n powers and roots - pearson · pdf fileuse squares, positive and negative square roots, ......

Powersandroots

Previous learningBefore they start, pupils should be able to:

recognise and use multiples, factors (divisors), common factor, highest common factor, lowest common multiple and primes

use squares, positive and negative square roots, cubes and cube roots, and index notation for small positive integer powers.

Objectives based on NC levels � and � (mainly level �)In this unit, pupils learn to:

represent problems and synthesise information in diff erent forms

use accurate notation

calculate accurately, selecting mental methods or a calculator as appropriate

estimate, approximate and check working

record methods, solutions and conclusions

make convincing arguments to justify generalisations or solutions

review and refi ne own fi ndings and approaches on the basis of discussions with others

and to:

use index notation for integer powers

know and use the index laws for multiplication and division of positive integer powers

extend mental methods of calculation with factors, powers and roots

use the power and root keys of a calculator

use ICT to estimate square roots and cube roots

use the prime factor decomposition of a number.

Lessons 1 Workingwithintegerpowersofnumbers

2 Estimatingsquareroots

3 Primefactordecomposition

About this unit Sound understanding of powers and roots of numbers helps pupils to generalise the principles in their work in algebra. It also helps pupils to be aware of the relationships between numbers and to know at a glance which properties they possess and which they do not.

Assessment This unit includes: an optional mental test which could replace part of a lesson (p. 00); a self-assessment section (N5.1 How well are you doing? class book p. 00); a set of questions to replace or supplement questions in the exercises

or homework tasks, or to use as an informal test (N5.1 Check up, CD-ROM).

Common errors and misconception

Look out for pupils who: think that n2 means n 2, or that √

__ n means n __ 2;

wrongly apply the index laws, e.g. 103 1 104 107, or 103 104 1012; think that 1 is a prime number; include 1 in the prime factor decomposition of a number; confuse the highest common factor (HCF) and lowest common

multiple (LCM); assume that the lowest common multiple of a and b is always a b.

N5.1

2 | N5.1 Powers and roots

Key terms and notation problem, solution, method, pattern, relationship, expression, solve, explain, systematic, trial and improvement

calculate, calculation, calculator, operation, multiply, divide, divisible, product, quotient positive, negative, integer

factor, factor pair, prime, prime factor decomposition, power, root, square, cube, square root, cube root, notation n2 and √

__ n , n3 and 3 √

__ n

Practical resources scientific calculators for pupilsindividual whiteboards

computers with spreadsheet software, e.g. Microsoft Excel, or graphics calculators

Exploring maths Tier 5 teacher’s book N5.1 Mental test, p. 00 Answers for Unit N5.1, pp. 00–00

Tier 5 CD-ROMPowerPoint files N5.1 Slides for lessons 1 to 3 Excel file N5.1 SquareRoot Tools and prepared toolsheets Calculator toolTier 5 programs Multiples and factors quiz HCF and LCM Ladder method

Tier 5 class book N5.1, pp. 00–00

Tier 5 home book N5.1, pp. 00–00

Tier 5 CD-ROM N5.1 Check up

Useful websites Topic B: Indices: simplifying www.mathsnet.net/algebra/index.html

Factor tree nlvm.usu.edu/en/nav/category_g_3_t_1.html

Grid game www.bbc.co.uk/education/mathsfile/gameswheel.html

N5.1 Powers and roots | �

Learning points

A number a raised to the power 4 is a4 or a a a a.

The number that expresses the power is its index, so 2, 5 and 7 are the indices of a2, a5 and a7.

To multiply two numbers in index form, add the indices, so am an am1n.

To divide two numbers in index form, subtract the indices, so am an amn.

1 Working with integer powers of numbers

Starter

� | N5.1 Powers and roots

Main activity

Use slide1.1 to discuss the objectives for the unit. This lesson is about finding positive and negative integer powers of numbers.

Remind pupils that when a number is multiplied by itself the product is called a power of that number. So a a, or a squared, is the second power of a, and is written as a2, a a a, or a cubed, is the third power of a and is written as a3, and so on. For a4 we say a to the power of 4, and similarly with higher powers.

