KS3 Maths · Year 8 · Chapter 9: Decimal numbers · Lesson 2

Standard form

Year 8About 60 minutesNo calculator48 marks

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Quick questions to start.

A1
Needed today
Write $5.06 \times 1 0^{4}$ as an ordinary number.
[1 mark]
Answer
A2
Last lesson
Work out $0.06 \times 1 0^{- 2}$ .
[1 mark]
Answer
A3
From chapter 8
Expand and simplify $(x + 4) (x - 6)$ .
[1 mark]
Answer
A4
Year 7
Round $0.04637$ to $2$ significant figures.
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Write each number in standard form.

(a) $0.0072$

(b) $0.000305$

Hint Move the digits until the first non-zero digit is in the units column. The number of places is the power; moving the digits to the left gives a negative power.

B1
Write each number in standard form.
(a) $0.064$
(b) $0.0000091$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 2

Write each number as an ordinary number.

(a) $3.8 \times 1 0^{- 4}$

(b) $6.05 \times 1 0^{5}$

Hint A negative power makes the number smaller: there are

4

zeros in front of the

3

, counting the one before the decimal point.

B2
Write each number as an ordinary number.
(a) $9.17 \times 1 0^{- 3}$
(b) $2.4 \times 1 0^{7}$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 3

These numbers are not in standard form. Write each one in standard form.

(a) $27 \times 1 0^{3}$

(b) $0.63 \times 1 0^{- 2}$

Hint Write the first number in standard form, then put the two powers of

10

together.

B3
These numbers are not in standard form. Write each one in standard form.
(a) $840 \times 1 0^{- 6}$
(b) $0.059 \times 1 0^{5}$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 4

Work out each answer. Give your answers in standard form.

(a) $3.2 \times 1 0^{4} + 6.5 \times 1 0^{3}$

(b) $4.1 \times 1 0^{- 3} - 7 \times 1 0^{- 4}$

Hint The powers are different, so you cannot just add the front numbers. Write both as ordinary numbers first.

B4
Work out each answer. Give your answers in standard form.
(a) $5.7 \times 1 0^{5} + 8.4 \times 1 0^{4}$
(b) $2.3 \times 1 0^{- 2} - 6 \times 1 0^{- 3}$
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 5

One speck of dust has a mass of $7 \times 1 0^{- 10}$ grams. Work out the mass of $3000000$ specks of dust. Give your answer in standard form.

Hint Write

3000000

3 \times 1 0^{6}

. Multiply the front numbers, then the powers of

10

B5
One sheet of gold leaf is $1.2 \times 1 0^{- 4}$ mm thick. A pile has $5000$ sheets. Work out the height of the pile, in mm. Give your answer in standard form.
[3 marks]
Answer

Practice

Year 8

C1
Work out each answer. Give your answers in standard form.
(a) $0.00 3^{2}$
(b) $0.01 2^{2}$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$
C2
Write these numbers in order of size. Start with the smallest.
$4.5 \times 1 0^{- 3}$ , $3.9 \times 1 0^{- 2}$ , $6 \times 1 0^{- 4}$ , $0.0052$
[2 marks]
Answer

C3
Which of these numbers are written in standard form?
P: $3.7 \times 1 0^{- 5}$ Q: $37 \times 1 0^{- 6}$
R: $0.37 \times 1 0^{- 4}$ S: $1 \times 1 0^{8}$
[2 marks]
Answer
C4
How many times bigger is $6 \times 1 0^{- 2}$ than $3 \times 1 0^{- 5}$ ?
[2 marks]
Answer
C5
Write each number in standard form.
(a) $25$ million
(b) three thousandths
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Exam-style questions

GCSE Foundation

D1
Answer each part.
(a)
Write $6.3 \times 1 0^{- 4}$ as an ordinary number.
[1 mark]
Answer
(b)
Write $0.0000802$ in standard form.
[1 mark]
Answer
(c)
Write $58$ million in standard form.
[1 mark]
Answer
Total for D1: 3 marks

D2
Mia says, " $34 \times 1 0^{- 3}$ is in standard form, because it is a number times a power of $10$ ."
Explain why Mia is wrong. Write $34 \times 1 0^{- 3}$ in standard form.
[2 marks]
D3
A red blood cell is $0.000008$ m wide. A human hair is $7.2 \times 1 0^{- 5}$ m wide.
(a)
Write $0.000008$ in standard form.
[1 mark]
Answer
(b)
How many times wider is the hair than the red blood cell?
[2 marks]
Answer
(c)
Work out the difference between the two widths. Give your answer in standard form.
[2 marks]
Answer
Total for D3: 5 marks
D4
The table shows the masses of four grains of sand.
(a)
Write the grains in order of mass. Start with the lightest.
[2 marks]
Answer
(b)
Work out the total mass of grains A and B. Give your answer in standard form.
[2 marks]
Answer
Total for D4: 4 marks
D5
A dust mite has a mass of $4 \times 1 0^{- 6}$ grams. Work out how many dust mites have a total mass of $1$ gram. Give your answer in standard form.
[3 marks]
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Two numbers add up to $6.1 \times 1 0^{- 3}$ . One of the numbers is $4$ times the other. Work out the two numbers. Give your answers in standard form.
[4 marks]
Hint 1 · What to try
Call the smaller number $n$ . Then the other is $4 n$ , and together they make $5 n$ .
Hint 2 · The first line
$5 n = 0.0061$ , so $n = 0.0061 \div 5$ .
Hint 3 · The full method
$n = 0.00122$ . Now work out $4 n$ , then write both numbers in standard form.
Answer $\dots\dots$ and $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write small and large numbers in standard form and change them back.
I can put a number such as $27 \times 1 0^{3}$ into standard form.
I can add, subtract and compare numbers in standard form.

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