Mathematics notes, unit 6

Length, Mass and Volume of Liquid

Textbook Mathematics Year 3 (DLP), Part 1 and Part 2, pages Part 2, 49–84

What this chapter covers

You can change metres to centimetres and centimetres to metres (6.1.1)
You can add, subtract, multiply and divide lengths in m and cm (6.1.2–6.1.5)
You can change kilograms to grams and grams to kilograms (6.2.1)
You can add, subtract, multiply and divide masses in kg and g (6.2.2–6.2.5)
You can change litres to millilitres and millilitres to litres (6.3.1)
You can add, subtract, multiply and divide volumes of liquid in ℓ and mℓ (6.3.2–6.3.5)
You can create stories and solve everyday problems about length, mass and volume of liquid (6.4.1–6.4.2)

In this unit we measure three things: how long something is (length), how heavy it is (mass) and how much liquid a container holds (volume of liquid). You will change between units and then add, subtract, multiply and divide with them.

Remember the rhyme (textbook p. 49): from centimetre to metre, divide by one hundred. From kilogram to gram, multiply by one thousand. From litre to millilitre, multiply by one thousand.

6.1Length: metres and centimetres

Length tells us how long, tall or far something is. We measure it in metres (m) and centimetres (cm).
1 m = 100 cm

A metre rule marked from 0 to 100 centimetres — A metre rule. From 0 to 100 there are 100 centimetres, and that is 1 metre.

A door is about 2 m tall. Your finger is about 1 cm wide. We use m for long things and cm for short things.

Changing units

From	To	What to do	Example
m	cm	multiply by 100	3 m = 3 × 100 = 300 cm
cm	m	divide by 100	400 cm = 400 ÷ 100 = 4 m

Worked examples

a) 9 m 20 cm = ? cm
Step 1: 9 m = 9 × 100 = 900 cm
Step 2: 900 cm + 20 cm = 920 cm

b) An orangutan is 150 cm tall. Write this in m and cm.
150 cm = 100 cm + 50 cm = 1 m + 50 cm = 1 m 50 cm

Careful with zeros. 808 cm = 800 cm + 8 cm = 8 m 8 cm, not 8 m 80 cm. In the same way, 6 m 5 cm = 600 + 5 = 605 cm, not 65 cm.

Adding and subtracting lengths

Add or subtract the same way as whole numbers. Keep m with m and cm with cm.

Path from the horse stable to the zebra enclosure (453 m) and on to the giraffe enclosure (390 m) — At the zoo: horse stable to zebra 453 m, zebra to giraffe 390 m (textbook p. 52).

Worked example: adding

Distance from the horse stable to the giraffe = 453 m + 390 m = 843 m

87 cm + 1 m 64 cm + 1 m 32 cm = ?
Step 1: add the cm: 87 + 64 + 32 = 183 cm
Step 2: add the m: 1 + 1 = 2 m
Step 3: 183 cm is more than 100 cm, so change 100 cm into 1 m: 183 cm = 1 m 83 cm
Step 4: 2 m + 1 m 83 cm = 3 m 83 cm

Worked example: subtracting

Gunung Murud in Sarawak is 2 423 m high. Gunung Ledang in Johor is 1 276 m high. Difference = 2 423 − 1 276 = 1 147 m

30 m 58 cm − 23 m 77 cm = ?
Step 1: 58 cm is less than 77 cm. Take 1 m from 30 m and change it to 100 cm.
Step 2: now we have 29 m 158 cm.
Step 3: cm: 158 − 77 = 81 cm. m: 29 − 23 = 6 m.
Answer: 6 m 81 cm

Did you know? Gunung Kinabalu in Sabah is the highest mountain in Malaysia. It is more than 4 000 m high!

Multiplying and dividing lengths

Five poles in a row, each 15 m apart — Five poles, 15 m apart. Count the gaps, not the poles: there are 4 gaps.

Worked examples

a) First pole to fifth pole: 4 gaps × 15 m = 60 m

b) A terrace house is 6 m 80 cm wide. How long is a row of 6 houses?
6 × 80 cm = 480 cm = 4 m 80 cm
6 × 6 m = 36 m
36 m + 4 m 80 cm = 40 m 80 cm

c) A horse stable 17 m 40 cm long has 3 equal sections. How long is each section?
Change to cm: 17 m 40 cm = 1 740 cm
1 740 ÷ 3 = 580 cm = 5 m 80 cm

When dividing, it is often easier to change everything to cm first, divide, and then change the answer back to m and cm.

a) 6 m 5 cm = ___ cm b) 430 cm = ___ m ___ cm
c) 13 m 78 cm − 6 m 90 cm d) 864 cm ÷ 8 (answer in m and cm)

Show answerHide answer

a) 600 + 5 = 605 cm
b) 4 m 30 cm
c) 1 378 cm − 690 cm = 688 cm = 6 m 88 cm
d) 864 ÷ 8 = 108 cm = 1 m 8 cm

6.2Mass: kilograms and grams

Mass tells us how heavy something is. We measure it in kilograms (kg) and grams (g).
1 kg = 1 000 g and half a kilogram = 500 g

A packet of sugar is 1 kg. A sweet is only a few grams. At the pasar, the seller puts fish on a weighing scale to find its mass.

