Chapter 1 of 8

All 8 chapters
  1. 1Changing metric units, readunderstood
  2. 2Perimeter, readunderstood
  3. 3Area of rectangles, readunderstood
  4. 4Area of triangles and parallelograms, readunderstood
  5. 5Trapeziums and joined shapes, readunderstood
  6. 6Circles, readunderstood
  7. 7Volume of cuboids, readunderstood
  8. 8Surface area, readunderstood

M3.6.1 Chapter 1

Changing metric units

How many metres are there in 3.5 kilometres?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

  • You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Change 3.5 km into metres and 450 mm into centimetres
  • Change between grams, kilograms and tonnes

A sign on a path says the river is 3.5 km away. How many metres is that? To find out, you convert the length. To convert is to change an amount from one unit to another.

We use metric units for length and mass: mm, cm, m and km, and g, kg and t. Metric units go up and down in tens, hundreds and thousands. That makes them quick to convert.

Draw it first

Climb down the ladder

  1. A ladder of length units, the biggest at the top: km, m, cm and mm. The arrows on the left go down and say × 1 000, × 100 and × 10. The arrows on the right go up and say ÷ 1 000, ÷ 100 and ÷ 10. Below the ladder it says Start: 3.5 km. Then the arrow from km to m is drawn heavy, and it says 3.5 × 1 000 = 3 500 m.kmmcmmm× 1 000÷ 1 000× 100÷ 100× 10÷ 10

    1 Write the units in a ladder with the biggest at the top: km, m, cm, mm.

  2. A ladder of length units, the biggest at the top: km, m, cm and mm. The arrows on the left go down and say × 1 000, × 100 and × 10. The arrows on the right go up and say ÷ 1 000, ÷ 100 and ÷ 10. Below the ladder it says Start: 3.5 km. Then the arrow from km to m is drawn heavy, and it says 3.5 × 1 000 = 3 500 m.kmmcmmm× 1 000÷ 1 000× 100÷ 100× 10÷ 10Start: 3.5 km

    2 Start with 3.5 km. To reach metres, go down one rung.

  3. A ladder of length units, the biggest at the top: km, m, cm and mm. The arrows on the left go down and say × 1 000, × 100 and × 10. The arrows on the right go up and say ÷ 1 000, ÷ 100 and ÷ 10. Below the ladder it says Start: 3.5 km. Then the arrow from km to m is drawn heavy, and it says 3.5 × 1 000 = 3 500 m.kmmcmmm× 1 000÷ 1 000× 100÷ 100× 10÷ 103.5 × 1 000 = 3 500 m

    3 Going down, you multiply by the number on the arrow: 3.5 × 1 000 = 3 500 m.

Going down the ladder, the unit gets smaller. You need more of them, so you multiply. Going up, the unit gets bigger. You need fewer of them, so you divide.

The length ladder has three steps. 1 km is 1 000 m. 1 m is 100 cm. 1 cm is 10 mm.

Worked example

Change 3.5 km into metres.

  1. 1 km is 1 000 m.
  2. Going from km to m is going down the ladder. The unit gets smaller, so multiply.
  3. 3.5 × 1 000 = 3 500 m.

Answer: 3 500 m

The length ladder with km, m, cm and mm. The arrow from km down to m is heavy and says × 1 000. The other arrows are thin. The arrows on the right go up and say ÷ 1 000, ÷ 100 and ÷ 10.kmmcmmm× 1 000÷ 1 000× 100÷ 100× 10÷ 10
From km to m you go down one rung and multiply by 1 000.

Multiplying by 1 000 moves every digit three places to the left, and zeros fill the gaps. So 3.5 becomes 3 500.

Worked example

Change 450 mm into centimetres.

  1. 10 mm is 1 cm.
  2. Going from mm to cm is going up the ladder. The unit gets bigger, so divide.
  3. 450 ÷ 10 = 45 cm.
  4. Check by going back down: 45 × 10 = 450 mm.

Answer: 45 cm

The length ladder with km, m, cm and mm. The arrow on the right from mm up to cm is heavy and says ÷ 10. The other arrows are thin.kmmcmmm× 1 000÷ 1 000× 100÷ 100× 10÷ 10
From mm to cm you go up one rung and divide by 10.

Mass has its own ladder. 1 kg is 1 000 g. The unit for a heavy load is the tonne (t). One tonne is 1 000 kg, about the mass of a small car.

A ladder of mass units, the biggest at the top: t, kg and g. The arrows on the left go down and say × 1 000 and × 1 000. The arrows on the right go up and say ÷ 1 000 and ÷ 1 000.tkgg× 1 000÷ 1 000× 1 000÷ 1 000
1 t is 1 000 kg. 1 kg is 1 000 g. Each step is 1 000.

Worked example

Change 3 600 g into kilograms.

