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?"
number 4 can be expressed as the sum of three positive integers in only one way:
4 = 1 + 1 + 2
the number 50 can be expressed as the sum of three positive integers in 200
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?
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
What if the prime sums for each row and column have to be different?
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
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?
you find another set of face values for A, B, and C that yield the same
properties? (Does such a set even exist?)
According to Waring’s theorem, any positive integer can be represented as the sum of nine
or fewer perfect cubes (not necessarily distinct).
instance, 89 can be represented as the sum of four perfect cubes: 27 + 27 + 27
+ 8 = 89.
you express 239 as a sum of nine or fewer perfect cubes?
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
How many siblings does Julie have?
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?
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.
many different six-unit paths are there from A to C?
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?
Perform the following steps:
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?