Multiplication and Division


 Alyson Matthews
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1 E Student Book 6 7 = 4 Name
2 Series E Contents Topic Multiplication facts (pp. 7) 5 and 0 times tables and 4 times tables 8 times table and 6 times tables Date completed Topic Using known facts (pp. 8 ) 9 times table 7 times table square numbers factors and multiples Topic Mental multiplication strategies (pp. ) multiplying by 0 and 00 doubling strategy split strategy compensation choose a strategy doubling and halving Topic 4 Division (pp. 8) division is sharing and grouping division is repeated subtraction linking multiplication and division facts remainders Copyright
3 Series E Contents Topic 5 Mental division strategies (pp. 9 ) dividing by 0 and 00 halving strategy split strategy strategy review Date completed Topic 6 Written methods (pp. 4 9) contracted multiplication extended multiplication short division short division with remainders short division with digit numbers Topic 7 Games and investigations (pp ) triple product apply factor bingo apply doubling strategy to 0 apply symbols solve Series Author: Nicola Herringer Copyright
4 Multiplication facts 5 and 0 times tables The 5 and 0 times tables are easier if you learn them together. Answer the 5 times table: Count in 5s down the ladders: 5 = a 5 b 75 c 40 5 = 5 = 4 5 = 5 5 = 6 5 = 7 5 = Fill in the missing number for each times _ table fact: 8 5 = 9 5 = 0 5 = a 5 = 5 c 5 = 0 e 5 = 5 b 5 = 45 d 5 = 50 f 5 = 40 Turnaround facts _ are the times tables_ turned around! 5 = 5 5 = 5 4 Complete the 5 times table turnarounds. a 5 8 = c 5 0 = b 5 = d 5 4 = E
5 Multiplication facts 5 and 0 times tables 5 Answer the 0 _ times table: 0 = 0 = 6 Write the missing numbers for these _ 5 times table facts: a 5 = 5 7 Write the missing numbers for these _ 0 times table facts: a 0 = 0 0 = b 5 5 = b 0 5 = 4 0 = c 5 = 0 c 0 = = 6 0 = d 5 45 d 0 9 = 7 0 = e 5 = 5 e 0 = = 9 0 = f 5 0 f 0 = = g 5 0 g 0 0 = 8 Follow the arrows by counting up in 0s: 0 9 Multiply each number in the top row by 5 and then by 0: What do you notice? E
6 Multiplication facts and 4 times tables The and 4 times tables are good facts to learn together. Complete the skip counting pattern of : 4 Answer the times table. One is in order, the other is mixed up. = 7 = = 0 = = 6 = 4 = 8 = 5 = = 6 = 9 = It is useful to be able to multiply numbers above 0 by. Try these: = = = 4 = 5 = 6 = 7 = 4 = 7 = 8 = = 8 = 9 = = 9 = 0 = 5 = 0 = 4 Complete these doubling wheels as quickly as you can. _ Multiplying by _ is the same as doubling. a b E
7 Multiplication facts and 4 times tables Now for the 4 times table. The 4 times table is just double the times table. This is handy to remember if you forget a 4 times table fact. 5 The times 6 = 4 = table should be easier, so complete it first. Then double each of the times table facts to get the 4 times table facts: = = 4 = 5 = 4 = 4 = 4 4 = 5 4 = 6 = 6 4 = Write the missing numbers for these _ 4 times table facts: a 4 = 8 b 4 = 6 c 4 = 40 d 4 = 4 7 = 8 = 9 = 0 = 7 4 = 8 4 = 9 4 = 0 4 = e 4 = f 4 = 6 g 4 = 0 h 4 = 8 7 Use the hint to get the answer. Then fill in the missing digit to make the 4 times table fact complete: a Hint: 6 4 = b Hint: 4 = c Hint: 8 4 = 8 Look at the numbers in the grid and circle numbers that would make a multiplication fact. Look for and 4 facts. They are either left to right or top to bottom. The first one has been done for you. There are 0 to find E
8 Multiplication facts 8 times table Here is the 8 times table. You can double the 4 times table to get the 8 times table. Complete the _ 4 times table as quickly as you can. Then after you have checked them, double them to complete the _ 8 times table facts: 4 = 4 = 4 = 4 4 = 8 = 8 = 8 = 4 8 = 5 4 = 5 8 = 6 4 = 6 8 = 7 4 = 7 8 = 8 4 = 8 8 = 9 4 = 9 8 = 0 4 = 0 8 = Use double, double and double again for these problems: a 6 8 = b 4 8 = c 9 8 = If you get stuck on the 8s, think double, double and double again. For example, 8_ Think: double is 6_ double 6 is _ double is 4 On Mia s calculator, the 8 key is broken. Show her the steps she could follow to find the answer to 6 8. Use a calculator to test the steps. E 5
