Properties of numbers

Transcription

1 N 4.1 Properties of numbers Previous learning Before they start, pupils should be able to: order, add and subtract positive and negative integers in context use simple tests of divisibility recognise square numbers to and the corresponding roots use the bracket keys and memory of a calculator. Objectives based on NC levels 5 and 6 (mainly level 5) In this unit, pupils learn to: identify the mathematical features of a context or problem try out and compare mathematical representations conjecture and generalise, identifying exceptional cases calculate accurately, selecting mental methods or a calculator as appropriate use accurate notation refine own findings and approaches on the basis of discussion with others record methods, solutions and conclusions and to: add, subtract, multiply and divide integers use the order of operations, including brackets, with more complex calculations use multiples, factors, common factors, highest common factor, lowest common multiple and primes find the prime factorisation of a number (e.g ) use squares, positive and negative square roots, cubes and cube roots, and index notation for small positive integer powers strengthen and extend mental methods of calculation use the function keys of a calculator for sign change, brackets, powers and roots, and interpret the display in context. Lessons 1 Order of operations 2 Adding and subtracting directed numbers About this unit Assessment 3 Multiplying and dividing directed numbers 4 Powers and roots 5 Multiples, factors and primes A good feel for number means that pupils are aware of relationships between numbers and know at a glance which properties they possess and which they do not. A sound understanding of the order of operations and powers and roots of numbers helps them to generalise the principles in their later work in algebra. This unit includes: an optional mental test that could replace part of a lesson (p. 14); a self-assessment section (N4.1 How well are you doing? class book p. 14); a set of questions to replace or supplement questions in the exercises or homework tasks, or to use as an informal test (N4.1 Check up, CD-ROM). 2 N4.1 Properties of numbers

2 Common errors and misconceptions Key terms and notation Look out for pupils who: disregard brackets, e.g. 7 (4 1) 7 4 1; wrongly apply the order of operations, including when using a calculator; confuse positive and negative integers with addition and subtraction operations, e.g. (3) (2) 5, (8) (6) 2; 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; think that n 2 means n 2, or that n means n. 2 problem, solution, method, pattern, relationship, expression, order, solve, explain, systematic calculate, calculation, calculator, operation, add, subtract, multiply, divide, divisible, sum, total, difference, product, quotient greater than (), less than (), value positive, negative, integer, odd, even, multiple, factor, prime, power, square, cube, root, square root, cube root, digit sum, notation n 2 and n, n 3 and 3 n Practical resources scientific calculators for pupils individual whiteboards Exploring maths Useful websites Tier 4 teacher s book N4.1 Mental test, p. 14 Answers for N4.1, pp Tier 4 CD-ROM PowerPoint files N4.1 Slides for lessons 1 to 5 Prepared toolsheets N4.1 Toolsheets for lesson 2 Tier 4 programs and tools Calculator tool Number line tool Directed numbers ( and ) quiz Directed numbers ( and ) quiz Target Ladder method Circle 0, Diffy, Factor tree nlvm.usu.edu/en/nav/category_g_3_t_1.html Grid game Multiplication square jigsaw nrich.maths.org/public/viewer.php?obj_id5573 Factor squares nrich.maths.org/public/viewer.php?obj_id5468 Tier 4 class book N4.1, pp N4.1 How well are you doing?, p. 14 Tier 4 home book N4.1, pp. 1 3 Tier 4 CD-ROM N4.1 Check up N4.1 Pupil resource sheets 2.1 One per pupil 3.1 One per pupil N4.1 Properties of numbers 3

