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1 . Whole Numbers, Frations, Deimals, and Perentages In this hapter you will learn about: multiplying and dividing negative numbers squares, ubes, and higher powers, and their roots adding and subtrating frations with different denominators and mixed numbers multiplying and dividing frations with whole numbers, frations, and mixed numbers multiplying and dividing deimals rounding deimals to deimal plaes and signifiant figures onverting between frations, deimals, and perentages reurring deimals finding perentages of quantities, inluding perentage inreases and dereases Get Ready You will need: red ounters 8

2 . Multiplying and Dividing Integers In this setion, you will multiply and divide with positive and negative numbers. Exerise. Warm Up Here are some arrays: A B C Look at array A. a How many rows in the first row? b How many rows in array A? How many in the first olumn? d How many olumns in array A? Here are some multipliations: olumns a Whih two of these multipliations math array A? First row: First olumn: b Whih multipliations math array B? Whih multipliations math array C? d Work out the answer to eah of the multipliations in part a, b and. Word fat: An array has pitures or numbers arranged in olumns and rows. We an use arrays to show multipliation. 9

3 Explaining When multiplying numbers, does it matter whih number is first? Explain. Look again at array A. a How many altogether? b Work out 8 6. How many Work out 8. How many in eah group? in eah group? 5 a Look again at array B. Write two division alulations for array B. Write answers to your alulations. b Look again at array C. Write two division alulations for array C. Write answers to your alulations. 6 Explaining When dividing numbers, does it matter whih number is first? Explain. 7 Copy and omplete this table of related multipliation and division alulations: Use your answers to Q to help you. Share 8 into 6 equal groups: Use your answers to Q5 to help you. Multipliation Division alulation Division alulation 6 = 8 8 = = 8 = 8 =?? = =?? 5 =?? 9 =? 7 0 =??? =??? =? 6 8 =??? =??? =? 8 a Work out: i 8 ii 8 iii ( 8) iv (8 ) Use BEDMAS: Brakets, Exponents (powers), Division and Multipliation (from left to right), Addition and Subtration (from left to right). 0

4 THINK b Compare your answers to part a. What do you notie? Explaining Compare your answers to parts a iii and a iv. Does the position of the brakets matter? 9 Here are some numbers: a Whih numbers are negative? b Whih numbers are positive? Write the numbers in order from smallest to largest. 0 We an use this symbol to mean : So is: What are these numbers? a b d Main Exerise Here is an array: a Look at the first row. What number is this? b How many rows in the array?

5 HANDS ON ACtiVITY HANDS ON ACtiVITY Write a multipliation alulation for all the ounters in this array. Write an answer to your alulation. d Look at the first olumn. What number is this? e How many olumns in the array? f Write another multipliation alulation for all the ounters in this array. Write an answer to your alulation. g Write two division alulations for this array. Use red ounters. Every red ounter is. a Make an array that has four ounters in the first row and seven rows altogether. b Write two multipliation alulations for this array. Write answers to your alulations. Write two division alulations for this array. Use red ounters. a Make two different arrays that give multipliation alulations with the answer. b For eah array write two multipliation alulations. Write answers to your alulations. For eah array write two division alulations. Reverse your multipliations from parts and f. Look at the first row. What number is this? How many rows in the array? Look at the first olumn. What number is this? How many olumns in the array? d Compare your alulations with others in your lass. Are you all orret?

6 Example Work out 5 and 5, and ompare the answers. Answer Step Use arrays. 5 5 Step Step Step Count the rows and olumns in eah array rows, 5 olumns 5 rows, olumns Write the multipliations and answers 5 = 0 5 = 0 Compare the answers. Both arrays give the same answer. It does not matter whih number is positive and whih is negative. a Work out: i 5 ii 5 iii 5 iv 5 b What do you notie about the answers to these alulations? Use red ounters to make arrays to help you. When multiplying positive and negative numbers, does it matter whih number is first? Explain. d What do you notie about the answers to: i a negative number a positive number? ii a positive number a negative number?