Weighing scale with the pointer halfway between 2 and 3 kg — Each small mark is 100 g. The pointer is halfway between 2 kg and 3 kg, so the fish is 2 kg 500 g.

Changing units

From	To	What to do	Example
kg	g	multiply by 1 000	4 kg = 4 × 1 000 = 4 000 g
g	kg	divide by 1 000	7 000 g = 7 000 ÷ 1 000 = 7 kg

Worked examples

a) 2 kg 500 g = 2 000 g + 500 g = 2 500 g
b) An ostrich egg is 1 400 g. 1 400 g = 1 000 g + 400 g = 1 kg 400 g

Grams need three places. 6 kg 8 g = 6 000 + 8 = 6 008 g, not 68 g or 6 800 g. Think of the gram part as 008.

Adding and subtracting mass

Worked example: adding

2 kg 450 g + 1 kg 950 g + 2 kg 780 g = ?
Step 1: g: 450 + 950 + 780 = 2 180 g
Step 2: kg: 2 + 1 + 2 = 5 kg
Step 3: 2 180 g = 2 kg 180 g, so add 2 kg to the 5 kg.
Answer: 7 kg 180 g

Worked example: subtracting

7 kg 90 g − 5 kg 780 g = ?
Step 1: 90 g is less than 780 g. Change 1 kg into 1 000 g: 7 kg 90 g = 6 kg 1 090 g
Step 2: g: 1 090 − 780 = 310 g. kg: 6 − 5 = 1 kg.
Answer: 1 kg 310 g

Multiplying and dividing mass

Worked examples

a) 9 × 2 kg 300 g
9 × 300 g = 2 700 g = 2 kg 700 g. 9 × 2 kg = 18 kg.
18 kg + 2 kg 700 g = 20 kg 700 g

b) 5 kg of mussels is packed equally into 8 packets.
5 kg = 5 000 g. 5 000 ÷ 8 = 625 g in each packet.

c) 8 kg 400 g ÷ 7
8 kg 400 g = 8 400 g. 8 400 ÷ 7 = 1 200 g = 1 kg 200 g

Did you know? The Rafflesia, the world's biggest flower, grows in the forests of Sabah, Sarawak and Perak. One flower can have a mass of about 7 kg!

a) 30 kg 510 g + 920 g b) 6 × 4 kg 129 g c) 25 kg 200 g ÷ 7

Show answerHide answer

a) 510 + 920 = 1 430 g = 1 kg 430 g, so 30 kg + 1 kg 430 g = 31 kg 430 g
b) 6 × 4 129 g = 24 774 g = 24 kg 774 g
c) 25 200 g ÷ 7 = 3 600 g = 3 kg 600 g

6.3Volume of liquid: litres and millilitres

Volume of liquid tells us how much liquid there is. We measure it in litres (ℓ) and millilitres (mℓ).
1 ℓ = 1 000 mℓ

A big carton of milk is 1 ℓ. A spoonful of medicine is about 5 mℓ. We use a measuring jug to measure liquids.

Measuring jug with water up to the 750 mℓ mark — Each small mark is 50 mℓ. The water is halfway between 700 and 800, so it is 750 mℓ. To make 1 ℓ we need 1 000 − 750 = 250 mℓ more.

Changing units

From	To	What to do	Example
ℓ	mℓ	multiply by 1 000	2 ℓ = 2 × 1 000 = 2 000 mℓ
mℓ	ℓ	divide by 1 000	8 000 mℓ = 8 000 ÷ 1 000 = 8 ℓ

Worked examples

a) 4 ℓ 360 mℓ = 4 000 mℓ + 360 mℓ = 4 360 mℓ
b) A bottle of fabric softener holds 2 350 mℓ. 2 350 mℓ = 2 000 mℓ + 350 mℓ = 2 ℓ 350 mℓ

Litres and millilitres work just like kilograms and grams: 1 000 small units make 1 big unit.

Adding and subtracting volumes

Worked examples

a) 5 ℓ 720 mℓ + 2 ℓ 80 mℓ + 980 mℓ
mℓ: 720 + 80 + 980 = 1 780 mℓ = 1 ℓ 780 mℓ. ℓ: 5 + 2 = 7 ℓ.
7 ℓ + 1 ℓ 780 mℓ = 8 ℓ 780 mℓ

b) A big bottle has 10 ℓ of water. 4 095 mℓ is poured into a small bottle. How much is left?
10 ℓ = 10 000 mℓ. 10 000 − 4 095 = 5 905 mℓ

Same units first! In 10 ℓ − 4 095 mℓ you cannot do 10 − 4 095. Change 10 ℓ to 10 000 mℓ before you subtract.

