Category Archives: Math Concepts

Crystal Capture

This weekend I made something fun and wanted to share it in case it provides fun for anyone else.

My daughter has a board game called Unicorn Glitterluck.

It’s super cute, but not the most engrossing game. She and I especially like the purple cloud crystals, so this weekend I started brainstorming a math game I could make for us to play together. I know number combinations is an important idea she’ll be working on in 1st grade, so I thought about how to build a game around that while also incorporating the crystals.

Introducing…Crystal Capture!

Knowing that certain totals have greater probabilities of appearing than others, I created a game board that takes advantage of this. Totals like 6, 7, and 8 get rolled fairly frequently, so those spaces only get 1 crystal each. Totals like 2, 3, 11, and 12, on the other hand, have less chance of being rolled, so I only put 1 space above each of these numbers, but that space has 3 crystals.

I mocked up a game board and we did a little play testing. I quickly learned a few things:

Play-Test

I originally thought we would play until the board was cleared. Everything was going so well until all we had left was the one space above 12. We spent a good 15 minutes rolling and re-rolling. We just couldn’t roll a 12!! That was getting boring fast which led me to introduce a special move when you roll a double. That at least gave us something to do while we waited to finally roll a 12.

That evening I made a fancier game board in Powerpoint and we played the game again this morning:

Since clearing the board can potentially take a long time, which sucks the life out of the game, I changed the end condition. Now, if all nine of the spaces above 6, 7, and 8 are empty, the game ends. Since these numbers get rolled more frequently, the game has a much greater chance of ending without dragging on too long.

I did keep the special move when you roll doubles though. This adds a little strategic element. When you roll a double, you can replenish the crystals in any one space on the board. Will you refill a space above 6, 7, or 8 to keep the game going just a little bit longer? Or will you replenish one of the three-crystal spaces in hopes of rolling that number and claiming the crystals for yourself?

All in all, my daughter and I had a good time playing the game, and I learned a lot about where she’s at in her thinking about number combinations. Some observations:

  • She is very comfortable using her fingers to find totals.
  • Even though she knows each hand has 5 fingers, she’ll still count all 5 fingers one-at-a-time about 75% of the time.
  • She is pretty comfortable with most of her doubles. She knows double 5 is 10, for example. She gets confused whether double 3 or double 4 is 8. We rarely rolled double 6, so I have no idea what she knows about that one.
  • In the context of this game at least, she is not thinking about counting on from the larger number…yet. She doesn’t have a repertoire of strategies to help her even if she did stop and analyze the two dice. If she sees 1 and 5, she’ll put 1 finger up on one hand and 5 on the other, then she’ll count all.
  • I did see hints of some combinations slowly sinking in. That’s one benefit to dice games like this. As students continue to roll the same combinations over and over, they’ll start to internalize them.

Several folks on Twitter expressed interest in the game, so I wanted to write up this post and share the materials in case anyone out there wants to play it with their own children or students.

You’ll have to scrounge up your own crystals to put in the spaces, but even if you don’t have fancy purple ones like we do, small objects like buttons, along with a little imagination, work just as well. Oh, and if you can get your hands on sparkly dice, that helps, too. My daughter loves the sparkly dice I found in a bag of dice I had lying around.

Have fun!

Represent! Part 2

In my previous post, I shared the first few questions I asked at a recent #ElemMathChat I hosted. Today I’d like to continue talking about using and connecting mathematical representations with a focus on fractions.

Let’s start with this question from the chat:

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Before reading on, pick one of the models yourself and analyze it.

  • How does it represent 2/3?
  • Where is the numerator represented in the model?
  • Where is the denominator represented in the model?
  • Can you answer these questions with all three models?

First it might help to differentiate the three models. The top left corner is an area model, the top right corner is a set model, and the bottom middle is a number line.

If you look at the area model, you’ll see that the whole rectangle – all of its area – has been partitioned into three equal parts, each with the same area. When we divide a shape or region into three parts with equal area, we actually have a name for each of those parts: thirds. Those thirds are countable. If I count all of the thirds in my area model, I count, “1 third, 2 thirds, 3 thirds.”

