In a four‑digit number, the sum of the digits is 10. All the digits are different. What is the largest such four‑digit number?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Num & Ops Base Ten
Make sense of problems and persevere in solving them.
Generalize place value understanding for multi-digit whole numbers.
4.NBT.A.2, CCSS.Math.Practice.MP1
Place a number in each of the following empty boxes so that the sum
of the numbers in any 3 consecutive boxes is 2013. What is the number
that should go in the box with the question mark?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Num & Ops Base Ten
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.A.2, 4.NBT.B.4, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
There are 4! = 24 ways to rank four objects. However, a friend told me that if ties
are allowed, the number increases to 75.
I
attempted to list all the possibilities by first listing the 24 orderings of
four objects, then using brackets to group ties involving two players, then
group ties involving three players, and finally the single case in which all
four objects are tied. But something has gone wrong; my list includes just 69
possibilities, not 75.
What
happened? Did I miss something, or was my friend mistaken?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Attend to precision.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
The rectangle shown consists of eight squares. The length of each side of each
square is 1 unit. The length of the shortest path from A to C using the lines
shown is 6 units.
How
many different six-unit paths are there from A to C?
Problems
Grades: 9th to 12th, 6th to 8th, 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
The length and width of a rectangle are whole numbers of centimeters. Neither is
divisible by 6. The area of the rectangle is 36 square centimeters.
What is the perimeter of the rectangle, in centimeters?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
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.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Gain familiarity with factors and multiples.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
3.MD.D.8, 4.OA.B.4, 3.MD.C.7b, 4.MD.A.3, CCSS.Math.Practice.MP1
A calendar year is typically referred to as a four‑digit number, as in 2008, or
as a two‑digit number, as in ’08. Sometimes, the two‑digit number divides
evenly into the four‑digit number, with no remainder.
How many times did this happen during the twentieth century?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
The Number System
Num & Ops Base Ten
Algebraic Thinking
Attend to precision.
Make sense of problems and persevere in solving them.
Compute fluently with multi-digit numbers and find common factors and multiples.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
Multiply and divide within 100.
3.OA.C.7, 5.NBT.B.6, 6.NS.B.2, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
What
is the smallest positive number with exactly ten positive integer divisors?
And what is the next one after that?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Algebraic Thinking
Make sense of problems and persevere in solving them.
Gain familiarity with factors and multiples.
4.OA.B.4, CCSS.Math.Practice.MP1
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: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Expression/Equation
Num & Ops Fractions
Algebraic Thinking
Attend to precision.
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
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: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Functions
Num & Ops Base Ten
Algebraic Thinking
Look for and make use of structure.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Interpreting Functions
Generalize place value understanding for multi-digit whole numbers.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.9, 4.NBT.A.2, HSF-IF.A.3, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP7
There are 5 houses on a street: house A, B, C, D and E.
The distance between any two adjacent houses is 100 feet. There are 2 children living in house A, 3
children living in house B, 4 children living in house C, 5 children living in
house D and 6 children living in house E.
If the school bus can only make one stop on that street, in front of
which house should the bus stop so that the sum of walking distance among all
children will be the least?
Problems
Grades: 9th to 12th, 6th to 8th, 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
Jada has 1 penny, 2 nickels, and 1 dime. How many different sums of money can she make, if she uses at least one coin?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Attend to precision.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6
If the sum
of three numbers equals zero, and the sum of their cubes equals 90, what is
their product?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Expression/Equation
The Number System
Attend to precision.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6
What is the smallest integer n > 1 for which 3^{n} > n^{9}?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Expression/Equation
Num & Ops Base Ten
Attend to precision.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6
Write the number 41 in a box. Now move in a counterclockwise direction, creating new
boxes and each time adding 1 to the number inside. This spiral starts out as
follows.
Note
that the numbers in bold — 41, 43, and 47 — are prime numbers
(numbers whose only divisors are themselves and 1), and they occur along a diagonal.
If you keep filling in the spiral, what is the first number that is not prime to
appear along this diagonal?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Algebraic Thinking
Make sense of problems and persevere in solving them.
Gain familiarity with factors and multiples.
4.OA.B.4, CCSS.Math.Practice.MP1
The letters of EAT can be
rearranged to become TEA by moving the third letter (T) to the first
position and by moving the other letters one position to the right. This
process could be described as 1→2, 2→3, 3→1.
When this same process is applied again, then TEA becomes ATE.
Similarly, the process 1→3, 2→2, 3→1, 4→4
will convert TONE to NOTE. When the process is applied again, NOTE returns to
TONE. (Not very interesting, is it?)
Your challenge begins with
the five letters A, E, M, S, and T. Use them to form a common English word.
Then, rearrange the letters to form a second common English word. Finally,
apply the same process of rearrangement to form a third common English word.
Can you do it?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
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
Mathematical Practices
The Number System
Expressing Geometric Properties with Equations
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, HSG-GPE.B.6, HSG-GPE.B.7
Students in Mrs. Walker’s classroom had an estimation
contest. The student whose estimate is
the closest to the number of marbles in a jar wins the contest. Vicki, who
estimated 135 marbles, won the contest. Timothy, who estimated 150 marbles, got
second place. Lyon, who estimated 152, got third place. And Quinn, who
estimated 131, got fourth place. What is the exact number of marbles in the
jar?
Problems
Grades: 9th to 12th, 6th to 8th, 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
Three lines cut rectangle ABCD into six congruent
(identical) squares.
The perimeter of rectangle ABCD is 30
cm. What is the area of the shaded region, in square centimeters?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Measurement & Data
Reason abstractly and quantitatively.
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.
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.C.7a, 3.MD.C.7b, 4.MD.A.3, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
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: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Ratio & Proportion
Num & Ops Base Ten
Algebraic Thinking
Look for and make use of structure.
Attend to precision.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
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, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6, CCSS.Math.Practice.MP7
How many different color
patterns can be created by placing the circles onto the 4 × 4 grid
such that each circle is placed on a square with the same number?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Attend to precision.
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6