Multiplying and dividing volumes

Worked examples

a) 8 × 2 ℓ 450 mℓ
8 × 450 mℓ = 3 600 mℓ = 3 ℓ 600 mℓ. 8 × 2 ℓ = 16 ℓ.
16 ℓ + 3 ℓ 600 mℓ = 19 ℓ 600 mℓ

b) 11 ℓ of air sirap is shared equally into 8 jugs.
11 ℓ = 11 000 mℓ. 11 000 ÷ 8 = 1 375 mℓ = 1 ℓ 375 mℓ in each jug.

a) 7 ℓ 50 mℓ = ___ mℓ b) 6 ℓ 730 mℓ + 9 ℓ 407 mℓ c) 6 ℓ ÷ 8 = ___ mℓ

Show answerHide answer

a) 7 000 + 50 = 7 050 mℓ
b) 730 + 407 = 1 137 mℓ = 1 ℓ 137 mℓ, so 15 ℓ + 1 ℓ 137 mℓ = 16 ℓ 137 mℓ
c) 6 000 mℓ ÷ 8 = 750 mℓ

6.4Creating stories and solving problems

Create a story

A number sentence can tell a story. Look at the numbers and units, then think of a real thing at home, at school or at the pasar malam.

Story examples (textbook p. 80)

3 × 250 g = 750 g
Mak uses 3 blocks of butter to bake Hari Raya biscuits. Each block is 250 g. She uses 750 g of butter altogether.

1 ℓ 200 mℓ ÷ 4 = 300 mℓ
There is 1 ℓ 200 mℓ of sugarcane juice in a jug at the pasar malam stall. It is poured equally into 4 glasses. Each glass has 300 mℓ.

A good story has: who, what is measured, the numbers with units, and a question or answer.

Solve the problems

Use four steps: Given (what do I know?), Find (what must I work out?), Method (which operation?) and Answer (with the unit). Drawing a bar model helps you see whether to add, subtract, multiply or divide.

Bar model: Lily's bar is 32 kg; Akmal's bar is shorter by 3 kg — Lily is 32 kg. She is 3 kg heavier than Akmal, so Akmal's bar is 3 kg shorter.

Worked example 1

Lily's mass is 32 kg. Her mass is 3 kg more than Akmal's. What is Akmal's mass?
Given: Lily 32 kg, 3 kg more than Akmal. Find: Akmal's mass.
Method: 32 kg − 3 kg = 29 kg
Answer: Akmal's mass is 29 kg.

"More than" does not always mean add. Lily is 3 kg more than Akmal, so Akmal is the lighter one. We subtract.

Worked example 2 (textbook p. 82)

Tini makes 3 ℓ 400 mℓ of barley drink. She makes 5 times as much bandung drink.
a) Bandung: 5 × 3 ℓ 400 mℓ = 5 × 3 400 mℓ = 17 000 mℓ = 17 ℓ
b) She pours the bandung equally into 8 jugs: 17 000 mℓ ÷ 8 = 2 125 mℓ = 2 ℓ 125 mℓ in each jug.
Check: 8 × 2 125 = 17 000. ✔

a) Lina pours 2 ℓ 520 mℓ of cocoa drink equally into 8 cups. How many mℓ are in each cup?
b) A yellow hose is 1 m 35 cm long. A green hose is 4 times as long. How long is the green hose?

Show answerHide answer

a) 2 520 mℓ ÷ 8 = 315 mℓ
b) 1 m 35 cm = 135 cm. 4 × 135 cm = 540 cm = 5 m 40 cm

Summary

Measure	Units	Big to small	Small to big
Length	1 m = 100 cm	m → cm: × 100	cm → m: ÷ 100
Mass	1 kg = 1 000 g	kg → g: × 1 000	g → kg: ÷ 1 000
Volume of liquid	1 ℓ = 1 000 mℓ	ℓ → mℓ: × 1 000	mℓ → ℓ: ÷ 1 000

Add, subtract, multiply and divide like whole numbers. Keep big units with big units and small units with small units.
When the small units reach 100 cm, 1 000 g or 1 000 mℓ, change them into 1 m, 1 kg or 1 ℓ.
When you cannot subtract the small units, change 1 big unit into small units first.
For division, it is often easier to change everything into the small unit first.
Always write the unit in your answer, and check whether the question wants cm, m and cm, g, kg and g, mℓ, or ℓ and mℓ.
Solve problems with Given, Find, Method and Answer. A bar model helps.