7 TALK 5 Work out: a 7 5 b 8 0 d 8 e 9 6 a Copy and omplete this table in your notebook: Calulation Answer 5????? b Identifying Patterns Copy this number line and irle all your answers. Are they getting bigger or smaller? Let s ontinue the pattern in the table. Add more rows to your table. What do you notie about the numbers in the left hand olumn? Complete the alulation olumn. Calulation Answer 5????? 0????????? d Complete the answers olumn of your table. e Identifying Relationships What do you notie about the answer to a negative number a negative number? Multiplying a positive number by a negative number is the same as multiplying a negative number by a positive number. The answer will always be negative. Use the pattern from part b to help you.

8 TALK 7 Work out: a 7 b d 9 e a Copy and omplete this table of multipliations and divisions: Multipliation Calulation Division Calulation Division Calulation = =? =? =?? =? = =?? =? = =?? =? = b What do you notie about the answer when: 9 Work out: i multiplying or dividing two numbers with different signs? ii multiplying or dividing two numbers with the same sign? a 8 5 b d 6 e f 0 0 Work out: a 6 6 b d 5 9 e 60 0 f 8 6 Work out: a 9 b d e 80 8 f 7 9 Work in pairs. You will need: negative number negative number = positive number,,, 6 digit ards negative sign ards and operation ards 6 x x x x 5

9 TALK Not a Write eight alulations and answers you an you make using only these ards. You must use at least one negative sign ard in eah alulation. Write all the alulations and answers in your notebook. b Compare your alulations with others in your lass. Work out: a b 5 5 d 6 e 6 f 8 6 Work out: You an make alulations like: =. You an also put two digits together to make numbers like or. a 5 b d 8 7 What alulation will you do first? Remember BEDMAS. With division and multipliation, work from left to right. e 6 6 f Jasem and Faisal work out the answer to Jasem s Method Step Divide first 0 8 = 5 Step Multiply by = 5 Faisal s Method Step Multiply first 8 5 = 0 Step Divide 0 by = for sale or Who is orret? Why? 6

10 . Powers and Roots In this setion you will find squares, ubes and higher powers, and their roots. Exerise. Warm Up a What is the square of? b What is the square of 5? What is the square of 6? Copy and omplete in your notebook: a 7 = 7 7 =? b 8 =?? =? 9 =?? =? a Use the multipliation button on your alulator to work out: i ii iii 5 Press buttons in order: b Use the square button on your alulator to hek your answers. Press buttons in order: x a What is the square root of 5? b What is the square root of 00? What is the square root of 6? 9 5 is 5 squared. The tells us to write the number down two times, and multiply them together: So 5 = is the square of. We say that is the square root of 9. square 9 square root?? =. What number multiplied by itself equals? Look at the squares in Q. 7

11 HANDS ON ACtiVITY 5 Work out: a 9 b 6 6 A square has an area of 00 m. What is the length of its sides? The length of the sides is in m. 7 How many ubes in eah shape? a b Main Exerise 8 Explaining Here are two numbers. a Whih number is the square of 9? Explain. b Whih is the square root of 9? Explain. 9 Work in pairs. You will need: 0 blank ards 8 x x a Write eah of these square roots and numbers on ards b Mix up the ards and plae them fae down on the table. Take turns to find mathing pairs. In your notebook write all the mathing pairs you have found The symbol means the square root of. 6 = 6 so 6 = 6 x x Write for eah pair? =? 8

