D
Donna Boucher
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Some lessons stick with you because of the mathematics students uncover when you give them time to think and talk. Years ago, I worked with a small group of fifth graders on perimeter, and what I remember most about the lesson is how much mathematical thinking came from simply asking questions and giving students space to reason together.
I started by asking students what they knew about perimeter. There were several reasons for that. First, it activated their prior knowledge and gave me a quick formative assessment of what they already understood. But just as importantly, I wanted to hear them talk about math. Mathematical discussions give students opportunities to organize their thinking, listen to the ideas of others, and develop the language they need to communicate mathematically.
I got lots of responses: It’s the distance all the way around the shape. You add all the sides to find it. There’s a formula for it. Rather than confirming an answer and moving on, I wanted to see what they could figure out using what they already knew.
So I drew a rectangle on the whiteboard and labeled it as shown.
I told students that mathematicians use what they already know to help them solve problems, and I asked them what they knew about this figure that might help them find the perimeter.
They knew it was a rectangle.
Great! So what do you know about a rectangle?
Two sides are the same and the other two sides are the same.
We added that information to the drawing. As we worked, I reminded them that mathematicians like to record their thinking so they can see the relationships they’re using and communicate their reasoning to others.
With the model drawn and labeled, I asked students to mentally calculate the perimeter. Then I went around the group and asked each student for their solution. I didn’t call on one student, hear 22 inches, and move on. Everyone had an opportunity to respond.
More importantly, I didn’t stop at the answer. I asked each student to explain how they had calculated it.
That’s where the interesting mathematics started.
Some students added the equal sides first:
7+7=147+7=14
4+4=84+4=8
14+8=2214+8=22
Others used multiplication:
7×2=147\times2=14
4×2=84\times2=8
14+8=2214+8=22
As students explained their thinking, I helped connect their informal language to academic vocabulary. They weren’t just getting the same answer in different ways—they were using what they knew about the structure of a rectangle and the relationships among its sides.
Then I joined the conversation.
You know how I did it? I added 4 + 7 to get 11. I knew if these two sides had a sum of 11, then the other two sides did, too. I like doubles, so I did 11 + 11 and also got 22.
They liked that strategy!
And I think there’s something important about sharing a teacher strategy this way. I wasn’t showing them how they should solve the problem. I was adding another idea to the collection of strategies the group had generated. Students could compare it to their own thinking and consider why it worked.
Later, as we worked through additional problems and students explained how they had found the perimeter, I started asking, How do you think I would solve it?
They could explain my doubling strategy.
That told me they hadn’t simply watched me perform a calculation. They understood the reasoning well enough to apply the strategy to a new figure.
Once students were comfortable finding the perimeter, it was time for more of a challenge. I drew another rectangle on the whiteboard, but this time I gave them the perimeter and left one side length unknown.
Again, I started with the same question:
What do you know?
They knew one side was 8 inches. They knew the figure was a rectangle, so the opposite side also had to be 8 inches. And through our discussion, they recognized that although we didn’t know the length of the two remaining sides, those sides had to be equal to each other.
So we already knew:
We also knew the perimeter was 22 inches. Most students reasoned that 16 plus something had to equal 22. I found that interesting—they naturally thought about adding up to 22 rather than subtracting 16 from 22.
But we weren’t finished. The remaining 6 inches couldn’t all belong to the missing side because there were two equal unknown sides to account for.
As we talked about getting our thinking down on paper, we connected what we knew about the rectangle to an equation:
Now students could see the structure of the problem. The two missing addends had to be equal, and together they had to total 6. That meant each missing side had to measure 3 inches.
What? You need more of a challenge? I can do that!
Talk about wide eyes!
So I brought students right back to the question we’d been asking throughout the lesson:
What do you know?
They knew it was a rectangle, so they knew the opposite sides had equal lengths. We used that information to label the two missing sides.
Then I asked students to mentally calculate the perimeter and, just as before, gave each student an opportunity to share an answer.
I got some wild answers!
Clearly, the fractions had changed things. So I told them I was going to do something that might seem almost magical.
I covered up the fractional parts of both mixed numbers.
Okay. What’s the perimeter now?
10 inches.
No problem.
Then I uncovered one of the halves.
How about now?
10 1/2 inches.
Then I uncovered the other half.
And now?
Big smiles.
11 inches!
What had initially looked like a completely different—and much harder—problem was suddenly manageable. Students already knew how to find the perimeter using the whole-number parts. Once they anchored their thinking in what they knew, they could account for the two additional halves.
