Multiplication and Division

Transcription

1 Series Student Multiplication and Division My name F

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3 Series F Contents Topic Mental mul plica on strategies (pp. 0) doubling strategy mul ply by 0s, 00s and,000s split strategy compensa on strategy factors and mul ples Date completed Topic Mental division strategies (pp. 9) use mul plica on facts divide by 0s, 00s and,000s halving strategy split strategy divisibility rules Topic Wri en methods (pp. 0 8) single-digit mul plica on mul -digit mul plica on division division with remainders problem solving Topic 4 Puzzles and inves ga ons (pp. 9 ) crack the code apply smart bu ons apply bugs inves gate puzzles apply Series Authors: Rachel Flenley Nicola Herringer Copyright

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5 Mental multiplication strategies doubling strategy Doubling is a useful strategy to use when mul plying. To mul ply a number by four, double it twice. To mul ply a number by eight, double it three mes. 5 4 double once = 0 8 double once = 6 double twice = 60 double twice = 5 double three mes = 04 Warm up with some doubling prac ce: a b c D D D Finish the doubling pa erns: a b 96 c 5 40 d 5 50 e f Choose a number and create your own doubling pa ern. How high can you go? What pa erns can you see within your pa ern? 4 Two sets of twins turn. They decide to have a joint birthday party with giant cake but they all want their own candles. How many candles will they need? F

6 Mental multiplication strategies doubling strategy 5 Use the doubling strategy to solve these: 4 a b 6 4 c 4 4 d 5 4 e 4 f 4 g 5 4 To mul ply by 4, double twice. To mul ply by 8, double three mes. 6 Use the doubling strategy to solve these: 4 8 a b 4 8 c 5 8 d 8 84 e 8 f Work out the answers in your head using the appropriate doubling strategy. Use a table like the one above if it helps. a 8 4 = b 6 4 = c 6 4 = d 4 8 = e 5 8 = f 8 = 8 Nick s dad offered him two methods of payment for helping with a 5 week landscaping project. Method : $4 a week for 5 weeks. Method : $8 for the first week, then double the payment each week. Which method would earn Nick the most money? Why? F

7 Mental multiplication strategies multiply by 0s, 00s and,000s When we mul ply by 0 we move the number one place value to the le. When we mul ply by 00 we move the number two place values to the le. When we mul ply by,000 we move the number three place values to the le. Look at how this works with the number 45: Ten Thousands Thousands Hundreds Tens Ones ,000 Mul ply the following numbers by 0, 00 and,000: a T Th Th H T O 7 b T Th Th H T O ,000,000 c T Th Th H T O d T Th Th H T O ,000,000 Try these: a 4 0 = b 4 00 = c 4,000 = d 9 0 = e 9,000 = f 9 00 = g,000 = h 00 = i 0 = You ll need a partner and a calculator for this ac vity. Take turns giving each other problems such as Show me The person whose turn it is to solve the problem, writes down their predic on and you both check it on the calculator. 0 points for each correct answer, and the first person to 50 points wins. F

8 Mental multiplication strategies multiply by 0s, 00s and,000s It is also handy to know how to mul ply mul ples of 0 such as 0 or 00 in our heads. 4 helps us work out 4 0: 4 = = 80 We can express this as 4 0 = 80 How would you work out 4 00? 4 Use pa erns to help you solve these: a b c 6 $4 6 $40 6 $400 d e $7 $70 $700 f g Answer these problems: a Jack runs 50 km per week. How far does he run over 0 weeks? If you re struggling with your tables, get onto Live Mathle cs and prac ce! b Hugo earns an allowance of $0 per week. If he saves half of this, how much will he have saved at the end of 8 weeks? c The sum of two numbers is 8. When you mul ply them together, the answer is 60. What are the numbers? 6 Finish these coun ng pa erns: a b c d e f g ,00 4 F

9 Mental multiplication strategies split strategy Some mes it s easier to split a number into parts and work with the parts separately. Look at 64 8 Split the number into 60 and 4 Work out (60 8) and then (4 8) Add the answers together = 5 Use the split strategy to answer the ques ons: a 46 4 (40 4) + (6 4) + = b 74 5 ( ) + ( ) + = c 48 4 ( ) + ( ) + = d 7 7 ( ) + ( ) + = e 6 8 ( ) + ( ) + = f 9 5 ( ) + ( ) + = Use the split strategy to answer the ques ons. This me see if you can do the brackets in your head: a 48 8 = + = b 5 7 = + = It's a good thing I know how to work with mul ples of ten in my head! c 9 4 = + = d 8 9 = + = e 86 7 = + = These problems have been worked out incorrectly. Circle where it all went wrong. a 7 6 (0 6) + (7 6) 80 + = 9 b 7 5 (0 5) + (7 5) = 05 c 9 (0 9) + ( 9) = 45 F 5