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Two of them have been shaded orange. So if I count only the parts that are orange, “1 third, 2 thirds,” I can say that 2 thirds, or 2/3, of the whole rectangle is shaded orange.

If you look at the set model, you might think at first that this model is the same as the area model, but this representation actually has some different features from the area model. In the set model, the focus is on the number of objects in the set rather than a specific area. I used circles in the above image, which are 2D and might make you think of area, but I could have just as easily used two yellow pencils and one orange sharpener to represent the fraction 2/3.

two-third-pencil-sharpener

I can divide the whole set into three equal groups. Each group contains the same number of objects. When we divide a set of objects into three groups with the same number of objects in each group, we actually have a name for each of those groups: thirds. Those thirds are countable. If I count all of the thirds in my set model, I count, “1 third, 2 thirds, 3 thirds.”

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Two of the groups contain only pencils. So if I count only those groups, “1 third, 2 thirds,” I can say that pencils make up 2 thirds, or 2/3, of the objects in this set.

Finally, we have the number line model which several people in the chat said is the most difficult for them to make sense of. While we have a wide amount of flexibility in how we show 2/3 using an area model or set model, the number line is limited by the fact that 2/3 can only be located at one precise location on the number line. It is always located at the same point between 0 and 1.

In this case, our whole is not an area or a set of objects. Rather, the whole is the interval from 0 to 1. That interval can be partitioned into three intervals of equal length. When we divide a unit interval into three intervals of equal length, we actually have a name for each of those intervals: thirds. Those thirds are countable. If you start at 0, you can count the intervals, “1 third, 2 thirds, 3 thirds.”

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However, what’s unique about the number line is that we label each of these intervals at the end right before the next interval begins. This is where you’ll see tick marks.

  • So 1/3 is located at the tick mark at the end of the first interval after 0.
  • 2/3 is located at the tick mark at the end of the second interval after 0, and
  • 3/3 is located at the tick mark at the end of the third interval that completes the unit interval. We know we have completed the unit interval because this is the location of the number 1.

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This quote sums up what I was aiming for with this discussion of representations of 2/3:

“Helping students understand the meaning of fractions in different contexts builds their understanding of the relevant features of different fraction representations and the relationships between them.” – Julie McNamara and Meghan Shaughnessy, Beyond Pizzas and Pies, p. 117

The bold words are very important to consider when working with students. What is obvious to adults, who presumably learned all of these math concepts years and years ago, is not necessarily obvious to children encountering them for the first time. What children attend to might be correct or it might be way off base. One common problem is that children tend to overgeneralize. A classic example is shared in Beyond Pizzas and Pies. Students were shown a model like this:

Rep2-06.JPG

They overwhelmingly said 1/3 is shaded. The relevant features to the students were shaded parts (1) and total parts (3). They weren’t attending to the critical feature of equal parts (equal areas).

I’ll close this post with a Which One Doesn’t Belong? challenge that I shared during the #ElemMathChat. (Note: I revised the image of the set model from what was presented during the chat.) As you analyze the four images, think about the relevant features of the area model, set model, and number line; look for relationships between them; and then look for critical differences that prove why one of the models doesn’t belong with the other three.

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Trick or Treat!

Now that I’ve completed sets of numberless word problems for all of the addition and subtraction CGI problem types, I wanted to do something fun.

This school year, my co-worker Regina Payne and I have been visiting the teachers in our Math Rocks cohort. One of the things they’ve been graciously letting us do is model how to facilitate a numberless word problems. In addition to making this a learning experience for the teachers, we’ve made it a learning experience for ourselves by putting a twist on the numberless word problem format.

Instead of your usual wordy word problem, we’ve been trying out problems that include visuals, specifically graphs. Instead of revealing numbers one at a time, we’ve been revealing parts of the graph. Let me walk you through an example I made tonight.

Here’s the graph I started with. I created it with some data I found on the Internet.

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If I threw this graph at a 4th or 5th grader along with a word problem, they would probably ignore what the graph is all about and just focus on getting the numbers they need for doing whatever computations they’ve decided to do. They would probably also ignore a vital piece of information – the scale that says “In Millions” – which means their answer is going to be about 1,000,000 times too small.

But what if we could change that by starting with something a little more accessible like this?

graph01

What do you notice? What do you wonder?

