Topic 3 Looking for Patterns in a Decimal Chart Lesson 1 Understand it! There are patterns in decimal number charts. Continue the pattern to label the other squares. Learning how and when to look for a pattern can help when solving problems. 0·01 0·02 0·03 0·08 0·15 0·16 0·1 0·19 0·29 0·32 0·34 0·37 Another Example Using the same system as above, you could fill in the diagonals of a decimal number chart. ge s In this decimal number chart, what is the pattern in the diagonals? 0·55 pa 0·55 m pl 2 more tenths than in 0·75 Sa Guided Practice Find the pattern for each of the following and then complete. 1 2 1 more tenth 1 more hundredth than in 0·55 0·86 1 more tenth 1 more hundredth than in 0.75 0·75 e 2 more tenths than in 0·55 0·66 0·95 Independent Practice Find the pattern for each of the following and then complete. 3 0·12 0·14 4 0·61 0·34 0·81 0·42 0·16 0·84 5 0·55 0·67 46 forty-six ACMNA124: Investigate everyday situations that use integers. Locate and represent these numbers on a number line. ACMNA133: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence. What are the missing decimals? 0·01 As you work with vertical columns, you will see the tenths increase by 1 and the hundredths stay the same as you move down. What are the missing decimals? 0·01 0·29 0·11 Moving from left to right, tenths are the same in each row except for the last number; the hundredths increase by 1. 0·21 0·31 0·26 0·27 0·28 0·29 0·30 Continue each pattern. 70·39, 0·40 , ??? , ??? , ??? , , 2·931, 2·932, ??? ??? 8 ??? 10 , ??? ??? , , ??? ??? , 1·306, 1·406 , 3·43, 3·23, ??? Problem Solving m pl e 9 ??? pa ge s 6Dane drew a grid of five cells in a row. The number 0·075 was in the middle cell. What did Dane’s grid look like? Sa 11a Some of the numbers are hidden. Fill in the missing numbers. 0·10 0·28 b Describe the pattern. forty-seven 47 Topic 3 Estimating Sums and Differences Lesson 2 How can you estimate with decimals? Understand it! To estimate means to find an approximate answer or solution. Changing numbers to other numbers that are easy to compute can help you estimate the sums or differences of decimals. The 10-second barrier in the 100-metre sprint was broken in the 1968 Olympics, when the winning time was 9·95 seconds. Mrs Johnson, the physical education teacher, ran the 100 metres in 14·7 seconds. About how much faster was the 1968 Olympic time than Mrs Johnson’s time? 14·7 seconds 9·95 seconds Guided Practice 11·769 + 0·686 = 1· ??? ??? 220·45 – 13·15 = m START plFINISH e pa ge s Complete each estimate by rounding to the nearest tenth. + 0· ??? ??? = ·5 ??? – 13·2 = Round each number to the nearest whole number to estimate the answer. 31·456 + 5·4 + 14·08 = 472·43 – 59·8 = ??? ??? – Reasoning ??? + ??? = ??? + = ??? ??? Sa 5Kirra and Jeremy want to go to a movie and have popcorn and a drink. A movie ticket costs $7.75 and the popcorn and drink combo costs $7.85. Kirra says if they bring $15, it will be enough. Jeremy says they each need more than $15. Who is correct? Why? Independent Practice Round each number to the nearest whole number to estimate each answer. 620·791 + 5·25 + 3·84 = 7$10.10 – $3.65 = 48 forty-eight ??? ??? – + ??? + = ??? = ??? ??? ACMNA128: Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers One Way Another way Use rounding to quickly estimate sums and differences. Round each number to the same place value. Use just the whole numbers to make an estimate, and then adjust the estimate using the remaining digits. Round each number to the nearest whole number. 14·7 14 2 9·95 29 5 14·7 15 2 9·95 2 10 5 The difference is about 5 seconds. Since 0·7 , 0·95, the difference is less than 5. The 1968 Olympic time is less than 5 seconds faster than Mrs Johnson’s time. Use just the whole numbers to estimate each answer. ??? – + ??? ??? = + ??? 