Skittles and middles (a 3 act investigation)

Some really quick notes about a notice & wonder/3-act activity we did, piggy-backing on Graham Fletcher and Mike Wiernicki's nice presentation to a regional NCTM conference.

In particular, I was struck the trouble that the teachers were having calculating the midpoint of a number line and wanted to see what my older two would have to say. I thought this could prompt some interesting "notice and wonder" from the two older J's. Overall, there was a lot to discuss from this presentation with many great mini-conversations.

Quick background
If you don't watch the video, they go through an estimation problem involving skittles poured from snack-sized packages into a large glass jar:

To be clear, these are the same jar!

Notice and wonder

My guys' notice and wonders:

• that's a lot of candy
• how many bags is that?
• why did they pour it in a jar? It is easier to take as a snack when it is in the little packs!
• how many skittles is that?
• who is talking (giving this presentation)?
• are those chocolate?
• it was a man pouring the candy.
• How do we know it was a man?
• how big is the jar?
• are there the same number of candies in each pack?
• why does he pat the candy at the top?
• could we have some candy? (quickly correct to: could we have some candy, please?)

Their high/low/estimates

1. J2 made guesses first: 30 (low), 600 (best guess), 1000 (high)
2. J1 said 50 (low), 800 (best guess), 1000 (high), but then smiled and changed to 0 (low), 800 (best guess), and 5000 (high).

I forgot to ask them to draw a number line and put their best guesses on it. However, we did talk a lot at this part of the presentation:

They quickly agreed on 525 as the middle of this number line. Then, we talked about where the teacher's answer 475 might have come from. This led to a bit of confusion and frustration, basically centering on the distinction between the number at the middle and the distance to the middle.

Student work and a quick multiplication

They did the multiplication calculation a couple of ways. First, they used a calculator. They would have been satisfied with this, but wanted to understand the student work, so were inspired to try some other approaches. It was really interesting for me to see how they responded to something when they were told it came from other kids:

Starting with the bottom right, we drew split rectangles/area model with 50 + 8 along the side and 10 + 4 on the top, then talked about how the student had just calculated the areas of the top left and bottom right rectangles. Some extra comments/mini conversations:

• this is like the students who thought 11 x 11 = 101
• Oh, I see someone else did their rectangles the other way: 50 + 8 on top and 10 +4 on the side
• It doesn't matter which way we do it
• like addition: 5 + 4 = 4+5
• Does that always work? No, for subtraction 4-5 isn't 5-4. Also division. Right, 100 candies shared by 2 boys is really different from 2 candies shared by 100 boys (both then fall on the floor laughing)
• Hey, look at that student who added fourteen 58s in a tree!

This grand reveal (which actually came before the student work) also gave them a nice counting challenge:

Contextualizing

At that point, we'd caught up to the part of the video I had watched and didn't know what came next. They got surprisingly excited about this slide:

Hey dad, there's a diagram!

They wanted to understand what "contextualized" and "decontextualized" mean. I gave them the following comparison:

• 5+4 = ? is a calculation without a context
• I'm measuring the length of this room. My tape measure is 4 meters long. I've measured twice, 4 meters and then another 5 cm. What is the total length?

The context helps them understand what we are doing, allows them to use their intuition to estimate the likely answer, and determine the appropriate relationship between the numbers.