Week 28 · Compound solids, volume stories and a 3-D city · 3 lessons
3
Designing a 3-D city
Discover Mathematics P5: Term 2, Unit 11, Lessons 7–9
GoalPlan a model city from boxes: choose dimensions, calculate each volume and the total, and stay within a limit.
New words
plan · a drawing from above that shows where each building goestotal · all the volumes added togetherlimit · the most you are allowed to use, such as 200 cubes
Explanation
Let us design a model city. Each building is a box built from unit cubes. First draw a plan from above on squared paper: each building covers a rectangle. The plan shows the length and the width. Then choose a height for each building. Volume = length × width × height tells how many cubes it needs. Add the volumes to find the total number of cubes, and check it against the limit. Our plan has a school 8 × 5 and height 2 (80 cubes), a library 4 × 4 and height 3 (48), a market 6 × 3 and height 2 (36) and a tower 2 × 2 and height 8 (32). The total is 80 + 48 + 36 + 32 = 196 cubes. If we have only 200 cubes, the plan fits with 4 cubes to spare. If a plan needs too many cubes, make a building lower or smaller and calculate again.
Total: 80 + 48 = 128, then 128 + 36 = 164, then 164 + 32 = 196 cubes. With a limit of 200 cubes, 200 − 196 = 4 cubes are left over.
Each cube of modelling clay costs 2 LE. The whole city costs 196 × 2 = 392 LE. If the tower is made only 4 high, it needs 2 × 2 × 4 = 16 cubes, which is 16 fewer, so the total is 196 − 16 = 180 cubes.
Class activity
Our model city. Work in a group with 100 or more sugar cubes or blocks. Draw a plan on squared paper with at least four buildings and write the length and width of each. Choose heights, calculate the volumes and the total, build the city, and count the cubes to check. Give your city a name.
Practice
1
A building is 5 cubes long, 3 wide and 4 high. How many cubes does it need?
5 × 3 × 4 = 60 cubes.
2
Three buildings need 60, 48 and 36 cubes. How many cubes in total?
60 + 48 + 36 = 144 cubes.
3
A tower has a base of 2 × 2 and needs 40 cubes. How high is it?
Base 2 × 2 = 4 cubes per layer. 40 ÷ 4 = 10 layers, so it is 10 high.
4
You have 150 cubes. Building A is 6 × 5 × 2 and building B is 4 × 4 × 3. How many cubes are left, and could a 7 × 3 × 2 building also fit?
A: 60. B: 48. Used: 108. Left: 150 − 108 = 42. The new building needs 7 × 3 × 2 = 42 cubes, so it fits exactly.
5
The library covers 4 × 4 squares of the plan and is 3 cubes high. How many cubes are in its roof layer, and how many in the whole library?
The roof layer has 4 × 4 = 16 cubes. The library has 16 × 3 = 48 cubes.