9 Multiplication facts and 6 times tables Here are the times and 6 times tables together. Can you think of why it s better to learn these facts together? Use the picture of the dice above to complete both the _ times _ table and the 6 times table: = = = 4 = 6 = 6 = 6 = 4 6 = Now try these mixed up: a 6 = b 4 = c 8 = 5 = 5 6 = d 9 6 = 6 = 6 6 = e 4 6 = 7 = 7 6 = f 5 = 8 = 8 6 = g 8 6 = 9 = 9 6 = h 9 = 0 = 0 6 = i 5 6 = Fill in the missing digits to make these times table facts complete: a = b = 6 c = 8 d 6 6 e 4 f 6 = 60 g 9 = 7 h 6 4 i 9 54 j 5 0 k 6 = 48 l 7 6 E
10 Multiplication facts and 6 times tables 4 Match the answers to the questions. Each answer has two matching questions Complete the cross number puzzle: Across Down What number am I? I am in the times table, 4 times table and 6 times table. _ I m not. I am E 7
11 Using known facts 9 times table If you get stuck on a 9 times table fact, you can use the 0 times table facts and then build down. 9 =? 0 = 0 So, 9 = 7 Think of the 0 facts and build down to get the 9 facts. The first one is done for you. 0 table Build down by 9 table 0 = 0 9 = 9 0 = 0 0 = = = = = = = = 00 Complete the 9: E
12 Using known facts 7 times table If you get stuck on a 7 times table fact, remember the 8 times table fact and build down. Think of the 8 table fact and build down to get the 7 table fact. 8 table Build down by 7 table 8 = 8 7 = 8 = 6 7 = 8 = 4 7 = 4 8 = 4 7 = 5 8 = = 6 8 = = 7 8 = = 8 8 = = 9 8 = = 0 8 = = Add the missing numbers to each fact: a 7 = 8 b 7 = 5 c 7 = d 7 = 4 e 7 = 49 f 7 = 4 Use the 8 to complete the 7: E 9
13 Using known facts square numbers A square number is a number multiplied by itself. = = 4 = 9 = = 4 = 9 Show these square numbers on the grid and write what they are equal to: a 4 = b 6 = c 5 = d = e 7 = Shade the square numbers on this multiplication grid: E
14 Using known facts factors and multiples When numbers are multipled together, the answer is called a multiple. The first multiples of are, 4, 6. = = 4 = 6 5, 0, 5, 0, 5, 0, 5, 40, 45, 50 are the first 0 multiples of 5. List the first ten multiples of each number: a 6 6 b c 0 d e 4 Write these numbers in the correct spots on the Venn diagram: Multiples of Multiples of The space in the_ diagram where the_ circles overlap is where_ you put numbers that are_ both multiples of and. Can you think of any other numbers up to 60 that could go into the overlapping space in the Venn diagram above? E
15 Using known facts factors and multiples Factors are numbers that you multiply together to give a multiple. 6 = 8 9 = 8 These arrays show some of the factors of 8:, 6, and 9. Can you think of any other factors of 8? Complete the number sentence for each set of arrays and then list the factors. a b c d The factors of are: Complete each diagram to show the factors of the number in the middle circle: a 4 c 0 b 6 E
16 Mental multiplication strategies multiplying by 0 and 00 When we multiply any number by 0, a zero goes in the units column and the digits all move one space along to the left. When we multiply any number by 00, a zero goes in both the units and the tens columns and all the digits move two spaces along to the left. Thousands Hundreds Tens Units Use the place value tables to multiply these numbers by 0 and 00: a Th H T U b Th H T U c Th H T U 7 Can you see a pattern in each of the tables? 0 00 Use patterns to solve these: a 4 = 4 0 = 4 00 = b 5 = 5 0 = 5 00 = c 8 = 8 0 = 8 00 = E