I’m guessing at least one student in the class would comment that it looks like a bar graph. Interesting. What do you think this bar graph could represent?

Oh, and you think a bar is missing in the middle. Interesting. What makes you say that?

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What new information was added to the graph? How does it change your thinking?

Oh, so there is a bar between Hershey’s and M&M’s. How tall do you think the bar for Snickers might be? Why do you say that?

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Now we know how tall the bar for Snickers is. How does that compare to our predictions?

Considering everything we know so far, what do you think this bar graph is about? What other information do we need in order to get the full story of this graph?

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What new information was added to the graph? How does it change your thinking about what this graph is about?

What are Sales? How do they relate to candy?

What does “In Millions” mean? How does that relate to Sales?

I know we don’t have any numbers yet, but what relationships do you see in the graph? What comparisons can you make?

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What new information was added? How does it change your thinking?

Hmm, how many dollars in sales do you think each bar represents? How did you decide?

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How do the actual numbers compare to your estimates?

What were the total sales for Reese’s in 2013? (I’d sneak in this question if I felt like the students needed a reminder about the scale being in millions.)

What are some other questions you could use answer using the data in this bar graph?

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What is this question asking?

How can you use the information in the graph to help you answer this question?

*****

I may or may not actually show that last slide. After reading this blog post by one of our instructional coaches Leilani Losli, I like the idea of letting the students generate and answer their own questions. In addition to being motivating for the students, it makes my time creating the graph well spent. I don’t want to spend a lot of time digging up data, making a graph, and then asking my students a whopping one question about it! That doesn’t motivate me to make more graphs. I  also want students to recognize that we can ask lots of different questions to make sense of data to better understand the story its telling.

Some thoughts before I close. This takes longer than your typical numberless word problem. There are a lot more reveals. Don’t be surprised if this takes you at least 15-20 minutes when you take into account all of the discussion. When I first do a graphing problem like this with a class, it’s worth the time. I like the extra scaffolding. Kids without a lot of sense making practice tend to be pretty terrible about paying attention to details in graphs, especially if their focus is on solving an accompanying word problem.

If I were to use this type of problem more frequently with a group of students, I could probably start to get away with fewer and fewer reveals. Remember, the numberless word problem routine is a scaffold not a crutch. My hope is that over time the students will develop good habits for attending to features and data in graphs on their own. If you’re looking for a transition to scaffold away from numberless and toward independence, you might start by showing the full graph and then have students notice and wonder about it before revealing the accompanying word problem.

If you’d like to try out this problem, here’s a link to a slideshow with all of the graph reveals. You’ll notice blank slides interspersed throughout. I’ve found that if you have a clicker or mouse that has a tendency to jump ahead a slide or two, the blank slide can prevent accidental reveals. It also helps with graphs because when I snip the pictures in they aren’t always exactly the same size. If the blank slides weren’t there, you’d probably notice the slight differences immediately, but clearing the screen between reveals mitigates that problem.

Happy Halloween!

Math with More Bad Drawings

In my last post, I shared some abominable strip diagrams. Last night, my friend messaged me again about some different models. Also pretty terrible.

“Sorry to hit you up for math help but I can’t find any like this on the internet.”

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There are two reasons for this, the second of which I’ll get to later in this post. The first is because this model is too bloated and trying to show competing ideas.

Here’s a cleaned up version of the model.

 

areamodel1

Any (good) area model should simultaneously represent multiplication and division. They’re inverses of each other. If you understand the components of the model, you should be able to write equations related to the model using both operations.

If I look at this model in terms of multiplication, I know I can multiply the length (7) times the width (13) to find the area (91). This area model represents 7 × 13 using the partial products of 7 × 10 and 7 × 3.

If I look at this model in terms of division, I know I can divide the area (91) by the width (7) to find the length (13). This area model represents 91 ÷ 7 using the partial quotients of 70 ÷ 7 and 21 ÷ 7.

All that from this one model. I don’t need all the “noise” included in the original model. For example, what is the purpose of writing the dimensions along the top as “10|70” and “3|21”? Knowing how an area model works, the only place 70 and 21 appropriately appear are inside the rectangle to show they represent area. Putting them along the top edge creates confusion about their meaning. Our students don’t need more confusion in their lives.

