It’s been a couple years since I did my first Hundred Face Challenge with my daughter’s grade 3/4 class. Today I did it again, in the same classroom, with the same teachers, but with wildly different kids. What I mean is, the first challenge happened in the middle of the year so they were further down the “making hundreds” road. This class is only three weeks into the school year; some kids were still working on “making 10” and others were way past that point. Despite this, today’s session showed me and the classroom teachers that this kind of visual/number/making activity can meet all kids where they’re at and move them a little further down the math learning path.
Since the Cuisennaire rods were new to the kids, I first asked them explore the materials with their partner. They spent a little time, most everyone building with them; some teams immediately started comparing the rods. “Let’s talk about what you’re noticing,” I asked. Here are some responses:
These are all smaller than each other so if you compare the biggest one to the medium size and then the other sizes and then you put this one up (1 block) then they’re all smaller than each other.
Different sizes and different colors.
They all count up. (Can you explain?) This one is a one-piece, and then if you put two ones it makes a two, and then if you put a two and a one it makes a three…and it goes all the way up.
There are four or more of each color [rod].
If you take a 10 and cut it up into squares it would be like ten squares.
Each color has a different size.
Then I asked: “What did we learn about these blocks?” They all equal something. “Okay, ” I said, “so let’s put them in some kind of order.” I modeled my own process of figuring out how much each rod was worth, and most of them followed along. Once that was done I gave them the challenge: “Your challenge is: To make a face that equals….dun dun DUNNNN…”
Random Kid: Seven!
Me: One hundred.
Whole class: WHAT?!!
Me: And here’s something to consider – how are you going to know for certain that you’ve made a face that equals 100?
The strategies I observed included calculators (!) which I nixed immediately with the statement that being a mathematician is about having reasons for how you got your answer and I confiscated calculators when I saw them 🙂
I also observed kids using their initial noticings of the stair-step-ness of the rods as a kind of key as to what amount/number they represented.
Some kids easily used mental math as a strategy and could tell me right away how they knew their face was 100. Many others made huge faces and then struggled to get started figuring out the count, or lost track of where they were in the counting process. This is when I noticed that at least half the class had a disconnect between the amount itself and the spatial representation of the amount. Because this was a once-off lesson with the rods, and they hadn’t yet had enough time to familiarize themselves this new math tool, I came to consider that success for some of these students was when they were able to accurately add up their face, even if it didn’t get all the way to 100.
About half way through the first challenge I reminded them again that they needed to prove they had a face that equaled 100 and handed out paper and pencils. This served to support the kids who had found making and computing the faces easy enough and moved them toward generalizing the arithmetic activity. It also served other kids by supporting their process of keeping track of how much of 100 they had and how much they still needed. Here’s one approach to proving 100:
A. explains: So I drew a picture of the face but we figured out how much each piece equals and then we did the math.
Me: How did you kept track with all that adding. Did you add up each area to keep track of it?
A: I started at this (bottom of picture) then I went to this (moving up the page).
Me: So you went in an organized way?
A: Yeah. And here’s our winning face!
Across the room, without knowing it, T and his partner used almost the exact same strategy and abstracted it down a little further with the prompting of their teacher.
The following “proof” was my absolute favorite because it was so hard won.
One of the teammates in this group initially just lay down on the ground and wouldn’t participate. But, she and her teammate did eventually make a face that was almost 100 and there was a good amount of (fairly) constructive conversation between them. The girl had initially counted the number of each kind of block, represented it visually/numerically and then circled each group. When I got over to them her partner was communicating quite energetically that in order for it to prove it it “had to equal something.” When I think about all the conceptual, social, and personal leaps both students needed to make this all happen I get a bit dizzy.
(MORE PROOFS AT THE END OF THIS POST)
CHALLENGE 2 & GALLERY WALK
The first time I did this activity in 2014 most of the class strategically (and unprompted) looked for and found all sorts of strategies for using the least number of rods in a 100 face, and had very little challenge with the computational aspect. On that day I decided the second challenge would be to create a face that was balanced on both sides like a human face, and it was the perfect challenge/task for them. Today, however, the kids had engaged in really productive struggle in the first round, and I thought we all might need something small and satisfying on which to end our day.
I asked the teams to pick a number of rods that added to 100. When they had done that (and I noticed that, now, at the end of the session, most of the teams did this much more easily) I asked them to see what kind of face they could make just with the blocks they picked. I did start double guessing myself and thinking I should have switched the order of the challenges but their teacher thought it actually reinforced the “making hundred” focus of the activity for those who had been very challenged in the first round, and everyone seemed satisfied with their faces at the end of the session.
“I think all the faces were really cool and just by looking at these blocks you wouldn’t be able to tell that you’d be able to make huge structures, cool faces…it would just look like simple blocks unless you really got into them and actually started building with them.”
“It’s really neat, because I thought we were doing something else at first. (What did you think it would be?) I thought we were actually just building structures, but it’s really neat that you can come up with something that’s not just a blocky face, you can come up with detail! Like I made triangle ears, lipstick, eyebrows…”
Classroom Teacher: “I noticed lots of different ways that groups and individuals have for adding. Like some groups just added each block as they went, and some groups would group their blocks, like think about all the reds and count them as a group, and then all the greens, and all the oranges. I thought it was neat to watch all the strategies.”
FINAL QUESTIONS & FAVORITES
“Why did we have to do faces? We could have built a house or something with 100.”
“The last thing was my favorite. (Why was it your favorite?) Because we already had a 100 ready to make the face.” (They used the same blocks but made a different face).
“I made a little monster out of the blocks.”
“My favorite thing was I like making the beard because it reminds me of my stepdad.”
“My favorite part was both of the faces because they both turned out really good. And I’m proud of them.”
“I wonder if anyone noticed that if you looked each way my face looks like a face both ways? It’s a flip face.”
If you want to see other faces, check out the Hundred Face Blog!
MORE PROOFS & MORE STRATEGIES!