10 Mental multiplication strategies split strategy 4 Each trail contains mul plica on problems and steps to solve them. Only one trail has been solved correctly. There are errors in the other two. Find and color the winning trail. FINISH (0 9) + ( 9) (0 6) + ( 6) (0 9) + (6 9) (40 7) + ( 7) START 6 F

11 Mental multiplication strategies compensation strategy When mul plying we can round to an easier number and then adjust. Look how we do this with 4 9: 9 is close to 0. We can do 4 0 in our heads because we know 4 = 4 0 = 0 We have to take off 4 because we used one group of 4 too many: 0 ( 4) = = 6 Use the compensa on strategy to answer the ques ons. The first one has been done for you. a 9 = 0 = 57 b 8 9 = = c 8 6 = = d 7 9 = = e 8 5 = = We can also adjust up. Look how we do this with 6 6: 6 is close to 60. We can do 6 60 in our heads because we know 6 6 = = 60 We have to then add more lots of 6: 60 + = = 7 Use the compensa on strategy and adjust up for these. The first one has been done for you. a 4 = 40 + = Would I use the compensa on strategy with numbers such as 56 or 84? Why or why not? b 8 4 = + = c 9 = + = d 9 = + = e 7 6 = + = F 7

12 Mental multiplication strategies compensation strategy In this ac vity you ll work alongside a partner. You ll each need two dice and your own copy of this page. For each line, roll the dice to find the tens digit and then roll it again to find the mul plier. Your partner will do the same. Use the compensa on strategy to mentally work out the answers to the problems. copy Tens Ones Mul plier Answer = 9 = = = 8 = = 9 = 8 = = = a Check each other s calcula ons. You may want to use a calculator. b Now, use the calculator to add your answers. The person with the highest score wins. 8 F

13 Mental multiplication strategies factors and multiples Factors are the numbers we mul ply together to get to another number: factor factor = whole number How many factors does the number have? 4 =, 6 =, = 4,, 6,, and are all factors of. List the factors of these numbers: a 8 b 5 c 4 d 9 e 6 f 5 g 0 h 4 Fill the gaps in these sentences. The first one has been done for you. a or 6 or or 8 or 4 people can share 6 pieces of candy evenly. b or or or or or people can share 0 slices of pizza evenly. c or or or or or or or people can share 4 cherries. d or or or or or or or people can share 0 pencils. e or people can share 5 balls evenly. Use a calculator to help you find as many factors of 84 as you can: A factor divides into a number evenly with no remainder. F 9

14 Mental multiplication strategies factors and multiples Mul ples are the answers we get when we mul ply factors. Think about the mes tables where is always a factor. What are the mul ples of?, 6, 9,, 5, 8,, 4, 7, 0, and 6 factor = mul ple 4 Fill in the gaps on these mul ple boards: a 4 b 5 c 9 d Numbers can be either factors or mul ples depending on where they sit in the number sentence. 5 Choose numbers between and 5 and put them in the first frame as factors. Your answer is the mul ple. Now take that mul ple and make it a factor in another number sentence. Write in the other factor and solve the problem. Then make the answer a factor again. Can you fill the grid? Use a calculator for the larger problems. The first one has been done for you. a 4 = = 4 4 = 48 b = = = c = = = d = = = 0 F

15 Mental division strategies use multiplication facts Knowing our mul plica on facts helps us with division as they do the reverse of each other. They are inverse opera ons. 5 = = Use your knowledge of mul plica on facts to help answer these division ques ons: a = = b = = c = = d = = e = = f = = g 08 = = Now try these: a 8 9 = c = e 4 7 = b 40 5 = d 54 6 = f 6 9 = Doing math without knowing your mul plica on facts is hard. Learning them makes your life much easier. It s worth persevering to conquer them! g 6 4 = h 45 9 = i 9 = j 4 6 = Fill in the division wheels. Use mul plica on facts to help you. a b c F

16 Mental division strategies use multiplication facts Knowing our fact families is also helpful. 5 = 5 5 = = 5 = 5 4 Complete the following pa erns. How many more mul plica on and division facts can you find, given the first fact? a 7 8 = 56 b 8 9 = 7 c 7 9 = = 9 8 = 9 7 = 56 = 8 7 = 9 6 = 9 8 = 7 9 = 8 9 = 7 5 Write down another mul plica on fact and two division facts for each ques on. a 6 7 = 4 b 5 9 = 45 c 9 6 = 54 d 7 8 = 6 e 8 = 96 f = 6 Look at these two division facts: 0 5 = 4 and 0 4 = 5 Imagine you re explaining to a younger child how they re related yet different. How would you do it? What would you say/write/draw? F