  1. 1 kg is 1 000 g.
  2. Going from g to kg is going up the ladder. The unit gets bigger, so divide.
  3. 3 600 ÷ 1 000 = 3.6 kg.
  4. Check by going back down: 3.6 × 1 000 = 3 600 g.

Answer: 3.6 kg

The mass ladder with t, kg and g. The arrow on the right from g up to kg is heavy and says ÷ 1 000. The other arrows are thin.tkgg× 1 000÷ 1 000× 1 000÷ 1 000
From g to kg you go up one rung and divide by 1 000.

Tonnes work the same way. To change 2.5 t into kilograms, go down one rung: 2.5 × 1 000 = 2 500 kg.

See it three ways

With things

Lay a ruler beside a pencil. Read its length in cm. Each cm has 10 small marks, so you can also say it in mm.

As a picture

A ruler marked in cm from 0 to 10, with a pencil along it. The pencil reaches the half way mark between 4 and 5, so it is 4.5 cm long. The word cm is on the ruler.012345678910cm

In symbols

4.5 cm × 10 = 45 mm. And 45 mm ÷ 10 = 4.5 cm.

Try it

Measure in two units

You need: a ruler with cm and mm marks, a pencil, a sheet of paper.

  1. Measure the length of your pencil in cm. Write it as a decimal, such as 15.2 cm.
  2. Multiply by 10 to change it into mm. Write the answer.
  3. Count the small marks on the ruler to read the length in mm. Compare the two answers.

What you will see: The two answers are the same length in two units. A pencil that is 7.5 cm long is also 75 mm long.

If you cannot try it, this is what happens.
A ruler marked in cm from 0 to 10 with a pencil along it that is 7.5 cm long. The word cm is on the ruler.012345678910cm

A wrong idea many people have

When you change to a smaller unit, the number gets smaller.

Why it feels right: The word "smaller" makes people think of a smaller number.

The test that breaks it: Take a piece of string 1 m long. Measure it in cm. You count 100 cm. The unit is smaller, but you count more of them.

What is true: A smaller unit fits more times into the same length. So the number gets bigger. 1 m is 100 cm.

Key words

convert
to change an amount from one unit to another.
metric units
units that go up and down in tens, hundreds and thousands, such as m, cm and kg.
tonne
a unit of mass. 1 tonne is 1 000 kg.

Back to your guess

The answer is D: 3 500 m

One kilometre is 1 000 m. So 3.5 km is 3.5 × 1 000 = 3 500 m. A metre is a much smaller unit than a kilometre, so you need many more of them.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A path is 4.2 km long. How long is it in metres?

    The length ladder with km, m, cm and mm. The arrow from km down to m is heavy and says × 1 000.kmmcmmm× 1 000÷ 1 000× 100÷ 100× 10÷ 10
    Tap your answer.

    1. Not this one. You multiplied by 10. Between km and m the number on the arrow is 1 000. Go to M3.6.1. Try another.

    2. Not this one. You multiplied by 100. That is the step from m to cm. From km to m you multiply by 1 000. Go to M3.6.1. Try another.

    3. Yes, that is right. 1 km is 1 000 m. Going down the ladder you multiply: 4.2 × 1 000 = 4 200 m.

    See the answer

    4 200 m

    1 km is 1 000 m. Going down the ladder you multiply: 4.2 × 1 000 = 4 200 m.

    • If you chose A: You multiplied by 10. Between km and m the number on the arrow is 1 000. Go to M3.6.1.
    • If you chose B: You multiplied by 100. That is the step from m to cm. From km to m you multiply by 1 000. Go to M3.6.1.

  2. A rope is 650 mm long. How long is it in centimetres?

    The length ladder with km, m, cm and mm. The arrow on the right from mm up to cm is heavy and says ÷ 10.kmmcmmm× 1 000÷ 1 000× 100÷ 100× 10÷ 10
    Tap your answer.

    1. Yes, that is right. 10 mm is 1 cm. Going up the ladder you divide: 650 ÷ 10 = 65 cm.

    2. Not this one. You divided by 100. From mm to cm the number on the arrow is 10, not 100. Go to M3.6.1. Try another.

    3. Not this one. You multiplied. A cm is bigger than a mm, so you need fewer cm. Go up the ladder and divide. Go to M3.6.1. Try another.

    See the answer

    65 cm

    10 mm is 1 cm. Going up the ladder you divide: 650 ÷ 10 = 65 cm.

    • If you chose B: You divided by 100. From mm to cm the number on the arrow is 10, not 100. Go to M3.6.1.
    • If you chose C: You multiplied. A cm is bigger than a mm, so you need fewer cm. Go up the ladder and divide. Go to M3.6.1.

  3. A bag of rice has a mass of 2 400 g. What is its mass in kilograms?

    The mass ladder with t, kg and g. The arrow on the right from g up to kg is heavy and says ÷ 1 000.tkgg× 1 000÷ 1 000× 1 000÷ 1 000
    Tap your answer.

