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    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 real-world and mathematical problems.
    Solve real-world 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 real-life and mathematical problems involving angle measure, area, surface area, and volume.
    7.G.B.4, HSG-GMD.A.1
    A rectangular wooden block (not necessarily a cube) is painted on the outside and then divided into one-unit 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 real-world 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 student-centered 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 real-world 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, 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 n2 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
    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
    Use class schedules to create time-distance graphs.
    Lesson Plan
    Grades: 6th to 8th
    Ratio & Proportion
    Stats & Probability
    Functions
    Expression/Equation
    Analyze proportional relationships and use them to solve real-world 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  \frac{3}{5} of the seats in the classroom, how many students would occupy  \frac{2}{3} 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, 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
    A rectangular wooden block (not necessarily a cube) is painted on the outside and then divided into one-unit 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 real-world 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 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
    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
    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 real-life and mathematical problems involving angle measure, area, surface area, and volume.
    Draw construct, and describe geometrical figures and describe the relationships between them.
    Classify two-dimensional 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 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

    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 real-world 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
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