Foundation Chapter 19 Simple perimeter, area and volume

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1 Foundation Chapter 19 Simple perimeter, area and volume Background This chapter shows how to find the perimeter and area of simple composite 2 D shapes, and the volume of a cuboid. Surface areas of simple prisms are also covered so that students can tackle everyday problems involving area and volume. Conversion between metric units of area and of volume is included and there is a brief look at working with two units at the same time, i.e. distance and time. Before starting this chapter, students should be familiar with/have experience of: Names of triangles, quadrilaterals and polygons Shapes that tessellate Conversion of units of length Concepts of area and volume Nets of simple solids Constructing simple 3 D shapes Teaching objectives: 19.1: Perimeters of shapes made up of straight lines and circumferences of circles SSM4f 19.2: Finding areas by counting squares SSM4f 19.3: Finding volumes by counting cubes SSM4g 19.4: Using formulae to find areas of rectangles, triangles and composite shapes SSM2e/4f 19.5: Using the formulae for the volume of a cuboid SSM4g 19.6: Finding the surface area of cuboids and prisms SSM4f 19.7: Division of volumes SSM4g 19.8: Solving problems involving area and volume SSM4f/g 19.9: Changing from one unit to another SSM4a/4i 19.10: Relationship between speed, time and distance and assorted questions SSM4c 19.11: Maximising areas using a spreadsheet SSM4f/NA5f Common difficulties Students often confuse the definition and formulae for perimeter with those for area when they have moved on from counting squares. Use string to walk the sides of a 2 D shape and measure it against a ruler to consolidate understanding of perimeter as a length and so avoid the common mistake of counting the number of squares that fit along the sides of the shape rather than each side of each square that lies along the perimeter. The concept of surface area may be difficult. Unfolding 3 D shapes to reveal their nets can help students visualisation. F19.2 F19.3 Area and volume Links to the specification: Finding areas by counting squares Using formulae to find areas of rectangles Finding volumes by counting cubes Links to the Student book: Pages SSM4f SSM2e/4f SSM4g Links to the Teaching and Learning software: F19 Volume of cuboids

2 Links to the Practice book: Exercise 19.2 Key words: Perimeter, area, volume, square, cubic Introduction This lesson provides basic, practical reminders of the concepts of area and volume. For the starter draw a page of five or six different 2 D shapes, e.g. rectangles, triangles, irregular shapes like leaves, and copy for each student. They also need a set of identical tiles or counters each, enough to cover the largest shape. You need access to an enlarged copy of the shapes and a transparent square grid. Key teaching points: The area of a 2 D shape is a measure of the amount of space it covers. Find the number of squares that cover a 2 D shape to find its area. The volume of a 3 D shape is a measure of the amount of spaces it occupies. Find the number of cubes that fill a 3 D shape to find its volume. Prior knowledge: Names of 2 D shapes 2 D shapes that tessellate Basic idea of area and volume Starter Give each student a copy of five or six different 2 D shapes and a set of small, identical tiles/counters. Ask them to measure the area of each shape with their tiles/counters. Point out that the sets of measuring tiles are not made from a standard shape. Ask questions such as: Which shape has the biggest/smallest area? What is its area? (Answers will vary.) What was difficult? How did you count tiles that did not fit exactly? (For example, use rules such as: count a tile if at least one half of it is within the shape, try and match up part squares to make whole ones.) How can we compare answers? (All use the same tiles/units to measure each area.) Which are the best tiles to use? (Shapes that leave no space uncovered are most accurate, e.g. squares.) Main activity Remind students about square kilometres, square metres, square centimetres and square millimetres. Ask for examples of situations where each would be used. Discuss which measure is being used in Exercise19C. Model how to use a cm 2 grid to estimate the area of an irregular shape. Ask pairs to find the areas of the shapes in Exercises 19C and 19D. Review students answers to the exercises. Ask for any quick methods that they used for the rectangles (19C Q1 and 4). Introduce the formula for the area of a rectangle and model how to use it. Give additional examples, asking students to check their answers by drawing the rectangles and counting squares. Variation/extension

3 Investigate how many different rectangles there are with an area of 12 cm 2. Which rectangle has the longest perimeter? Plenary Stress that the area of a 2 D shape is a measure of the amount of space it covers and that you can use a cm 2 grid to work out the area of a shape. Discuss the measure of space for a 3 D shape (volume) and cubic units (cm 3 /m 3 ). Use Teaching and Learning software Volume of cuboids to illustrate how to fill a cuboid layer by layer with centimetre cubes and how to find the total number of cubes. If time permits, ask students for oral answers to Exercise 19C (page 350). Exercise hints: Each square of a cm 2 grid has an area of one square centimetre (1 cm 2 ). It is a good idea to number the squares as you count to make sure that you don t miss a square or part of a square. A cube with sides of length 1 cm has a volume of one cubic centimetre (1 cm 3 ). Homework: Practice book Exercise 19.2 Follow up work: Student book Sections Lesson plan F19.4 F19.4 Finding areas using formulae Links to the specification: Using formulae to find areas of rectangles, triangles and composite shapes SSM2e/4f Links to the Student book: Pages Links to the Practice book: Exercise 19.3 Key words: Area, formula(e) Introduction This lesson builds on the idea of using a formula for the area of a rectangle to find formulae for the areas of triangles and parallelograms. For the main activity you need several paper rectangles with known dimensions, two identical paper scalene triangles, a paper parallelogram and a pair of scissors. Key teaching points: Area formulae for 2 D shapes: Area of a rectangle = l w Area of a square = l l

4 Area of a triangle = 1/2b h Area of a parallelogram = b h Prior knowledge: Names of triangles and quadrilaterals Formula for area of a rectangle Conversion of units of length, e.g. cm to mm and vice versa Starter The area of a rectangle is 18 cm 2. Ask students to draw all the different rectangles this might be. Encourage them to include part squares/decimals. Collect answers. Make the link between length and width, and multiplications that make 18. Repeat for other target areas/products, e.g. 21, 25, 12, 42, 72, 130. Main activity Display some paper rectangles. Give their dimensions, including one with mixed units, e.g. mm and cm. Ask students to use the formula for the area of a rectangle to work out the area of each. Go through answers, emphasising the need for matching units. Cut a rectangle in half into triangles and ask students for the area of each right angled triangle. Lead them to the formula for finding this (1/2l w). Would this formula work for a scalene triangle? Show two identical scalene triangles, each with its base (b) and vertical height marked (h). Cut one along its vertical height and use the two parts with the second scalene triangle to make a rectangle. Ask students for the formula for the area of this rectangle (b h) and then the triangle. See diagram on Student book page 352. Show a paper parallelogram with the base and the vertical height from one corner marked. Cut along the vertical height and move the resulting triangle to the opposite side to model how to use the formula for the area of this rectangle to find the area of the parallelogram. See diagram on Student book page 353. Discuss Example 6 on page 353 with students and ask them to complete Exercise 19F, laying out their work in the same way. Discuss ways of multiplying two decimals before you start. Variation/extension Convert the dimensions of the shapes in Exercise 19F into mm and calculate the area in mm 2. What is the link between the area in cm 2 and in mm 2? Plenary Draw a mixture of shapes on the board. Write an area calculation containing a mistake beside each. Ask students to spot the mistake and advise how to calculate the area properly. Exercise hints: A formula is a relationship between quantities, written using either letters or words. Homework: Students draw several trapeziums on paper, cut each out, then draw and measure its vertical height. Challenge them to work out the area of each by cutting it into a rectangle and triangle(s).

5 Follow up work: Student book Exercise 19G, Section 19.6 and Chapter 26