Assuming that the circumference of each circle below passes through the
centers of the other two, and that the radius of each circle is 1, what
is the total gray area?
Problems
Grades: 9th to 12th, 6th to 8th
Geometry
Geometric Measurement and Dimension
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.4, HSG-GMD.A.1
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
The triangle at left lies on a flat surface and is pushed at the top vertex. The
length of the congruent sides does not change, but the angle between the two
congruent sides will increase, and the base will stretch. Initially, the area
of the triangle will increase, but eventually the area will decrease,
continuing until the triangle collapses.
What
is the maximum area achieved during this process? And, what is the length of
the base when this process is used to create a different triangle whose area is
the same as the triangle above?
Problems
Grades: 9th to 12th, 6th to 8th, 3rd to 5th
Geometry
Measurement & Data
Similarity, Right Triangles, and Trigonometry
Understand and apply the Pythagorean Theorem.
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.5a, 3.MD.C.5b, 3.MD.C.6, 3.MD.D.8, 8.G.B.7, HSG-SRT.C.8
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
Use research on the Amazon deforestation to determine whether it is occurring exponentially.
Lesson Plan
Grades: 9th to 12th
Stats & Probability
Interpreting Categorical and Quantitative Data
HSS-ID.B.6a, HSS-ID.B.6c, HSS-ID.C.7
The diagram at left shows the top of a regular pentagon with the top of a square
inscribed in it. The shapes share a vertex at the top, and the other two
vertices of the square lie on the sides of the pentagon. If the diagram were
continued to include the entire pentagon and the entire square, which shape
would extend below the other?
In
other words, does the whole square fit inside the pentagon, does the square
protrude at the bottom, or do the square and pentagon meet at a single point?
Problems
Grades: 9th to 12th
Geometry
Circles
HSG-C.A.3
Two
chess players compete in a best-of-five match. If Chekmatova has a 60% chance
of winning any particular game, what is the likelihood that she will win the
match?
Problems
Grades: 9th to 12th, 6th to 8th
Stats & Probability
Conditional Probability and the Rules of Probability
Investigate chance processes and develop, use, and evaluate probability models.
7.SP.C.5, 7.SP.C.7a, 7.SP.C.8a, 7.SP.C.8b, HSS-CP.B.8, HSS-CP.B.9
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
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
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
Create and use linear models to make projections about what impact increased portion sizes may have on weight.
Lesson Plan
Grades: 9th to 12th
Functions
Algebra
Linear, Quadratic, and Exponential Models
Building Functions
Interpreting Functions
Reasoning with Equations and Inequalities
Creating Equations
HSA-CED.A.2, HSA-CED.A.3, HSA-REI.A.1, HSF-IF.C.7a, HSF-BF.A.1a, HSF-LE.A.1a, HSF-LE.A.2, HSF-LE.B.5
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
circle of radius 1 unit is inscribed inside a right triangle that has height a and base b. If b is an integer,
what are the possible values of a?
Problems
Grades: 9th to 12th
Geometry
Circles
HSG-C.A.3
Examine and draw representations of cubes and then learn how to analyze these representations using complex numbers.
Lesson Plan
Grades: 9th to 12th
Number & Quantity
The Complex Number System
HSN-CN.A.1, HSN-CN.A.2, HSN-CN.B.5, HSN-CN.B.6
develop a delicious new drink by mixing various concentrations of a two-fold dilution series.
Lesson Plan
Grades: High School, 9th to 12th
Functions
Linear, Quadratic, and Exponential Models
HSF-LE.A.1c
Explore the concept of genetics and inheritance using probability.
Lesson Plan
Grades: 9th to 12th, 6th to 8th
Stats & Probability
Making Inferences and Justifying Conclusions
Investigate patterns of association in bivariate data.
Investigate chance processes and develop, use, and evaluate probability models.
Summarize and describe distributions.
6.SP.B.5b, 7.SP.C.6, 7.SP.C.7a, 7.SP.C.7b, 8.SP.A.1, HSS-IC.A.2
Discover how to determine the appropriate number of digits that should be reported.
Lesson Plan
Grades: High School, 9th to 12th
Number & Quantity
Quantities
HSN-Q.A.3
Investigate the properties of regression lines and correlation using an online interactive tool.
Lesson Plan
Grades: High School, 9th to 12th
Stats & Probability
Interpreting Categorical and Quantitative Data
HSS-ID.B.6a, HSS-ID.B.6b, HSS-ID.B.6c, HSS-ID.C.7, HSS-ID.C.8
Discover the pattern of Pick’s Theorem using physical or virtual manipulatives.
Lesson Plan
Grades: 9th to 12th
Geometry
Congruence
HSG-CO.D.12
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