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: 9th to 12th, 6th to 8th, 3rd to 5th
Stats & Probability
Mathematical Practices
Using Probability to Make Decisions
Conditional Probability and the Rules of Probability
Make sense of problems and persevere in solving them.
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.7a, 7.SP.C.8a, 7.SP.C.8b, HSS-CP.B.8, HSS-CP.B.9, CCSS.Math.Practice.MP1, HSS-CP.A.2, HSS-MD.B.5a
A man has to take a wolf, a goat, and some cabbage across a river. His rowboat has enough room for the man plus either the wolf or the goat or the cabbage. If he takes the cabbage with him, the wolf will eat the goat. If he takes the wolf, the goat will eat the cabbage. Only when the man is present are the goat and the cabbage safe from their enemies. All the same, the man carries wolf, goat, and cabbage across the river. How?
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
Write 2014 with the first four prime numbers, with the
aid of the operations addition, multiplication and exponentiation.
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th, High School
Mathematical Practices
Expression/Equation
Algebraic Thinking
Make sense of problems and persevere in solving them.
Apply and extend previous understandings of arithmetic to algebraic expressions.
Gain familiarity with factors and multiples.
Multiply and divide within 100.
3.OA.C.7, 4.OA.B.4, 6.EE.A.1, CCSS.Math.Practice.MP1
A pocket watch is placed next to a digital clock. Several times a day,
the number of minutes shown by the digital clock is equal to the number
of degrees between the hands of the watch. (The watch does not have a
second hand.) As you can see, 10:27 is not one of those times — the
angle between the hands is much greater than 27°. If fractional minutes
aren’t allowed, at what times does this happen?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Measurement & Data
Model with mathematics.
Make sense of problems and persevere in solving them.
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3.MD.A.1, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP4
On the map shown, begin at Start. Travel the roads along any path you
like, following typical traffic laws, and each time you pass a number,
add it to your current sum. However, you are not allowed to pass any
number more than once. Can you reach End with a sum of 91?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Num & Ops Base Ten
Attend to precision.
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.MP6
Let A = 1, B = 2, and so on, with each letter equal to its position in the alphabet. The lexivalue of a word is the sum of the values of its letters. For example, ROMANS has a lexivalue of 18 + 15 + 13 + 1 + 14 + 19 = 80.
Now, do the following:
- Pick a number.
- Convert it into its representation in Roman numerals.
- Find the lexivalue for that Roman numeral.
For example, if you choose 11, that becomes XI in Roman numerals, and XI has a lexivalue of 24 + 9 = 33.
Are there any numbers for which the lexivalue is equal to the original number?
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
Equations to solve in your
head:
Is this a joke? Not if you can
multiply the first equation by 6,751 and the second by 3,249 in your head, and
not if you use a second, simpler method.
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Algebra
Mathematical Practices
Expression/Equation
Reasoning with Equations and Inequalities
Attend to precision.
Make sense of problems and persevere in solving them.
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.8b, HSA-REI.C.6, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP6, HSA-REI.C.5
Wheels A, B, C, and D are connected with belts as shown. If wheel A starts to rotate clockwise as the arrow indicates, can all 4 wheels rotate? If so, which way does each wheel rotate?
Can all the wheels turn if all 4 belts are crossed? If 1 or 3 belts are crossed?
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
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, 3rd to 5th, High School
Geometry
Mathematical Practices
Geometric Measurement and Dimension
Circles
Make sense of problems and persevere in solving them.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.4, CCSS.Math.Practice.MP1, HSG-C.A.3, HSG-GMD.A.1
In 2008, September and
December both began on a Monday. But every year, there are two months that do
not begin on the same day of the week as any other month. What are those two
months?
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
How
do I love thee? Let me graph the ways!
Can
you come up with one or more equations to graph a heart on the coordinate
plane? The equations can be rectangular, polar, or parametric.
Bonus:
Can you shift your heart so the graph or its interior includes the point (2, 14)?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Functions
Algebra
Make sense of problems and persevere in solving them.
Building Functions
Interpreting Functions
Reasoning with Equations and Inequalities
Creating Equations
HSA-CED.A.2, HSA-REI.D.10, HSF-IF.B.4, HSF-IF.C.7b, HSF-BF.B.3, CCSS.Math.Practice.MP1
When 68 is divided by a certain number, the remainder is 4. Find the sum of all possible divisors.
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Algebraic Thinking
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Gain familiarity with factors and multiples.
Multiply and divide within 100.
3.OA.C.7, 4.OA.B.4, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
How do I love thee? Let me plot the ways!
A heart is drawn on a coordinate plane by plotting the following points and connecting them:
- The coordinates of the points are:
(
n
,
n
), (
n
- 3,
n
+ 3), (
n
- 6,
n
), (
n
- 9,
n
+ 3), (
n
- 12,
n
), (
n
- 12,
n
- 3), (
n
- 6,
n
- 9), and (
n
,
n
-3).
- The coordinates of one point are (2, 14).
- All coordinates are positive integers.
What is the value of
n
?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
The Number System
Geometry
Make sense of problems and persevere in solving them.
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, 5.G.A.2, 6.NS.C.6b, 6.NS.C.6c, 6.NS.C.8, CCSS.Math.Practice.MP1
A magic rectangle is an m× n array of the positive integers from 1 to m× n such
that the numbers in each row have a constant sum and the numbers in
each column have a constant sum (although the row sum need not equal the
column sum). Shown below is a 3 × 5 magic rectangle with the integers
1-15.
Two of three arrays at left can be filled with the integers 1-24 to form a magic rectangle. Which one can't, and why not?
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
The following isosceles trapezoid is composed of 7
matches. Modify the position of three matches in order to obtain two
equilateral triangles.
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Geometry
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Draw construct, and describe geometrical figures and describe the relationships between them.
7.G.A.2, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
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, 3rd to 5th
Stats & Probability
Mathematical Practices
Using Probability to Make Decisions
Make sense of problems and persevere in solving them.
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.5, 7.SP.C.7a, CCSS.Math.Practice.MP1, HSS-MD.B.5a
A figure resembling a spiral is shown with 35 matches. Move 4 matches to form 3 squares.
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Geometry
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
Draw construct, and describe geometrical figures and describe the relationships between them.
7.G.A.2, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2
How do I love thee? Let me build the
ways!
Make
a heart using any of the shapes in the PDF file. You can change their size, but you
cannot change their shape. And you can use a shape more than once.
Can
you make a heart with just three shapes? What about five? Six? Ten? How many
different hearts can you make?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP1
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, 3rd to 5th
Ratio & Proportion
Mathematical Practices
Functions
Stats & Probability
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Attend to precision.
Reason abstractly and quantitatively.
Make sense of problems and persevere in solving them.
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, HSF-IF.A.3, CCSS.Math.Practice.MP1, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6, 7.RP.A.2a
Ask a friend to pick a number from 1 through 1,000. After asking him
ten questions that can be answered yes or no, you tell him the number.
What kind of Questions?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Mathematical Practices
Num & Ops Base Ten
Attend to precision.
Reason abstractly and quantitatively.
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, CCSS.Math.Practice.MP2, CCSS.Math.Practice.MP6