Not every group made the connection that quickly, and that was useful information, too. With some students, the discussion revealed that we needed to spend a little time counting by fractional amounts. What started as a lesson about perimeter gave me an unexpected opportunity to work on fraction number sense.
And that’s one of the reasons I value this kind of exploration so much. When students are talking and explaining their thinking, we learn far more than whether they can produce a correct answer. Their thinking tells us what they understand and where the gaps are, and we can respond in the moment.
Before we finished, I asked students why that last problem had stumped them. Why had it seemed so much harder than the others?
Of course, they immediately blamed the fractions.
And that gave me the perfect opportunity to return to the message we’d been building throughout the lesson:
They actually knew a lot about this problem. They knew the properties of rectangles. They knew perimeter is the distance around a figure. They knew how to find perimeter when the side lengths were whole numbers. The fractions were the unfamiliar part, so we temporarily set them aside and started with everything we did understand.
Once the familiar mathematics was in place, the unfamiliar part wasn’t nearly as intimidating.
Looking back, what I love most about this lesson isn’t that students learned how to find perimeter. It’s how much mathematics surfaced because I gave them opportunities to think and talk.
Throughout the lesson, I kept coming back to one simple question: What do you know? From there, students used the properties of rectangles, number relationships, and what they understood about perimeter to figure out increasingly challenging problems. They explained different strategies, listened to one another’s thinking, and even made sense of a strategy I contributed to the discussion.
And because they were doing so much of the talking, I was learning right along with them. Their explanations gave me valuable formative assessment information. When the fractions caused trouble, I didn’t have to guess where the difficulty was—their thinking showed me, and I could respond with the instruction they needed.
That’s one of the reasons math talk and exploration are such important parts of mathematics instruction. When we ask good questions and give students space to reason, we get to see what they know, what they’re wondering, and what connections they’re making. And students begin to see themselves as mathematicians who don’t need to be shown what to do every time they encounter something unfamiliar.
Mathematicians use what they know to figure out what they don’t know. That’s the message I wanted those fifth graders to leave with that day—and all these years later, it’s still one of the most important messages I want students to hear.
If you’d like to give your students more opportunities to reason about perimeter, I’ve put together a free set of perimeter task cards you can use for continued practice. The cards are a great way to keep the focus on thinking, discussing, and using what students know to solve problems.
Download the free perimeter task cards here.
The post Exploring Perimeter: Using What You Know appeared first on Math Coach's Corner.
I started by asking students what they knew about perimeter. There were several reasons for that. First, it activated their prior knowledge and gave me a quick formative assessment of what they already understood. But just as importantly, I wanted to hear them talk about math. Mathematical discussions give students opportunities to organize their thinking, listen to the ideas of others, and develop the language they need to communicate mathematically.
I got lots of responses: It’s the distance all the way around the shape. You add all the sides to find it. There’s a formula for it. Rather than confirming an answer and moving on, I wanted to see what they could figure out using what they already knew.
So I drew a rectangle on the whiteboard and labeled it as shown.
I told students that mathematicians use what they already know to help them solve problems, and I asked them what they knew about this figure that might help them find the perimeter.
They knew it was a rectangle.
Great! So what do you know about a rectangle?
Two sides are the same and the other two sides are the same.
We added that information to the drawing. As we worked, I reminded them that mathematicians like to record their thinking so they can see the relationships they’re using and communicate their reasoning to others.
Make Space for Different Strategies
With the model drawn and labeled, I asked students to mentally calculate the perimeter. Then I went around the group and asked each student for their solution. I didn’t call on one student, hear 22 inches, and move on. Everyone had an opportunity to respond.
More importantly, I didn’t stop at the answer. I asked each student to explain how they had calculated it.
That’s where the interesting mathematics started.
Some students added the equal sides first:
7+7=147+7=14
4+4=84+4=8
14+8=2214+8=22
Others used multiplication:
7×2=147\times2=14
4×2=84\times2=8
14+8=2214+8=22
As students explained their thinking, I helped connect their informal language to academic vocabulary. They weren’t just getting the same answer in different ways—they were using what they knew about the structure of a rectangle and the relationships among its sides.
Then I joined the conversation.
You know how I did it? I added 4 + 7 to get 11. I knew if these two sides had a sum of 11, then the other two sides did, too. I like doubles, so I did 11 + 11 and also got 22.
They liked that strategy!
And I think there’s something important about sharing a teacher strategy this way. I wasn’t showing them how they should solve the problem. I was adding another idea to the collection of strategies the group had generated. Students could compare it to their own thinking and consider why it worked.
Later, as we worked through additional problems and students explained how they had found the perimeter, I started asking, How do you think I would solve it?
They could explain my doubling strategy.