17 Mental multiplication strategies multiplying by 0 and 00 How do you multiply by other multiples of 0? Let s look at 8 0. We can use known times tables facts and write this as place value amounts: 8 tens = 6 tens So, 8 0 = 60 Draw lines from the numbers written as place value amounts to the times tables facts: 0 tens 4 tens 6 tens 7 tens tens 6 tens 4 tens 4 4 tens 5 tens 7 tens 6 6 tens 9 tens Write the digit that represents each place value amount: a 0 tens = d 5 tens = g 9 tens = b 6 tens = e tens = h 6 tens = c tens = f 8 tens = i 8 tens = First complete the hints and then use them to write the facts: Hints: Facts: a 4 6 tens = tens 4 60 = b 9 tens = tens 9 0 = c 7 tens = tens 70 = 4 Complete the number wheels: a b E
18 Mental multiplication strategies doubling strategy There are many double facts that you should know. This includes numbers outside the times tables we have been working on. Here are double facts that are handy to know: double 5 is 0 double 50 is 00 Can you think of more? Complete these function machines: a IN OUT 5 0 b double IN OUT 5 60 Can you see what_ doubledouble is the_ same as? Yes, that s right,_ it s the same as Complete these doubling wheels: 7 8 a 9 b E 5
19 Mental multiplication strategies doubling strategy We also use doubling when we multiply by 4 and by 8. To multiply a number by 4, double it twice. To multiply a number by 8, double it times. 0 4 = 40 0 once 0 0 twice 40 8 = 88 once twice 44 three times 88 Keep doubling to get the 4 and 8 facts. Here are some tables to help you. The first one has been done for you. a 4 = 48 b 5 4 = once 4 twice 48 5 once 5 twice c 8 4 = 8 once 8 twice d 4 = once twice e 6 8 = 6 once 6 twice 6 three times f 5 8 = 5 once 5 twice 5 three times g 8 = In this last table_ choose a digit_ number to multiply_ by 8 and double it _ three times. once twice three times 6 E
20 Mental multiplication strategies split strategy The split strategy is when we multiply numbers in pairs and then add the parts. Let s use the split strategy for 6 4. Split 6 into 0 and 6. Multiply each part. Add the answers together = 04 So, 6 4 = 04 Use the split strategy to answer these: a = So, 4 = b = So, 45 5 = c = So, 5 4 = E 7
21 Mental multiplication strategies compensation Use the compensation strategy to make it easier to multiply digit numbers that are close to a ten. Look at is close to 0, so we can multiply by the next multiple of ten which is 0. Then we build down because we have an extra group of = 80 4 So, 9 4 = 76 Use the compensation strategy to answer these: a So, 5 9 = b 49 So, 49 = c So, 4 9 = Use the compensation strategy to answer these questions. _ This time you need to look for more than one extra group _ to subtract: a We have rounded up_ to 0. So instead of_ 4 8 we have 4 0._ This is more groups_ of 4. So we subtract 8. So, 4 8 = b 7 So, 7 = 8 E
22 Mental multiplication strategies choose a strategy Roll a die to get the missing number, then use either the split or compensation strategy to get the answer. You can place the numbers rolled on the die in any question. a 5 So, 5 b 6 So, 6 c 49 So, 49 d 58 So, 58 E 9
23 Mental multiplication strategies doubling and halving We can change the factors of a multiplication question to make it easier. Look at 6. If we halve the largest factor and double the smaller factor, we make an array on the grid that is the same size. Both arrays have the same amount of squares. Count the squares, are they equal to 8 6? 6 Halve 8 6 = 48 Make these problems easier by using doubling and halving. Shade an array for each: a 8 Halve b 4 4 Halve 0 E
24 Mental multiplication strategies doubling and halving Use the doubling and halving strategy to solve these: a 4 b 48 5 Halve Halve c 6 5 d 64 5 Halve Halve Follow this doubling and halving trail through to the bottom: a Halve b Halve c Halve 8 56 =? 8 5 =? 8 45 =? So, 8 56 = So, 8 5 = So, 8 45 = d What do you notice? E
