# Powers and roots 5.1. Previous learning. Objectives based on NC levels and (mainly level ) Lessons 1 Working with integer powers of numbers

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1 N 5.1 Powers and roots Previous learning Before 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 different 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 refine own findings 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 Working with integer powers of numbers About this unit Assessment Estimating square roots 3 Prime factor decomposition 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. 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 N5.1 Powers and roots Look out for pupils who: think that n means n, or that n means n ; wrongly apply the index laws, e.g , or ; 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.

2 Key terms and notation Practical resources Exploring maths Useful websites 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 n and n, n 3 and 3 n scientific calculators for pupils individual whiteboards Tier 5 teacher s book N5.1 Mental test, p. 00 Answers for Unit N5.1, pp Tier 5 CD-ROM PowerPoint files N5.1 Slides for lessons 1 to 3 Excel file N5.1 SquareRoot Tools and prepared toolsheets Calculator tool Tier 5 programs Multiples and factors quiz HCF and LCM Ladder method computers with spreadsheet software, e.g. Microsoft Excel, or graphics calculators Tier 5 class book N5.1, pp Tier 5 home book N5.1, pp Tier 5 CD-ROM N5.1 Check up Topic B: Indices: simplifying Factor tree nlvm.usu.edu/en/nav/category_g_3_t_1.html Grid game N5.1 Powers and roots

3 1 Working with integer powers of numbers Learning points A number a raised to the power 4 is a 4 or a a a a. The number that expresses the power is its index, so, 5 and 7 are the indices of a, a 5 and a 7. To multiply two numbers in index form, add the indices, so a m a n a m1n. To divide two numbers in index form, subtract the indices, so a m a n a m n. Starter Use slide 1.1 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 a, a a a, or a cubed, is the third power of a and is written as a 3, and so on. For a 4 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 a 5 and a 7. When the index is 1, it is usually omitted: we write a, rather than a 1. Ask pupils to calculate mentally powers of positive and negative integers, e.g. ( 8), 8, ( ) 5, 5, ( 3) 4, 3 4, ( 5) 3, 5 3 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) 13? What is ( 1) 14? Extend to decimals, e.g. ask pupils to calculate mentally: (0.1) 3, (0.7), ( 0.) 4 TO Main activity Use the Calculator tool to show pupils how to use the x y 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: Similarly 1 1, 1 4 and Show the table of powers of on slide 1.. 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. N5.1 Powers and roots

4 What do you notice about the indices in these calculations? What is a quick way of multiplying powers of? Why does it work? What is this calculation in index form? [ ] [ ] [ ] What do you notice about the indices in these calculations? What is a quick way of dividing one power of by another? Why does it work? Repeat with the powers of 4 on slide 1.3. Now generalise. Write on the board m m 3. What will this simplify to? Explain why. [m m 3 (m m) (m m m) m m m m m m 5 ] Stress that the indices have been added, so that: m m 3 m 1 3 m 5 Repeat with m 5 m. Stress that for division the indices are subtracted, so that: m 5 m m 5 m 3 Discuss negative indices, e.g. m 3 m 7 m 3 7 m 4 = 1 m 4 m 5 m 3 m 5 3 m. Select individual work from N5.1 Exercise 1 in the class book (p. 00). Review Homework Show slide 1.4. 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 slide 1.5. Homework Ask pupils to do N5.1 Task 1 in the home book (p. 00). N5.1 Powers and roots

5 Estimating square roots Learning points n is the square root of n, e.g n is the cube root of n, e.g , Trial and improvement can be used to estimate square roots when a calculator is not available. Starter 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 a, or more simply as a, and that a square root of a positive number can be positive or negative, e.g. if a 5 9, a Show the grid on slide.1. Write on the board: x 51, z 5 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 5 9 and z 5 5. Main activity 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 Since 70 is closer to 64 than to 81, we expect 70 to be closer to 8 than to 9, perhaps about 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 and 3, because 7 lies between and 3. Try Try.6 = Try.7 = 7.9. Try.65 = Try.64 = Try.645 = Too low Too low Too high. Very close but a little bit too high Very close but too low Still too low The answer lies between.645 and.65. All numbers between.645 and.65 round up to.65. So correct to two decimal places. Ask pupils to work in pairs and, using only the key on their calculator, to find 1 to two decimal places [answer: 3.46]. Establish first that it must lie between 3 and 4. N5.1 Powers and roots

6 Show the class how they could use a spreadsheet for this activity, without using the square-root function, e.g. use the Excel file N5.1 SquareRoot. 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 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. XL Select individual work from N5.1 Exercise in the class book (p. 00). Review Introduce root notation. Explain that if 79 is the cube of 9, then 9 is the cube root of 79, which is written as 3 _ The cube root, fourth root, fifth root, of a are denoted by 3 a, 4 a, 5 a, Use the Calculator tool 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 _ 16, key in 1 6 x 3. [Answer: 6] Ask pupils to work out 3 64 and Explain that the cube root of a positive number is positive, and the cube root of a negative number is negative. TO Sum up the lesson by reminding pupils of the learning points. Homework Ask pupils to do N5.1 Task in the home book (p. 00). N5.1 Powers and roots

7 3 Prime factor decomposition 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. QZ Starter 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 Multiples and factors quiz. Ask pupils to answer on their whiteboards. Use Next and Back to move through the questions at a suitable pace. Main activity Write on the board three products such as: 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 0 3 4, then continue factorising any number in the product that is not a prime. Repeat with SIM Launch Ladder method. 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 Repeat with SIM Show how to find the highest common factor (HCF) and lowest common multiple (LCM) of a pair of numbers. Launch HCF and LCM. N5.1 Powers and roots

