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    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
    Enforce the skills of identifying equivalent trigonometric expressions using puzzles.
    Lesson Plan
    Grades: High School, 9th to 12th
    Geometry
    Functions
    Similarity, Right Triangles, and Trigonometry
    Trigonometric Functions
    HSF-TF.A.3, HSG-SRT.C.6, HSG-SRT.C.7, HSG-SRT.C.8, HSG-SRT.D.10, HSG-SRT.D.11
    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

    Equations to solve in your head:

    \begin{array}{l}
 6,751x + 3,249y = 26,751 \\ 
 3,249x + 6,751y = 23,249 \\ 
 \end{array}

    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
    Offers a method for finding the slope of a line from its graph. 
    Lesson Plan
    Grades: 6th to 8th, High School, 9th to 12th
    Ratio & Proportion
    Functions
    Expression/Equation
    Analyze proportional relationships and use them to solve real-world and mathematical problems.
    Interpreting Functions
    Use functions to model relationships between quantities.
    Understand the connections between proportional relationships, lines, and linear equations.
    8.EE.B.6, 8.F.B.4, HSF-IF.B.6, 7.RP.A.2b

    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, \frac{3}{2} , \frac{5}{3} , \frac{8}{5} , \frac{{13}}{8} , 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
    Extend the Fibonacci Sequence through an algebra exploration.
    Lesson Plan
    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

    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 \frac{C}{\pi }, where C is the circumference, but please do not make a mistake. The diameter of the log is not \frac{{45}}{\pi }.

    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

    Construct perpendicular bisectors, find circumcenters, calculate area, and use proportions to solve a real world problem.

    Lesson Plan
    Grades: High School, 9th to 12th
    Geometry
    Congruence
    HSG-CO.D.12
    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

    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
    Answers the questions, "When will I ever need to solve a system of 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

    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
    Physical and virtual manipulatives are used to explore and discover conic sections by cutting a cone with a plane.
    Lesson Plan
    Grades: 9th to 12th, High School
    Geometry
    Expressing Geometric Properties with Equations
    HSG-GPE.A.1, HSG-GPE.A.2, HSG-GPE.A.3
    Make the connection between trigonometric rations and graphs of sine and cosine functions.
    Lesson Plan
    Grades: 9th to 12th, High School
    Functions
    Geometry
    Trigonometric Functions
    Similarity, Right Triangles, and Trigonometry
    Interpreting Functions
    HSF-IF.B.4, HSF-IF.B.5, HSF-IF.C.7e, HSF-TF.A.3, HSG-SRT.C.7, HSF-TF.A.2, HSF-TF.A.4, HSF-TF.B.5
    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 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
    Mara has 3 times as many dollars as her brother, Timmy. If Mara is given $20 by their mother, she will have 7 times as many dollars as Timmy. How many dollars does Timmy have?
    Problems
    Grades: 6th to 8th, 9th to 12th, 3rd to 5th
    Ratio & Proportion
    Functions
    Algebra
    Expression/Equation
    Algebraic Thinking
    Analyze proportional relationships and use them to solve real-world and mathematical problems.
    Linear, Quadratic, and Exponential Models
    Building Functions
    Reasoning with Equations and Inequalities
    Creating Equations
    Use functions to model relationships between quantities.
    Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
    Represent and analyze quantitative relationships between dependent and independent variables.
    Reason about and solve one-variable equations and inequalities.
    Multiply and divide within 100.
    3.OA.C.7, 6.EE.B.6, 6.EE.C.9, 7.EE.B.4a, 8.F.B.4, HSA-CED.A.2, HSA-CED.A.3, HSA-REI.A.1, HSF-BF.A.1a, HSF-LE.A.2, 7.RP.A.2c
    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

    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
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