Powers and roots 6.1. Previous learning. Objectives based on NC levels and (mainly level ) Lessons 1 Squares, cubes and roots.

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1 N 6.1 Powers and roots Previous learning Before they start, pupils should be able to: use index notation and the index laws for positive integer powers understand and use the order of operations, including brackets use the power and root keys of a calculator multiply and divide integers and decimals by any positive integer power of 10. Objectives based on NC levels and (mainly level ) In this unit, pupils learn to: and to: compare and evaluate representations of problems or situations use accurate notation calculate accurately, using mental methods or calculating devices as appropriate explore the effects of varying values examine, refine and justify arguments, conclusions and generalisations use a range of forms to communicate findings effectively to different audiences use index notation with negative powers, recognising that the index laws can be applied to these know that n 1/2 n and n 1/3 3 n for any positive number and estimate square roots and cube roots use standard index form, expressed in conventional notation and on a calculator display, and know how to enter numbers in standard index form convert between ordinary and standard index form representations. Lessons 1 Squares, cubes and roots About this unit Assessment 2 Equivalent calculations using powers of 10 3 Standard form A sound understanding of powers and roots of numbers helps pupils to generalise the principles in their work in algebra. This unit introduces pupils to expressing numbers in standard form. The ability to use numbers in standard form helps pupils to calculate and is essential for work in science. This unit includes: an optional mental test which could replace part of a lesson (p. 00); a self-assessment section (N6.1 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 (N6.1 Check up, CD-ROM). Common errors and misconception Look out for pupils who: think that n 2 means n 2, or that n means n ; 2 wrongly apply the laws of indices, e.g , or ; forget that any number raised to the power 0 equals 1; confuse the exponent and power keys on their calculators. 2 N6.1 Powers and roots

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, power, root, square, cube, square root, cube root, notation n 2 and n, n 3 and 3 n, index, indices, standard form scientific calculators for pupils individual whiteboards Tier 6 teacher s book N6.1 Mental test, p. 00 Answers for Unit N6.1, pp Tier 6 CD-ROM PowerPoint files N6.1 Slides for lessons 1 to 3 Excel file N6.1 CubeRoot Tools and prepared toolsheets Calculator tool Tier 6 programs Squaring quiz Powers of 10 quiz Equivalent calculations Standard form computers with spreadsheet software, e.g. Microsoft Excel, or graphics calculators Tier 6 class book N6.1, pp N6.1 How well are you doing? p. 00 Tier 6 home book N6.1, pp Tier 6 CD-ROM N6.1 Check up Hole in one uk.knowledgebox.com/index.phtml?d=23364 Topic B: Indices: simplifying Semi-detached, Sissa s reward N6.1 Powers and roots

3 1 Squares, cubes and roots Learning points n is the square root of n. The square root can be positive or negative. 3 n is the cube root of n. The cube root of a positive number is positive, and of a negative number is negative. You can estimate square roots and cube roots using trial and improvement. QZ Starter You may want this lesson to take place in a computer room, with enough computers for one between two pupils, so that pupils can use a spreadsheet. Alternatively, they could use graphics calculators. Use slide 1.1 to discuss the objectives for the unit. This lesson is about estimating the value of square and cube roots. Launch Squaring quiz. Ask pupils to answer questions on their whiteboards. Use Next and Back to move through the questions at a suitable pace. Main activity Explain that the square root of a can be positive or negative and is written as a or a 1/2. The cube root of a is 3 a or a 1/3. The cube root of a positive number is positive, and of a negative number is negative. The fourth root of a is 4 a or a1/4, the fifth root is 5 a or a 1/5, and so on. Discuss how to estimate the cube root of a number that is not a perfect cube. For example, 3 70 must lie between 3 64 and 3 125, so Since 70 is closer to 64 than to 125, we expect 3 70 to be closer to 4 than to 5, perhaps about TO Use the Calculator tool to remind pupils 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. Show that the calculator gives a value of for the cube root of 70. Show how to find 3 15 using trial and improvement. Establish first that it must lie between 2 and 3, because and Try Try Try Try Try too high too low very close but too high too low very close but too low The answer lies between and But numbers between and 2.47 all round up to So 3 15 is 2.47 correct to two decimal places. Ask pupils in pairs to use trial and improvement to find 3 50 to two decimal places [answer: 3.68]. Establish first that it must lie between 3 and 4. N6.1 Powers and roots

4 Use the Excel file N6.1 CubeRoot to show how to use a spreadsheet for this activity. Estimate other cube roots by overtyping 4 and 4.6. If possible, pupils should develop similar spreadsheets, using either a computer or a graphics calculator. XL Select individual work from N6.1 Exercise 1 in the class book (p. 00). Review Discuss a problem for pupils to work on in pairs. The square of 55 is When you add the two parts 30 and 25 together the sum is 55, the number itself. Find another two-digit number with this property. [Solution: 45 or 99] Ask pupils to remember the points on slide 1.2. Homework Ask pupils to do N6.1 Task 1 in the home book (p. 00). N6.1 Powers and roots

