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: 3rd to 5th, 9th to 12th, 6th to 8th
Num & Ops Base Ten
Mathematical Practices
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
3.NBT.A.2, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, 4.NBT.B.4
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: 3rd to 5th, 9th to 12th, 6th to 8th
Measurement & Data
Mathematical Practices
Algebraic Thinking
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Make sense of problems and persevere in solving them.
Gain familiarity with factors and multiples.
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.7b, 3.MD.D.8, 4.OA.B.4, CCSS.Math.Practice.MP1, 4.MD.A.3
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
Algebraic Thinking
The Number System
Num & Ops Base Ten
Attend to precision.
Make sense of problems and persevere in solving them.
Multiply and divide within 100.
Compute fluently with multi-digit numbers and find common factors and multiples.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.B.6, 6.NS.B.2, 3.OA.C.7, 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: 6th to 8th, 3rd to 5th, 9th to 12th
Expression/Equation
Algebraic Thinking
Mathematical Practices
Num & Ops Fractions
Apply and extend previous understandings of arithmetic to algebraic expressions.
Multiply and divide within 100.
Attend to precision.
Make sense of problems and persevere in solving them.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.5a, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6, 3.OA.C.7, 6.EE.A.1
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, 3rd to 5th, 6th to 8th
Functions
Num & Ops Base Ten
Mathematical Practices
Algebraic Thinking
Interpreting Functions
Generalize place value understanding for multi-digit whole numbers.
Look for and make use of structure.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.D.9, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP7, 4.NBT.A.2, HSF-IF.A.3
Can you really trisect an angle with a carpenter's square?
Problems
Grades: High School, 9th to 12th
Geometry
Congruence
HSG-CO.B, HSG-CO.C, HSG-CO.D
How much should the manager of a theater raise his prices in order to maximize his profit?
Problems
Grades: High School, 9th to 12th
Functions
Building Functions
Interpreting Functions
HSF-IF.B.4, HSF-IF.B.5, HSF-IF.C.7, HSF-BF.A.1
Find two transformations that will move one triangle onto the other.
Problems
Grades: High School, 9th to 12th
Geometry
Congruence
HSG-CO.A.5
A Ninja apprentice needs your help to use the density of titanium to calculate the angle of the spike on his throwing star!
Problems
Grades: High School, 9th to 12th
Geometry
Circles
Similarity, Right Triangles, and Trigonometry
HSG-SRT.D, HSG-C.B
Find the dimensions of a window designed to let in the maximum amount of sunshine.
Problems
Grades: 9th to 12th
Geometry
Functions
Modeling with Geometry
Interpreting Functions
HSF-IF.B.4, HSG-MG.A.3
Which of two planes is gaining altitude more quickly?
Problems
Grades: High School, 9th to 12th
Geometry
Similarity, Right Triangles, and Trigonometry
HSG-SRT.C.8
Find the radius of the inscribed circle.
Problems
Grades: High School, 9th to 12th
Geometry
Circles
Similarity, Right Triangles, and Trigonometry
HSG-SRT.C.8, HSG-C.A.2
A girder must be carried along a corridor that has a 90-degree turn at the end. How wide must the corridor be in order for the girder to fit around the corner?
Problems
Grades: 9th to 12th, High School
Geometry
Functions
Similarity, Right Triangles, and Trigonometry
Interpreting Functions
HSF-IF.B.5, HSG-SRT.B.5
Why can't this figure, which includes a circle and a rhombus, exist with the given measurements?
Problems
Grades: High School, 9th to 12th
Geometry
Circles
Similarity, Right Triangles, and Trigonometry
HSG-SRT.C.8, HSG-C.A.2
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