25 Division division is sharing and grouping Division can mean sharing or grouping. There are lollies shared between 4 kids. How many are in each share? 4 = There are 6 apples and 4 go into each basket. How many baskets do I need? 6 4 = 4 Solve these sharing and grouping questions: a There are 9 cupcakes and kids are sharing. How many are in each share? = b lollies are shared between a group of kids so they each get. How many kids are sharing? = c There are 4 pencils and 6 pencil pots. How many pencils go into each pencil pot? = E 4
26 Division division is sharing and grouping Draw pictures to show these division questions. Then write the division fact and decide whether it is a sharing or a grouping question. If you need to find out_ how many items there_ are in each share, it s_ a sharing question. If_ you need to find out the number of equal shares,_ it s a grouping question. a Divide 6 lollies between 4 girls. How many does each girl get? = sharing / grouping b From a packet of 4 pencils, each person will get 6. How many people are sharing the pencils? = sharing / grouping c 48 eggs are laid by 6 hens. If they all laid the same amount, how many did each hen lay? = sharing / grouping E 4
27 Division division is repeated subtraction Division can also be thought of as repeated subtraction. Look at 0 5 =? This question is asking how many groups of 5 there are in 0. Jump in 5s along the number line and then count the jumps So, 0 5 = 6 Show these division facts as repeated subtraction. First label the number lines and then show the jumps. a 6 6 = 0 6 b = 0 Write a division fact to match these number lines. Show the jumps. a = b = E 4
28 Division linking multiplication and division facts Knowing multiplication facts will help with division facts. This is because they are opposites. Look at how we can describe this array: 6 4 = 4 6 groups of 4 is = 4 4 groups of 6 is = 6 4 divided into 4 shares is = 4 4 divided into 6 shares is 4. Describe each of these arrays using two multiplication and two division facts: a = = b = = c = = d = = Draw an array of 6 rows of then describe it with multiplication and division facts. This is also called_ a fact family. = = E 4 5
29 Division linking multiplication and division facts Write a fact family for each set of numbers in the triangle. The first one has been done for you. a 5 7 = = = = 5 b 7 = 9 = c 48 = 6 8 = d 40 = 8 5 = 4 For these problems, think of a multiplication fact to help write the division fact: a 5 is shared between 5 people. How much does each person get? = b 45 people get into 9 cars. How many people are in each car? = 6 E 4
30 Division remainders Sometimes division is not exact. From, we can make fair shares of 6 with left over. We call the left over the remainder. 6 = remainder In each array, ring the fair shares to see the remainder: a 5 = remainder b 9 6 = remainder c 7 = remainder d 9 = remainder E 4 7
31 Division remainders Now use your multiplication facts. 5 6 =? Think 4 6 = 4 + is 5 So, 5 6 = 4 remainder Use your multiplication facts to write the division facts and the remainder: a 0 =? Think + is So, = remainder b 0 4 =? Think + is So, = remainder c 7 9 =? Think + is So, = remainder Complete each word problem: a 9 pencils were shared between 6 kids. How many did each kid get? = remainder b 4 fish were divided between 6 tanks. How many fish are in each tank? = remainder c From 7 flowers, 5 flowers were arranged in each vase. How many vases were used? = remainder 4 Write in the missing digit to make this statement true: 6 = 8 remainder 8 E 4
32 Mental division strategies dividing by 0 and 00 When we divide any number by 0, we move the number one place value space to the right. When we divide any number by 00, we move the number two place value spaces to the right. Thousands Hundreds Tens Units Use the place value tables to divide these numbers by 0 and 00. a Th H T U b Th H T U c Th H T U d Th H T U Use patterns to solve these: a 400 = = = b = = = b 500 = = = Use a calculator to solve these: a = b 49 0 = E 5 9