3 1 Order of operations Learning points Deal with brackets first. When there are no brackets, multiply and divide before you add and subtract. Slide 1.1 Slide 1.2 Starter Show slide 1.1. Discuss the objectives for the first three lessons. Say that this lesson is about working out calculations in the correct order. Show slide 1.2. Say that three darts have landed on different numbers. What is the least possible score? [11] What is the greatest possible score? [24] Ask pupils to write the numbers 11 to 24 in a list in their books. Record next to 11, and next to 24. What other scores are possible? Give pupils a minute or two to find the scores that they can make with three darts on different numbers. They should discover that 13, 15, 20 and 23 are not possible. Can you make the missing scores if two darts land on the same number? [e.g ; ; ; ] Main activity Revise the order of operations: brackets first, then squares, then multiplication and division, then addition and subtraction. Demonstrate some examples. Example Work out the square: Then the division: Then the multiplication: 54 5 Finally, the subtraction: 49 Example 2 (2 4) Work out the bracket: Then the square: Then the multiplication: Finally, the division: 30 Introduce nested brackets (usually round brackets inside square ones). Example 3 45 [11 (5 3)] Work out the inside bracket: 45 [11 2] Then the remaining bracket: 45 9 Finally, the division: 5 Example 4 50 [50 (20 2)] Work out the inside bracket: 50 [50 10] Then the remaining bracket: 50 5 Finally, the division: 10 4 N4.1 Properties of numbers

4 Remind pupils of the meaning of the square-root sign ( ). As a class, work through N4.1 Exercise 1 questions 1 and 2 in the class book (p. 2). Ask pupils to write answers on their whiteboards. Use the Calculator tool to show or remind pupils how to use their calculator bracket keys. Repeat examples 2, 3 and 4 above using a calculator. TO Ask pupils to do the rest of N4.1 Exercise 1 in the class book (p. 2). Review Give pupils a target number, say 16. Ask them to make a calculation using all the numbers 2, 4, 7 and 10 once, with 16 as the answer, e.g. (2 7 10) 4. Launch Target to give further problems. Use the operation signs, brackets and the given single-digit numbers to make the target number. Remind pupils of the learning points for the lesson. ITP Homework Ask pupils to do N4.1 Task 1 in the home book (p. 1). N4.1 Properties of numbers 5

5 2 Adding and subtracting directed numbers Learning points When you are adding or subtracting positive and negative numbers, two signs together can be regarded as one sign: is is is is Two signs that are the same can be regarded as. Two signs that are different can be regarded as. TO Starter Say that this lesson is about adding and subtracting positive and negative numbers. Use Toolsheet 2.1, a number line from 5 to 5. Remind pupils that positive numbers have a sign in front of them, although we don t always write it. Negative numbers have a sign in front of them. We always write the sign. Point to zero. Give an instruction, such as add 3. Ask pupils as a whole class to say where this would take them on the line. Repeat with another instruction, such as subtract 5. Ask again where this would take them. Give more instructions, including some that land on numbers beyond the line on the board. If you wish, use the Number line jumps tool to show the effect of add 3 by clicking on the line on the start and end values of the jump. TO Main activity Discuss examples of adding positive and negative numbers. You could interpret the first number as a starting temperature and the second as a rise or fall, for example: 6 (2) 8 Start with 6 C, and add a rise of 2 degrees. 2 (7) 5 Start with 2 C, and add a fall of 7 degrees. (3) (4) 1 (5) (3) 8 Start with 3 C, and add a rise of 4 degrees. Start with 5 C, and add a fall of 3 degrees. Not all calculators have a sign-change key. If there is a sign-change key on your pupils calculators (e.g. +/ or ( ) ), point it out and explain that it can be used for calculations with negative numbers. Modify the instructions below to suit your calculators. Use the Calculator tool to demonstrate how to input the negative number 2 by typing 2 + /, which should give a display 2. Repeat the calculations already on the board using the calculator: Key in = The display should show 8. Key in / = The display should show 5. Key in 3 + / + 4 = The display should show 1. Key in 5 + / / = The display should show 8. Refer pupils to the first set of calculations on slide 2.1. Get them to call out the answers as you run through them. Display the second set of questions. This time ask pupils to use their own calculators to key in the calculation. 6 N4.1 Properties of numbers

