Place value and fractions. Explanation and worked examples We read this number as two hundred and fifty-six point nine one.

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1 3 3 Place vale and ractions Exlanation and worked examles Level Yo shold know and nderstand which digit o a nmer shows the nmer o: ten thosands thosands 000 hndreds 00 tens 0 nits As well as the nmer o: tenths 0 hndredths We read this nmer as two hndred and ity-six oint nine one. It means two hndreds, ive tens, six nits, nine tenths and one hndredth. Ordering and comaring nmers How do yo know i a nmer is igger or smaller than another nmer? To t nmers in order o size yo comare the digits y lace vale. Which is the larger nmer, 86 or 406? To work it ot, line the nmer in a tale y lace vale. Thosands (Th) Hndreds (H) Tens (T) Units (U) Hodder and Stoghton 04

2 3 3 Place vale and ractions Level Start rom the let-hand side with the digit which has the largest lace vale. The digit in the thosands colmn is the same, so yo look at the next digit, the hndreds. Here the is greater than the 4, so 86 is the larger nmer. Notice that yo do not need to comare the rest o the digits once yo have ond one lace vale where the digits are dierent. Which is the larger nmer, 3.6 or 3.9? Tens (T) Units (U). Tenths Hndredths Here again the digit with the largest lace vale, the tens, is the same or oth nmers, as is the nits digit. Moving to the tenths colmn, 9 is greater than, so 3.9 is the larger nmer. Learner ti Be carel i there are a dierent nmers o digits when yo are comaring size and make sre yo comare the digits y lace vale. Let s look at another examle. Which is the larger nmer, 99. or 0 90? Arrange the nmers y lace vale. Ten Thosands Hndreds Tens Units. Tenths thosands In this examle it is clear that 0 90 is the larger nmer, as the irst digit or 99. is in the thosands colmn, whereas the irst digit or 0 90 is in the ten thosands colmn. Hodder and Stoghton 04

3 3 3 Place vale and ractions Level Negative nmers I yo cont down rom ten to zero, what haens i yo want to carry on conting down rom zero? Yo can do this, t yo need to se a negative symol (also called a mins sign) to show it is elow zero, and say negative (or mins ), negative, negative 3, and so on Rememer the negative sign is art o the nmer and it is not asking yo to take away. Think what this means: 6 is lower down the nmer line than. So i it was temeratre, 6 C wold e colder than C. Negative nmers have many ses in everyday lie, or examle, to show money withdrawal rom a ank accont, to indicate temeratres elow zero and to show height elow sea level. Fractions At Level yo need to nderstand and se eqivalences etween common ractions, decimals and ercentages. This is covered in more deth in the Fractions, Decimals, Percentages and Rationit. Here is a reminder aot simle ractions. Hodder and Stoghton 04 3

4 3 3 Place vale and ractions Level Look at this izza. It is sliced into 6 eqal ieces, so as a raction o the whole izza we write that one slice is 6. The to nmer o the raction is called the nmerator and tells yo the nmer o eqal arts. The ottom nmer is called the denominator and tells yo how many o those arts make one whole. So i yo eat 4 slices o the izza yo have eaten 4 o the whole izza. 6 Notice the denominator stays the same, ecase the izza is still slit into 6 eqal ieces. When yo are learning aot ractions to start with its really sel to have a ictre o a izza in yor head, ct into the right nmer o eqal ieces the denominator with the nmer yo want to eat searated ot the nmerator. Hodder and Stoghton 04 4

5 3 3 Place vale and ractions Level The izza aove shows three-eighths, 3 8. The izza elow shows one qarter, 4. By looking at the diagrams aove, yo can see that three-eighths o the izza is more than one-qarter. What aot one-third and two-iths, which is igger? The diagrams show that is more than 3. Hodder and Stoghton 04