12 TALK TALK HANDS ON ACtiVITY d Whih square roots were easy to remember? Whih square roots are more diffiult to remember? With a friend, try to learn all these square roots. 0 a Write down the first 0 square numbers. b Work out the next 0 square numbers. Use Identifying Patterns Look at the last digit in eah of your answers in parts a and b. Can you see a pattern? x + d What do you think the last digit is for? Explain. e Chek your answer to part d using a alulator. You will need: multilink ubes a Use multilink ubes to make models of the first ube numbers, shown here. x x = x x = 8 Write =?, =? 0 =? x on your alulator to work out =?, =? 0 =? G8-8-N.-fig6 Cirle the last digit in eah answer. b Make the ube. What is the ube of? =, and = 8. and 8 are alled ube numbers. How many multilink ubes did you use for your ube? d What is the ube of? 9

13 TALK TALK Copy and omplete in your notebook: a 5 = =? b 0 =???=? Say 0 in two different ways. d Find the ube button on your alulator. What do you think this button does? e Use the ube button to hek your answers to part a and b. Use the ube button on your alulator to work out: a 7 b 5 is the ube root of 8, beause = 8. a Copy and omplete: i is the ube root of? beause??? =? ii is the ube root of? beause??? =? b What is the ube root of 000? 5 Explaining Find a number that is both a square and a ube. Explain. 6 What do you notie about eah set of three ards, A, B and C? Set A Set B Press buttons in order:??? = 000. What number will fill the spaes to make 000? G8-8-N. - - alulator keys is read as to the power of, or ubed. The tells us to write the number down three times, and multiply them together: = 6 0 means the ube root of... ube 5 5 ube root x 5 x = x x x 8 x Set C 8 0

14 HANDS ON ACtiVITY You will need: 8 blank ards 7 Work in pairs. a Make two sets of ards like Set A in Q6:? x? For?, use any number between and 0. Do not use. On the last ard write the answer of the first ard b Make two sets of ards like set B in Q6:? x? x? For?, use any number between and 5. Do not use. Make two sets of ards like set C in Q6: number For the number ard, use,, 5 or 0. d Play the game. You have made 8 ards. Mix up your ards. Put all the ards fae down.?? fae up? Person : Turn over three ards. Are your ards equal? If yes, keep the ards. If no, put them fae down again. Person : Turn over three ards. Are your ards equal? If yes, keep the ards. If no, put them fae down again. 6 number? 6 number turn over x x x x fae down Repeat until you find all sets of three ards. The winner is the person with the most ards.

15 TALK TALK THINK 8 a Write the answer to: i 5 ii 7 b Find these buttons on your alulator and x i What do you think the button does? Use your alulator to hek your answer to part a i. ii What do you think the orange part of this button x You have to press the shift button Use your alulator to hek your answer to part a ii. 9 Explaining A ube has a volume of 5 m. What is the length of eah of its sides? Explain. 0 a means a number written down two times and multiplied together.. does? means a number written down three times and multiplied together. Continue the pattern to opy and omplete these statements: i means a number written down? times and multiplied together. ii 5 means a number written down? times and multiplied together. iii 0 means a number written down? times and multiplied together. b Identifying Patterns Copy and omplete this pattern: i? = =? ii? = =? iii? = =? iv? = =? Write to the power of. Write a alulation to show what it means. d Write 6 to the power of 5. Write a alulation to show what it means. Find this button on your alulator. a Use your alulator to hek your answers to Q0b. b Use your alulator to find the answer to: i ii 6 5 G8-8-N. -8- alulator keys x SHIFT to use the orange funtion. The length of the sides is in m. G8-8-N-summary-figa G8-8-N-summary-figa -alulator keys -al G8-8-N-summary-figa -alulator key SHIFT SHIFT x x G8-8-N-summary-figa -alula x x Calulators have a square button and a ube button. x x x =?? SHIFT x SHIFT x SHIFT x x For higher powers, use the button: type the number then the button and then press the number for the power. SHIFT SHIFT x

