Multiplication and division. Explanation and worked examples. First, we ll look at work you should know at this level.

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1 x Mltilication and division Exlanation and worked examles Level First, we ll look at work yo shold know at this level. Work ot these mltilication and division sms: a) ) 96 c) 8 d) 6 Soltions: a) 9 6 Yo mltily irst the nits and then the tens y = 6 and = 9 ) 96 Yo irst divide the tens and then the nits y 9 = and 6 = c) = 6 Yo write the 6 in the nits colmn and carry tens over. = ; + carried over = d) 8 6 into does not go so yo look at the next digit. into is remainder Hodder and Stoghton 04

2 Mltilication and division Level x Look at the remainder together with the next digit. into 6 is 8 Mltilication tales Yo can ild the mltilication tales rom what yo already know Yo shold know the times tale. The 4 times tale is twice those nmers Yo can now dole each o these or the 8 times tale Yo can do something similar with the times tale Hodder and Stoghton 04

3 Mltilication and division Level x The vales in the 6 times tale are twice as mch as those in the times tale The vales in the 9 times tale are three times the times tale, or yo can add the vales in the and 6 times tales It is very sel to e ale to mltily y. Yo shold rememer the times tale. This is more easily rememered as hal the 0 times tale Finally the times tale. There is no easy way to rememer this. Yo cold add the times tale and times tale together, t it is roaly est to jst learn this y heart Hodder and Stoghton 04

4 Mltilication and division Level x Now yo have seen all o the tales rom to 0. Try to make yor own tale with all the mltiles in them. Coy and comlete this mltilication grid. From yor tale, check that 8 and 8 give the same answer. x Yo will ind it sel to learn these tales and the rles or making them. Learner ti The order o the nmers does not matter when mltilying. For examle, 8 and 8 give the same answer. Rememer that i, or examle, yo know 8 = 6, yo also know 6 8 = and 6 = 8 Sqare nmers I yo look at yor comleted mltilication grid, yo shold have a diagonal line that gives yo the answers to,, to 0 0. These are the sqare nmers, 4, 9, 6,, 6, 49, 64, 8 and 00. These nmers are worth rememering. Hodder and Stoghton 04 4

5 Mltilication and division Level x Mltiles and actors Mltiles The nmers in the ive times tale are, 0,, 0, etc. These nmers are called mltiles o. Yo shold know yor ive times tale to ives are 60 t the mltiles o ive do not sto at 60 they go on: 6, 0,. In act there is no end to the list o mltiles. Any nmer that ends in 0 or is a mltile o. Let s try some worked examlesing mltiles: a) List the mltiles o etween 0 and ) List the mltiles o 6 etween 0 and 00 c) List the mltiles o 9 etween 0 and 00 Soltions: To work these ot, yo shold list the times tales or, 6 and 9 and carry on ntil yo get to or 00, as instrcted. a), 4, 6, 8, 0,, 4, 6, 8, 0,, 4, 6, 8, 0,, 4 ) 6,, 8, 4, 0, 6, 4, 48, 4, 60, 66,, 8, 84, 90, 96 c) 9, 8,, 6, 4, 4, 6,, 8, 90, 99 Look again at or examles. Notice that 8, 6, 4, and 90 are in the list o mltiles or oth 6 and 9. These nmers are called common mltiles o 6 and 9 ecase they are in, or common to, oth lists o mltiles. Factors A nmer that will divide into another nmer exactly is called a actor o that nmer. For examle: is a actor o 8 is a actor o 0 is a actor o 00 is a actor o 6 9 is a actor o 9 Hodder and Stoghton 04

6 Mltilication and division Level x Notice that is a actor o every nmer, and that every nmer is a actor o itsel. The actors o 0 are,,,, 6, 0, and 0. The actors o 0 are,,, 0, and 0. Notice that,, and 0 are in oth lists o actors. We say that,, and 0 are the common actors o 0 and 0 ecase they are common to oth lists. Learner ti Yo can check yo have listed all the actors y writing down all the actor airs. For examle, or the actors o 0 (,,,, 6, 0,, 0). 0 = 0 = 0 0 = 0 6 = 0 Mltilying and dividing y 0 and 00 Here are two entries in the 0 times tale: 0 = 0 0 = 0 Yo can see that to mltily y 0 yo move the nits into the tens colmn, the tens into the hndreds colmn and so on. Yo t a zero in the nits colmn. So, or examle: 0 = = = 00 To mltily y 00 yo move the nits into the hndreds colmn, the tens into the thosands colmn and so on. Yo t zeros in the nits and the tens colmns. Hodder and Stoghton 04 6

7 x Mltilication and division So, or examle: 4 00 = = 600 Let s try some worked examles. Write down the answers to these a) 6 0 ) 4 00 c) Soltions: a) 60 ) 400 c) Level Dividing y 0 and 00 Dividing is the reverse o mltilying, so to divide y 0 yo move the digits one lace to the right and take a zero o. This only works i the nmer ends in 0. To divide y 00 yo move the digits two laces to the right and take two zeros o. This will work i the nmer ends in 00. Let s try some worked examles: a) 80 0 ) c) Soltions: a) 80 8 ) c) Hodder and Stoghton 04