33 Mental division strategies halving strategy When you halve numbers you are dividing them by. In this function machine, numbers go IN, have the rule applied and come OUT again. IN 8 4 RULE: Halve OUT 4 6 Complete the halving function machines. Halve the number going IN the machine and write the answer in the OUT column: a IN OUT b IN OUT 80 RULE: 70 RULE: 40 Halve 4 Halve 0 6 c IN OUT d IN OUT 4 RULE: 8 RULE: 90 Halve 50 Halve Below is a halvinghalving function machine. The number goes IN and is halved and then halved again and comes OUT. IN OUT RULE: Halve RULE: Halve 0 E 5
34 Mental division strategies halving strategy We also use halvinghalving to divide by 4. Look at these diagrams: Halve 6 once Halve 6 twice Use the tables for halvinghalving to divide by 4: a 80 4 = Halve 80 once Halve 80 twice b 48 4 = Halve 48 once Halve 48 twice c 64 4 = Halve 64 once Halve 64 twice d 0 4 = Halve 0 once Halve 0 twice e 44 4 = Halve 44 once Halve 44 twice f 88 4 = Halve 88 once Halve 88 twice 4 Complete the division wheels: a b E 5
35 Mental division strategies split strategy Division problems can be much easier to solve if you split the number. Look at 5 5. Can we split the number into two multiples of 5? Yes, we can split 5 into 00 and We divide each part by 5 and then add the two answers together = 5 Use the split strategy to divide these by 5: a 5 5 b = = Use the split strategy to divide these by 4: a 64 4 b = = Use the split strategy to divide these by : a 0 b 6 + = + = E 5
36 Mental division strategies strategy review Review your division strategies. Use either the halving strategy or the split strategy to complete the tables. The first one has been done for you. a Use the split strategy: b Use the halving strategy: 48 = = 48 is = 0 and 8 = = 6 c Use the split strategy: d Use the halving strategy: = 40 4 = Solve this riddle by matching the letter to the answer. Use a mental division strategy for each problem. What is it that the more you take, the more you leave behind? 68 4 = s 90 6 = p 5 5 = e 00 0 = f 40 4 = o 8 4 = t E 5
37 Written methods contracted multiplication H T U Start with the units. 4 = units. Rename this as ten and units. Put the in the units column and regroup the to the tens column. 5 plus the regrouped is 6 tens. Rename this as hundred and 6 tens. Practise these problems: a H T U b H T U c H T U d H T U e H T U f H T U Use contracted multiplication to solve these word problems: a On a farm, 6 lambs were born every day over 5 days. How many lambs were born in total? H T U b For my school fete day, I baked 9 trays of cupcakes. If there are 4 cupcakes on each tray, how many did I bake in total? H T U 4 E 6
38 Written methods extended multiplication H T U 4 ( 4) 9 0 ( 0) 0 In extended multiplication, we multiply the units and tens separately, then add the answers together. Practise these problems: a H T U b H T U (4 ) (5 6) (4 0) (5 0) c H T U d H T U ( ) ( ) ( ) ( ) Use extended multiplication to solve this word problem: H T U In a pet store, there are 7 tanks of tropical fish with 4 fish per tank. How many fish are there altogether? ( ) ( ) E 6 5
39 Written methods short division Another way to represent division is with the division symbol. T U This is the same as 6 6 = 6 If the answer is a single digit, it should go in the units column. Solve these division problems using the division symbol: a b c d e f g h i Use the division symbol to solve each problem: a 4 cupcakes were iced by 7 kids. If they each iced the same amount, how many did they ice each? b How many pots were used if 6 seeds were planted in each pot from a packet of 54? c I run the same distance each day. Over 9 days the total distance is 7 km. How far did I run each day? 6 E 6