The repeated subtraction underneath isn’t terrible, but it’s unnecessary if you just want to know what multiplication or division sentence this model represents. Now, if a student were building the area model while using the partial quotients strategy, then the subtraction might be a useful recording strategy, but that’s not the same as being part of the model itself. I think it’s important to distinguish between those two things: features of the model itself and recording strategies a person might use as they build the model.

So the first problem my friend shared wasn’t great, but of course there was a second problem.

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And it’s worse.

Holy cow! Bring on the tears.

I get that a student solving 46 ÷ 2 might think about and possibly even jot down potential options for partial quotients, but there is no reason this needs to be shown to children on their homework. And there’s still the problem of there being two numbers side-by-side along the length. Does someone think interpreting bad models is a sign of rigorous math instruction? I don’t.

Here’s the cleaned up version.

areamodel3

While the original model was terrible, the question wasn’t bad at all. I’d probably revise it slightly though. I might say, “Gina found partial quotients to solve 46 ÷ 2. She recorded her work in the area model shown. Circle the number(s) in Gina’s model that shows the quotient of 46 ÷ 2. Convince me you circled the right numbers in the model.”

So earlier in the post I mentioned there are two reasons my friend couldn’t find anything like this on the internet. The first is because these were bad drawings. I tried looking for videos of someone solving a division problem using partial quotients and an area model which led me to the second problem. So many videos out there of varying quality. And by varying, I mean it’s easy to find videos that aren’t all that great. Many demonstrate either a limited view of partial quotients or a limited understanding of the area model.

One of the great things about using partial quotients to divide is the flexibility in how you can choose to decompose the dividend. In the first problem in this post, for example, the dividend (91) was decomposed into 70 and 21, which are both easy to divide by 7. It could just as easily been decomposed in to

  • 90 and 1
  • 35, 35, and 21
  • 63 and 28

While looking for videos to share with my friend, I found these (Video 1 | Video 2). What I noticed is that the partial quotients method is carried out in a rigid way that maps closely to the long division algorithm. In one of the videos, the presenter even connects the area model to long division notation.

areamodel4

The emphasis on place value is appreciated, but students deserve to know that they do have choice in how they decompose the dividend. Place value isn’t the only way.

These were the good videos. They might have missed out on sharing the power of this strategy, but at least the math is good. (I still didn’t share them with my friend.)

Sadly, there were also the bad videos. My major beef with these is that if you aren’t familiar or comfortable with partial quotients, you could just as easily watch a bad video and think you’re getting good information. These videos are so bad because, intentionally or not, they demonstrate big misunderstandings about the area model.

Example 1

In this example, the students are writing the numbers in the wrong place on the model. The partial areas (800, 370, and 23) should all be inside the rectangle while the lengths (100, 70, and 4) should all be outside along the top. I’m not blaming the kids. From what I can tell, they invented this strategy in their class (Cool!) but their teacher helped them make this video to share their strategy far and wide on the internet (Not cool!). Rather, as a teacher, I would have noted the students’ misunderstandings, helped them develop a better understanding of the area model, and then helped them create a video to show off their strategy.

Example 2

This one doesn’t even try to represent the values of the numbers. For whatever reason, the long division algorithm is carried out in boxes. Which, by the way, I don’t care if your video calls this the “box method” or “rectangle method.” It does not excuse you from misrepresenting the area model, because that’s what you’re doing. So many people believe math is confusing enough. Don’t add fuel to the fire.

Considering the time and effort that goes into building an understanding of area as a model for multiplication and division, we shouldn’t be making or showing these bad models to our students. We shouldn’t be showing them to our parents either. Seriously, if you share YouTube videos with your parents, please preview them and make sure the mathematics is good. Make sure they model the kinds of thinking, reasoning, and representing we want our own students to be developing.

Remember, the only people who should be making bad drawings are our students because they’re still figuring all of this out. Our job is to help them so that over time they get better.

 

 

We Can Make Shapes!

In my previous post I shared one of two mathematical conversations I had with my daughter this morning. Here’s the second.

“Look I made a triangle.”

