Higher Chapter 13 Measure and mensuration Background This chapter is about measurement of lengths, areas and volumes - mainly of geometrical shapes. It gathers together and formalises previous experience of finding perimeters, areas and volumes, including the circumference and area of a circle and the volume of prisms. Formulae are developed as appropriate and used to solve increasingly complex problems. Sections 13.1, 13.2 and 13.5 consider the accuracy of measurement, its effect and how to choose the right degree of accuracy. The chapter finishes with a look at how dimension theory can help to avoid confusion between the various formulae for perimeters, areas and volumes. Before starting this chapter, students should be familiar with/have experience of: Concept of perimeter, area and volume Finding perimeters and areas Changing the subject of a formula Finding the coordinates of the mid­point of a line on a 2­D grid Teaching objectives 13.1 Understanding that accuracy depends on the accuracy of the measuring instrument 13.2 Measurements to the nearest unit may be inaccurate by up to 0.5 in either direction 13.3 Calculating perimeters and areas, using formulae as appropriate 13.4 Using formulae; includes finding the radius 13.5 Volume and surface area of cuboids and prisms 13.6 Using coordinates in three dimensions 13.7 Identifying formulae for length, area and volume SSM4a SSM4a SSM2e/4f SSM4d SSM2i/4d SSM3e SSM3d H13.5 Volume and surface area of 3­D shapes Links to the specification: Volume and surface area of cuboids and prisms SSM2i/4d Links to the Student book: Pages 244-248 Links to the Teaching and Learning software: H13 Volume of cuboids H13 Volume of prisms Links to the Practice book: Exercise 13.3 Key words: Volume, surface area Introduction This lesson reminds students about finding the volume and surface area of a cuboid and leads on to finding formulae to apply to a general prism.
Key teaching points For a general prism: Surface area = 2 × area of base + total area of vertical faces Volume = area of base × vertical height Prior knowledge: Concept of perimeter, area and volume Finding perimeters and areas Changing the subject of a formula Starter Use Teaching and Learning software Volumes of cuboids to remind students about the volume of a cube and a cuboid. Ask students to write down the formula for the volume of each shape. Give examples for them to calculate quickly, some with mixed units, e.g. cubes with sides 6 mm, 110 cm, 0.4 m; cuboids measuring 3 cm × 4 cm × 5 cm, 5 mm × 6 cm × 4 cm, 0.1 m × 0.75 m × 2.1 m. Main activity Check that students can find the surface area of a cube, e.g. with side 4 cm, and a cuboid, e.g. 3 cm × 4 cm × 5 cm. Ask them to write down a formula for each. Remind students that a cube and a cuboid are special cases of a prism. Look at the H­and L­ shape prisms on Teaching and Learning software Volume of prisms. Elicit that the volume can be calculated by multiplying the area of the base by the vertical height. Is this true for any prism? Why? Repeat this process for finding the surface area of the H­ and L­shaped prisms. Lead students towards finding a general statement for the surface area of any prism: Surface area = 2 × area of base + total area of vertical faces. Set questions from Student book page 247 Exercise 13D. Variation/extension Discuss Student book page 245-6 Worked examination questions 2 and 3, then try questions 16 and 17 from Student book page 550. Plenary Discuss students’ answers to questions from Student book page 247 Exercise 13D and identify any algebraic problems. For which questions did you need to change the subject of the formula? Challenge students to find the dimensions of a cuboid with, for example, volume 168 cm 3 and surface area 202 cm 2 (3 cm × 7 cm × 8 cm). Homework: Student book pages 252-253 Mixed exercise 13 questions 1-8. Follow­up work: Student book pages 248-254 Lesson plan H13.6 Chapters 17,19 Links to other subjects: Science
H13.6 1­D, 2­D or 3­D? Links to the specification: Using coordinates in three dimensions SSM3e Links to the Student book: Pages 248-250 Links to the Teaching and Learning software: H13 3­D coordinates on shapes H13 Plotting 3­D coordinates Links to the Practice book: Exercise 13.3 Key words: Dimensional, 1­/2­/3­D Introduction This lesson reminds students about ways of representing 1­/2­/3­D objects on a flat plane and gives them practice in plotting 3­D coordinates. They move on to apply what they know about finding the coordinates of the mid­point of a line to problems on a 3­D grid. Four­quadrant coordinate grids are useful for the Starter. Students need isometric paper for the Main activity. Key teaching points: The number line is 1­dimensional or 1­D. Flat shapes are 2­dimensional or 2­D. Solid shapes are 3­dimensional or 3­D. If A is (x, y, z), B is (p, q, r), the mid­point of AB is (x + p)/2, (y + q)/2, (z + r)/2. Prior knowledge: Finding the coordinates of the mid­point of a line on a 2­D grid Starter Call out four coordinates and ask students to plot them, e.g. (0, 2), (4, 0), (0, -4), (-2, 0). What shape have you plotted? (trapezium) What properties does the shape have? How can you find the coordinates of the point where its diagonals cross? Give students a shape and coordinates for all its vertices except one. Ask them to find the missing coordinate, e.g. (-2, 3), ( -4, 0), (-2, -3), rhombus (0, 0) (2, -1) , (5, 1), (6, 3), kite (4, 2) (3, -3), (1, -3), (3, -5), (4, -5), (4, -4), hexagon Possibilities include ((3, -4), (5,
-2). Discuss strategies and patterns used. Main activity Remind students that:
the number line goes in one direction (either horizontally or vertically) and is 1­ dimensional or 1­D; coordinate grids, as used in the starter, and flat shapes go in two directions and are 2­ dimensional or 2­D; solid shapes go in three directions and are 3­dimensional or 3­D, so need three coordinates and three axes to describe points on them. Show some shapes from Teaching and Learning software 3­D coordinates on 3­D shapes. Begin with a cube where coordinates of the points in the xy plane are all (x, y, 0). Ask students if they can write down these four sets of coordinates. Check answers by selecting Show/Hide coordinates as you arrive at each one. Repeat for the other four vertices, adding the xz or yz grids to assist where needed. Use Teaching and Learning software Plotting 3­D coordinates. Plot each point, asking an individual to describe how it has been reached, e.g. you go 2 units along the x­axis, 3 units parallel to the y­axis and 5 units parallel to the z­axis. Refer to Student book page 249 Example 7. Go through stages (a)-(f) quickly. For (g), discuss how to find the 3­D coordinates of the mid­point of a line when you know the coordinates of each end­point. Ask students to draw a simple 3­D shape on isometric paper, label the vertices with letters and their coordinates, choose a line within it and find its midpoint. Repeat for several examples. Variation/extension Using isometric paper, draw 3­D axes. Ask students to draw a cuboid and write down the 3­D coordinates. Can you see a pattern in the coordinates? Make up some ‘Find the missing coordinate’ problems for different 3­D shapes. Plenary Write up the set of coordinates (0, 0, 0), (3, 0, 0), (3, 2, 0), (0, 2, 0), (0, 0, 5), (0, 2, 5), (3, 0, 5), (3, 2, 1). Which one is a mistake? How can you tell? Exercise hints: Remember to give the coordinates in the order (x, y, z). If A is (x, y) and B is (p, q), then the mid­point of AB = x + p)/2, (y + q)/2 Homework: Student book page 249 Exercise 13E Follow­up work: Section 13.7 Chapter 15