That told me they hadn’t simply watched me perform a calculation. They understood the reasoning well enough to apply the strategy to a new figure.
Turn the Thinking Around
Once students were comfortable finding the perimeter, it was time for more of a challenge. I drew another rectangle on the whiteboard, but this time I gave them the perimeter and left one side length unknown.
Again, I started with the same question:
What do you know?
They knew one side was 8 inches. They knew the figure was a rectangle, so the opposite side also had to be 8 inches. And through our discussion, they recognized that although we didn’t know the length of the two remaining sides, those sides had to be equal to each other.
So we already knew:
We also knew the perimeter was 22 inches. Most students reasoned that 16 plus something had to equal 22. I found that interesting—they naturally thought about adding up to 22 rather than subtracting 16 from 22.
But we weren’t finished. The remaining 6 inches couldn’t all belong to the missing side because there were two equal unknown sides to account for.
As we talked about getting our thinking down on paper, we connected what we knew about the rectangle to an equation:
Now students could see the structure of the problem. The two missing addends had to be equal, and together they had to total 6. That meant each missing side had to measure 3 inches.
Use What You Know
What? You need more of a challenge? I can do that!
Talk about wide eyes!
So I brought students right back to the question we’d been asking throughout the lesson:
What do you know?
They knew it was a rectangle, so they knew the opposite sides had equal lengths. We used that information to label the two missing sides.
Then I asked students to mentally calculate the perimeter and, just as before, gave each student an opportunity to share an answer.
I got some wild answers!
Clearly, the fractions had changed things. So I told them I was going to do something that might seem almost magical.
I covered up the fractional parts of both mixed numbers.
Okay. What’s the perimeter now?
10 inches.
No problem.
Then I uncovered one of the halves.
How about now?
10 1/2 inches.
Then I uncovered the other half.
And now?
Big smiles.
11 inches!
What had initially looked like a completely different—and much harder—problem was suddenly manageable. Students already knew how to find the perimeter using the whole-number parts. Once they anchored their thinking in what they knew, they could account for the two additional halves.
Not every group made the connection that quickly, and that was useful information, too. With some students, the discussion revealed that we needed to spend a little time counting by fractional amounts. What started as a lesson about perimeter gave me an unexpected opportunity to work on fraction number sense.
And that’s one of the reasons I value this kind of exploration so much. When students are talking and explaining their thinking, we learn far more than whether they can produce a correct answer. Their thinking tells us what they understand and where the gaps are, and we can respond in the moment.
Mathematicians Use What They Know
Before we finished, I asked students why that last problem had stumped them. Why had it seemed so much harder than the others?
Of course, they immediately blamed the fractions.
And that gave me the perfect opportunity to return to the message we’d been building throughout the lesson:
Mathematicians use what they know to help figure out what they don’t know.
They actually knew a lot about this problem. They knew the properties of rectangles. They knew perimeter is the distance around a figure. They knew how to find perimeter when the side lengths were whole numbers. The fractions were the unfamiliar part, so we temporarily set them aside and started with everything we did understand.
Once the familiar mathematics was in place, the unfamiliar part wasn’t nearly as intimidating.
Let Students Do the Thinking
Looking back, what I love most about this lesson isn’t that students learned how to find perimeter. It’s how much mathematics surfaced because I gave them opportunities to think and talk.
Throughout the lesson, I kept coming back to one simple question: What do you know? From there, students used the properties of rectangles, number relationships, and what they understood about perimeter to figure out increasingly challenging problems. They explained different strategies, listened to one another’s thinking, and even made sense of a strategy I contributed to the discussion.
And because they were doing so much of the talking, I was learning right along with them. Their explanations gave me valuable formative assessment information. When the fractions caused trouble, I didn’t have to guess where the difficulty was—their thinking showed me, and I could respond with the instruction they needed.
That’s one of the reasons math talk and exploration are such important parts of mathematics instruction. When we ask good questions and give students space to reason, we get to see what they know, what they’re wondering, and what connections they’re making. And students begin to see themselves as mathematicians who don’t need to be shown what to do every time they encounter something unfamiliar.
Mathematicians use what they know to figure out what they don’t know. That’s the message I wanted those fifth graders to leave with that day—and all these years later, it’s still one of the most important messages I want students to hear.
Keep the Perimeter Thinking Going
If you’d like to give your students more opportunities to reason about perimeter, I’ve put together a free set of perimeter task cards you can use for continued practice. The cards are a great way to keep the focus on thinking, discussing, and using what students know to solve problems.
Download the free perimeter task cards here.
The post Exploring Perimeter: Using What You Know appeared first on Math Coach's Corner.