40 Written methods short division with remainders This is the way we write remainders when using the division symbol. 6 5 r This is the same as 5 6 = remainder. Check your work with the closest multiplication fact: 6 = Then add on the remainder: + = 5 Solve these division problems and then check them. a r Check with the multiplication fact and add the remainder: b r Check with the multiplication fact and add the remainder: c r Check with the multiplication fact and add the remainder: d r Check with the multiplication fact and add the remainder: What is the question if I am checking with this multiplication fact? r 5 6 = 0 + E 6 7
41 Written methods short division with digit numbers In short division with digit numbers we split the number: 468 is divided by is 00, so we put a in the hundreds place. 60 divided by is 0, so we put a in the tens place. 8 is divided by is 4, so we put a 4 in the units place. H T U Practise splitting these: a 68 is + + c 567 is + + b 445 is + + d 5 is + + Now put these split numbers back together: a is c is b is d is Solve these division problems with digit numbers: a b 6 9 c 8 4 d Here are two division problems with missing numbers in the questions. Find out the missing numbers by using the numbers that are part of the answer as clues. a 4 4 b 6 8 E 6
42 Written methods short division with digit numbers Sometimes we need to split the number a different way, for example: 55 = divided by 5 is 00, so we put a in the hundreds place. 5 divided by 5 is, so we put a in the units place. What goes in the tens place? A zero does. The zero has the very important job of keeping the other numbers in their place! H T U Practise these problems. We have put the zero in to remind you: 0 0 a 4 8 b c 9 d Practise these problems. This time, you need to remember the zero! a 9 8 b 6 6 c 4 8 d E 6 9
43 Triple product apply Getting ready This is a game for players. You will need a copy of this page, 6 counters each and dice. copy What to do Player rolls all dice and chooses of the numbers to multiply. If the player can see the answer in the grid, they claim this number by placing a counter over the number. Then Player has a turn. The winner is the first to place all 6 counters on the grid E 7
44 Factor bingo apply Getting ready This is a game for three players. Each player needs a copy of this page. The caller needs a pile of the numbers from to 9. copy What to do Each multiplication grid contains all the answers, while the factors are missing. Remember factors are the numbers that you multiply to get the answer. The aim of the game is to be the first player to fill their grid with the factors. One hint is provided in each grid to start you off. Choose one person to be the caller and the other two play the round. The caller picks a number without looking and reads it out to the players. The players write it on the grids, if it fits as a factor. The first to fill in one of the grids completely is the winner E 7 4
45 Doubling strategy to 0 apply Getting ready This is a game for two players. You will need a copy of page 4, a die and a pencil to write down your scores. You may like to make extra copies of page 4 to play again later. copy What to do The aim of this game is to score the highest number of points each time without going over 0. Roll the dice and choose which strategy you will use. From the Strategy column, circle for double, for doubledouble or for doubledoubledouble. For example, Player has rolled a 5 and has chosen strategy doubledoubledouble. This makes a score of 40 but because it is over 0 it doesn t count. Look at the rest of the sample game to see how the game turned out. Strategy Strategy Strategy Sample game Player Die Strategy Score Total 56 Player Die Strategy Score Total 5 4 E 7
46 Doubling strategy to 0 apply Strategy Strategy Strategy Player Die Strategy Score Total Player Die Strategy Score Total E 7 4
47 Symbols solve What to do Can you work out the value of each symbol? The values are,, 4, 6, 8, 9 and. Remember, the same symbol means that it s the same number. = = = = = = = 44 E 7
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