How many do you see? How do you see them?

This summer, as Gwyneth and I were packing up Othello, I started playing with different arrangements of discs – mostly arrays – and asked her “How many?” I remembered the following arrangement, taken from AIMS’ Cookie Combos activity.

3^2 + 4 * 4

“Sixteen plus nine, so nineteen plus six… twenty, twenty-five,” she said. (I don’t think that she actually said “twenty” aloud. That came after my clarifying question: “Wait. Huh?”)

There’s a lot happening in Gwyneth’s bridging through twenty strategy – partitioning of quantities, place value, commutative property, breaking apart to make (a multiple of) ten. All within a three count, standard algorithm be damned.

This invented strategy discussion was a happy accident. The goal of this problem when we pose it to teachers is to see different ways to visualize the group and represent these using expressions. It’s about valuing different methods; the solution – counting 25 cookies – is easy enough.

How many do you see? How do you see them? How many different ways can you find?


Some popular solutions:

7 + 2 * 5 + 2 * 3 + 2 * 1
3 * 5 + 2 * 4 + 2
4 * 5 + 5

If you look just right, you can see two arrays:

4 * 4 + 3 * 3

A creative solution that involves counting what’s not there:

7^2 – 4 * 6

And moving what is:


If you plan on using these images with your students, I recommend displaying the photo with just white discs. This leaves the problem open. Two colours were used above to illustrate various visualizations. This can steer student thinking. (See how the use of colour is intended to be helpful here.) If students miss one of the visualizations above, display that photo and ask for the expression (or vice versa).

Survivor: 100 Chart Challenge

I don’t watch Survivor. Stopped watching after Richard Hatch, often competing naked, won the first season.

Channel surfing last week, this grabbed my attention:

Host Jeff Probst:

“Alright, let’s get to today’s duel. For today’s duel you’re gonna race across a balance beam, collecting bags of numbered tiles. You must then place the tiles in order, one to one hundred.”

(Aside: If there are three opponents, is it still called a duel?)

The reaction online was swift and harsh:

“It is seriously the most idiot-proof puzzle in the history of puzzles. You basically have to know how to count and that’s pretty much it.” (source)

But that’s not pretty much it. I mean, it is counting from one to one hundred (and that is how the contestants solved the “puzzle”), but it could be more than that. A better strategy involves comparing numbers, understanding place value, and identifying patterns found in tables.

At 1:49 and 2:02, we see two contestants, Laura and Brad, respectively, place 25 from the second bag (11 to 30).


A literal translation of “You gotta put ’em in order”? Each competitor places 25 only after placing 24. Then, he/she tries to find 26 in his/her pile o’ tiles. Some tiles are facing down. Suppose a player turns over a tile and finds 28 rather than 26. He or she should take advantage of another pattern and place it under 18.

At 3:11, it’s down to Brad and John for the last spot. At 3:18, Brad places 87 after 86.


He could have caught John if he had an understanding of place value. Suppose Brad turns over 94 before finding 87. Should he drop 94 and continue looking for 87 or just place 94 in the 9th row (9 tens) and 4th column (4 ones)?

This challenge reminds me of an activity I’ve used in Grade 3 classrooms. Take some 100 charts. Cut each chart into “puzzle” pieces. Place in a Ziploc bag. In pairs, have students reassemble. Ask students to describe how they solved their puzzle. This activity is much more engaging (and puzzling) than it has a right to be.

100 Chart Puzzle

Don’t be surprised if you see some completed 100 charts that look like this:

100 Chart Puzzle 2