10 Circle the best estimate for 0·375 + 2·46. A 2·5 B3 ??? = ??? ge s 991·26 – 30·463 = ??? pa 87·12 + 2·501 + 9·2 = C3·5 D4 Sa m pl e 11Rachel is shopping and needs to buy bread, ham and apples to make lunch. She has a ten-dollar note. Will she have enough money for her purchases? Use estimation to find whether she will have enough money. Explain your answer. Problem Solving 12a The difference between the heights of two sisters is about 0·25 m. How tall might each sister be? b Measure the difference in height between you and a classmate. Express this as both a fraction and a decimal of a metre. forty-nine 49 Topic 3 Multiplying Decimals by 10, 100 or 1 000 Lesson 3 Understand it! $0.45 per kg What is the pattern when multiplying decimals by 10, 100 or 1 000? Patterns can be used when mentally multiplying decimals by multiples of 10, 100 and 1 000. $2.89 per kg A baker buys some of the ingredients he uses in bulk. He needs to purchase 10 kg of pecans and 100 kg of flour. How much will the baker spend for each amount? Choose an Operation Multiply to join equal groups. Mental Computation ??? 2 3·1 ??? ??? Multiply each number by 10. 50·006 ??? 6 2·3 7 3·08 m pl 9In the problem at the top of the page, how much would the baker spend if he buys 10 kg of flour? ??? 4 7·4 8 1·34 ??? ??? ??? 100 kg of flour? ??? Sa Reasoning ??? e Guided Practice 3 0·45 pa 10·009 ge s Multiply each number by 100. 10 Marcy and David each multiplied 5·6 × 10 and 0·721 × 100. Marcy got 0·56 and 7·21 for her products. David got 56 and 72·1 for his products. Which student multiplied correctly? Explain your answer. Independent Practice Type of coin Number Saved The table shows the coins saved by Tina and her sister for one year. Use it to answer each of the following. 5c 1 000 11Find the total value of each type of coin the girls have saved. 10c 100 20c 1 000 $2 10 5c ??? 10c ??? 20c ??? $2 ??? 50 fifty ACMNA130: Multiply and divide decimals by powers of 10 Cost of the pecans $2.89 per kg $0.45 per kg Cost of the flour $2.89 × 10 = $28.90 $0.45 × 100 = $45 When multiplying by 10, each number moves to the next place value. When multiplying by 100, each number increases 100 times, therefore it moves 2 place values. 12 Find the total value of the coins saved by the sisters. ??? Complete the following. ??? × 34·2 = 342 17 ??? × 68 = 68 000 15 90·3 × Sa 14 m pl e pa ge s 13 The principal of Middleton School has a big glass jar of marbles. The empty jar weighs about 400 grams, and each of the 1 000 marbles weighs 6·7 grams. Find the total weight in grams of the jar of marbles. ??? 18 347·2 × 1 000 = ??? = 9 030 ??? 16 7·04 × 100 = 19 603·2 × ??? ??? = 6 032 Problem Solving 20a Marian multiplied a number by 100 and got an answer between 280 and 289. What might the number have been? ??? b What is the smallest and largest number it could have been, to two decimal places? smallest largest ??? ??? fifty-one 51 Topic 3 vv Multiplying Decimals Lesson 4 How can you multiply whole numbers and decimals? Understand it! Barry displayed four paintings side-by-side in one row. Each painting has the same width. What is the total width of the four paintings? Choose an Operation Multiply to find the total width of the four paintings. Multiplying decimals is similar to multiplying whole numbers. Placevalue relationships help you determine where to place the decimal point in the product. ge s Each is 0.36 metres wide. Another Example How can you multiply a decimal by a decimal? pa Find 0·5 × 0·3. Use what you know about the numbers to help you calculate the answer. The same operation can be said in different ways. Another Way e One Way 0·5 is the same as one half. 0·5 = m pl Use a model to show 0·5 by shading the first 5 columns. 1 2 0·3 = 0·30 or 30 hundredths The product is the area where the shading overlaps. Half of 0·3 or half of thirty hundredths is 15 hundredths. Sa Show 0·3 by shading the first 3 rows. 0·5 × 0·3 = 0·15 0·5 × 0·3 = 0·15 Guided Practice Multiply the parts and then add the parts to calculate the answer for each of the following. 