8 Mltilication and division Level x Mltilying and dividing decimals y 0 and 00 Now that yo know how to mltily and divide integers y 0 and 00, yo can se lace vale tales to hel yo mltily and divide decimals y 0 and 00. Let se a worked examle to see how to do this. Work ot these mltilication and division sms: a).6 0 ).6 00 c) 4. 0 d) e) Soltions: a) Hndreds 0 Tens 0 Units Tenths 6 Hndredths To mltily y 0 yo move each digit one lace to the let..6 0 = 6 ) Thosands 0 Hndreds 0 0 Tens Units Tenths 6 Hndredths To mltily y 00 yo move each digit two laces to the let. Notice that yo have to t a zero in the nits colmn otherwise the nmer wold read 6 rather than = 60 c) Thosands Hndreds Tens Units Tenths Hndredths To divide y 0 yo move each digit one lace to the right = Hodder and Stoghton 04 8

9 Mltilication and division Level x d) Thosands Hndreds 4 0 Tens Units Tenths 0 Hndredths thosands To divide y 00 yo move each digit two laces to the right = 4. e) Tens Units.. Tenths 9. Hndredths Thosandths 0 9 Ten Thosandths To divide y 00 yo move each digit two laces to the right. Yo sally write a zero when there are no digits in ront o the decimal oint. This makes the nmer easier to read = 0.9 Learner ti It is worth rememering that yo do not have to draw a lace vale tale. Yo can simly move the digits in yor head i yo are ale to. Mltilying a decimal y an integer Comare these two mltilications or inding the cost o three CDs at 4.9 each. Working in ence Yo know that 4.9 = = 4.8 Mltily irst the nits, then the tens and then the hndreds y. Write the nitnder the nits, the tennder the tens and so on. Hodder and Stoghton 04 9

10 x Mltilication and division Convert yor answer rom ence to onds y dividing y 00. Working in onds To mltily a decimal y an integer, t the decimal ointnder each other. Make sre yo line yor work carelly. Pt the irst digit yo work ot nder the last decimal lace. In each case, the digits are the same. Level Written methods or mltilication Yo need to e ale to do sms like 8 and 8 6 withot a calclator. There are several ways o doing this. Choose a method yo are hay with and se it. Long mltilication 8 90 ( 0) 4 4 ( 8) 04 Add This is the traditional method, called long mltilication. The second method ses a grid. Hodder and Stoghton 04 0

11 Mltilication and division Level x Grid method a) ) Lattice method In this method, a lattice is drawn to hel s work ot the sm. Let s try a worked examle: 6 4. Rememer to add along the diagonal (red) arrows. 6 x Note that when the sms add to more than 0, the is taken into the next colmn. Practical rolem aroach Sales rices are oten set at 4.99, 9.9, etc. This aroach can e sed in this tye o examle. Bracelets are riced at.9 in a sale. Ashley ys racelets or her riends as resents. What is the total cost o the racelets? Hodder and Stoghton 04

12 x Mltilication and division Soltion.9 =.9 = ( ) = = 60 = 9. Level Division and remainders I one nmer does not divide exactly into another nmer yo will have a remainder. In a ractical sitation yo may need to decide what to do with the remainder, which will need to e related to the sitation. For examle, say yo have a 00 cm length o wood. How many lengths o 60 cm can e ct o rom this length? When yo do yo get whole 60 cm lengths which measres 80 cm in total and so yo have a remainder o 0cm let. What i yo have a 40cm length o wood? How many lengths o 4 cm can e ct o rom this length? Yo may notice that 0 4 = 40 which is 0 cm more than 40 cm. So rom this yo can work ot 9 4 = 40. This leaves = cm. So yo can ct 9 lengths o 4 cm and have cm let over. Mltilying a decimal y a decimal What do we see i we work ot the ollowing sms: 0 4, 4,. 4? Soltions: 0 4 = = = 4.8 For each o the answers, the digits are the same. The only things that are dierent are the lace vales o the digits. Hodder and Stoghton 04

13 Mltilication and division Level x This will hel yo in working ot the answer to a calclation sch as Look at these calclations: = 6 0. = 0.6 Mltilying tenths y means the answer is 6 tenths. 0. = 0. = 0.6 Mltilying y tenths means the answer is 6 tenths. These hel yo to see that: = 0.06 Mltilying tenths y tenths gives 6 hndredths. There are stes yo take when mltilying decimals. Carry ot the mltilication t take notice o the decimal oints. The digits in this answer will e the same as the digits in the inal answer. Cont the total nmer o decimal laces in the two nmers eing mltilied. Pt the decimal oint in the answer yo got in ste so that the inal answer has the same nmer o decimal laces as yo ond in ste. Let s try an examle: Soltion: First work ot 8 = 6 The total nmer o decimal laces in 0.8 and 0. = + = The answer is 0.6 Hodder and Stoghton 04

14 Mltilication and division Level x Learner tis, and all mean divided y. A qick way to work ot a calclation sch as is to do 4 =. Then cont the nmer o zeros in the calclation, three in this case, and add them to yor answer. So the answer is wold give the same answer as 8 6 t it isally easier to have the smaller nmer on the ottom. When mltilying more diiclt nmers, it is a good idea to work ot an estimate to see i yor answer is sensile. For examle, i yo are estimating the vale o three items that cost 4.86 each, it is hell to rememer that =, so yor answer mst e less than. When yo mltily sing decimals, mltilying y a nmer etween 0 and makes the answer smaller than the nmer yo started with. For examle, 0. = 0.. Hodder and Stoghton 04 4