16 THINK Copy and omplete in your notebook: a 5 = =?. This means that 65 =? b 6 =?????? = 6. This means 6? = 5 =????? =. This means? =? d 8 =? = 56. This means? = e 5 =? =?. This means? =? f 0 6 =? =?. This means? =? Find this button on your alulator: a Use your alulator to hek your answers to Q. b Use your alulator to find the answer to: What do you notie about your answers to part b and Qb ii? Copy and omplete: a 00 =? beause?? = 00. b 000 =? beause??? = 000. Press =? beause? = d What is 5? SHIFT 5 a Use your alulator to work out: i 8 ii 5 0 iii 7 8 b Rewrite the questions and answers from parts a, b and as multipliation sums. Look bak at Q. x x????? = Calulators have a square root button and a ube root SHIFT button. x For higher roots, use SHIFT x.

17 . Adding and Subtrating Frations In this setion you will add and subtrat frations with different denominators and mixed numbers. Get Ready You will need: fration tiles Exerise. Warm Up fration irles A whole number has no fration part, e.g. 0, 5, A fration is part of a whole. It has a numerator (top number) and a denominator (bottom number). E.g. numerator 8 denominator A mixed number has a whole number and a fration part. E.g. whole fration number

18 HANDS ON ACtiVITY Look at these frations. a Whih fration is losest to? b Whih fration is losest to? Write the frations in asending order. Look at these whole numbers, frations, and mixed numbers. In your notebook, write the: a whole numbers b frations One way is to use a number line. mixed numbers Look at these frations and improper frations. In your notebook, write the improper frations. Use fration irles a Count seven green 5 piees. These make the improper fration 7 5. What mixed number do they make? Word fat: Asending order means from smallest to biggest. An improper fration has a numerator bigger than the denominator, e.g 5 0 or 9. How many whole irles an you make? How many 5 piees left over? 5

19 HANDS ON ACtiVITY HANDS ON ACtiVITY b Count eleven 5 piees. i What improper fration do they make? ii What mixed number do they make? Write the improper fration as a mixed number. d Write the mixed number as an improper fration. e Explaining Is loser to or? Explain. 5 Use 6 fration tiles. a Use fration tiles to show 6 : b Add 6. Work out: Work out: a b d Use fration tiles.. Write your answer as a whole number.. Write your answer as a mixed number.. Write your answer as a mixed number. a Use fration tiles to show 7. b Subtrat. Write the answer to 7. 8 Work out: a b a What fration of the whole bar is eah: 6 6 i pink piee ii green piee iii purple piee whole 6 6 Use fration irles to help you. Use fration tiles to help you. whole = 5 5 6

20 HANDS ON ACtiVITY b Copy and omplete in your notebook: 5 is equivalent to? This bar is the same total size as the bars in part a. What fration is eah small piee of this bar? d In your notebook, opy and omplete: i ii Look at the green 5 bar in part a. Whih two tiles are the same size as one green bar? 5 is equivalent to?. 0 is equivalent to?. 0 Use sets of, 6, 8, and fration tiles. a Math pairs of equivalent frations. b Find all the fration tiles that are equivalent to. Copy and omplete to find equivalent frations: a b d??? 8 =?? 8 6 =? 5 = 8? 0 =??? Word fat: Equivalent means equal or the same value as. Look at the green bar in part a and ompare it to the piture in part. Whih fration is the same as?? 00 7

21 THINK HANDS ON ACtiVITY TALK Main Exerise a Faisal wants to work out 8 +, but he sees that the denominators are different. What an he do? b Copy and omplete: 8 + = 8 +? 8 Use 6 and fration tiles. =?? a Use fration tiles to show b Hind wants to work out What an she do? Copy and omplete: =? =?? + 5 d How do you hange 6 to? without using fration tiles? Work out: a d g j b + 6 e + 9 h Here are some fration ards: f i 0 a Predit whih two ards add to give the smallest answer. Explain why you think this is. Can you use tiles to make 6? Think about equivalent frations. 9 0 Close your eyes and imagine eah fration in tiles. Whih two tiles are the smallest? You an only add frations with the same denominator. Word fat: Predit means make a good guess. TALK b Add the two frations you predited in part a. Compare your answer to part b with others in your lass. Did you get the smallest answer? 8