The number that expresses the power is its index. So 5 and 7 are the indices of a5 and a7. When the index is 1, it is usually omitted: we write a, rather than a1.

Ask pupils to calculate mentally powers of positive and negative integers, e.g.

(8)2, 82, (2)5, 25, (3)4, 34, (5)3, 53

Record answers on the board, and ask pupils what they notice. Draw out that for negative numbers even powers are positive, and odd powers are negative.

What is (1)123? What is (1)124?

Extend to decimals, e.g. ask pupils to calculate mentally:

(0.1)3, (0.7)2, (0.2)4

Use the Calculatortool to show pupils how to use the xy keys of their calculators.

Use the key to explore raising a number to the power 0. Explain that this always has the answer 1.

Discuss negative indices. Ask pupils to consider this pattern.

How is each number found from the one above it?

What is the pattern of the indices?

What are the next few lines of the pattern?

Establish that:

0.1 1⁄10 101

0.01 1⁄100 102

0.001 1⁄1000 103.

Similarly 1⁄2 21, 1⁄4 22 and 1⁄8 23.

Show the table of powers of 2 on slide1.2. Ask pupils in pairs to make up and record some multiplications using the table. Ask questions to help pupils to discover for themselves the rules for calculations with indices.

TO

10 000 104

1000 103

100 102

10 101

1 100

Homework

Review


Select individual work from N5.1Exercise1 in the class book (p. 00).

Homework

What do you notice about the indices in these calculations?

What is a quick way of multiplying powers of 2? Why does it work?

What is this calculation in index form?

32 256 8192 [25 28 213]

4096 512 [212 29 23]

16 384 64 [214 26 28]

What do you notice about the indices in these calculations?

What is a quick way of dividing one power of 2 by another?Why does it work?

Repeat with the powers of 4 on slide1.3.

Now generalise. Write on the board m2 m3.

What will this simplify to? Explain why. [m2 m3 (m m) (m m m) m m m m m m5]

Stress that the indices have been added, so that:

m2 m3 m2 1 3 m5

Repeat with m5 m2. Stress that for division the indices are subtracted, so that:

m5 m2 m5 2 m3

Discuss negative indices, e.g.

m3 m7 m3 7 m4 = 1m4

m5 m3 m53 m2.

Show slide1.4.

Point to two different powers of 10. Ask pupils to multiply or divide them and to write the answer on their whiteboards.

Stress that the rules for multiplying and dividing numbers in index form apply to both positive and negative indices.

Ask pupils to remember the points on slide1.5.

Ask pupils to do N5.1Task1 in the home book (p. 00).

Learning points

√ __

n is the square root of n, e.g. √ ___

81 69

3 √

__ n is the cube root of n, e.g. 3 √

____ 125 5, 3 √

_____ 27 3.

Trial and improvement can be used to estimate square roots when a calculator is not available.

Starter

Main activity

Tell pupils that in this lesson they will be estimating the value of square roots.

Remind them that the square root of a is denoted by 2 √

__ a , or more simply as √

__ a ,

and that a square root of a positive number can be positive or negative, e.g. if a2 9, √

__ a 63.

Show the grid on slide2.1. Write on the board: x 1, z 4.

Point to an expression on the grid. Ask pupils to work out its value mentally and to write the answer on their whiteboards. Ask someone to explain how they calculated it. After a while, change the values for x and z to: x 9 and z 25.

Discuss how to estimate the positive square root of a number that is not a perfect square, e.g. √

___ 70 must lie between √

___ 64 and √

___ 81 , so 8 √

___ 70 9. Since 70 is closer

to 64 than to 81, we expect √ ___

70 to be closer to 8 than to 9, perhaps about 8.4.

Show the class how they could find √7 if they had only a basic calculator with no square-root key. Tell them that the process is called trial and improvement.

Explain that √7 must lie between 2 and 3, because 7 lies between 22 and 32.

Try 2.52 6.25. Too low

Try 2.62 = 6.76. Too low

Try 2.72 = 7.29. Too high.

