# No More Mathematical Matchmaking: The Return of the Inaba Place Value Puzzles

When I first started teaching, I believed in being a mathematical matchmaker. I thought that differentiation meant giving students different tiered assignments — it seemed much better than pushing every student through the same thing. I thought it was up to me to select the perfect match for my students. I talked a lot with my class about how it’s “okay” that we are all working on different things. “We all get what we need! Let’s celebrate our diversity!”

Falsely, I believed that this was being responsive.

Despite my good intentions as an early career teacher, my tiered assignments were communicating messages around the students worth as a thinker. Who was I to tell Destiny that bigger numbers and more complex problem solving wasn’t what she “needed”?

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Today, in second grade, we returned to the Inaba Place Value Puzzles, at the end of our unit on place value.

The original puzzles (found here) gives a series of one digit numbers and a sum, and asks the student to multiply the one digit numbers by 1, 10 or 100 to balance the equation. For example

1 + 2 + 3 = 213
10 + 200 + 3 = 213
ten + 2 hundreds + 3 ones = 213

The set of problems gets increasingly more difficult (more regrouping, more digits, etc.). It works up to problems like this.

We launched the lesson with 3 sample problems — lots of turn and talks. Anticipation built when I revealed the third problem: 9___ + 3___ + 1___ = 400

The first student offered 9 + 300 + 100. “300 + 100 equals 400!” Miles stated.

“You… take it away…?” Miles shrugged.

Liora raised her hand to add on. “Well, what if we made it 90 and 10, so that’s the 100, and then we have 300, so that’s 400, too.”

“How should I record the 90 in words?” I asked.

“9 tens,” Liora stated crisply.

From across the rug, Miles thought aloud: “Oh, I was going to say 90 ones.”

I recorded it under the 9 tens. “Are those amounts worth the same? 9 tens… 90 ones…”

The students were unanimous: yes.

We then talked about what other ways we could record the 3 hundreds. Could we write it as 10s? 1s?

Then it was time to explore.

We gave everyone:

• Access to manipulatives that could support and push thinking, e.g. base 10 blocks.

What I observed:

• Students making choices about where to start in the problem set
• Students using fingers to keep track of some problems
• Students looking upwards at the ceiling, mouthing the names of some numbers — deep in thought
• Students using base 10 blocks to support their thinking, e.g. visualize a quantity, add numbers to evaluate their a potential solution,
• The teacher conferring with individual students — crouching down to meet them, or sitting in a small chair next to them.
• Pairs of students huddled to discuss a problem.
• And, yes, some competitive students gazing at other students’ work…

I conferred with students, too. I love listening to students talk about their thinking.

Gio was working on this problem:

6______ + 9______ + 2______ = 35

“There’s no 3 for the 3 tens in 35,” Gio observed.

“Hmm… I wonder if there are other numbers we can use for tens.”

Gio picked up 6 ten blocks. “This is too much.”

He started to pick up 9 blocks, but shook his head as he held only 2. “Nine will be way too big! That’s 90. I only want 35.”

Gio then pulled out the 2 tens.

“What do you think we should do with the 6 and the 9?”

“Maybe they will be tens, too,” he mused, as if he had already forgotten his thinking. “But I think I will try ones.” He plucked them from the bin, and then started counting. “26… 27, 28, 29, 30, 31, 32, 33, 34, 35. 35! That’s it! They’re all ones!”

I watched to see how students combined the different values. Some added the first two numbers (e.g. 20 + 6) and then either added or counted on the last value. Others combined all the numbers in the same place value (e.g. 20 and 6+9 which is 15) and then added them together (20 + 15 = 35). In some conferences, I gently nudged students to attend to place value. In others, we talked about the challenges of adding three numbers, and how we can keep track of the sum of the first two in order to count on.

Jaxon tried to explain a pattern he noticed. “A lot of them need to be minus. You think because it’s 31 you need the 30 but you actually minus and use the lower number.”

Miles looked on quizzically. I pushed Jaxon to explain again.

“You know, so for the 31 I tried it with 3 tens but that was too big. The 8 and the 2 make another 10. So I needed to use the 2 and then there are extras to make it up to 31,” Jaxon continued.

“Do you see other problems where that’s true?”

Both boys glanced down the page. “Maybe this one?”

5______ + 9______ + 6______ = 65
tens + 9 ones + 6 ones = 65
50 + 9 + 6 = 65

“It’s not the 6, it’s the 5 for 50 and then there’s more,” Jax continued. He was starting to notice a pattern.

