Predict, estimate and then calculate the path that results in the fastest time to travel between two points when different terrains affect the fastest path.
Lesson Plan
Grades: 6th to 8th
Geometry
Ratio & Proportion
Understand and apply the Pythagorean Theorem.
Analyze proportional relationships and use them to solve realworld and mathematical problems.
Solve realworld and mathematical problems involving area, surface area, and volume.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.2, 6.RP.A.3b, 6.G.A.3, 7.RP.A.1, 8.G.B.8
The number groups below are the last five digits of the fifth powers
of the numbers 31 through 39. However, the groups aren't in the right
order to represent the fifth powers of 31 through 39 sequentially. Using
only these digits, and without using a calculator, can you place the
groups in the correct order?
35393 

35424 

29151 
24199 

21875 

35168 
54432 

43957 

66176 
Problems
Grades: 6th to 8th, 3rd to 5th
Expression/Equation
Num & Ops Fractions
Apply and extend previous understandings of arithmetic to algebraic expressions.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.B.5a, 6.EE.A.1
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 reallife and mathematical problems involving angle measure, area, surface area, and volume.
7.G.B.4, HSGGMD.A.1
A rectangular wooden block (not necessarily a cube) is painted on the
outside and then divided into oneunit cubes. It turns out that exactly
half of the cubes have paint on them. What were the dimensions of the
block before it was painted?
Problems
Grades: 3rd to 5th, 6th to 8th
Measurement & Data
Geometry
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Solve realworld and mathematical problems involving area, surface area, and volume.
5.MD.C.3a, 5.MD.C.3b, 6.G.A.2, 6.G.A.4, 5.MD.C.4, 5.MD.C.5a
Explore the concept of slope through a studentcentered problem of data collection and evaluation.
Lesson Plan
Grades: 6th to 8th
Ratio & Proportion
Functions
Expression/Equation
Geometry
Analyze proportional relationships and use them to solve realworld and mathematical problems.
Use functions to model relationships between quantities.
Understand the connections between proportional relationships, lines, and linear equations.
Draw construct, and describe geometrical figures and describe the relationships between them.
7.G.A.1, 8.EE.B.6, 8.F.B.4, 7.RP.A.2b
One invention saves 30% on fuel; a second, 45%; and a third, 25%.
If you use all three inventions at once can you save 100%? If not, how much?
Problems
Grades: 6th to 8th
The Number System
Ratio & Proportion
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.3c, 7.NS.A.3
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, HSGSRT.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, HSGSRT.C.8, HSGC.A.3
Two
chess players compete in a bestoffive 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, HSSCP.B.8, HSSCP.B.9
Use class schedules to create timedistance graphs.
Lesson Plan
Grades: 6th to 8th
Ratio & Proportion
Stats & Probability
Functions
Expression/Equation
Analyze proportional relationships and use them to solve realworld and mathematical problems.
Summarize and describe distributions.
Use functions to model relationships between quantities.
Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.5, 8.EE.B.6, 8.F.B.4, 8.F.B.5, 6.SP.B.5b, 7.RP.A.2b, 7.RP.A.2d
If 18 students occupy
of the seats in the classroom, how many students would occupy
of the seats in the room?
Problems
Grades: 3rd to 5th, 6th to 8th
Num & Ops Fractions
Ratio & Proportion
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Understand ratio concepts and use ratio reasoning to solve problems.
5.NF.B.4a, 6.RP.A.1, 6.RP.A.3a, 4.NF.B.4b, 4.NF.B.4c, 5.NF.B.7a, 5.NF.B.7b, 5.NF.B.7c
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, HSAREI.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, HSACED.A.3, HSAREI.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, HSGC.A.3
A
rectangular wooden block (not necessarily a cube) is painted on the outside and
then divided into oneunit cubes. As it happens, the total number of painted
faces equals the total number of unpainted faces. What were the dimensions of
the block before it was painted?
Problems
Grades: 3rd to 5th, 6th to 8th
Measurement & Data
Geometry
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Solve realworld and mathematical problems involving area, surface area, and volume.
5.MD.C.3a, 5.MD.C.3b, 6.G.A.2, 6.G.A.4, 5.MD.C.4, 5.MD.C.5a
Label the ten points in the grid shown with the letters AJ 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 realworld 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 realworld and mathematical problems.
5.G.A.1, 6.NS.C.8, 6.G.A.3, 4.G.A.1, 8.G.B.8, HSGGPE.B.6, HSGGPE.B.7
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, HSSIC.A.2
Construct the 12 pentomino figures, and utilize them to explore area and perimeter.
Lesson Plan
Grades: 3rd to 5th, 6th to 8th, PreK to 2nd, Pre K to 2nd
Measurement & Data
Geometry
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Solve reallife and mathematical problems involving angle measure, area, surface area, and volume.
Draw construct, and describe geometrical figures and describe the relationships between them.
Classify twodimensional figures into categories based on their properties.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Reason with shapes and their attributes.
Analyze, compare, create, and compose shapes.
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
K.G.A.2, K.G.B.4, K.G.B.5, K.G.B.6, 1.G.A.1, 1.G.A.2, 2.G.A.1, 3.MD.C.5a, 3.MD.C.5b, 3.MD.C.6, 3.MD.D.8, 5.G.B.4, 7.G.A.2, 7.G.B.6, 3.MD.C.7d
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 345 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, HSGSRT.C.8, HSGGPE.B.7
The odometer of the family
car shows 15,951 miles. The driver noticed that this number is palindromic: it
reads the same backward as forward.
“Curious,” the driver said
to himself. “It will be a long time before that happens again.”
But 2 hours later, the
odometer showed a new palindromic number.
How fast was the car
traveling in those 2 hours
Problems
Grades: 6th to 8th
Ratio & Proportion
Analyze proportional relationships and use them to solve realworld and mathematical problems.
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.2, 6.RP.A.3b, 7.RP.A.1