5 2 Equivalent calculations using powers of 10 Learning points Use facts that you know to work out new facts. If you multiply the numerator or divide the denominator of a fraction by a power of 10, the answer is multiplied by the same power of 10. If you divide the numerator or multiply the denominator of a fraction by a power of 10, the answer is divided by the same power of 10. QZ Starter Tell pupils that in this lesson they will be multiplying and dividing by powers of 10 and using the facts that they know to work out new facts. Write on the board a number such as Ask pupils to explain how to multiply it by 10, 100 or 1000, and then divide it by the same numbers. Launch Powers of 10 quiz. Ask pupils to answer on their whiteboards. Use Next and Back to move through the questions at a suitable pace. From time to time, ask a pupil to explain how they worked out the answer. Main activity Show slide 2.1. Which, if any, of the calculations are equivalent? Explain why. Show that Show that SIM Repeat with slide 2.2. Show that and that Launch Equivalent calculations. There are four options for the type of calculation to be displayed: a b, a b, a b c or (a b) c. To start with, the type is a b. The range and decimal places in each value in the starting calculation can be specified using the drop-down menus. A calculation with its answer is shown, with four related randomly generated questions created by multiplying or dividing numbers in the calculation by 10, 100 or What happens to the answer when we multiply/divide one of the numbers by 10, 100, 1000? How can you work out the solutions to the four questions using the given fact? N6.1 Powers and roots

6 Click on the answer boxes to reveal the answers. Click New question to generate a new calculation and four related questions. Repeat for a variety of different calculations. Select individual work from N6.1 Exercise 2 in the class book (p. 00). Review Using examples, discuss what happens to the value of a fraction if: the numerator is multiplied by 100; the denominator is multiplied by 1000; the numerator is divided by 10; the denominator is divided by 100. Sum up using the points on slide 2.3. Homework Ask pupils to do N6.1 Task 2 in the home book (p. 00). N6.1 Powers and roots

7 3 Standard form Learning points 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. To raise the power of a number to a power, multiply the indices, so (a m ) n a m n. A number in standard form is of the form A 10 n, where 1 A 10 and n is an integer. Starter Tell pupils that in this lesson they will be using the index laws and expressing very large and very small numbers in a special way called standard form. Recap the rules for multiplying and dividing powers: to multiply powers of a number, add the indices, so a m a n a m1n ; to divide powers of a number, subtract the indices, so a m a n a m n. Show the target board on slide 3.1. Point to two numbers and ask a pupil to multiply or divide them. Make sure that the class know which way round to do the division. Repeat several times. Discuss what happens when the power of a number is raised to a power, e.g. two squared all cubed. Show that: (2 2 ) Repeat with other examples, and draw out the generalisation: to raise the power of a number to a power, multiply the indices, so (a m ) n a m n. Refer again to the target board on slide 3.1. This time point to a number and ask a pupil to square it, cube it, or raise it to the power 4. Repeat several times. Select individual work from N6.1 Exercise 3A in the class book (p. 00). SIM Main activity Explain that a way is needed to express and use very large and very small numbers without writing out many zeros. For example can be written as , and can be written as Demonstrate how to convert ordinary numbers to standard form, e.g What is the connection between the original number and the power of 10? [The power is the number of places that the digits move.] Explain the definition of a standard form number: A 10 n, where 1 A 10 and n is an integer. Now show how to convert standard form numbers to ordinary numbers, e.g Launch Standard form. Click Play to start, and Pause to stop. Control the animation by dragging the playback control. N6.1 Powers and roots

8 Stress the connection between multiplying, moving digits and standard form. (In Examples 3 and 4, while all the expressions are equal, only the final expression is in standard form.) Select individual work from N6.1 Exercise 3B in the class book (p. 00). Review Explain how to enter a number in standard form into your pupils calculators. The first way uses the EXP key. For example, is entered by pressing: 2 4 EXP 5 There are other variations of the EXP key, such as x10x Possible displays, depending on the type and age of the calculator, are 2.4E5, , or The second way uses the key is entered by pressing: With either method, negative powers need either the negative key or the sign change key 1/. Pressing after the entry of a standard form number converts it to an ordinary number. If it matches your pupils calculators, you could use the Calculator tool to show pupils how to express numbers in standard form. Write a couple of standard form numbers on the board for pupils to enter into their calculators and convert to ordinary numbers. Then write a couple of numbers on the board in the form of a calculator display of a standard form number, and ask pupils to write them first as standard form numbers, and then as ordinary numbers. Sum up the lesson by stressing the points on slide 3.2. TO Round off the unit by referring again to the objectives. Suggest that pupils find time to try the self-assessment problems in N6.1 How well are you doing? in the class book (p. 00). Homework Ask pupils to do N6.1 Task 3 in the home book (p. 00). N6.1 Powers and roots