22 TALK TALK TALK HANDS ON ACtiVITY d Predit whih two ards add to give the biggest answer. e Add the two frations you predited in part d. f Compare your answer to part e to others in your lass. Did you get the biggest answer? g Here are four more fration ards: Predit whih two ards will add to give: i the smallest answer. ii the largest answer. h Make your own four fration ards give them to your partner to find the smallest and largest totals. Chek their answers. 6 a Explaining Estimate the answer to Explain. b Work out: Give your answer as a mixed number. Write as 8?. How good was your estimate in part a? 7 a Explaining Estimate the answer to Explain. b Work out Give your answer as a mixed number. How good was your estimate in part a? 8 Use and 6 fation tiles. a Can you use tiles to make 6? b Can you use 6 tiles to make? Look bak at your answers in Q0. 5 What easy fration is 5 8 lose to? You an use fration tiles to hek. Then think about adding this easy fration to the. Make the denominators the same. Word fat: To estimate hoose lose easy numbers or frations. Then do the alulation. These will give a lose answer to help you know if you are orret. G8-8-N.-fig8 - blak headsarves What is 0 9 lose to? i What fration is equivalent to 6? ii What fration is equivalent to? 9

23 HANDS ON ACtiVITY THINK d Identifying Relationships How an you use your answers to part to work out 6 +? e Copy and omplete in your notebook. 6 + =?? +?? =?? 9 Use a set of fration tiles. a Can you use: i tiles to make? ii tiles to make 6? b What fration tiles an you use to work out +? Copy and omplete in your notebook: + =?? +?? =?? 0 Use fration tiles to work out: a + b d Give your answer to part as a mixed number. Example Work out +. Answer Step We need to find a ommon denominator. Step Step Think about what size fration tiles you The ommon denominator ould use to show both quarters and will be. thirds. Rewrite the frations with the same denominator. Add the frations. = 9 and = 8 + = = 7 Step Write your answer as a mixed number. = 5 Work out. Give your answers as mixed numbers. Set out your working like in the Example. You an only add frations with the same denominator. =?? and =?? same denominator a + 5 b + + d 5 + 0

24 Ali, Hassan and Mubarak enter a ompetition. If they win, they agree to share the prize like this: Ali gets 5. Hassan gets. Mubarak gets 0. They will give the rest to harity. a What fration would Ali and Hassan have together? b What fration would they all have? What fration would they give to harity? Example Work out + 5. Answer Step Add the whole numbers together. + 5 = 9 Step Step Step Step 5 Work out: Change the frations so they have the same denominator. Add the frations = 6 and = 6 + = = 5 6 Add the whole number and the sum of the frations together to give a mixed number = 95 6 Chek your answer by making an estimate. and a bit + 5 and a bit = 9 and a bigger bit. a b d e + 8 f g h Work out: a + b d a Explaining Estimate the answer to + 5. Explain. b Work out + 5. Give your answer as a mixed number. How good was your estimate in part a? If the fration parts add to more than a whole number, you need to add this to the other whole numbers.

25 HANDS ON ACtiVITY THINK 6 Use 8 fration irle piees. a Use fration irles to show 8. b Work out 8 8. Use your fration irles to work out: d Chek your answer to part by estimating. Example Work out. Answer Step Subtrat the whole numbers = Step Step Change the frations so they have the same denominator. Subtrat the fration parts. We an only take 8 away, so we have an extra to take. = 8 and = 9 = 8 and = 9 Step Subtrat the extra fration from the whole number. = Step 5 You annot take 7 8 away from 8. You will have to take some of the eighths from the whole number part. Chek your answer by making an estimate. and a little bit and a big bit = a bit less than. 7 Here are three alulations: i 5 6 ii 5 iii 5 a Estimate the answers to eah alulation. b Work out the answer to eah alulation. I am taking away less than, so the answer will be... Follow the steps in the example, and hek using fration irles. TALK How good were your estimates?