8 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 4 and 18, then drag factors to the answer boxes. Which numbers are both factors of 4 and factors of 18? Which is the highest number that is both a factor of 4 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 ( ); all the prime factors give the LCM ( ). Repeat with 18 and Select individual work from N5.1 Exercise 3 in the class book (p. 00). Review Sum up the lesson by stressing the points on slide 3.1. Round off the unit by referring again to the objectives. Suggest that pupils find time to try the self-assessment problems in N5.1 How well are you doing? in the class book (p. 00). Homework Ask pupils to do N5.1 Task 3 in the home book (p. 00). N5.1 Powers and roots

9 N5.1 Mental test Read each question aloud twice. Allow a suitable pause for pupils to write answers. 1 What number is five to the power three? 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. 005 KS3 4 What is the next number in the sequence of square numbers? 004 KS3 One, four, nine, sixteen,... 5 Look at the numbers. Write down each number that is a factor of one hundred Y7 [Write on board ] 6 Write two factors of twenty-four which add to make eleven. 005 KS 7 What is the square root of eighty-one? 001 KS3 8 What number is five cubed? 003 KS3 9 The volume of a cube is sixty-four centimetres cubed. 00 KS3 What is the length of an edge of the cube? 10 What is the square of three thousand? 001 KS3 11 To the nearest whole number, what is the square root of 004 KS3 eighty-three point nine? 1 I think of a number. I square my number and get the answer 007 KS3 one thousand six hundred. What could my number be? Key: KS3 Key Stage 3 test KS Key Stage test Y7 Year 7 optional test (1999) Questions 3 to 7 are at level 5; 8 to 11 are at level 6; 1 is at level 7 Answers 1 15, 3, or , 0, 5, and cm N5.1 Powers and roots

10 N5.1 Check up and resource sheets Check up Answer these questions by writing in your book. Powers and roots (no calculator) N level 6 Which two of the numbers below are not square numbers? David says that What is 10? 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? 5 Year 8 optional test level 6 Terry has 4 centimetre cubes. He uses them to make a cuboid that is one cube high. Tina has 4 centimetre cubes. She uses them to make a solid cuboid that is two cubes high What could the dimensions of her cuboid be? 1 cm high 4 cm wide 6 cm long 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 1 and 7? Powers and roots (calculator allowed) level 6 Mary thinks of a number. First I subtract 3.76 Then I find the square root of what I get My answer is 6.80 Which number did Mary think of? Pearson Education 008 Tier 5 resource sheets N5.1 Powers and roots N5.1 N5.1 Powers and roots 11

11 N5.1 Answers Class book Exercise 1 1 a 3 b 4 5 c 3 8 d ( 1) 4 e 5 f 6 1 a 64 b 43 c 56 d 18 e 1 f 1 g 1 h a 401 b c 1331 d e 104 f g h a 8 b 3 5 c d a 8 e 5 4 f 1 4 g 8 0 h b a b c d e f Rachel and Hannah are 14 and 11 years old. Extension problem 8 The smallest whole numbers are 6 and 10: Exercise 1 a x 3 b x 7 c x 1 d x 1 a 1.41 to d.p. b.15 to d.p. c 4 d 0. e 5 f 1. to d.p. g 1.73 to d.p. h 1 3 a b 7 c 11 d 8 4 Each answer is correct to 1 d.p. a.4 b 6.7 c 10.7 d Between 700 and 750 slabs will be used. 6 6 is 678, which is too few, and 8 8 is 784, which is too many. So the exact number of slabs is a a 9.7 to 1 d.p. b a 1.3 to 1 d.p. c a 0.4 to 1 d.p metres to d.p. Extension problem 8 98 = is not a perfect square. There is no whole number between the square root of 8880 and the square root of 8889 but 98 lies between the and Exercise 3 1 a 3 b 3 5 c 3 7 d 3 3 e 3 3 f 3 3 a 1,, 5, 10, 5, 50 b 5 3 a 1, 3, 5, 9, 15, 45 b e.g. 63 (with factors 1, 3, 7, 9, 1, 63) 5 a 7 and 30: HCF 6, LCM 360 b 50 and 80: HCF 10, LCM 400 c 48 and 84: HCF 1, LCM HCF 15, LCM HCF 10, LCM N5.1 Powers and roots

12 8 a and 3 b 6 c a 8 and 40: HCF 4, LCM 80 b 00 and 175: HCF 5, LCM 1400 c 36 and 64: HCF 4, LCM Extension problem 11 a 4 (with factors 1, and 4) b 4 16 (with factors 1,, 4, 8, 16) c 7 factors: factors: factors: factors: days from now, since 40 is the lowest common multiple of 1,, 3, 4, 5, 6 and 7. N5.1 How well are you doing? 1 a 3, 4, 5, 3 3 or 9, 16, 5, 7 b a is the largest b 5 and 7 are not square numbers. 3 a 3 9 b 7 18 c 3 = 18 4 a a 3 b b 5 a HCF is 1 b LCM is Suzy s number is Home book TASK 1 1 a 3 11 b 6 c 11 1 d x 6 e 4 3 f 10 4 g 8 10 h z a b c a (15) 5 (5) 65 b (11) 11 (19) 361 (1) 441 (9) 841 (31) 961 c (6) 3 16 d (13) TASK 1 a 19 b 9 c 9 d 5 e 4 f g 8.67 to d.p. h 4.4 to d.p cm to 1 d.p. 3 a 3 b 4 c 6 d 10 TASK 3 1 a b 3 5 a 3 5 b , 13 and 17 CD-ROM CHECK UP 1 5 and 7 are not square numbers B: 11 5 cm high by 1 cm wide by 1 cm long cm high by cm wide by 6 cm long cm high by 3 cm wide by 4 cm long N5.1 Powers and roots 13

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