6 3 3 Place vale and ractions Level Decimals in money and measre Yo shold already e amiliar with sing decimals when writing amonts o money and recognise that the decimal oint searates the onds rom the ence. So, or examle, 3.4 means 3 and 4. When working with lengths, yo need to rememer: 0 mm = cm 00 cm = m 000 m = km Assessment ti In Fnctional Skills assessments yo need to e ale to write yor answer to a sensile degree o accracy. For examle, in money calclations yo wold need to write yor answer correct to two decimal laces. In measrement qestions involving calclations in centimetres, yo may want to write yor answer correct to one decimal lace, which wold e to the nearest millimetre. Percentages 0 Percent means ot o 00, or examle 0% means so to write a 00 ercentage as a raction yo write it with a denominator o 00. This is covered in more detail in the Fractions, Decimals, Percentages and Ratios toic at Level. Yo may ind the ollowing tale sel; it shows eqivalences etween common ractions, decimals and ercentages. Fraction Decimal Percentage 0. 0% % 0. 0% 0.0 % Hodder and Stoghton 04 6

7 3 3 Place vale and ractions Level For the ractions yo may still want to think o a ictre o a izza eing ct into eqal ieces. The irst raction is one-hal and the eqivalent decimal 0., hal o a hndred is 0 so these mean the same as 0%. Hal o one-hal is one-qarter which is the same as 0. or %. The next line starts with. Yo know the irst nmer ater the 0 decimal oint reresents the nmer o tenths, so is the same as 0. 0 which is the same as 0%. Learner tis Notice that to change a decimal to a ercentage, yo mltily y 00. This means that to change a ercentage to a decimal yo divide y 00. Yo need to know and qickly recognise what the digits o a nmer reresent sing their lace vale. I a nmer has decimal arts, the decimal oint is to the right o the nits. The to nmer o a raction is called the nmerator and tells yo the nmer o eqal arts. The ottom nmer is called the denominator and tells yo how many o those arts make one whole. Fractions, decimals and ercentages are jst dierent ways o showing arts o whole nmers. It sel to know that 0., and 0% are the same; 0., 4 and % are the same. Hodder and Stoghton 04

8 3 3 Ronding and estimating Exlanation and worked examles Level For very large nmers it isal to aroximate to the nearest hndred, thosand, ten thosand, etc. The distance rom the Earth to the Sn variet it isally given as 93 million miles, that is , to the nearest million. Let s look at an examle o ronding. A newsaer reort says that eole attended the new Leeds concert vene on the evening it oened. This does not mean that eole were actally at the irst concert, it is jst easier to reort. Actally there were 8 63 eole there. Conting in thosands, 8 63 is etween and It is nearer to So 8 63 to the nearest thosand is Hodder and Stoghton 04

9 3 3 Ronding and estimating Level Here is a qick method o ronding to the nearest thosand. Ste Pt a ring rond the thosands digit. For examle, Ste Look at the next digit to the right. I it is less than, leave the thosands digit as it is. I it is or more, add to the thosands digit. Ste 3 Relace the remaining digits y zeros. For the examle aove, A similar method can e sed to rond to the nearest 00, 0 000, etc. Look at the examles elow. a) Rond 4 40 to the nearest 00. ) Rond to the nearest c) Rond 63 to the nearest 0. Soltion a) 4 40 is the 00s digit is less than, so yo rond down. ) is the 0 000s digit is greater than, so yo rond. c) 63 is the tens digit is or more, so yo rond. Look at this nmer line... is nearer to than. So. to the nearest whole nmer is..4 is nearer to than 3. So.4 to the nearest whole nmer is. Hodder and Stoghton 04

10 3 3 Ronding and estimating Look at this nmer line is hal way etween 6 and. Yo always rond any nmers that are in the middle. So 6. to the nearest whole nmer is. Level To rond to the nearest whole nmer, look at the irst decimal lace. I it is less than, leave the whole nmer as it is. I it is or more, add to the whole nmer. Yo take no notice o any digits in the second decimal lace and rther to the right. Rond these to the nearest whole nmer. a) 4.9 ).3 c) 9. d) 4.03 e) 46.9 Soltion a) The irst decimal lace is 9 so yo add to 4. ) The irst decimal lace is 3 so yo leave as it is. c) 9 The irst decimal lace is so yo add to 9. d) 4 The irst decimal lace is 0 so yo leave 4 as it is. e) 4 The irst decimal lace is 9 so yo add to 46. Ronding to a given nmer o decimal laces When a nmer is written in decimal orm, the digits on the right-hand side o the decimal oint are known as decimal laces. Nmers can have many dierent decimal laces. 6.3 is written to decimal lace.. is written to decimal laces is written to 3 decimal laces. And so on. Yo can shorten decimal lace to d... Hodder and Stoghton 04 3