    1. Not this one. You divided by 10. From g to kg the number on the arrow is 1 000. Go to M3.6.1. Try another.

    2. Yes, that is right. 1 kg is 1 000 g. Going up the ladder you divide: 2 400 ÷ 1 000 = 2.4 kg.

    3. Not this one. You multiplied. A kg is bigger than a g, so there are fewer kg. Go up the ladder and divide. Go to M3.6.1. Try another.

    See the answer

    2.4 kg

    1 kg is 1 000 g. Going up the ladder you divide: 2 400 ÷ 1 000 = 2.4 kg.

    • If you chose A: You divided by 10. From g to kg the number on the arrow is 1 000. Go to M3.6.1.
    • If you chose C: You multiplied. A kg is bigger than a g, so there are fewer kg. Go up the ladder and divide. Go to M3.6.1.

  4. A lorry carries 4 500 kg of sand. How many tonnes is that?

    The mass ladder with t, kg and g. The arrow on the right from kg up to t is heavy and says ÷ 1 000.tkgg× 1 000÷ 1 000× 1 000÷ 1 000
    Tap your answer.

    1. Not this one. You divided by 100. From kg to t the number on the arrow is 1 000. Go to M3.6.1. Try another.

    2. Not this one. You divided by 10. From kg to t the number on the arrow is 1 000. Go to M3.6.1. Try another.

    3. Yes, that is right. 1 t is 1 000 kg. Going up the ladder you divide: 4 500 ÷ 1 000 = 4.5 t.

    See the answer

    4.5 t

    1 t is 1 000 kg. Going up the ladder you divide: 4 500 ÷ 1 000 = 4.5 t.

    • If you chose A: You divided by 100. From kg to t the number on the arrow is 1 000. Go to M3.6.1.
    • If you chose B: You divided by 10. From kg to t the number on the arrow is 1 000. Go to M3.6.1.

  5. Which of these lengths is the longest?

    Tap your answer.

    1. Yes, that is right. Change all three to metres. 1 500 m, then 1.2 × 1 000 = 1 200 m, then 140 000 ÷ 100 = 1 400 m. The longest is 1 500 m.

    2. Not this one. 1.2 km is 1.2 × 1 000 = 1 200 m. That is shorter than 1 500 m. Go to M3.6.1. Try another.

    3. Not this one. 140 000 ÷ 100 = 1 400 m. That is shorter than 1 500 m. A bigger number does not mean a longer length when the units differ. Go to M3.6.1. Try another.

    See the answer

    1 500 m

    Change all three to metres. 1 500 m, then 1.2 × 1 000 = 1 200 m, then 140 000 ÷ 100 = 1 400 m. The longest is 1 500 m.

    • If you chose B: 1.2 km is 1.2 × 1 000 = 1 200 m. That is shorter than 1 500 m. Go to M3.6.1.
    • If you chose C: 140 000 ÷ 100 = 1 400 m. That is shorter than 1 500 m. A bigger number does not mean a longer length when the units differ. Go to M3.6.1.

  6. You change 5 m into centimetres. What happens to the number?

    The length ladder with km, m, cm and mm. The arrow from m down to cm is heavy and says × 100.kmmcmmm× 1 000÷ 1 000× 100÷ 100× 10÷ 10
    Tap your answer.

    1. Not this one. A small unit fits many times into 5 m, so you count more of them. 5 m is 500 cm. Go to M3.6.1. Try another.

    2. Yes, that is right. Going down the ladder the unit gets smaller, so the number gets bigger: 5 × 100 = 500 cm.

    3. Not this one. Both are lengths, but the units are different sizes. The number changes: 5 m is 500 cm. Go to M3.6.1. Try another.

    See the answer

    It gets bigger, because a cm is smaller than a m.

    Going down the ladder the unit gets smaller, so the number gets bigger: 5 × 100 = 500 cm.

    • If you chose A: A small unit fits many times into 5 m, so you count more of them. 5 m is 500 cm. Go to M3.6.1.
    • If you chose C: Both are lengths, but the units are different sizes. The number changes: 5 m is 500 cm. Go to M3.6.1.

  7. A girl is 1 m 62 cm tall. How tall is she in centimetres?

    See the answer

    162 cm

    1 m is 100 cm. So 1 m 62 cm is 100 + 62 = 162 cm.

  8. A wall is 2.8 km long. How long is it in metres?

    See the answer

    2 800 m

    1 km is 1 000 m. So 2.8 × 1 000 = 2 800 m.

Keep it

Copy this into your exercise book.

Down the ladder: multiply. Up the ladder: divide. 1 km = 1 000 m. 1 m = 100 cm. 1 cm = 10 mm. 1 t = 1 000 kg. 1 kg = 1 000 g.

Draw: The length ladder: km, m, cm, mm. Write × 1 000, × 100 and × 10 on the arrows going down. Write ÷ on the arrows going up.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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