Try 2.652 = 7.0225. Very close but a little bit too high

Try 2.642 = 6.9696. Very close but too low

Try 2.6452 = 6.986025 Still too low

The answer lies between 2.645 and 2.65. All numbers between 2.645 and 2.65 round up to 2.65. So √7 2.65 correct to two decimal places.

Ask pupils to work in pairs and, using only the × key on their calculator, to find √

___ 12 to two decimal places [answer: 3.46]. Establish first that it must lie between 3

and 4.


8

√70

9

√64 √81

2 Estimating square roots

Homework

XL



Review

Show the class how they could use a spreadsheet for this activity, without using the square-root function, e.g. use the Excel file N5.1SquareRoot.

Point out that the strategy here is different. We work systematically in tenths from 3 to 4, then in hundredths from 3.4 to 3.5, then in thousandths from 3.46 to 3.47.

You can use this file to estimate other square roots by overtyping 3 and 3.4. If possible, pupils should develop similar spreadsheets, using either a computer or a graphics calculator.

Introduce root notation. Explain that if 729 is the cube of 9, then 9 is the cube root of 729, which is written as 3

√ ____

729 9. The cube root, fourth root, fifth root, … of a are denoted by 3

√ __

a , 4 √

__ a , 5

√ __

a , …

Use the Calculatortool to demonstrate how to find a cube root. You may need to explain that some calculators have a cube root key 3√ . Others have a key like x√ , or other variations.

For example, to find the value of 3 √

____ 216 , key in 2 1 6 x√ 3 .

[Answer: 6]

Ask pupils to work out 3 √

___ 64 and 3

√125. Explain that the cube root of a positive number is positive, and the cube root of a negative number is negative.

Sum up the lesson by reminding pupils of the learning points.


TO

Starter

Main activity

Learning points

Writing a number as the product of its prime factors is called the prime factor decomposition of the number.

You can use a tree method or a ladder method to find a number’s prime factors.

To find the highest common factor (HCF) of a pair of numbers, find the product of all the prime factors common to both numbers.

To find the lowest common multiple (LCM) of a pair of numbers, find the smallest number that is a multiple of each of the numbers.

Tell pupils that in this lesson they will be finding the prime factors of numbers and using them to find common factors and multiples of a pair of numbers.

Remind them of the definitions of multiple, factor, factor pair and prime number.

Launch Multiplesandfactorsquiz. Ask pupils to answer on their whiteboards. Use ‘Next’ and ‘Back’ to move through the questions at a suitable pace.

Write on the board three products such as:

11 5 3 2 2 13 2 3 5 5

What do you notice about the numbers in these products?

Establish that they are all prime numbers. Explain that when a number is expressed as the product of its prime factors it is called the prime factor decomposition of a number. Stress that because 1 is not a prime number it is not included in the decomposition.

How can we find the prime factor decomposition of 80?

First explain the tree method, i.e. split 80 into a product such as 20 4, then continue factorising any number in the product that is not a prime. Repeat with 300.

LaunchLaddermethod. Use it to show the alternative method, where the number is repeatedly divided by any prime that will divide into it exactly. Demonstrate with 63, dragging numbers from the grid to the relevant positions. Continue to divide by prime numbers until the answer is 1. Express the answer as 63 3 3 7 32 7. Repeat with 245.

Show how to find the highest common factor (HCF) and lowest common multiple (LCM) of a pair of numbers. Launch HCFandLCM.

QZ


SIM

SIM

3 633 217 7

1

20

80

5 22

4

2

4

2

� Prime factor decomposition

10 1530

300

52

325

Homework

Review

Choose ‘lowest common multiple’. Select 8 and 6 using the arrows by the numbers. Drag multiples of 8 and multiples of 6 from the 100-square to the answer boxes. (Numbers snap back to the 100-square if dragged from answer box.) Numbers common to both boxes change colour to blue.

Which numbers are both multiples of 8 and multiples of 6?

Which is the lowest number that is both a multiple of 8 and 6?

Drag the HCF into the box below the 100-square.