I look over and she’s sitting cross-legged on the floor. It takes me a moment, but I realize she’s talking about the square tile she’s sitting on and the triangle she can see in the corner. Here’s a re-creation of it since I didn’t take any photos.

triangle01

The third side of the triangle looked a lot cleaner with her crossed legs. This graphic of a child doesn’t quite work, but you get the idea.

“Oh! I see. How do you know it’s a triangle?”

As usual when I ask that question about a geometric shape – How do you know it’s a ___? –  she didn’t really say anything back. I turned around to put something in my lunchbox.

“Look! The triangle is smaller!”

I turned back around to look and she had scooted up on the tile. “So it is!”

triangle02

With pure delight she exclaimed, “We can make shapes!”

She started scooting back on the tile and stopped when she got here.

triangle03

“Is that a triangle, too?” I asked.

She looked down and thought for a moment. She slowly started scooting up until she got to the diagonal. Then she stopped and looked up at me.

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She doesn’t yet know how to articulate what a triangle is, but she is clearly grappling with and making judgments about the “triangleness” of her shapes. It’s fascinating.

Even better, her exclamation, “We can make shapes!” makes me so happy. It’s such a simple statement, but it felt so empowered. She came to the realization all on her own as she moved her body back and forth on our tile floor.

Counting Down to the Weekend

“Do I go to music class and swim class today?”

“No, today is Monday. Remember, I said you go to work for 5 days before you go to music class and swim class.” I hold up my fingers one by one as I call out, “Monday, Tuesday, Wednesday, Thursday, Friday.”

I put down all five fingers and continue, “So far we went to work on Monday and we’ll go today on Tuesday.” I put those two fingers back up as I talk.

Without skipping a beat she says, “Three more days! Today it will be 3, and then 2, and then 1.”

This was completely unexpected and so fascinating to hear! If only I hadn’t been in the middle of rushing to get dressed and ready to walk out the door to work. Looking back, I would have loved to ask, “How did you know there are three days left?”

In thinking about this conversation throughout the day, I’ve thought about all the play we’ve done with counting over the past several months. Fingers are a favorite of mine since they’re always close at hand.

In the car, one of the games we’ll play is that I hold up some number of fingers at my chest and ask, “Guess how many fingers I’m holding up.” She makes a guess and then I hold them up so she can see if she got it right. Nothing fancy, but it gives her a lot of opportunities to count and see quantities from 1 to 5.

Another game I like to play is, “Do you want me to show you 5 really fast?” She says, “Yes.” I put my hand behind my back and say, “Ready, set, go!” And then I whip out my hand with all my fingers out. She counts my fingers every time to prove there are 5 fingers, but I’m beginning to wonder if the counting is really necessary.

So I’m curious about how she knew it was 3 days until Saturday. The way I held my hand, she couldn’t see the three fingers that were down. Did she see them in her mind? Did she subitize? Did she count one by one super fast? There was hardly a heartbeat between what I said and her response. The counting back from 3 was really fast also.

Things to explore as we talk more.

I love being a parent and getting to have these kinds of conversations with my daughter. When she surprises me with a new understanding or insight, it’s like a wonderful gift. I treasure each and every one.

(Side note: Her Montessori school calls their learning time “work periods” so we’ve been calling it “going to work” since she started there a year ago. She likes the idea that she goes to work everyday like Daddy and Papa do. If I accidentally say something about going to school she’ll usually correct me, “No, I go to work!”)

[UPDATE 10/5/2016] This morning she asked a question she asks pretty much everyday without fail, “Is today a work day?”

“What did I say when you asked me last night?”

“It is a work day.”

I go back to eating my breakfast.

“We went on this day and this day, and this is today.” I look over and she’s holding up three fingers in front of her face. She’s grabbing the tip of her middle finger as she’s saying that this is today. She tells herself, “There’s two days left!”

Clearly our conversation yesterday wasn’t a fluke! She wasn’t even talking to me at the end. She was talking it out and making the observation all to herself. How cool!

A little later she’s in the kitchen and I ask her, “Can you show me how many workdays we’ve had on the Math Rack?” (By the way, we’ve had fun counting on the Math Rack, but I’ve never asked her to do anything like this before.)