The classroom teacher pulled the students back together for a close to share some of their strategies, and talk about strategies for getting “unstuck.”

Lila is on an IEP, and the special educator frequently takes her from class for individualized instruction, missing out on the experience and content the whole class is exploring. Today, Lila stayed with us the entire time.

“I could do this all day!” Lila exclaimed, collapsing into her spot on the rug. “That was so much fun!”

“Me, too!” Aaron agreed. (His mother has been e-mailing us about making certain he is “appropriately challenged.”)

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At no point, did we decide for kids where they should start, or put them off on a different path. We did not cut students off from opportunities based on a single assessment. Instead, we listened to students, and then used what we knew about them to nudge them to new understandings. Some of the students surprised us! My favorite feeling!

While everyone was working on the same task, with the same paper in front of them, the thinking varied. Some modeled directly, and others employed a wide range of additive strategies. Listening to the student gave us some ideas about where to go next with this task, or future learning experiences we would like to create.

The classroom teacher and I met briefly afterwards. “Everyone had such great thinking during the lesson,” the teacher mused. “I feel bad that we didn’t do centers or games, but I feel really good that I had the ability to walk around and listen to kids. Maybe this is a better structure for me.”

I don’t think it’s about the structure, per se, but I do think this lesson had some beautiful elements: choice, agency, listening, feedback…

Most importantly, I no longer feel like I need to be the matchmaker to address everyone’s “needs.” I did not need to pick out many different tasks. I did not label papers with little symbols in the corner (star, circle, square) that were my unfortunate shorthand for “high, medium, low” — as if the students couldn’t tell what I thought of their math abilities.

Instead, we gathered as a community, and everyone pushed their thinking in new directions. We opened opportunities instead of directing students down a path of my choosing. The students are excited to continue tomorrow.

1. (Now that I’ve gotten wordpress to stop swallowing all my comments)

I enjoyed reading this post. Its both a topic I often think think about and a structure I often use in my math circle. I had some followup questions for you that I still grapple with.

1. I’m assuming since you let kids choose where to go in the problem set they also must be loosely grouped i.e. a group has to by definition work on the same problem. How do you manage this? For me this automatically means groups will split by at least gender.

2. How does group discussion fit into this especially when the room has made very different choices on which problems to work through.

3. This is less important to me: but how do you ensure minimum coverage and mastery? It seems like a risk is consistently having a kid hang out in the easier section of the problems.

4. On the other end, how do you manage the initial whole group material discussion. Does this mean only the introductory/basic parts can be done as a whole? (This is basically what I end up usually doing) Extension which may be reached by only some of the room have to be talked about in smaller groups. This can be quite tricky to manage as the various paths the kids on branch out)

Thanks

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1. Thanks for your comment! Some quick responses:

(1) For this particular lesson, we did not group at all. Students chose their own starting places. The problems that we used in our launch were mostly from the first page (no regrouping), and so we did say that if the students like wrestling with that last problem (9 ___ + 3 ____ + 1 ___ = 400) that they might want to start somewhere other than the first page. Some students even started midway down the page.

Because students were encouraged to choose their own working space around the room, this means that some students did cluster and work on similar problems.

(2) For this particular lesson, we only had group discussions at the beginning/launch and the close/share. The classroom teacher led this particular close, and she opted to have two kids come up and explain their strategies for solving two problems. She wanted them to describe their thinking clearly enough so that students who had not worked on those particular problems would be have at least one access point. There was some light discourse around it, although the close was short that day.

(3) Some days, I’m perplexed by that dilemma of making certain that students get to specific, high leverage points within a problem. With this particular puzzle, this didn’t bother me. Every students we conferred with had pushed their thinking, and hit a productive challenge. (Honestly, it’s a little bit magical when it lines up for every student we talk to!)

(4) Yes, I tend to launch with an entry point. I agree that, depending on the problem(s) involved, it can be tricky to manage the various paths as the kids branch out! My personal style is to have a single, whole group close, but sometimes that just doesn’t work well — or I attempt to do multiple lesson closes, e.g. pull a small group that got to a far point, and then discuss something more accessible for the whole class. My closing tends to be a consolidation, naming the most salient points from the day’s experience, which often doesn’t include those “far reaches,” sadly. It depends on my mathematical goal.
I also find this particular dilemma difficult to discuss in the abstract. It’s easier for me to show what happened at the end of different lessons, with the acknowledgment that it’s not “perfect.” It’s a challenge!

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