Repeat several times with different numbers, then change to ‘highest common factor’, which works similarly. Select 24 and 18, then drag factors to the answer boxes.

Which numbers are both factors of 24 and factors of 18?

Which is the highest number that is both a factor of 24 and 18?

Repeat with different numbers.

Show how to use a Venn diagram to find the HCF and LCM of a pair of numbers such as 36 and 30. Explain that:

the overlapping prime factors give the HCF (2 3 2 3 6);

all the prime factors give the LCM (2 2 3 3 5 22 32 5 180).

Repeat with 18 and 24.

Sum up the lesson by stressing the points on slide3.1.

Round off the unit by referring again to the objectives. Suggest that pupils find time to try the self-assessment problems in N5.1Howwellareyoudoing? in the class book (p. 00).




3

2

3

5

2

36 30

Readeachquestionaloudtwice.

Allowasuitablepauseforpupilstowriteanswers.

1 What number is five to the power three?

2 Write all the prime factors of forty-two.

3 Write down a factor of thirty-six that is greater than ten and less than twenty. 2005 KS3

4 What is the next number in the sequence of square numbers? 2004 KS3 One, four, nine, sixteen, ...

5 Look at the numbers. Write down each number that is a factor of one hundred. 1999 Y7 [Write on board 10 15 20 25 30 35 40 45 50]

6 Write two factors of twenty-four which add to make eleven. 2005 KS2

7 What is the square root of eighty-one? 2001 KS3

8 What number is five cubed? 2003 KS3

9 The volume of a cube is sixty-four centimetres cubed. 2002 KS3 What is the length of an edge of the cube?

10 What is the square of three thousand? 2001 KS3

11 To the nearest whole number, what is the square root of 2004 KS3 eighty-three point nine?

12 I think of a number. I square my number and get the answer 2007 KS3 one thousand six hundred. What could my number be?

Key:

KS3 Key Stage 3 test KS2 Key Stage2 test Y7 Year 7 optional test (1999)Questions 3 to 7 are at level 5; 8 to 11 are at level 6; 12 is at level 7

Answers 1 125 2 2, 3, 7

3 12 or 18 4 25

5 10, 20, 25, 50 6 3 and 8

7 9 8 125

9 4 cm 10 9 000 000

11 9 12 40

N�.1 Mental test


N�.1 Check up and resource sheets

Answer these questions by writing in your book.

Powers and roots (no calculator)

4 2001 level 6

Which two of the numbers below are not square numbers?

2 David says that 211 � 2048.

What is 210?

3 To the nearest whole number, what is the square root of 93.7?

4 If √ ___

81 � n � √ ____

144, then n could be which of the following numbers?

9 11 12 13

5 Year 8 optional test level 6

Terry has 24 centimetre cubes.He uses them to make a cuboid that is one cube high.

Tina has 24 centimetre cubes.She uses them to make a solid cuboid that is two cubes high.

What could the dimensions of her cuboid be?

6 What is the biggest number that is a factor of both 105 and 135?

7 What is the smallest number that is a multiple of both 12 and 27?

Powers and roots (calculator allowed)

8 1996 level 6

Mary thinks of a number.

Which number did Mary think of?

Check up

Pearson Education 2008

N5.1

Tier 5 resource sheets | N5.1 Powers and roots | N5.1

24 25 26 27 28

1 cm high

4 cm wide

6 cm long

First I subtract 3.76Then I � nd the square

root of what I get My answer is 6.80

N5.1 Powers and roots | 11

N�.1 Answers


Class book

Exercise 11 a 23 b 45 c 38

d (1)4 e 52 f 61

2 a 64 b 243 c 256

d 128 e 1 f 1

g 116

h 1125

3 a 2401 b 15 625 c 1331

d 19 683 e 1024 f 11.39

g 32 157.43 h 11 272.96

4 a 28 b 35 c 104

d a8 e 54 f 124

g 80 h b3

5

11 6 21

31 2 1

5 42 7

6 a 19 32 1 32 1 12 b 41 62 1 22 1 12

c 50 52 1 42 1 32 d 65 62 1 52 1 22

e 75 72 1 52 1 12 f 94 72 1 62 1 32

7 Rachel and Hannah are 14 and 11 years old.

Extension problem8 The smallest whole numbers are 6 and 10:

102 62 100 36 64 43

103 63 1000 216 784 282

Exercise 21 a x 63 b x 67

c x 612 d x 61

2 a 61.41 to 2 d.p. b 2.15 to 2 d.p.

c 4 d 60.2

e 5 f 61.22 to 2 d.p.

g 61.73 to 2 d.p. h 1

3 a 2 b 7 c 11 d 8

4 Each answer is correct to 1 d.p.

a 2.4 b 6.7 c 10.7 d 8.4

5 Between 700 and 750 slabs will be used. 26 26 is 678, which is too few, and 28 28 is 784, which is too many. So the exact number of slabs is 27 27 729.

6 a a 9.7 to 1 d.p.

b a 12.3 to 1 d.p.

c a 20.4 to 1 d.p.

7 3.87 metres to 2 d.p.

Extension problem8 2982 = 88804

888 is not a perfect square. There is no whole number between the square root of 8880 and the square root of 8889 but 298 lies between the √

______

88800 and √ ______

88800.

Exercise 31 a 3 22 b 3 5 c 3 7

d 23 3 e 33 f 2 33

2 a 1, 2, 5, 10, 25, 50 b 2 × 52

3 a 1, 3, 5, 9, 15, 45 b 5 32

4 e.g. 63 (with factors 1, 3, 7, 9, 21, 63)

5 a 72 and 30: HCF 6, LCM 360

b 50 and 80: HCF 10, LCM 400

c 48 and 84: HCF 12, LCM 336

6 HCF 15, LCM 600

7 HCF 10, LCM 360

22

5

35

2

120 75

2 2

5 3

3

2

40 90

N5.1 Powers and roots | 1�

8 a 2 and 3 b 6 c 378

9 a 28 and 40: HCF 4, LCM 280

b 200 and 175: HCF 25, LCM 1400

c 36 and 64: HCF 4, LCM 576

10 1050

Extension problem11 a 22 4 (with factors 1, 2 and 4)

b 24 16 (with factors 1, 2, 4, 8, 16)

c 7 factors: 26 64

9 factors: 28 256

11 factors: 210 1024

13 factors: 212 4096

12 420 days from now, since 420 is the lowest common multiple of 1, 2, 3, 4, 5, 6 and 7.

N5.1 How well are you doing?1 a 32, 24, 52, 33 or 9, 16, 25, 27

b 57 55 × 52 3125 × 25 78 125

2 a 34 81 is the largest. 34 92

b 25 and 27 are not square numbers.

3 a 32 9 b 27 128

c 32 2 = 18

4 a a 3 b b 2

5 a HCF is 12 b LCM is 144

6 Suzy’s number is 4.9.

7 5 × 11 × 19 1045

Home book

TASK 11 a 311 b 26 c 111 d x6

e 43 f 104 g 810 h z

2 a 28 33 + 13

b 72 43 + 23

c 1125 103 + 53

3 a (15)2 225 (25)2 625

b (11)2 121 (19)2 361

(21)2 441 (29)2 841

(31)2 961

c (6)3 216

d (13)3 2197

TASK 21 a 19 b 9

c 9 d 5

e 24 f 2

g 8.67 to 2 d.p. h 4.24 to 2 d.p.

2 18.8 cm to 1 d.p.

3 a 3 b 4 c 6 d 10

TASK 31 a 23 3 7 b 35

2 a 2 32 52 b 5 7 17

3 45

4 936

5 7, 13 and 17

CD-ROM

CHECK UP1 25 and 27 are not square numbers.

2 2048 2 1024

3 10

4 B: 11

5 2 cm high by 1 cm wide by 12 cm long2 cm high by 2 cm wide by 6 cm long 2 cm high by 3 cm wide by 4 cm long

6 15

7 108

8 50

n powers and roots - pearson · pdf fileuse squares, positive and negative square roots, ......

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