17 Mental division strategies divide by 0s, 00s and,000s When we divide by 0 we move the number one place value to the right. When we divide by 00 we move the number two place values to the right. When we divide by,000 we move the number three place values to the right. Look what happens to 45,000 when we apply these rules: Ten Thousands Thousands Hundreds Tens Ones ,000 Divide the following numbers by 0, 00 and,000: a T Th Th H T O b T Th Th H T O ,000,000 c T Th Th H T O d T Th Th H T O ,000,000 Draw lines to match the answers with the ques ons: a What number is one thousand mes smaller than,000? 9,500 b What number is one hundred mes smaller than,000? 88 c What number is one hundred mes smaller than 95,000? 950 d What number is ten mes smaller than 95,000? 880 e What number is one hundred mes smaller than 8,800? 0 f What number is ten mes smaller than 8,800? F

18 Mental division strategies halving strategy When the two numbers seem too large to work with in our heads, we can halve them un l we get to a division fact we recognize. Both numbers must be even for this to work. 6 4 (half 6) (half 4) 6 7 = 9 Prac se your halving. The first one has been done for you. a 6 b 4 c halve 7 halve,000 halve Halve each number to get to a recognizable division fact. The first one has been done for you. a = 8 b 44 6 = c 96 = d 0 4 = e 6 8 = Match the problems with their halved equivalents. Then solve the problem. The first one has been done for you. a = 5 b = c = d = e = f = 4 F

19 Mental division strategies halving strategy Some mes we need to keep halving un l we reach an easy division fact = 4 4 Keep halving un l you get to a fact you can work with. If you can do it in your head, just fill in the last box. Otherwise, use the lines to help you. a 6 6 = = = b 96 8 = = = c 4 = = = d 68 4 = = = e 44 6 = = = f 88 7 = = = 5 Draw lines to connect numbers that could be doubled or halved to reach each other Work with a partner to solve this problem using halving: You have an a er school job at the local candy shop, making up the mixed candy bags. Today, you have to evenly share 88 ski les among 48 bags. How many ski les will you put in each bag? Show each halved sum. F 5

20 Mental division strategies split strategy Division problems also become easier if you split the number to be divided into recognizable facts. Look at the problem 44 9 Can we divide 44 into mul ples of 9? We can divide it into 54 and 90. These are both easily divided by 9. Then we add the two answers together = 6 Use the split strategy to divide these numbers. Use the clues to guide you: a 8 b 85 5 c = = = d 64 4 e 9 7 f = + = + = Now try these: 60 6 a 90 6 = 0 6 b = Hmmm 9 7. The ones digit helps me here. What mul ple of 7 ends in? I know,. So that makes the other number 70! c 7 4 = 4 d 44 8 = 96 6 F

21 Mental division strategies split strategy Play this game with a partner. Use one copy of this page between you. Cut out the problems on the le and stack them face up. Cut out and spread the other cards face up. Work together (or race) to find two numbers you could divide to solve the problem on the top card of the pile. One card in the pair will be grey, the other white. For example, if the problem was 76 4, you could locate 6 and 40. copy F 7

22 Mental division strategies divisibility rules Divisibility rules tell us if a number can be divided evenly by another (that is, with no remainders). Use the rules to test out the numbers in the last column. The first two have been done for you: Divisible by Rule Test A number is divisible by if it s even (ends in 0,, 4, 6, or 8). Is 458 divisible by? Yes, because it ends in an even number. 4 5 A number is divisible by if the sum of its digits is divisible by. A number is divisible by 4 if the number made by the last digits is divisible by 4. A number is divisible by 5 if there s a 0 or 5 in the ones place. Is 7,8 divisible by? = 8 Yes, because 8 is divisible by. Is,9 divisible by 4? Is 455 divisible by 5? 6 A number is divisible by 6 if it is divisible by AND. Is 74,60 divisible by 6? Is it divisible by? Is it also divisible by? 9 0 A number is divisible by 9 if the sum of its digits is divisible by 9. A number is divisible by 0 if there is a zero in the ones place. Is 6,45 divisible by 9? Is 5,680 divisible by 0? 8 F

23 Mental division strategies divisibility rules These numbers can all be divided with no remainders. Work with a partner to find the rule or rules that can be used to divide them. Fill in the tables ,05,486 6,99,56 9,050 0, Numbers may go onto more than table! F 9