9 N6.1 Mental test Read each question aloud twice. Allow a suitable pause for pupils to write answers. 1 What number is five cubed? 2003 KS3 2 Divide twenty-four by minus six KS3 3 To the nearest whole number, what is the square root of 2004 KS3 eighty-three point nine? 4 Estimate the value of nine point two multiplied by two point nine KS3 5 Round three point seven nine five to one decimal place KS3 6 Look at the calculation KS3 Write down an approximate answer. [Write on board: ] 7 m squared equals one hundred KS3 Write down the two possible values of m plus fifteen. 8 What is the square root of forty thousand? 2006 KS3 9 Work out the value of two to the power six divided by two squared KS3 10 Nine multiplied by nine has the same value as three to the power what? 2006 KS3 11 What is the square root of nine twenty-fifths? 2006 KS3 12 What would be the last digit of one hundred and thirty-three to the 2003 KS3 power four? Key: KS3 Key Stage 3 Mental test Questions 1 to 5 are at level 6. Questions 6 to 12 are at level 7. Answers and / N6.1 Powers and roots

10 N6.1 Check up and resource sheets Answer these questions by writing in your book. Powers and roots (no calculator) level 6 a Check up Put these values in order of size with the smallest first b Look at this information. 5 5 is 3125 N6.1 Check up [continued] level 7 Look at the diagram of a cuboid. xcm Not drawn accurately What is 5 7? level 7 a The table below shows values of x and y for the equation y x2 x 5. Copy and complete the table. x y b The value of y is 0 for a value of x between 1 and 2. Find the value of x, to one decimal place, that gives the value of y closest to 0. You may use trial and improvement. 3 Given that a , work out: 35 x y b The volume of the cuboid is 100 cm 3. What could the values of x and y be? Give two possible pairs of values level 8 a One of the numbers below has the same value as Which number is it? (3.6 10) a One of the numbers below has the same value as Which number is it? c ( ) ( ) can be written simply as Write these values as simply as possible: ( ) ( ) xcm ycm Powers and roots (calculator allowed) level 7 Some numbers are smaller than their squares. For example: Which numbers are equal to their squares? Pearson Education 2008 Tier 6 resource sheets N6.1 Powers and roots N6.1 Pearson Education 2008 Tier 6 resource sheets N6.1 Powers and roots N6.1 N6.1 Powers and roots 11

11 N6.1 Answers Class book Exercise 1 1 a x 4 b x 3 c x 5 d x 1 2 a 2.24 b 4.64 c 4.31 d a 3 b 4 c 8 d 10 4 a 3.42 b 4.40 c 8.19 d a 2.4 b 19.1 c cm 7 There are 8 different ways to write 150 as the sum of four square numbers: There are 3 different ways to write 150 as the sum of three square numbers: Exercise 2 1 a 363 b c 36.3 d a 48.8 b c d a b 230 c 56 d a 2.1 b 21 c 0.72 d a b 340 c d a 69.6 b 6.96 c 69.6 d a 9.6 b c 4.2 d 420 Exercise 3A 1 a c b 1 9 d 1 2 a 3 1 b 4 1 c 10 2 d 2 2 e 4 5 f 3 3 g 5 7 h a 2 8 b 5 4 c 10 6 d 1 4 a 2 1 b 5 c 4 2 d 1 5 a n 5 b n 3 c n 6 d n 2 Exercise 3B 1 a b c d e f g h a b c 7800 d e f g h N6.1 Powers and roots

12 3 a b c d Extension problem 5 When n is odd 3n 1 7n is divisible by 10. Powers of 3 (i.e. 3 1, 3 2, 3 3, 3 4 ) end in 3, 9, 7, 1 in a repeating cycle. Powers of 7 end in 7, 9, 3, 1 in a repeating cycle. When the odd powers of 3 and 7 are added the last digit is 0 and is therefore divisible by ( ) ( ) a b c d e f N6.1 How well are you doing? 1 a 4.36 b c 5.31 d a k 3, m 6 b For integer values of (x, y) choose any four from: (2, 6), (4, 3), (8, 2), ( 2, 6), ( 8, 2) 5 m 12, n Home book TASK 1 1 a 3.6 b , 17, 18, 26 and 27 TASK 2 1 a 1792 b 1792 c d a 0.32 b 0.32 c 32 d a b 320 c 26 d a 645 b 6.45 c 64.5 d 6450 TASK 3 1 a b c d e f g h a b c 9400 d e f g h a b c d CD-ROM CHECK UP 1 a 3 2, 2 4, 5 2, 3 3 b a b The value of x, to one decimal place, that gives the value of y closest to 0, is x a b and 1 x y Possible integer values of (x, y) are any two of: (1, 100), (2, 25), (5, 4), (10, 1). 6 a b c 6, N6.1 Powers and roots 13