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
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
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 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
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