Fixed area rectangles and perimeters

Measurement progression, 5th step

The purpose of the activity

In this activity, the learners develop an understanding of how to calculate the area and perimeter of rectangles from the measurement of side lengths. They learn that the perimeters of rectangles of fixed area do not remain the same.

The teaching points

• The learners understand that the perimeter of a rectangle is calculated by adding together the four side lengths or by adding together two adjacent side lengths and doubling.
• The learners understand that the area of a rectangle is calculated by multiplying together the lengths of two adjacent sides.
• The learners understand that the rectangle with the shortest perimeter for a fixed area is a square.
• The learners measure length using centimetres and millimetres, and area using square centimetres (cm²) and square millimetres (mm²).
• Discuss with the learners relevant or authentic situations where measuring the perimeter or area of rectangles is applicable.

Resources

• Rulers and measuring tapes (marked in centimetres, millimetres).
• 1 centimetre grid paper (Material Master).
• Scissors.
• A selection of rectangular and square objects.

The guided teaching and learning sequence

1. Give each of the learners the grid paper and ask them to cut out a 4 centimetre x 4 centimetre square. Ask:

“What is the area of the square?”
“How do you record that?” (16 cm²)
“How did you calculate the area?”

If the learners counted squares, ask if they could do it another way. Encourage them to notice that you multiply the side lengths together, which in the case of a square is the same as squaring the side length: 4 x 4 = 42 = 16).

2. Next ask:

“What is the perimeter of the square?”
(Check the learners all understand that the perimeter is the distance around the sides of the square)
“How do you record that?” (16 cm)
“How did you calculate the perimeter?”

If the learners added the four side lengths, encourage them to notice that you can calculate the perimeter by multiplying one of the side lengths by four (as the four sides of a square are of equal length).

3. Ask the learners to cut the square into two pieces with a single straight scissor cut, so that the two pieces when moved and rearranged form a rectangle with a perimeter of 20 centimetres.

Ask for a volunteer to demonstrate how they formed a rectangle with a 20 centimetre perimeter from the square.

4. Ask:

“What is the area of this rectangle?”
“How did you know or work that out?”

Encourage the learners to notice that the area is unchanged.

5. Ask:

“Can you make any other rectangles that have an area of 16 cm²?” (16 x 1 rectangle)
“What is the perimeter of this rectangle?” (34 centimetres)

6. Next ask the learners to cut a 12 centimetre x 3 centimetre rectangle from squared paper and record its area and perimeter on the following chart:

7. Have the learners construct as many rectangles as they can with an area of 36 cm² and record their findings on the chart.

“Which rectangle has the largest perimeter?”
“Which rectangle has the smallest perimeter?”
“When might it be useful to know how to find the smallest perimeter for a given area?”
(One possibility is when making a garden that requires a boundary fence or wooden border. A square garden will require the smallest amount of fencing or border material.)

Follow-up activity

Ask the learners to work in pairs to find the area and perimeter of the rectangular faces of a variety of objects (books, shoe box, room, desk top, computer screen).

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