24 Written methods single-digit multiplication H T O Single-digit mul plica on is one way to solve a mul plica on problem. First we use our mental strategies to es mate an easier problem: 50 = 450. The answer will be around 450. We start with the ones. 6 is 8 ones. We rename this as ten and 8 ones. We put 8 in the ones column and carry the to the tens column. 5 plus the carried is 6 tens. We rename this as hundred and 6 tens. We put 6 in the tens column and carry the to the hundreds column. plus the carried is 4 hundreds. We put 4 in the hundreds column. Solve these problems using single-digit mul plica on. Es mate first: e: e: e: a H T O b H T O c H T O e: e: e: d H T O e H T O f H T O Solve these word problems. Show how you worked them out: a Dan s dad has stooped to bribery to curb Dan s PlaySta on addic on. For every evening, Dan spends away from the PlaySta on, his dad pays him $. So far, Dan has racked up an impressive 7 nights (though he looks like he may give up any day now). How much money does this equate to? b Dan s mom thinks she might get in on the ac on too and pays Dan $4 for every week that he puts his dishes in the dishwasher and his dirty clothes in the basket. Dan is less excited about this plan but does manage weeks in year. How much has he made out of this scheme? 0 F

25 Written methods single-digit multiplication Below are Jess and Harry s tests. Check them and give them a grade out of 5. If they made mistakes, give them some feedback as to where they went wrong. Jess Harry F

26 Written methods multi-digit multiplication Mul plying larger numbers is a process of mul ple steps. First, you mul ply: 54 6 =,5 You have now used up all your ones. Cross out the 6 in the ones place and place a zero in the next column to show this. Now you mul ply: 54 0 = 0, , 0 9 Add up your work:,5 + 0,840 = 4,09 Use a calculator to help you work out the values you could expect when mul plying the following. Check the columns: a a one by a one 9 7 b a ten by a one 4 5 c a hundred by a one 6 7 d a ten by a ten 7 e a ten by a hundred 55 0 T TH TH H T O would give me a one only. But 8 6 would give me tens and ones. I ll check both columns. Prac ce mul plying mul -digit numbers. Es mate first: e: e: e: a 6 7 b 8 9 c F

27 Written methods multi-digit multiplication More prac ce with mul plying mul -digit numbers: a 8 9 b c d 4 4 e f Use mul plica on to solve these problems: a The sixth grade teachers are taking their classes to the Museum of Natural History. There is a total of 5 students in sixth grade and admission to the museum is $ per student. What is the total cost for all sixth graders? b Allison has a steady babysi ng job and makes $00 a month. She has been on the job for 8 months. How much money has she earned? c Tommy and his family want to move to Brazil and stay with their grandparents for 6 months. They aren t sure if the visit will work with their budget. They calculate it will cost $567 per month. How much will the whole trip cost? F

28 Written methods division In division, we use our knowledge of mul plica on to help us. We can split 96 into divided by is 00, so we put a in the hundreds place. 0 divided by is 0, so we put a in the tens place divided by is, so we put a in the ones place. 96 = Divide these numbers: a b c d e f g h i Some mes it s easier to split the numbers differently. We can also split 96 into divided by is 00, so we put a in the 9 6 hundreds place 6 divided by is. We put the in the tens place and the in the ones place. 96 = In these problems, if there are no tens in a number we put a 0 in to show this and also to hold the place of the other numbers! Decide how you ll split these numbers and then divide. Remember to put in zeros as needed. a b c d e F

29 Written methods division with remainders Some mes numbers don t divide evenly. The amount le over is called the remainder. Look at 57 divided by divided by 5 is r 7 divided by 5 is 5 with le over (this is the remainder) This can be wri en as r = 05 r. Divide these digit numbers. Each problem will have a remainder. a r b r c r d r e r f r Divide these digit numbers. Each problem will have a remainder. a r b r c r d r e r f r Solve these problems: a Giovanni s grandma has given him a bag of gold coins to share among himself and his two sisters. There are 47 gold coins altogether. How many does each child get if they re shared evenly? How would you suggest they deal with the remainder? b You have 59 Airheads to add to party bags. Each bag gets 5 Airheads. How many full party bags can you make? F 5

30 Written methods division with remainders There are ways of expressing remainders. How we do it depends on how we d deal with the problem in the real world. Look at: 0 5 r One way is to write r as in the example above. We use this when we don t care about being absolutely precise and when the remainder can t be easily broken up. An example would be sharing 57 jelly beans among 5 people. Solve these problems expressing the remainders as r. a Share 6 blue pencils among 4 people. b Share 5 paper clips among 7 people. 5 We can also express a remainder as a frac on. We do this when we can easily share the remainder. For example, 9 cakes shared among people is 6 and one third each. Solve these problems expressing the remainder as a frac on: 6 9 a Share pizzas among 4 people. b Share 50 sandwiches among people. 6 We express remainders as decimals when we must be absolutely precise. Sharing dollar amounts is a good example of this. We add the cents a er the decimal point to help us. Try these: a Share dollars among 4 people. b Share 7 dollars between people. 7 divided by is. Now we have one dollar le. How how many cents is half of one dollar? F

