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 5^{2}. 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
of the seats in the classroom, how many students would occupy
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
777^{2}
means 777 × 777,
777^{3} means 777 × 777 × 777,
and so on.
Suppose 777^{7} 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 3^{n} > n^{9}?
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
. 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
?
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