6 What do you notice about the two sets of answers? Draw out the rules that two signs together can be thought of as one sign. Establish that is equivalent to, and is equivalent to. Give out N4.1 Resource sheet 2.1. Complete the first addition table as a whole class, asking pupils to fill in the blank boxes as you go. Ask pupils to complete the second addition table in pairs. For answers, see p. 16. RS Say that it is best to think of subtraction as a difference. Use the context of temperature differences to illustrate, recording each calculation on the board: 5 (3) 8 From 3 C to 5 C is a rise of 8 degrees. (3) (8) 5 From 8 C to 3 C is a rise of 5 degrees From 9 C to 6 C is a fall of 3 degrees. (3) 4 7 From 4 C to 3 C is a fall of 7 degrees. Alternatively, use Toolsheet 2.2 showing a vertical number line from 8 to 8. As before, repeat the calculations on the board using calculators. TO Refer pupils to the first set of calculations on slide 2.2. Get them to call out the answers as you run through them. Display the second set of questions. This time ask pupils to key in the calculation and use the sign-change key to obtain the answers. What do you notice about the two sets of answers? Draw out that is equivalent to, and is equivalent to. Refer pupils again to Resource sheet 2.1. Complete the first subtraction table as a class. Ask pupils to complete the second table on their own. For answers, see p. 16. RS Select further individual work from N4.1 Exercise 2 in the class book (p. 3). Review Launch Directed numbers ( and ). Use Next and Back to move through the questions. Ask pupils to answer on their whiteboards, or refer a question to an individual pupil to respond. Pause now and then to ask pupils how they worked out the answer. Discuss and rectify errors and misunderstandings. Summarise the rules for adding and subtracting directed numbers. QZ Homework Ask pupils to do N4.1 Task 2 in the home book (p. 1). N4.1 Properties of numbers 7

7 3 Multiplying and dividing directed numbers Learning points For addition or subtraction of directed numbers, two signs together can be regarded as one sign: is is is is For multiplication or division of directed numbers, two signs that are the same results in and two signs that are different results in. is is is is Starter Say that the lesson is about multiplying and dividing positive and negative numbers. Discuss the signs and. Give some pairs of directed numbers (e.g. 4 and 2, 3 and 5) and ask pupils to insert or between them on their whiteboards. Introduce the and signs. Explain that if n is an integer and 2 n 1, then the possible values for n are 2, 1, 0 or 1, and that if 5 n 7, then n could be 5, 6 or 7. Tell pupils that N is an integer lying between 4 and 6, and that N 1. Ask pupils to decide in pairs on some statements that describe the possible values of N (e.g. 4 N 0 and 2 N < 6). Main activity Remind pupils that Similarly, (2) (2) (2) (2) 3 6. Explain that we can also write this as 3 (2) 6. Develop the multiplication table on the right. (3) (2) 6 Point out the patterns. The left-hand column is (2) (2) 4 decreasing by 1 and the right-hand column is (1) (2) 2 increasing by 2. ( 0 ) (2) 0 (1) (2) 2 Use the patterns to continue the table. (2) (2) 4 (3) (2) 6 RS Give out N4.1 Resource sheet 3.1. Complete the first multiplication table as a whole class, asking pupils to fill in the blank boxes as you go along. Ask pupils if they can see a quick way of working out products such as: (2) (3) or (3) (2) or (1) (3) They should notice that multiplication where the two signs are the same results in and multiplication where the two signs are different results in. Demonstrate the use of the sign-change key on the calculator, adapting the instructions below for your calculators: Key in / = The display should show 12. Key in 5 + / 8 + / = The display should show N4.1 Properties of numbers

8 Refer again to N4.1 Resource sheet 3.1. Complete the first multiplication table as a whole class, asking pupils to fill in the blank boxes as you go. Ask pupils to complete the second and third tables in pairs. For answers, see p. 16. RS Remind pupils that if we know that , we also know that and Show slide 3.1 and complete the patterns with the class. Link to division, for example: if (3) (5) 15, then (15) (3) 5 and (15) (5) 3 Repeat with: (3) (6) 18, so (18) (6) 3 and (18) (3) 6 (4) (7) 28, so (28) (7) 4 and (28) (4) 7 Pupils should notice that division where the two signs are the same results in and division where the two signs are different results in. Write on the board one or two questions for pupils to answer on whiteboards, for example: c (7) c 8 Now ask them to evaluate some expressions, such as: (7) 2 5 [(6) 2] (4 1) 2 [(4) 2] (3 7) Ask pupils to do N4.1 Exercise 3 in the class book (p. 7). Review Go through solutions to Exercise 3. Invite individual pupils to explain their methods. Launch Directed numbers ( and ). Use Next and Back to move through the questions. Ask pupils to answer on their whiteboards, or refer a question to an individual pupil. Discuss and rectify errors and misunderstandings. Summarise the learning points from this and the previous lesson using slide 3.2. QZ Homework Ask pupils to do N4.1 Task 3 in the home book (p. 2). N4.1 Properties of numbers 9

