Areas of Rectangles

Session 1

In this session students are introduced to the idea of using multiplication to find the area of a rectangle.

1. Show the students a large rectangular piece of paper measuring 30cm by 60cm and a pile of smaller squares each measuring 10cm by 10cm. Tell the students you want to know how many 10 by 10 squares are needed to cover the large piece of paper.
2. Ask a volunteer to place the 10 by 10 squares side by side on the rectangle.
3. Count with the students the number of squares needed to cover the rectangle. (18). Explain that the area is 18 squares.



4. Ask the students:
   How many squares are in one row? (6)
   So how many would be in two rows? (6 + 6 = 12)
   How many would be in three rows? (6 + 6 + 6 = 18)
   Record for the students:
   1 row: 6
   2 rows: 6 + 6 = 12
   3 rows: 6 + 6 + 6 = 18
5. Ask the students:
   How else can 6 + 6 + 6 be written? (3 x 6)
6. Ask the students:
   How many squares are in one column? (3)
   So how many would be in two columns? ( (3 +3 = 6)
   How many would be in three columns? ( (3 +3 +3 = 9)
   How many would be in four columns? ( (3 +3 +3 +3 = 12)
   How many would be in five columns? ( (3 +3 +3 +3 + 3 = 15)
   How many would be in six columns? ( (3 +3 +3 +3 + 3 + 3 = 18).
   Record for the students:
   1 column: 3
   2 column: 3 + 3 = 6 etc
7. Ask the students:
   How else can 3 +3 + 3 + 3 +3 + 3 be written? (6 x 3)
8. Ask the students:
   How could you work out the area without counting every square?
9. Draw a rectangle on the board and mark the sides as 12cm and 10cm.
   
    
   Ask the students:
   How could you work out the area using the lengths of the sides of the rectangle?
10. Explain to the students that the answer is written as 120 followed by the units cm².
11. Give students pictures of rectangles to find the area of.

Sessions 2 and 3

In the next two sessions students continue to solve problems that involve finding the area of rectangles. Here are three types of problems that students can solve.

1. Students can continue to use the formula to find the area of rectangles, or if the area is known then finding the missing side length. For example:



2. Students can find the area of composite shapes by finding the area of the rectangles. For example:
    
   
   This shape can be seen to be comprised of two 2cm by 4 cm rectangles, or a 2cm by 6cm rectangle and a 2 cm by 2 cm rectangle, or 4 cm by 6 cm rectangle with a 4 by 2 rectangle missing. There are different ways to solve composite shapes.
3. Students can find areas of rectangles by measuring the lengths of the sides of rectangles then applying the multiplication formula.

Session 4

In this session students explore using proportional reasoning to find areas of rectangles.

1. Pose the problem: Sam’s family was shopping for ground sheet to take camping. The first one they looked at measured 2 by 3m. Sam said if they wanted one twice as big they should get the 4 by 6m size. Is Sam right?
2. Ask the students to draw pictures of the ground sheets and to help them decide if Sam is correct.
3. Work with students to establish that doubling the area only involves doubling one side of the rectangle. Doubling both sides of the rectangle increases the area by four times.



4. Using this proportional reasoning students will be able to solve problems without recalculating from side lengths. Here are some example problems:
   
   • The recipe made enough icing to cover the top of a 20cm by 20cm cake, what size cake can you ice if you double the amount of icing?
   • The birthday card had a front cover measuring 15cm by 10cm, what is area of the piece of cardboard used to make it?
   • The garden had two areas that needed paving. Each area measured 5m by 8m. What is the total area to be paved?
   • The gardener charged his customers by the area of their lawn. If the bill was $20 to mow a lawn that was 6m by 20m, what should the bill be for a 20m by 12m lawn?

Session 5

In the session students design a pattern with pieces of coloured paper.

1. Students need to start with a rectangle. Then make a second rectangle by doubling the height of the original. Then a third rectangle by doubling the base of the original. Then a fourth rectangle by doubling both the base and the height of the original.
2. Discuss with students how the areas of these rectangles can be calculated.
3. Students then make multiple copies of each rectangle and then glue them side by side to make a pattern. For example: