Mathematics notes, unit 7
Coordinates, Ratio and Proportion
Textbook Mathematics Year 4 (DLP), pages 215–232
What this chapter covers
- You can name the origin, the horizontal axis (x-axis) and the vertical axis (y-axis) of a Cartesian plane (7.1.1)
- You can read and write the coordinates of a point in the first quadrant, x first and then y (7.1.1, 7.1.2)
- You can mark a point on a Cartesian plane from its coordinates (7.1.2)
- You can state the ratio of two quantities of the same kind, with the same unit (7.2.1)
- You can find the value of one item and use it in simple proportion (7.3.1)
- You can solve problems about coordinates, ratio and proportion (7.4.1)
7.1Coordinates
A map on a grid lets us say exactly where a place is. A Cartesian plane has two number lines that cross each other at right angles.
The horizontal axis is the x-axis (across). The vertical axis is the y-axis (up). They cross at the origin, O. The coordinates of a point say where it is: (x, y).
Idea from Mathematics Year 4 (DLP), p.215: places of interest on a Cartesian plane. Figure drawn for this site.
Worked example: write the coordinates of the Lake
Step 1. Start at the origin O.Step 2. Go across to the right until you are under the Lake: 4 units. This is the x-coordinate.
Step 3. Go up to the Lake: 5 units. This is the y-coordinate.
Step 4. Write the x-coordinate first, then the y-coordinate: the Lake is at (4, 5).
| Point | Across (x) | Up (y) | Coordinates |
|---|---|---|---|
| Origin | 0 | 0 | (0, 0) |
| Waterfall (on the x-axis) | 4 | 0 | (4, 0) |
| Market | 1 | 3 | (1, 3) |
Order matters. (4, 5) and (5, 4) are two different places. Always across first, then up. Also, the origin is written O (a letter), not the number 0.
A point on the x-axis has y = 0, like (4, 0). A point on the y-axis has x = 0, like (0, 5).
Look at the Cartesian plane above. What are the coordinates of the Temple? Which place is at (1, 3)?
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The Temple is at (5, 2). The place at (1, 3) is the Market.
Marking a point
Worked example: mark P (4, 2) and Q (0, 5)
Step 1. For P: from the origin move 4 units to the right, then 2 units up. Mark the point and write P (4, 2).Step 2. For Q: do not move right at all (0 units). Move 5 units up the y-axis. Mark the point and write Q (0, 5).
A butterfly is 3 units to the right of the origin. It flies 6 units up and lands on a flower. What are the coordinates of the flower?
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(3, 6): 3 across, then 6 up.
7.2Ratio
A ratio compares two quantities of the same kind. We write it with a colon: a : b, read "a to b".
Idea from Mathematics Year 4 (DLP), p.218: ingredients of a kuih recipe. Figure drawn for this site.
Worked example: pens and books
Aina has 2 pens and 7 books.Step 1. Ratio of pens to books: 2 : 7.
Step 2. Ratio of books to pens: 7 : 2. The order follows the question.
Say the first thing in the question first. "Pens to books" is 2 : 7, not 7 : 2. Do not write the units (cm, kg, RM) in the ratio.
Same unit first
Both quantities must be in the same unit before you write the ratio.
| Compare | Change to the same unit | Ratio |
|---|---|---|
| RM1 and RM100 | already the same | 1 : 100 |
| 1 kg and 1 000 g | 1 kg = 1 000 g | 1 000 : 1 000 = 1 : 1 |
| 1 mℓ and 1 ℓ | 1 ℓ = 1 000 mℓ | 1 : 1 000 |
| 1 cm and 1 m | 1 m = 100 cm | 1 : 100 |
State the ratio of the mass of 1 g sugar to the mass of 1 kg flour.
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1 kg = 1 000 g, so the ratio is 1 : 1 000.
7.3Proportion
Things are in proportion when one item always has the same value, for example, the same price for each apple. To solve, use the unitary method: first find the value of one (divide), then find the value of the number you want (multiply).
Idea from Mathematics Year 4 (DLP), p.223: price of curtain cloth. Figure drawn for this site.
Worked example: heart rate
An adult's heart beats 360 times in 5 minutes. How many times in 3 minutes?Step 1. In 1 minute: 360 ÷ 5 = 72 times.
Step 2. In 3 minutes: 3 × 72 = 216 times.
Worked example: how many metres?
RM6 buys 1 m of cloth. How long a piece can be bought with RM84?RM84 ÷ RM6 = 14 m.
Do not just double or halve without thinking. Always find the value of 1 first. Check: if 5 buys more than 3, the answer for 5 must be bigger.
A car travels 240 km in 3 hours at a steady speed. How far does it go in 5 hours?
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In 1 hour: 240 ÷ 3 = 80 km. In 5 hours: 5 × 80 = 400 km.
7.4Solving problems
Understand the problem, plan a strategy, solve, then check. Many problems join two ideas.
Worked example: which pair of items?
A shop has a television (RM1 750), a refrigerator (RM1 800) and a water heater (RM220). Ayub pays RM2 020 for two items.Step 1. Try: 1 750 + 220 = 1 970. Not 2 020.
Step 2. Try: 1 800 + 220 = 2 020. Correct, so he bought the refrigerator and the water heater.
Worked example: ratio from "more than"
Halim's durian is 1 kg. Rekha's is 5 kg heavier.Step 1. Rekha's durian: 1 + 5 = 6 kg.
Step 2. Ratio of Halim's to Rekha's: 1 : 6.
Worked example: the best buy
Shop A: 4 books for RM4.80. Shop B: 2 books for RM2.60. Shop C: 3 books for RM3.30.Step 1. One book: A RM1.20, B RM1.30, C RM1.10.
Step 2. The cheapest is shop C. For 12 books: 12 × RM1.10 = RM13.20.
A tap fills 5 buckets in 20 minutes. How many buckets does it fill in 36 minutes at the same rate?
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20 ÷ 5 = 4 min for 1 bucket. 36 ÷ 4 = 9 buckets.
Summary
- A Cartesian plane has an x-axis (across), a y-axis (up) and the origin O at (0, 0).
- Coordinates are written (x, y): across first, then up.
- A ratio a : b compares two quantities of the same kind in the same unit; the order follows the question.
- Proportion: find the value of 1 (divide), then the value you want (multiply).
- For problems: understand, plan, solve, check.