11 3 3 Ronding and estimating Level When working with nmers, yo may e asked to rond a nmer to an aroriate nmer o decimal laces. For examle, money calclations shold e written correct to two decimal laces. Rememer this rle. Cont the decimal laces rom the decimal oint and look at the irst digit yo need to remove. I this digit is less than, jst remove all the nwanted laces. I this digit is or larger, add to the digit in the last decimal lace yo want and then remove the nwanted decimal laces. Rond these nmers to the nmer o decimal laces given. a) 6.33 to decimal lace ).334 to decimal laces c) to decimal lace d) 3.6 to decimal laces Yo may ind it hell to draw yor own nmer line. For a nmer ronded to decimal lace, the nmer line shold show nmers to decimal laces and so on. Soltion a) 6. The second decimal lace is less than so yo jst remove the nwanted decimal laces. ).33 The third decimal lace is less than so yo jst remove the nwanted decimal laces. c) 88. The second decimal lace is so yo add to the digit in the irst decimal lace. d) 3.6 The third decimal lace is larger than so yo add to the digit in the second decimal lace. Ronding to a given nmer o decimal laces is oten sed in everyday sitations. Hodder and Stoghton 04 4

12 3 3 Ronding and estimating Level Estimating and rolem solving In Fnctional Skills assessments at Level yo need to show evidence o checking a calclation. One way to do this is to rond each nmer in the calclation to signiicant igre and work ot the calclation again. Yor ronded answer shold e aot the same size as yor original calclation. To rond a nmer to signiicant igre yo rond it to the lace vale o the largest digit. For examle, when ronding to signiicant igre the lace vale o its largest digit is thosands, so yo rond it to the nearest thosand which ecomes Ronding 4 to signiicant igre, the lace vale o the largest digit is hndreds, so yo rond it to the nearest hndred, which gives yo 00. Worked examles Jim ys gol alls at 3.6 each. What is the total cost? Show how yo can check yor answer. Soltion 3.6 = Check: Ronding each nmer to signiicant igre = 80 This is close to so hay to accet answer. Estimate the reslt o each o these calclations y ronding to signiicant igre. a) 3.. ) 4 63 c) d) e) 8 38 ) g) Hodder and Stoghton 04

13 3 3 Ronding and estimating Level Soltion a) 4 = 8 ) 0 60 = 3000 c) 4 9 = 36 d) = = 9 and there is decimal lace. e) is the same as 9 4 = =. or.3 or ) = = 8 and there are decimal laces. g) = = 0. or 0.3 Assessment ti An alication o estimation is to check i the answer yo have got ater a calclation is aot right. For examle, i yor answer is in the hndreds, t when yo estimate it is in the ten thosands, something has gone wrong, so yo will need to do the calclation again. Think aot which o the nmers in these statements are likely to e exact, and which have een ronded. a) Yesterday, I sent 4.6. ) My height is 80 cm. c) Her new dress cost 40. d) The attendance at the ootall match was e) The cost o ilding the new school is million. ) The vale o i is 3.4. g) The Olymic Games were held in Beijing in 008. h) There were 8 eole at the meeting. Hodder and Stoghton 04 6

14 3 3 Ronding and estimating Level Learner ti Ronding whole and decimal nmers Have a ictre o a nmer line in yor head. Pt the nmer yo are ronding on it. This hels yo decide whether to rond or down. Estimating A common mistake is to t the wrong nmer o zeros when aroximating. Rememer it mst e aot the same size as the nmer yo started with. A calclation can e checked sing estimation y ronding the nmers in the calclation to signiicant igre. This means yo rond to the lace vale o the largest digit in each nmer. Assessment ti When writing onds and ence sing a decimal, do not t a at the end o the decimal, t se the ond sign,, at the eginning. For Fnctional Skills yo shold always inclde the sign yo may lose marks i yo do not write the nits in the answer. Hodder and Stoghton 04