1. 270,000,000 Polos are produced each week. Students could discuss how many would be produced in a year. How many packets does this represent? How many packets are produced per head of population in the UK? We also heard that the factory produces 3,700,000 Kit-Kats per day, and that the disc at the centre of a Smartie weighs 0.0005kg.

2. 55,000,000 is 5.5 x 10⁷ in standard form. The other examples given above were converted to 2.7 x 10⁸, 3.7 x 10⁶ and 5 x 10^-4.

3. A negative power means that the number is small. What situations in real life might give rise to very small numbers? Micronutrients are discussed in the programme; other examples would arise in the contexts of drug testing in athletics, water pollution, chemical and scientific processes, micro-organisms, and medical research.

4. 2.1 x 10¹⁰ Smarties are produced each year. The programme invites students to focus on the need, when converting to standard form, to use a front number that is at least 1 but less than 10. Students could discuss the reasons for this rule. They could increase their appreciation of the size of this number by considering questions like:

If a year's production of Smarties were laid in a line, how long would it be? Would it go round the equator (about 40,000km)?
How much would these Smarties weigh?
If these Smarties were piled one on top of another, how high would they reach?

5. 1g of chocolate contains 0.000007g of vitamin E. Katie works out how much there would be in 1kg. Most food packaging gives information on ingredients per 100g: this could be scaled up or down to investigate other examples.

6. (For light relief) the speed limit is 15 - presumably miles per hour.

7. Pioneer 10 passed Pluto in 1983. It is now (in the year 2000) 11 thousand million kilometres away. How far has it travelled in the intervening years, and what is its speed? Pluto is 5,900 million kilometres from the Sun, and Earth is 150 million kilometres from the Sun.

8. Light travels at 3 x 10⁸ metres per second. The example used the relation

(time) = ^(distance) / _(speed)

to calculate the time taken for light from Karen's galaxy to reach the Earth. Students could collect information to work out times from other stars, planets and so on, using the same method. A light year is the distance travelled by light in one year: it is 9.5 x 10¹⁵ metres.

9. The time taken was found to be 3 x 10¹⁴seconds. Students could discuss how to convert this to years, as Karen did in the programme.

10. Katie did not convert her final answer to standard form. This is a common mistake.

Worksheet 2: Tick or Trash

Question 1

Student B is correct.

Student A has added the indices for the powers of 10 instead of subtracting.

Students should note the impossibility of Student A's answer for 'calculations per second' being greater than the value for 'calculations per hour' given in the question.

Question 2

Student B is correct.

Student A has made a mistake in converting the final answer to standard form.

Students could discuss appropriate rounding of the result here. Neither student has made an error in this and either could reasonably be accepted. (The mark scheme gives 2.69 x 10^-3).

Worksheet 3: Exam Practise Questions (Edexcel)

This is a selection of past exam questions from Edexcel, targeted at the middle ability range. You can use the sequence as it stands or select individual questions to suit your needs.

Given below is the mark scheme for each question, which will help you to see how marks are allocated and what the examiner is especially looking out for. The mark schemes use the following abbreviations:
oe
or equivalent
cao
correct answer only
ft
follow-through marks
dep
dependent
indep
independent
M
method marks
A
accuracy marks
B
benefit-of-doubt mark
SC
special case

Question 1

(a) 1 ÷ 220 (or any power of 10) ÷ 220

4.5 (4.54...) x 10^?? or 4 x 10^-3 or 5 x 10^-3

Answer = 4.55 x 10^-3

(b) 6 ÷ 0.8 (= 7.5)

'ans.(a)' x 10⁶ or ¹ / ₂₂₀ x 6 x 10⁶

'6 ÷ 0.8' x 'ans.(a)' x 10⁶ = 34090.9...

Answer = 34100 cm³ or better

Notes:

(a) M1 (can be implied by the next A1). A1. A1 (accept rounded or truncated ie 4.55 or 4.54 or better).

(b) M1 (reverse %) (accept 7.5 seen). M1 (indep change m³ to cm³). A1 cao (accept 3.41 x 10⁴ or better).

Question 2

(a) 9.1 x 10^-25

(b) 9.1 x 10^-25 x 5000000 = 4.55 x 10^-18 g

Notes:

(a) M1 for 9.1 x 10^k. A1 k = -25.

SC: B1 for 9 x 10^-25.

(b) M1 '(a)' x 5000000. M1 '4.55' x 10ⁿ dep. A1 n = -18.

SC: B2 for a x 10^-18 where a = 4, 5, 4.5, 4.6.

Question 3

152 000 - 46 000 = 106 000

106 000 ÷ 0.0456 = 2.32 x 10⁶

Question 4

(a) 9.35 x 10⁷

(b) 250 000

Question 5

(a) 3.62 x 10⁸

(b) From 5.11 x 10⁸ to 5.12 x 10⁸

Question 6

3.27 x 10⁴

Question 7

(a) 10⁶⁰⁰ ÷ 10³⁰³ = 10^{600 - 303} = 1 x 10²⁹⁷ or 10²⁹⁷

(b) 40 x 10³⁰³ = 4 x 10³⁰⁴

Question 8

(a) 3 x 10^-8

(b) (a) ÷ 100 = 3 x 10^-10