14 × 0·32 = (4 × 0·3 + 4 × 0·02) = 27 × 2·1 = (7 × ??? +7× ??? ??? )= + ??? ??? + = ??? ??? = ??? Independent Practice Mentally calculate the answer for each of the following. 310 × 0·32 = 52 fifty-two ??? 4 100 × 0·129 = ??? 5 2·36 × 1 000 = ??? ACMNA129: Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies What You Think Find 0·36 3 4. Multiplying 0·36 m 3 4 is like adding 0·36 four times on a decimal model. Unit n n Unit Ud n d Frac M Simp What Write Fac FYou D Clear d Fix Multiplying the Op parts is one way to calculate Op1 2 % the answer. Int Mode ? MR/MC 1000. n 100. % Taking notes helps keep track of the parts. n n πd Ud Frac( ) 0·3 × 4 = 1·2 M F D 7 × 4Fac=80·24 9 0·06 MR/MC Simp d Fix % 1. 1000. % 0.1 π 0.01 Op1 Op2 Add 1·2 4 the parts: 5 Int 6 + 0·24. – The total width is 1·44 metres. 2 multiply 3 ( 1 can) also + a calculator. You using 7 0.001 80 9. Press: 0·36 4 5 6 Display: 10. 1. 0.1 0.01 10. 1 2 (–) – 4 ENTER = 1,44 TI-15 3 keys + ge s 100. The product is the total area shaded. 0·36 m 3 4 5 1·44 ? Mode 75 TI-15 display Multiply the parts and then add the parts to calculate the. answer for ENTER each of the following. (–) 0.001 0 = ??? 7 6 × 0·28 = ??? 8 3 × 0·134 = ??? pa 60·7 × 12 = m pl e 9Mary is making a patchwork quilt. Each panel is TI-15 keys 12·5 cm wide. She has 8 panels for the first row. How wide is her quilt so far? ??? Sa 10If Mary’s quilt is to be 1·25 m wide, how many more panels does she need for the first row? ??? Problem Solving 11a Jen solved this number sentence: 9 × 0·989 = 89·01. How can you use estimation to show that Jen’s answer is incorrect? What is the correct answer? ??? b Explain where Jen went wrong. fifty-three 53 Topic 3 Dividing Decimals by 10, 100 or 1 000 Lesson 5 Understand it! How can you divide decimals by 10, 100 and 1 000? Patterns can be used when mentally dividing decimals by 10, 100 or 1 000. Sandra wants to cut a cloth into 10 strips. All the strips should be exactly the same size. How wide will each strip be? Choose an Operation Divide to find equal parts of a whole. Mental Computation Mentally calculate the answer for each of the following. 472·5 ÷ 10 = ??? ??? 2 126·4 ÷ 100 = ??? pa 1370·2 ÷ 10 = ge s 89·5 cm 5 28·14 ÷ 100 = ??? 6 42·5 ÷ 1 000 = ??? e Guided Practice ??? 3 684·5 ÷ 1 000 = m pl 7If Sandra wanted to cut the 89·5 cm cloth into 100 strips, how wide would each strip be? ??? Sa 8Using the table at the top of the page, why was it necessary to place a zero in the tenths place when dividing by 1 000? Independent Practice Pacific School posted the winning swimming times. Use the table to answer each of the following. 9What was the time per metre of the swimmer who swam the butterfly? 50-metre freestyle 22·17 seconds 100-metre backstroke 53·83 seconds 100-metre butterfly 58·49 seconds ??? 10 If the 50-metre freestyle swimmer could swim the 100-metre freestyle in exactly double the 50-metre time, what would be the time per metre? 11What was the time per metre of the swimmer who swam the backstroke? 54 fifty-four ACMNA130: Multiply and divide decimals by powers of 10 ??? ??? Notice the patterns in the table. Find 89·5 4 10. The quotient of a number divided by 10, 100 or 1 000 is less than the number. Divide by Examples Numbers move to the right 1 12·5 4 1 5 12·5 0 places Dividing by 10 decreases the value of each digit. 10 12·5 4 10 5 1·25 1 place 100 12·5 4 100 5 0·125 2 places Since place value is based on 10, dividing by 10, 100 or 1 000 gives the same result as moving the numbers to the right. 1 000 12·5 4 1 000 5 0·0125 3 places So, 89·5 4 10 5 8·95 Each cloth strip will be 8·95 centimetres wide. pa ge s 12 How is dividing 360 by 10 similar to dividing 3 600 by 100? Explain. Problem Solving Sa m pl e 13a Helen saved for 10 weeks to buy a gym membership. The membership fee was $315.00. How much did she save each week? ??? b If she goes to the gym every day for 100 days, how much has her membership cost her per day? ??? fifty-five 55

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