If
x^{2} + y^{2} = 36, xy = 32,
what is the positive value of x + y?
Problems
Grades: 9th to 12th
Functions
Algebra
Interpreting Functions
Reasoning with Equations and Inequalities
HSA-REI.B.4b, HSF-IF.C.8a
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
Functions
Algebra
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
In the chart, color each square according to the clues below.
- Two positive odd numbers that have a sum of 40 and the largest possible product.
- The smallest square number that is the sum of two non‑zero square numbers.
- The next five numbers in the arithmetic sequence 8, 19, 30, __, __, __, __, __.
- The maximum possible number of givens in a standard 9 × 9 Sudoku grid that does not render a unique solution.
- Two different odd numbers, one of whose digits are the reverse of the other, whose sum is 154.
- The two prime numbers whose product is 4 less than 5
^{2}
.
- In a normal distribution, the percent of values within one standard deviation of the mean.
- The 43
^{rd}
positive even number.
- The first four positive multiples of 4.
- The integer lengths of three sides of a right triangle whose area is 600 square units.
- The value of the sum 2
^{0}
+ 2
^{1}
+ 2
^{2}
+ 2
^{3}
.
- The value of the sum 2
^{0}
+ 2
^{1}
+ 2
^{2}
+ 2
^{3}
+ 2
^{4}
.
Problems
Grades: 6th to 8th, 9th to 12th, 3rd to 5th
Expression/Equation
Functions
Stats & Probability
Num & Ops Base Ten
Algebraic Thinking
Apply and extend previous understandings of arithmetic to algebraic expressions.
Interpreting Functions
Summarize and describe distributions.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Generate and analyze patterns.
Gain familiarity with factors and multiples.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Multiply and divide within 100.
3.OA.C.7, 3.OA.D.9, 3.NBT.A.2, 4.OA.B.4, 4.OA.C.5, 4.NBT.B.4, 4.NBT.B.5, 6.SP.B.5c, HSF-IF.A.3, 6.EE.A.1
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
Ratio & Proportion
Functions
Stats & Probability
Analyze proportional relationships and use them to solve real-world and mathematical problems.
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, 7.RP.A.2a
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
Expression/Equation
Functions
Num & Ops Base Ten
Algebraic Thinking
Apply and extend previous understandings of arithmetic to algebraic expressions.
Interpreting Functions
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.
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, 3.OA.D.9, 4.OA.A.2, 4.OA.A.3, 4.NBT.B.5, 5.NBT.B.5, HSF-IF.A.3, 6.EE.A.1
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
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, HSA-REI.C.5
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
Stats & Probability
Using Probability to Make Decisions
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.5, 7.SP.C.7a, HSS-MD.B.5a
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
To the left is a circle
with an inscribed square. Obviously, there isn’t room for another
nonoverlapping square of the same size within the circle. But suppose that you
divided the square into n^{2}
smaller squares, each with side length 1/n.
Would one of those smaller squares fit in the space between the large square
and the circle? As shown to the left, this works if n = 16 and the large square were divided into 256 smaller
squares. But it would work for smaller values of n, too.
What is the smallest value
of n such that one of the smaller
squares would fit between the larger square and the circle?
Problems
Grades: 9th to 12th, 6th to 8th
Geometry
Circles
Similarity, Right Triangles, and Trigonometry
Understand and apply the Pythagorean Theorem.
8.G.B.7, HSG-SRT.C.8, HSG-C.A.3
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
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
Geometry
Geometric Measurement and Dimension
Circles
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.4, HSG-C.A.3, HSG-GMD.A.1
What is the smallest integer that can be the hypotenuse of two
different right triangles, each of which has legs whose lengths are also
integers?
Problems
Grades: 9th to 12th, 6th to 8th
Geometry
Similarity, Right Triangles, and Trigonometry
Understand and apply the Pythagorean Theorem.
8.G.B.7, HSG-SRT.C.8
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
Solve puzzles to strengthen understanding of expanding and factoring polynomials.
Lesson Plan
Grades: 9th to 12th, High School
Algebra
Arithmetic with Polynomials and Rational Functions
Seeing Structure in Expressions
HSA-SSE.A.2, HSA-SSE.B.3a, HSA-APR.A.1
Use proportions and similar figures to adjust the size of the New York City Subway Map so that it is drawn to scale.
Lesson Plan
Grades: 9th to 12th, 6th to 8th, High School
Geometry
Number & Quantity
Ratio & Proportion
Similarity, Right Triangles, and Trigonometry
Congruence
Understand congruence and similarity using physical models, transparencies, or geometry software.
Quantities
Draw construct, and describe geometrical figures and describe the relationships between them.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1, 6.RP.A.3a, 7.G.A.1, HSN-Q.A.3, 8.G.A.1a, 8.G.A.1b, 8.G.A.1c, 8.G.A.5, HSG-CO.A.1, HSG-SRT.A.2, HSG-SRT.A.3, HSG-SRT.B.4, HSG-SRT.B.5
Model linear functions using Barbie dolls and rubber bands.
Lesson Plan
Grades: High School, 6th to 8th, 9th to 12th
Stats & Probability
Interpreting Categorical and Quantitative Data
Investigate patterns of association in bivariate data.
8.SP.A.1, 8.SP.A.2, 8.SP.A.3, HSS-ID.B.6a, HSS-ID.B.6c, HSS-ID.C.7
Manipulate and fold three paper circles to explore four circle theorems.
Lesson Plan
Grades: 9th to 12th
Geometry
Circles
HSG-C.A.2
Use polydrons to build nets and create the most appealing fish tank.
Lesson Plan
Grades: 9th to 12th, High School, 6th to 8th
Geometry
Congruence
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, 6.G.A.4, 8.G.C.9, HSG-GMD.A.3, HSG-CO.A.1
Participate in a modeling activity where they will learn the rules for translations, rotations, and reflections.
Lesson Plan
Grades: 9th to 12th, 6th to 8th
Geometry
Congruence
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.3, HSG-CO.A.2, HSG-CO.A.4, HSG-CO.A.5, HSG-CO.B.6
Use problem-solving skills to find the solution to a multi-variable problem that is solved by manipulating linear equations.
Lesson Plan
Grades: High School, 9th to 12th
Algebra
Reasoning with Equations and Inequalities
Creating Equations
HSA-CED.A.3, HSA-REI.C.6, HSA-REI.D.11