9 4 Powers and roots Learning points The square of a number n is n 2 or n n. Examples: , (9) The square root of n is n. Example: 81 9 The cube of a number n is n 3 or n n n. Examples: , (5) The cube root of n is 3 n. Example: When a negative number is raised to an even power, the result is positive. When a negative number is raised to an odd power, the result is negative. TO Starter Say that this lesson is about powers and roots of numbers. Remind pupils that (2) (2) 4 and So every square number such as 4 has two square roots, one positive and one negative. Ask the class: What is the square root of? using some of the square numbers up to Discuss how to estimate the positive square root of a number that is not a perfect square. For example, 60 must lie between 49 and 64, i.e , but as 60 is closer to 64 than to 49, 60 must be closer to 8 than to 7, perhaps about Use the Calculator tool to show how to use the square and square-root keys. You may need to explain that on some calculators the square-root key is pressed before the number and on others afterwards, and that most calculators give only the positive square root. Explain that the cube root of 125 is 5, and that we write The fourth root of 1296 is 6 or 6, and we write Some calculators have a cube root key 3 2. For other roots, keys vary from calculator to calculator. Demonstrate how these keys work on your calculators. Main activity Explain the notation , or 13 cubed. In general, a 3 means a a a, a 4 means a a a a, a 5 means a a a a a, and so on. Explain that a n is usually read as a to the power n and means a multiplied by itself n times. If a 5, what is a 3? If a 2, what is a 3? If a 3, what is a 4? If a 1, what is a 4? Draw out that when a negative number is raised to an even power the result is positive, and when a negative number is raised to an odd power the result is negative. 10 N4.1 Properties of numbers

10 Use the Calculator tool to show pupils how to use the x y keys of their calculators. Discuss the powers of 10 (10 0 1, , , , and so on) and their importance in the decimal number system. Repeat with 3 5 and 4 6, then use the calculator to explore what happens when a number is raised to the power 0. Explain that this is always has the answer 1. TO Write on the board: c c c Say that each box represents the same number. Let pupils try to find a solution with their calculators. After a couple of minutes stop them and explain that using reasoning will lead more quickly to the answer. Could the number in each box be a negative number? Is the number greater than 10? Is it greater than 20? How do you know? What would be a good number to try next? Agree that trying to see whether it is too big or too small cuts the possibilities by half. The remaining possibilities are 11, 12, 13 and 14. If the box represents an even number, is the answer odd or even? Say that there are now two possibilities: 11 and 13. Ask which is more likely? Get pupils to consider the last digit in each case. Confirm that 13 is correct by getting pupils to find using their calculators. Select individual work from N4.1 Exercise 4 in the class book (p. 9). Review Pose the problems on slide 4.1. Establish that each person has birth parents, grandparents, great grandparents,, great great great grandparents. 500 years is about 20 generations, so each person had about ancestors living 500 years ago. (The population of England in 1500 was roughly 1 million, so they could all be your ancestors, provided that all your ancestors were living in England at that time and families didn t intermarry too much.) Ask pupils to remember the points on slide 4.2. Homework Ask pupils to do N4.1 Task 4 in the home book (p. 2). N4.1 Properties of numbers 11

