS.Math.Practice.MP1, 5.MD.C.3a, 5.MD.C.3b, 5.MD.C.4, 6.G.A.2, 6.G.A.4
In each of the following 5 boxes, there are some letter
cards. Joseph takes exactly one letter card out of each box and spells the word
‘QUEST’. Which letter card did he take out from Box 4?
Problems
Grades: 6th to 8th, 3rd to 5th, 9th to 12th
Mathematical Practices
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
In the Sudoku grid at left, only the numbers along the perimeter are shown.
Suppose
the grid was completed according to the standard Sudoku rules — each row,
column and 3 × 3 square contains the numbers 1–9. Would the sum
of all the missing numbers be divisible by 3?
Problems
Grades: 6th to 8th, 3rd to 5th, 9th to 12th
Mathematical Practices
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Attend to precision.
CCSS.Math.Practice.MP6, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
In our Mathematics Circle
we diagrammed 16 blocks of our city. How many different routes can we draw from A to C
moving only upward and to the right?”
Different routes may, of
course, have portions that coincide (as in the diagram).
“This problem is not easy.
Have we solved it by counting 70 different routes?”
What answer should we give
these students?
Problems
Grades: 6th to 8th, 9th to 12th, 3rd to 5th
Mathematical Practices
Attend to precision.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6