# Facilitating Number Boxes, Part 2 – Supporting Reasoning with Representations (Learning With Students)

“Hmm.. 8 + 3. Let’s put the 8 in the ten frame. If we add 3, will it be more than, less than, more equal to 10?”

Kyleigh missed parts of kindergarten and first grade, and it’s my pleasure to work with her one-on-one for a few sessions. We have been working on addition within 20, specifically bridging across a ten. After a few experiences working with sums to 10, I decided we’d play Number Boxes — Close to 10 Edition.

Number Boxes is a flexible game that invites strategy and supports powerful discourse around mathematical ideas. I’m being purposefully vague about the ideas because the game is a bit of a chameleon: it can be adapted to fit a range of content in K-12 mathematics. I’m always impressed by how adaptable it is. I recently spoke with James Myklebust-Hampshire (@MrJames_MH) and Adina (@adinam225) about how they had adapted it to their own contexts, and it was all brilliant.

Anyway, here is the essence of the game:

In a post last week, I wrote about some ways to press for thinking during facilitation of Number Boxes. This post is going to focus on another facilitation move: supporting student thinking with mathematical representations.

And now… let’s talk about Kyleigh.

## Close to 10

I set up the game board for me and Kyleigh to play the game.

I asked her what she noticed and what she wondered about the set up. “There’s adding,” she said. “And… more adding?” She pointed to the throwaway boxes in the top right corner. I explained their function, and that once a number has been thrown away, it can’t be brought back into the expression.

“And we want to get as close to 10 as we can,” I continued. “That might mean that you add two numbers and get exactly ten.”

“Like 5 + 5!” Kyleigh crowed.

“Exactly! Or it might mean that you add the two numbers and you get close to ten. Can you think of some numbers that are close to ten?”

“Eleven is really close. Or twelve.”

With that, I cued up the 10-sided dice on my phone, and we were off.

I let Kyleigh click “roll” to give her ownership. She tapped it aggressively several times — bam bam bam bam — as if that would make the outcome more random.

Three.

She immediately to throw is away. “Three isn’t close to ten,” Kyleigh stated crisply. I wondered if she understood the rules of the game. It’s not that a single digit needs to be close to ten, but the sum of two numbers.

“What if you used the three? How many more to make exactly…?”

“Time to roll!” Kyleigh interrupted.

Bam bam bam bam bam. The next number: four.

“Throw away!” Kyleigh sang out triumphantly. I have seen Kyleigh engage in some brilliant mathematical thinking during our sessions together — I just needed her to slow down.

“Okay, why are you throwing away the 4, too?”

“Because it’s small.”

“And then you add some more onto it. Do you think you could get close to ten? Or mayve exactly ten?”

“Time to roll!” Kyleigh announced, as she picked up my phone. This game was going nowhere fast.

## Slowing Down

By this point, Kylie had thrown away a 3 and a 4. The next number: seven.

As someone who is excellent at primary grade mathematics, my heart sank: she had thrown away the 3, but if she had kept it, she could have paired it with the 7 to get exactly ten! That would win!

But it was too late. Once the numbers are thrown away, they’re gone. Kyleigh and no spaces left for throw away numbers, so she had to place the 7 as one of the addends.

“Which one should I put it in?” Kyleigh asked, while comically shifting her gaze from one box to the other.

I bit my tongue. I wanted to tell her, “it doesn’t matter,” but I was worried a conversation about the commutative property of addition would push us completely off the rails. instead, I asked her, “where would you like to put the 7?” Calm. Measured. Quietly seething.

Kyleigh chose the box on the right. It was literally as good a choice as any.

“Okay, so let’s say you want to get exactly ten. What number are you hoping for?”

“A ten!”

“A ten and some extra would push you way over,” I explained. “Hey, we’ve been using ten frames. I wonder if a ten frame could help.”

## Enter: The Ten Frame

I drew a crooked ten frame in the corner of the dry erase board.

“Okay, so you have a seven.” We needed to recalibrate Kyleigh’s thinking. “We want to get close to ten. How much would we need to get exactly ten?”

Kyleigh turned the board around so that the problems were upside down for her, and then started filling in the ten frame with 7s. 7… 7… 7…

She stopped when she had written seven 7s.

“We need three… oh, no!” Kyleigh had realized the consequence of her action. She looked at me at first with panic — round, wide eyes — and then with an impish smile. She went to erase the 3, as if I hadn’t seen it or didn’t understand the rules.

“Oh, we both know you already threw away that three!” I laughed, lighthearted.

Kyleigh offered an angry pout, but she drew the 3 back in the throwaway box.

“Okay, so a three would be perfect. Maybe you’ll get something close to that. Something that won’t make it exactly a ten, but it could get close.”

“Like a four?!” Kyleigh wailed, throwing her body from the chair, ready to crawl under the table.

“Come back! Let’s keep playing.”

She relented. “Fine.”

She finished the game with a 9, and lamented all of her previous choices. “Let’s try it again,” I suggested.

## Another Round

Kyleigh was more strategic in our next round. The first number was a six, and she decided not to waste her throwaway boxes. She proudly placed the 6 as the first addend, and then filled up her ten frame with six 6s. The next roll was a 9.

“Where, where should I put it!” she moaned dramatically, keeping watch on me out of the corner of her eye. I laughed.

“Well, if you keep it… what’s 6 + 9? Is that close to ten?”

“I don’t know!”

“What do you know?’

“I know 9 + 9 is 18,” she offered.

I must have looked taken aback, because I didn’t expect that ot be her go to strategy. Most of the time, she prefers to use manipulatives (including her fingers) to calculate sums and differences.

“Okay, so if you know 9 + 9 is 18… can you figure out 9 + 8?”

“Take away one… it’s 17!”

I recorded the problems on the side of the board.

“And 9 + 7?” 16. And 9 + 6…

“Wait!” Kyleigh threw her hands up to stop me. “This isn’t 9 + 6. This is 6 + 9! What do we do now!”

She eyed the underside of the table, looking for her escape.

I was surprised that someone who could fluidly move from 9 + 9 to 9 + 8 to 9 + 7 would get stuck on the commutative property, but I guess it shouldn’t have been a shock: she had hinted that this was an unfamiliar idea earlier in the session, when she couldn’t decide where to place an addend.

The next roll was a 2. “Does that get us close?” Kyleigh whispered to me.

## How close to 10?

Kyleigh hemmed and hawed about the 2.

“Should I add it?” She whispered again.

“What is 6 + 2?”

Kyleigh used her fingers to arrive at 8.

“Is that close to 10? I was hoping for an 11,” she confessed.

I drew a number line on the board, and we marked off our goal of ten. I also showed the sum if 6 + 9, to show that, yes, we could in fact get closer, and do better.

Kyleigh marked off the 8. “That’s pretty close,” she mused. “It’s only two away.”

I insisted that we finish the fourth roll, to see if we had made the best choice. The final roll was another two. “Do we add that, too?”

“Nah, there’s only. a throwaway box left. But what would happen if we did add another 2 to 6 + 2?”

Kyleigh’s eyes glittered, and then she said, “did you know that my dog ate my sister’s homework?” She laughed. I’d lost her.

## Reflections

Kyleigh shared some great thinking during the game. While much of it felt supported rather than independent — I had nudged her with a visual representation, or created a problem string for her — I ended the session feeling optimistic some of the connections that Kyleigh had made. The game had opened up some new avenues of thought.

The next day, we were working again on some problems involving sums near 10.

“Can you get me a ten frame?” Kyleigh asked before we even began.