11 5 Multiples, factors and primes Learning points Writing a number as the product of its prime factors is called the prime factor decomposition of the number. 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. Starter Use slide 5.1 to discuss the objectives for this lesson. Remind pupils that: the factors of a number are all the numbers that divide into it exactly, so that the factors of 6 are 1, 2, 3 and 6, and the factors of 9 are 1, 3 and 9; factors can be paired (for 6, the factor pairs are 1 and 6, 2 and 3), except for square numbers, which have an odd number of factors (for 4, the factors are 1, 2 and 4); the number itself and 1 are always one of the factor pairs and, for prime numbers, they are the only factors. Show the target board on slide 5.2. Point at a number and ask pupils to write all its factor pairs on their whiteboards. Main activity Write on the board a list of the first few primes: 2, 3, 5, 7, 11, 13, What are all these numbers? Establish that they are all prime. Explain that when a number is expressed as the product of its prime factors we call it the prime factor decomposition of a number. How can we find the prime factor decomposition of 48? Explain the tree method, i.e. split 48 into a product such as 12 4, then continue factorising any number in the product that is not a prime. Repeat with SIM Launch Ladder method. Drag numbers from the grid to where you need them. For example, drag 75 from the grid to the box, then drag a prime factor of 75 (e.g. 3) to the circle, and so on. Continue to divide by prime numbers until the answer is 1. Express the answer as Repeat with another example, such as N4.1 Properties of numbers

12 Show how to find the highest common factor (HCF) and lowest common multiple (LCM) of a pair of numbers. Find the prime factors of and Represent the prime factors in a Venn diagram. Explain that: the overlapping prime factors give the HCF 3 2 (2 3 6); 5 3 all the prime factors give the LCM ( ). Repeat with another example, e.g. 10 and 24 (HCF 2; LCM 120). Ask pupils to do N4.1 Exercise 5 in the class book (p. 12). Review Finish with a game of Bingo. Show slide 5.3. Ask pupils to draw four boxes, choose four different numbers from the slide and write them in their boxes in any order. Read out the clues below in any order. If pupil have the answer in one of their boxes they can cross it out. The first player to cross out all four numbers calls out Bingo!. A multiple of 3 and of 4 less than 50 [36] An even multiple of 7 [84] The highest common factor of 81 and 18 [9] A multiple of 17 [51] The lowest common multiple of 12 and 20 [60] A multiple of 5 and of 7 [70] The highest common factor of 33 and 55 [11] A multiple of 13 [65] Sum up the lesson using points on slide 5.4. Round off the unit by referring again to the objectives. Suggest that pupils find time to try the self-assessment problems in N4.1 How well are you doing? in the class book (p. 14). Homework Ask pupils to do N4.1 Task 5 in the home book (p. 3). N4.1 Properties of numbers 13

13 N4.1 Mental test Read each question aloud twice. Allow a suitable pause for pupils to write answers. 1 What is the smallest whole number that is divisible by five and by three? 2003 KS3 2 What is the next square number after thirty-six? 2005 PT 3 The number one is a factor of both fifteen and twenty-four. What other number is a factor of both fifteen and twenty-four? 2007 KS3 4 Write down a factor of thirty-six that is greater than ten and less than twenty KS3 5 Write a multiple of nine that is bigger than seventy and smaller than eighty KS3 6 What is the next prime number after nineteen? 2002 KS3 7 What number should you add to minus three to get the answer five? 2003 KS3 8 I am thinking of a two-digit number that is a multiple of eight. The digits add up to six. What number am I thinking of? 2003 KS3 9 Subtract three from minus five KS3 10 Multiply minus four by minus five KS3 11 What number is five cubed? 2003 KS3 12 Divide twenty-four by minus six KS3 Key: KS3 Key Stage 3 test PT Progress test Questions 1 to 5 are at level 4; 6 to 9 are at level 5; 10 to 12 are at level 6. Answers or N4.1 Properties of numbers

