Working with partial units

1. Remind students how the area and perimeter of a rectangle can be found using the side lengths. Use a rectangle that is 20cm x 15cm as an example. Draw the rectangle on a large sheet of paper. Let students use calculators if needed.
   What is the perimeter of this rectangle? (2 x (20 + 15) = 2 x 35 = 70cm)
   What is the area of this rectangle? (20 x 15 = 300cm²)

2. Provide the students with copies of Copymaster Five and calculators.
   What do you notice about all these rectangles? (Students should see that many side lengths are decimals and fractions)
   Your task is to find the area and perimeter of each rectangle.
   Put the rectangles in order by area then order them by perimeter.
   Are the orders the same?

3. Let the students work in pairs to calculate areas and perimeters. Roam as they work and look for:
   • Do students collect part units to form whole units, such as two halves to make one unit?
   • Do students use whole number multiplication to calculate most of the area in each rectangle?
   • Do students convert fractions of units to decimals and use multiplication on the calculator to find both perimeters and areas?
4. At times during the investigation, you might gather the students to discuss important learning points above or discuss the same idea with pairs. After sufficient progress is made gather the group. Focus on strategies for dealing with partial units of both length and area.
   Teaching points might include:
   • Changing fractions to decimals to find areas by multiplication. For example:
      is the same as
     The area is 5 x 8.75 = 43.75 square units
   • Adding fractions and decimals to find perimeters. For example, the perimeter of this rectangle:
      equals 2×(6½+5½) = 2×13 = 26 units.
   • Finding fractions of fractions of a unit. For example, the shaded area is one half of one half (1/2×1/2 = 1/4).
     
5. The areas and perimeters are:

   Rectangle	A	B	C	D	E
   Area	38sq units	35.75 sq units	43.75sq units	40sq units	30sq units
   Perimeter	27units	26units	27.5units	30 2⁄3 units	29units

   Note that the orders are different for area and perimeter.

 Next steps

1. Provide rectangles with decimal side lengths for students to measure then work out areas and perimeters. Encourage students to estimate the areas before calculating. For example, the rectangle measures 12.8cm x 7.4cm so an estimate of 13 x 7 = 91cm2 is reasonable for area and 2 x (13 + 7) = 40cm is reasonable for perimeter.
   

2. Extend interpretation area and perimeter with problems where area or perimeter is given but a side length is not. Two examples might be:

   • The perimeter is 42.2 cm. What is the width of the rectangle?

     

   • The area is 99.54cm2. What is the width of the rectangle marked ?