26 THINK 8 Amal has 5 meters of material. She uses meters for a blind and 0 of a meter for ushions. a Estimate how muh material she has left. b Work out how muh material she has left. Use your estimate to hek your answer. 9 Copy this number puzzle into your notebook. 6 Use fration irles to help you. 6 This puzzle follows the rule: All the rows, olumns and diagonals add to the same total. Complete the number puzzle. Add the frations in the first olumn to find the total. 0 Defining Problems Make your own number puzzle using the same rule. Put mixed numbers and frations in the boxes. rows 6 Word fat: 6 olumns diagonals Copy your number puzzle, but leave five boxes blank (like in Q9). Challenge a friend to omplete your number puzzle.

27 . Multiplying Frations In this setion you will multiply frations by whole numbers, frations and mixed numbers. Get Ready You will need: fration tiles fration irles popsile stiks Exerise. Warm Up In your notebook, list all the fators of: a b 0 d 5 e f g 6 Whih fators are ommon to: a and b 0 and 5 and d and 6 a What fration of eah bar is blue? A B C lear plasti sheets Use your answers to Q. b What do you notie about the frations in part a? A fator is a whole number that divides exatly into another number, e.g. the fators of are, and. Word fat: Common to means they belong to two or more numbers, e.g. and are fators of both 9 and. They are the ommon fators. is the highest ommon fator.

28 Copy and omplete to simplify the frations: 5 a b 0 5 d???? =? 0 5 =? =?? 6 6 =?????? Use your answers to Q to help you find the highest ommon fator to divide by In your notebook, write the: 7 a Improper frations b Mixed numbers Math the improper frations to the equivalent mixed numbers. Use fration irles to help you. 6 a Use 5 and 0 fration tiles to work out Can you use 0 tiles to make 7 5? b Simplify your answer. Give it as a mixed number. You an simplify frations by dividing by the highest ommon fator. You an simplify and then hange your improper fration to a mixed number. Or you an hange your improper fration to a mixed number first, and then simplify. 5

29 Example Work out 6. Answer You an think of 6 as of 6, or as 6 lots of. You an use whihever is easier. Method Thinking about of 6: Start with 6 stiks and split them into groups: 6 = Method Thinking about : Put together six piees of fration irles: 6 = 7 a Use popsile stiks (Method ), to show these fration multipliations, and write the answers in your notebook. i 5 0 ii iii 6 8 b Use fration irles (Method ), to show the same multipliations and write the answers in your notebook. Do you get the same answers? 8 Use popsile stiks (Method ), then fration irles (Method ) to show, and then work out: a 6 b 5 0 For 6 you need two of the three groups of popsile stiks or you need six lots of irle piees. d

30 TALK HANDS ON ACtiVITY Main Exerise 9 Use fration irles to work out: a 5 b d 5 8 Give eah answer as an improper fration and then a mixed number. 0 Identifying Patterns Look again at your answers to Q9. a Look at the numerators and whole numbers in eah question. Then look at the numerator of the improper fration in eah answer. What do you notie? b Look at the denominators in eah question. Then look at the denominators of the improper fration in eah answer. What do you notie? Explaining What is 5 8? Explain. d Explain to a friend an easy way to multiply a fration by a whole number. e Identifying Relationships Use the words numerator and denominator to opy and omplete this general rule: When you multiply a fration by a whole number the? of the answer will be the same as the? of the fration. The? of the answer will be equal to the whole number the? of the fration. Use lear plasti sheets. a Draw a retangle measuring 6 m by m. For example, 5 = 5?. For example, 5 =??. Look at your answer to Q9d. Does it matter whih number is first when multiplying? The numerator is the top number in a fration. The denominator is the bottom number in a fration. Word fat: vertial line horizontal line 7