Areas of Rectangles

Purpose

In this unit students learn to use the multiplication formula to find the area of a rectangle. Using proportional reasoning students explore what happens to the area when the length and/or height of a rectangle is doubled.

Specific Learning Outcomes
  • use multiplication to calculate the area of a rectangle
  • measure the length of a side using a ruler
  • use proportional reasoning to find the area of a rectangle
Description of Mathematics

In this unit students explore how to find the area of a rectangle by using the lengths of the sides. The teaching sequence shows how to help students understand the formula by firstly counting the area in individual units and then skip counting the number of units in either the rows or the columns, before presenting the formula as length multiplied by height. Some time is given for students to revise measuring lengths of sides with rulers. Problems are illustrated that require students to use proportional thinking as they explore the relationship between changing the side lengths and the resulting change to the area.

Required Resource Materials
About 20 squares of card/paper (10cm by 10cm)

Large piece of paper (30cm by 60cm)

Coloured paper

Rulers

Activity

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 (like memo squares). Tell the students you want to know how many of these small squares are needed to cover the large paper rectangle. You can set a context such as this is the back garden and these are the concrete tiles we will be using to cover it. How many square tiles will cover this area.
  2. Ask a volunteer to place the 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 or 3 x 6 = 18
    How could you work out the area without counting every square? Do we get the same answer if we count/add/multiply columns instead of rows?
  5. Draw a rectangle on the board and mark the sides as 12cm and 10cm. Ask students to come forward and sketch in square centimeters to create one row (12 square cms across) and a column (10 square cms down)

    rectangle area 10cm by 12cm 
    Ask the students:
    How could you work out the area using the lengths of the sides of the rectangle?
  6. Explain to the students that the answer is written as 120 followed by the units cm2.
  7. Give students diagrams of rectangles of which to find the area or set up measuring tasks with rectangles around the class or school. 

Sessions 2 and 3

Discuss the idea of a formula and introduce the notation L x W = A. 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:
    Composite shape 

    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 a ground sheet to take camping. The first one they looked at measured 2 by 3m. Sam said if they wanted one with an area 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:

 

Colourful  rectange


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