31 Written methods problem solving We regularly come across mul plica on and division problems in our everyday life. It doesn t ma er which strategy we use to solve them, we can choose the one that suits us or the problem best. One real-life problem is comparing prices to find the best deal. It s easy if the prices and amounts are the same but what if the amounts are different? Use a strategy to help you find the best deal on these: a b oz oz 8 oz 6 oz $.95 $5.4 $.95 $8.50 Best deal is Best deal is c d 0 pack CD Single CD 8 fl oz fl oz $.90 $.75 $.40 $.80 Best deal is Best deal is You go to the gas sta on with your weekly allowance of $5. When you take a $.75 chocolate bar to the counter, they offer you the special of bars for $4.50. Which is a be er deal? Show why. F 7

32 Written methods problem solving You re planning a trip to the Wet and Wild theme park and there are many cket op ons. Use a strategy of your choice and the price list below to answer the following ques ons: Entry -day pass $ Extras 5-minute helicopter ride $4 -day pass $48 0-minute helicopter ride $74 Annual pass $99 0-minute helicopter ride $09 Individual rides $ Sunset cruise $ 0-ride pass $95 Lunch cruise $ Order online $5 discount Swim with the dolphins $75 a If you buy a -day pass, what is the cost per day? b How much cheaper is this op on than buying two -day passes? c If you bought an annual pass, how many mes would you need to visit to make it a be er op on than buying either a -day or -day pass? d What if you choose just the rides? How much would you save if you bought the 0-ride pass instead of the individual rides? e If you took a 5-minute helicopter ride, what would be the cost per minute? f What about if you chose the 0-minute flight op on? What would be the cost per minute? g Plan a day s i nerary for you and a partner. How much will this cost? 8 F

33 Crack the code apply What to do Use the code below to work out the hidden message A A = A A is F = H + L F = M M = M + M M is E = F E = Once I work out the first couple, the rest come easily! T M = A T is L = I I = T + T = H H is ( L) A = C C = H M = L L is F + A = N N = L = U U is T = S S = What to do Try this one: A A = A + A A is L + E = S S is If two le ers are together, we mul ply them to get the answer. A + A = T T is N N = I I is T = N N is U A = C C is AL T = E E is S ( T) = P P is E = L L is U P = O O is E + T = U U is S + E = R R is F 4 9

34 Smart buttons apply Ge ng ready In this ac vity, you ll use your knowledge of mul plica on, division, subtrac on and addi on to find as many number statements you can to create one number. What to do Using ONLY the number, +,, and keys on your calculator, find as many ways as you can to create the number. For example, you could make: + + = 6 = Record your statements on a piece of paper. Now, compare your answers with a partner s. How many did they find? Can you add to each other s lists? What s the longest statement? What s the shortest? What to do Choose another number to make and see how many statements you can find or challenge a partner to a compe on. Set a me limit and see who can find the most ways to make the number 5 within the me span. Bugs investigate Ge ng ready Use your knowledge of mul ples to help you work out how many boy bugs and girl bugs there are in the problem below. Lis ng all the mul ples is a strategy that would help. What to do Girl bugs have 4 dots on their backs, boy bugs have 9. Altogether there are 48 dots. Work out how many girl bugs and how many boy bugs there are. What to do next What if girl bugs have 8 dots and boy bugs have 6 and there are 0 dots altogether? How many different answers can you find? 0 F 4

35 Puzzles apply What to do Use your knowledge of mul plica on to work out the missing values: a 8 b 7 c d 8 e 6 8 f g 6 h 4 i What to do Fill in the mul plica on and division tables by working out the missing digits. The arrows show you some good star ng places F 4

36 Puzzles apply What to do Complete this crossnumber puzzle: Across Down What to do Test your speed and accuracy. Race against a partner or the clock to complete each table: Time: Time: Time: What to do Use the guess and check strategy to solve this problem. You could use a calculator to help if you wish. Tracey paid $.0 for 7 gummy snakes and 4 pieces of taffy. Madison paid $.95 for 4 gummy snakes and 7 pieces of taffy. How much does one gummy snake cost? How much does one piece of taffy cost? If the decimals are confusing me, I can change the amounts to 0 cents and 95 cents. F 4