☰ Year plan · Mathematics

Distances, routes and shapes on the plane

Week 25 Β· Distances on the coordinate plane and drawing shapes from points Β· 3 lessons
2

Distance on the coordinate plane: routes and maps

Discover Mathematics P6: Term 2, Unit 11, Lessons 4–6
GoalUse distances along grid lines to follow routes and find places on a grid map.
New words
route Β· the path from one place to another along grid linesgrid map Β· a map drawn on a coordinate planescale Β· how much real distance one unit on the map stands for
Explanation

A grid map is a coordinate plane in which each place is a point. Roads go along the grid lines, so a route between two places is made of horizontal and vertical pieces. To find the length of a route, find the distance of each piece (same x or same y) and add them. The scale tells what one unit means: if 1 unit is 100 m, then 12 units are 1,200 m. We can also look for places at a given distance: from (1, 2), the points 4 units away along the horizontal line are (5, 2) and (βˆ’3, 2), one on each side. We never need a distance formula; we only count along the axes.

β˜…Remember
A route = horizontal pieces + vertical pieces. Find each piece with absolute values, add them, then use the scale.
Worked examples
xyβˆ’6βˆ’5βˆ’4βˆ’3βˆ’2βˆ’1123456βˆ’6βˆ’5βˆ’4βˆ’3βˆ’2βˆ’11234560PLB
On a map 1 unit = 100 m. The post office P is at (βˆ’4, 3), the library L at (βˆ’4, βˆ’2) and the bakery B at (3, βˆ’2). P to L: same x, 3 + 2 = 5 units. L to B: same y, 4 + 3 = 7 units. The route P β†’ L β†’ B is 5 + 7 = 12 units = 1,200 m.
A school S is at (1, 2). Which points on the same horizontal line are 4 units from S? Go 4 right: (5, 2). Go 4 left: (βˆ’3, 2). There are two answers.
A walker goes from (βˆ’3, βˆ’3) to (4, βˆ’3) and then to (4, 2). First piece: 3 + 4 = 7 units. Second piece: 3 + 2 = 5 units. Total 12 units. If 1 unit is 50 m, the walk is 12 Γ— 50 = 600 m.
Class activity

Draw my town. On squared paper draw axes and place five places (home, school, shop, park, club) at points with whole-number coordinates. Choose a scale. Write the length of the route from home to school along the grid lines.

Practice
1
A house is at (2, 3) and a shop at (2, βˆ’4). How far apart are they?
2
A park is at (βˆ’5, βˆ’1) and a pool at (βˆ’5, βˆ’6). Find the distance.
3
Find the points 3 units from (βˆ’2, βˆ’1) on the vertical line through it.
4
Look at the map. How long is the route from P to L to B in units? If 1 unit is 100 m, how many metres?
xyβˆ’6βˆ’5βˆ’4βˆ’3βˆ’2βˆ’1123456βˆ’6βˆ’5βˆ’4βˆ’3βˆ’2βˆ’11234560PLB
5
Salma walks from (1, 1) to (1, 6) and then to (βˆ’2, 6). How many units does she walk?