She pulls over three beads, “One, two, three.” Then she holds up her thumb, touches it to the first bead and says, “One.” She holds up her pointer finger, touches it to the second bead and says, “Two.” Finally she holds up her middle finger, touches it to the third bead and says, “Three.”

“Can you show me how many days we have left down here?” I point to the bottom Math Rack.

She pulls over two beads, “One, two.” Then she puts her thumb, pointer, and middle fingers back up and moves her hand over to the two beads she just pulled over so that the two fingers that are still down are touching them.

I feel like she’s turned a corner developmentally and a whole new landscape has opened up. I’m so excited to explore it with her!

Decisions, Decisions

This week our Math Rocks cohort met for the fourth time. We had two full days together in July, and we had our first after school session two weeks ago. One of our aims this year is to create a community of practice around an instructional routine, specifically the number talks routine. We spent a full day building a shared understanding of number talks back in July. You can read about that here. We also debriefed a bit about them during our session two weeks ago.

This week we put the spotlight on number talks again. We actually broke the group up by grade levels to focus our conversations. Regina led our K-2 teachers while I led our 3-5 teachers. The purpose of today’s session was to think about the decisions we have to make as teachers as we record students’ strategies. How do you accurately capture what a student is saying while at the same time creating a representation that everyone else in the class can analyze and potentially learn from?

We started the session with a little noticing and wondering about various representations of 65 – 32:

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Very quickly someone brought up exactly what I was hoping for which is that some of the representations show similar strategies but in different ways. For example, the number line in the top left corner shows a strategy of counting back and so do the equations closer to the bottom right corner.

This discussion also led into another discussion about the constant difference strategy – what it is and how it works. It wasn’t exactly in my plans to go into detail about it this afternoon, but since my secondary goal for the day was to focus specifically on recording subtraction strategies, it seemed a worthwhile time investment.

After our discussions I shared the following two slides that I recreated from an amazing session I attended by Pam Harris back in May. (For the record, every session I attend with her is amazing.)

The first slide differentiates strategies from models. Basically, if you have students telling you their strategy is, “I did a number line,” and you’re cool with that, then you should read this slide closely:

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The second slide differentiates tools for building relationships from tools for computation. This slide is crucial because it shows that while we want students to use tools like a hundred chart to learn about navigating numbers within 100, the goal is to eventually draw out worthwhile strategies, such as jumping forward and/or backward by 10s and then 1s.

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The strategy on the right that shows 32 + 30 followed by 62 + 3 is totally the type of strategy students should eventually do symbolically after building relationships with a tool like the hundred chart.

After blowing their minds with those two slides, I led them in a number talk of 52 – 37. During my recording of their strategies, I stopped a lot to talk about why I chose to do what I did, to solicit their feedback, and even to make some changes on the fly based on our discussion.

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For example, in the top right corner of the board I initially used equations to represent a compensation strategy. Someone asked if this could be modeled on a number line because she thought it might make more sense, so I did just that in the top left corner. By the time we were done they were like, “Oh, hey! That ends up looking like a strip diagram!”

It was amusing that the first strategies they shared involved constant difference. They were so excited about learning how the strategy worked that they wanted to give it a try. I didn’t want to quash their excitement by telling them that the strategy tends to work better, especially for students, when you adjust the second number to a multiple of ten. I wanted to stay focused on my goals for the day. We’ll discuss the strategy more in a future session.

(Unless you’re in Math Rocks and you’re reading this! In which case, see if you can figure out why that’s the case and share it at our next meeting.)

After some great discussion about recording a variety of strategies, we watched Kristin Gray in action leading a number talk of 61 – 27.

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We talked about how she recorded the students’ strategies. We also talked about some really lovely teacher moves that I made sure to draw attention to.

We wrapped up our time together talking about what new ideas they learned that they wanted to try out with their students. I had asked one of the teachers to lead us in another number talk, but we ran out of time so I think I’m going to have her do that at the start of our next session together. Hopefully everyone will have had some intentional experiences with recording strategies between now and then to draw on during that number talk.

Oh, another thing we talked about at various points during the session was how to lead students in the direction of certain strategies. This gets into problem strings, which may or may not happen in number talks depending on whom you talk to. Regardless, here are some we came up with. Can you figure out what strategies they might be leading students to notice and think about?

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