Math Conversations: Perimeter

I am wary of whole group math conversations because I fear students tuning out and not paying attention. Having conversations as part of a mini-lesson with us all at our carpet spots seems to keep everyone engaged and participating. Well, I haven’t gotten to EVERYONE yet, but this conversation has at least 12 student voices included.

Today (I started writing this when we did the activity, then edited it over winter break) I wanted to revisit posing mathematical questions. We generate questions plenty when we have our larger math projects, but when I gave an assessment asking students to generate questions about a perimeter situation, they weren’t able to transfer the skills we had practiced. The other challenge was I wanted students to be able to distinguish between asking their fantastic open questions and more grade-level appropriate questions (not too easy, not crazy-challenging).

Anyway, I shared the following situation. Vincent and Seenar are building corrals for their horses. The store Vincent went to had 56 feet of fencing available, and the store Seenar went to had 48 feet of fencing available.

perimeter

I opened the floor to questions. I’ve bolded my comments because at some points I forgot which students were speaking.

“I want to know how many horses could fit around the edges of the fencing,” Millie said.

“I have a clarifying question for Millie,” Ivy said. “How big are the horses; how long are they?”

“Five feet.” “Okay, but then you’d have leftover space because if you count by 5’s, for Vincent’s fence, you’d get to 55 and then you’d have one extra.” “Well then just make sure there’s space for a door.”

“I have a question: If each fence were 10 feet wide, how long would the other sides be?”

“Miz Houghton, how did you know the perimeters would work out right?” “What do you mean?” “Like, how did you know the sides would match up no matter what we made the length to be?” “Well, I knew that my perimeter would have to be even because any time you double a number, what happens?” “……..” “……….” “Double 1 is? Double 2 is? Double 3 is?” “2, 4, 6, 8…” “They’re all even!” “You’re right, and because I double the length and double the width, as long as I’m using whole numbers, I knew I’d get an even perimeter.”

“You could have an odd perimeter if one of the sides were 14 and one were 15.” “Yes, but what shape would it be?” “Just a regular quadrilateral, not a rectangle or square.”

“It’s like when we play Double Barrel.” “How so?” “In Double Barrel Again we double odd numbers like 15 and 45 and they still end up even.”

“We could do this as a multiplication problem, couldn’t we?” Vincent asked. “Because you multiply both of the sides by two and add them?”

“If you were to generalize the perimeter of any rectangle, yes, you’d multiply the length by two and the width by two.”

“You might see perimeter of rectangles expressed like this in general.” “Or squares, because they’re rectangles.” “Well, squares could also be 4 times the length, because all the sides are the same.”

“I agree. These are all great thoughts on perimeter and generating questions around perimeter. Keep these different conditions in mind when you’re looking to develop a problem situation during Work on Writing.”

I’ll admit, I’m a constructivist skeptic. As mentioned above, I’m a whole-group-math-discussion skeptic. But this worked out pretty darned well. The conversation took place during one of our Daily 5 Math minilessons, and it was less than 15 minutes.

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