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    Chelsea Cutting from Mount Gambier, South Australia, tells us about the real-world connections her students are able to make after using Illumination resources.
    Success Story
    Jan Gebert is an Illuminations lesson plan reviewer and instructor of professional and secondary education at East Stroudsburg University. So she definitely knows a thing or two about quality lessons. Illuminations asked her for her favorite out of our 600+ lessons.
    Success Story

    Deeanna Golden, a teacher of 24 years at F.M. Golson Elementary School in Marianna, Florida, is a beloved Illuminations lesson plan writer. So we asked her, "Why do you think it is important to share resources?"

    Success Story

    The number 4 can be expressed as the sum of three positive integers in only one way:

    4 = 1 + 1 + 2

    However, the number 50 can be expressed as the sum of three positive integers in 200 ways.

    Somewhere in between, there is a number n that can be expressed as the sum of three positive integers in precisely n ways. Can you find n?

    Problems
    Grades: 3rd to 5th
    Num & Ops Base Ten
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    3.NBT.A.2, 4.NBT.B.4
    In the diagram at above, each "path" from top to bottom correctly spells the word MATCH.
    What is the total number of different
    paths in the diagram?
    Problems
    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 prime number is a natural number greater than 1 whose only factors are 1 and itself. Can you place the digits 1–9 into the nine boxes (one digit per box) so that the sum of every row and every column is a prime number?

    What if the prime sums for each row and column have to be different?

    Problems
    Grades: 3rd to 5th
    Algebraic Thinking
    Gain familiarity with factors and multiples.
    4.OA.B.4
    The factorial of n is the product of all positive integers less than or equal to n. It is represented as n!. An example with n = 8 is shown below. With that in mind, can you find three sets of numbers (a, b, c) such that a! × b! = c! and a < b < c < 25?
    Problems
    Grades: 3rd to 5th
    Algebraic Thinking
    Multiply and divide within 100.
    3.OA.C.7

    Consider three six-sided dice A, B, and C, with the following numbers on their sides:

    A: 2, 2, 4, 4, 9, 9

    B: 1, 1, 6, 6, 8, 8

    C: 3, 3, 5, 5, 7, 7

    What is the probability that:

    ·     A produces a higher number than B?

    ·     B produces a higher number than C?

    ·     C produces a higher number than A?

    Can you find another set of face values for A, B, and C that yield the same properties? (Does such a set even exist?)

    Problems
    Grades: 6th to 8th
    Stats & Probability
    Investigate chance processes and develop, use, and evaluate probability models.
    7.SP.C.5, 7.SP.C.7a
    A pocket watch is placed next to a digital clock. Several times a day, the product of the hours and minutes on the digital clock is equal to the number of degrees between the hands of the watch. (The watch does not have a second hand.) As you can see, 10:27 is not one of those times — the angle between the hands is not 270°. If fractional minutes aren’t allowed, find the times at which the product of the hours and minutes is equal to the number of degrees between the hands.
    Problems
    Grades: 3rd to 5th
    Measurement & Data
    Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
    3.MD.A.1

    According to Waring’s theorem, any positive integer can be represented as the sum of nine or fewer perfect cubes (not necessarily distinct).

    For instance, 89 can be represented as the sum of four perfect cubes: 27 + 27 + 27 + 8 = 89.

    Can you express 239 as a sum of nine or fewer perfect cubes?

    Problems
    If x2 + y2 = 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

    When Julie’s family travels, her father always drives, and her mother always sits in the front passenger seat. Julie and her siblings sit in the middle and back row of their van.

    Julie told her brothers and sisters, “Of all the ways that two of us can sit in the middle row, I’m involved in one‑third of those pairs.”

    How many siblings does Julie have?  

    Problems

    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
    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 rectangle shown consists of eight squares. The length of each side of each square is 1 unit. The length of the shortest path from A to C using the lines shown is 6 units. 

    How many different six-unit paths are there from A to C?

    Problems

    Assign each letter a value equal to its position in the alphabet (A = 1, B = 2, C = 3, …). Then find the product value of a word by multiplying the values together. For example, CAT has a product value of 60, because C = 3, A = 1, T = 20, and 3 × 1 × 20 = 60. 

     

    How many other words can you find with a product value of 60?

    Problems
    Grades: 3rd to 5th
    Num & Ops Base Ten
    Algebraic Thinking
    Use place value understanding and properties of operations to perform multi-digit arithmetic.
    Multiply and divide within 100.
    3.OA.C.7, 4.NBT.B.5
    Every day in a non‑leap year, John took a different path from home to his favorite store. He walked on the grid of streets shown at left, and he only walked north or east along each street. His home is in the lower left corner of the diagram. He started on January 1, and on December 31 he took the last possible path. At what intersection is his favorite store located?
    Problems
    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

    Perform the following steps:

    • Pick a positive integer, any positive integer at all.
    • Write out the number in English.
    • Count the number of letters in that word. The number of letters is now your new number.
    •  Write out your new number in English, and count the number of letters.
    • Repeat until it feels … um, well … repetitive.  

    When you can't go any further, what number are you at? And how can you be sure that you will always end at this number, no matter what number you chose at the beginning?

    Problems
    Grades: 3rd to 5th
    Num & Ops Base Ten
    Generalize place value understanding for multi-digit whole numbers.
    4.NBT.A.2
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