14 N4.1 Check up and resource sheets cc ccc N4.1 Properties of numbers 15

15 Class book N4.1 Answers Order of operations EXERCISE 1 1 a 24 b 0.5 c 10 d 50 e 12 f 10 g 12.5 h 4 i a 2 (9 1) 16 b (5 3) 2 16 c (2 3) (1 4) 25 d 3 [7 (4 1)] 24 e 10 [6 (3 2)] 1 f 90 [20 (13 2)] 10 g 5 (6 3) 2 13 h [9 (8 1)] 8 16 c d 2 a b a 4.41 b 12 c d 8.17 e 24.7 f 42.2 g a (37 21) b (756 18) c 27 (36 18) 675 d 31 (87 19) 2108 e (486 18) f (56 63) g 837 (46 12) 285 h 52 (96 16) 5824 Adding and subtracting directed numbers EXERCISE 2 1 a b c a 2 (5) 7 4 b (3) (8) 8 3 c 7 (2) (8) 1 d (5) 6 (3) 4 e (5) 7 (2) 0 f g 3 (8) 7 2 h 2 (2) (5) 1 i (2) (8) 10 j (2) 9 11 k (1) (8) 7 l Extension problem 4 A 4 B 2 C 0 D 2 E 5 F 1 G 5 H 1 I N4.1 Properties of numbers

16 Multiplying and dividing directed numbers EXERCISE 3 1 A B A B A B A B A B a 24 b 5 c 2 d 12 e 12 f 2.5 g 72 h 200 i 6 3 a 20 points b 3 points c 16 points d 8 correct e 8 wrong f 28 questions d 8 correct 4 a 2 [(5) 4] 2 b [(2) (6)] 3 24 c 9 (7 4) 6 d [(3) (4)] 6 6 e [8 (2)] (5) 2 f 14 (6 7) 15 5 a 14 F b 28.4 F c 68 F d 23 F e 40 F Extension problem 6 a 12 b 3 c 9 d 175 e 3 f 8 g 10 h 4 i 2 Powers and roots EXERCISE 4 1 a 169 b 256 c 128 d a 1296 b c 6561 d e 4096 f 24.1 g h a 56 b 9 c 9 d 19 e 24 f 2 g 8.7 h a b c d a b c d a 81 b 88 c 27 d 512 e 729 Extension problem Multiples, factors and primes EXERCISE 5 1 a b c d e f a b c d e f a 40 and 90 HCF 10 LCM 360 b 48 and 42 HCF 6 LCM 336 N4.1 Properties of numbers 17

17 4 a 60 and 150 HCF 30 LCM 300 b 126 and 210 HCF 42 LCM a 175 and 200 HCF 25 LCM 1400 b 112 and 140 HCF 28 LCM 560 c 42 and 105 HCF 21 LCM Consider units digits whose products fit the conditions. For example, in the first problem, the two units digits could be 0 and 1, or 4 and 5, or 5 and 6. The two numbers will also be close to a b c Extension problem 7 To give a zero on the end of a product, we need to consider a multiple of 10, or a multiple of 5 paired with a multiple of 2. 10! has 2 zeros at the end of it. 20! has 4 zeros at the end of it How well are you doing? 1 a 3 (2) 1, or 6 (5) 1 b (8) (2) 6 c (5) (1) 5 d 6 (2) and The last digit is the last digit of 4 4 4, or or 405 Home book TASK 1 1 a (56 38) b ( ) c 650 (48 35) 50 d 27 (13 15) 5265 TASK 2 1 a 2 (6) 4 b 4 (1) 5 c (4) (5) 9 d 2 (5) 7 e f (4) (2) 6 g 7 (7) 0 h 2 (8) 6 i 3 (5) 8 j (2) (3) 1 TASK 3 1 a 2 (6) 12 b 4 (1) 4 c (4) (5) 20 d 14 (7) 2 e (9) (4) 36 f 3 (1) 3 g (4) 6 24 h 0 (7) 0 i 2 (3) 6 j 10 (5) 2 k l (27) (3) 9 TASK 4 There are 8 different ways of writing 150 as the sum of four squares: There are 3 different ways of writing 150 as the sum of three squares: N4.1 Properties of numbers

18 TASK 5 1 a b a b a 100 and 150 HCF 50 LCM 300 b 78 and 91 HCF 13 LCM Teacher s book Check up 1 MENTAL TEST , 2, 5, and Check up 2 WRITTEN TEST 1 a 6 (4) 2 b (7) 4 3 c (7) (4) (3) 14 d (4) (3) 12 e (4) (1) a is the largest. b 43 has the same value as 26. Resource sheets RESOURCE SHEET 1 (LESSON 2) RESOURCE SHEET 2 (LESSON 3) first number Any three-digit multiple of 18 other than N4.1 Properties of numbers 19