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Can They Be Equal?

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

Sometimes area and perimeter of rectangles are taught separately, and are often confused. In this problem students consider the relationship between them and are being challenged to engage in some sophisticated mathematical thinking.

Possible approach

Show the students this image and ask them to work out the area and perimeter of each rectangle. 
Collect answers.
"That's interesting, the first rectangle has an area that is numerically greater than the perimeter, but the second one has an area that is numerically less than the perimeter. I wonder if you could find a rectangle whose area and perimeter are numerically the same?"
Set students to work on this challenge, perhaps encouraging them to work in pairs so they can share ideas on how to proceed.
"If you manage to find a rectangle that satisfies my conditions, see if you can find a few more."
 
Circulate and observe the methods and reasoning students are using. Look out for students who:
  • fix one attribute (side length, area, perimeter) and vary the others using trial and improvement
  • fix one attribute and use algebra to solve for the other attributes
  • write an algebraic expression for area and perimeter, equate them, and substitute values into the resulting equation
For students who are struggling to get started:
"What is the same about the two rectangles we started with?"
"What could you change?"
"How does the area and perimeter change as you change the height of the rectangle?"
 
 
Once everyone has had a chance to find a few rectangles that satisfy the condition, collect together the dimensions on the board.
Invite students to share any different strategies you observed them using as they were working.
"I'd like you to have a go at finding a few more rectangles, using several different strategies." 
"While you are working, think about how many different rectangles we could possibly find."
 
Finish off by asking students to share their ideas about how many different rectangles satisfy the criteria, together with convincing arguments about why there are infinitely many.

Possible extension

Ask students to consider other polygons with numerically equal areas and perimeters - those who have met Pythagoras' theorem could investigate right-angled and isosceles triangles, and those who have met trigonometry could work on regular polygons.
 
Students could be invited to consider cuboids whose surface area is numerically equal to their volume.

Possible support

A more scaffolded introduction to the problem:
 
Tell the students you are thinking of a rectangle. Ask them to work out its dimensions if:
the area is 24 and the perimeter is 20
the area is 24 and the perimeter is 22
the area is 24 and the perimeter is 28
the area is 24 and the perimeter is 50
 
Record the solutions on the board. Ask the students to comment on anything they notice. (This might be to do with the shape of the rectangles, or perhaps the evenness of the perimeters.) 
 
Repeat the process keeping the perimeter fixed this time, to 20.
Can they find the dimensions of rectangles with areas of 9, 16, 21, 24, 25? 
 
 
Another activity to help students to become fluent in working out the different attributes of rectangles:
Students could make up their own card matching game where each set contains three cards about a specific rectangle, one with area, one with perimeter and one with the dimensions. Students have to find all three in a set. Each student produces 8 sets, shuffles them and hands them on to their neighbour to sort.