Multiplication and Division

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1 Series Student Multiplication and Division My name E

2 Copyright 009 P Learning. All rights reserved. First edition printed 009 in Australia. A catalogue record for this book is available from P Learning Ltd. ISBN Ownership of content The materials in this resource, including without limitation all information, text, graphics, advertisements, names, logos and trade marks (Content) are protected by copyright, trade mark and other intellectual property laws unless expressly indicated otherwise. You must not modify, copy, reproduce, republish or distribute this Content in any way except as expressly provided for in these General Conditions or with our express prior written consent. Copyright Copyright in this resource is owned or licensed by us. Other than for the purposes of, and subject to the conditions prescribed under, the Copyright Act 968 (Cth) and similar legislation which applies in your location, and except as expressly authorised by these General Conditions, you may not in any form or by any means: adapt, reproduce, store, distribute, print, display, perform, publish or create derivative works from any part of this resource; or commercialise any information, products or services obtained from any part of this resource. Where copyright legislation in a location includes a remunerated scheme to permit educational institutions to copy or print any part of the resource, we will claim for remuneration under that scheme where worksheets are printed or photocopied by teachers for use by students, and where teachers direct students to print or photocopy worksheets for use by students at school. A worksheet is a page of learning, designed for a student to write on using an ink pen or pencil. This may lead to an increase in the fees for educational institutions to participate in the relevant scheme. Published P Learning Ltd For more copies of this book, contact us at: Designed P Learning Ltd Although every precaution has been taken in the preparation of this book, the publisher and authors assume no responsibility for errors or omissions. Neither is any liability assumed for damages resulting from the use of this information contained herein.

3 Series E Contents Topic Multiplication facts (pp. 4) 5 and 0 times tables and 4 times tables 8 times table and 6 times tables 6 times tables 7 times tables 9 times table times table times table Date completed Topic Using known facts (pp. 5 6) factors and multiples Topic Mental multiplication strategies (pp. 7 9) multiplying by 0 and 00 multiplying/dividing by 0 and multiplying numbers doubling strategy split strategy compensation strategy choose a strategy doubling and halving word problems Topic 4 Division (pp. 0 4) division is sharing and grouping division is repeated subtraction linking multiplication and division facts

4 Series E Contents Topic 5 Mental division strategies (pp. 5 4) dividing by 0 and 00 halving strategy split strategy word problem Date completed Topic 6 Written methods (pp. 4 46) short multiplication short division short division with -digit numbers Topic 7 Patterns and algebra (pp ) skip counting completing and describing patterns predicting repeating patterns predicting growing patterns function machines understanding equivalence balanced equations using + and using symbols for unknowns Topic 8 Games and investigations (pp. 59 6) Series Author: Nicola Herringer triple product apply factor bingo apply doubling strategy to 0 apply symbols solve Copyright

5 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 = 5 = a 5 b 75 c 40 5 = 4 5 = 5 5 = 6 5 = 7 5 = 8 5 = 9 5 = 0 5 = Fill in the missing number for each times table fact: a 5 = 5 b 5 = 45 5 = c 5 = 0 d 5 = 50 5 = e 5 = 5 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

6 Multiplication facts 5 and 0 times tables 5 Answer the 0 times table: 0 = 6 Write the missing numbers for these 5 times table facts: 7 Write the missing numbers for these 0 times table facts: 0 = a 5 = 5 a 0 = 0 0 = 4 0 = b 5 5 = b 0 5 = 5 0 = 6 0 = c 5 = 0 c 0 = = d 5 45 d 0 9 = 8 0 = 9 0 = e 5 = 5 e 0 = = f 5 0 f 0 = 70 0 = 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

7 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 = It is useful to be able to multiply numbers above 0 by. Try these: = = 0 = 4 = 4 = 5 = 6 = 8 = 5 = 6 = = 6 = 7 = 8 = 9 = 4 = 7 = 9 = = 8 = 0 = = = 5 = 9 = = = 0 = 4 Complete these doubling wheels as quickly as you can. Multiplying by is the same as doubling. a 6 0 Double 8 5 b Double E

8 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 = 4 = 6 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 = 6 = 7 = 4 = 4 = 4 4 = 5 4 = 6 4 = 7 4 = Write the missing numbers for these 4 times table facts: a 4 = 8 b 4 = 6 c 4 = 40 d 4 = 4 8 = 9 = 0 = = = 8 4 = 9 4 = 0 4 = 4 = 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: Double 6 4 = b Hint: Double 4 = c Hint: Double 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

9 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 = 5 4 = 8 = 8 = 8 = 4 8 = 5 8 = 6 4 = 6 8 = 7 4 = 7 8 = 8 4 = 8 8 = 9 4 = 9 8 = 0 4 = 0 8 = 4 = 8 = 4 = 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

10 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 = 5 = 6 = 6 = 6 = 4 6 = 5 6 = Now try these mixed up: a 6 = b 4 = c 8 = 6 = 6 6 = d 9 6 = 7 = 7 6 = e 4 6 = 8 = 9 = 0 = = 8 6 = 9 6 = 0 6 = 6 = f 5 = g 8 6 = h 9 = = 6 = i 5 6 = Fill in the missing digits to make these times table facts complete: a = d 6 6 g 9 = 7 j 5 0 b = 6 e 4 h 6 4 k 6 = 48 c = 8 f 6 = 60 i 9 54 l 7 6 E

11 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

12 Multiplication facts 6 times table You know more times tables facts than you + realise. For example, knowing your 5 can help with your 6. + The array shows rows of 5. If we add another + dot to each row we can change rows of 5 to rows of 6. This is called building up. 5 = = 8 Change these 5 arrays into 6 arrays. a = + 6 = b = = Complete this table to show how to change a 5 array to a 6 array by building up. The first one has been done for you. 5 Number to add 6 a 5 = 5 6 = 8 b 5 = 0 c 7 5 = 5 d 4 5 = 0 e 6 5 = 0 f 9 5 = 45 8 E

13 Multiplication facts 7 times table Practise your 7 times table. Use this array to complete 7 = the 7 times table: 7 = 7 = 4 7 = 5 7 = 6 7 = 7 7 = 8 7 = 9 7 = 0 7 = 7 = 7 = Fill in the missing numbers: a 7 = 6 b 7 = 4 c 7 = d 7 = 8 e 7 = 5 f 7 = 49 g 7 = 56 Complete these 7 facts. Look out for turnarounds. a 4 7 = b 7 7 = c 7 = d 7 5 = e 9 7 = f 7 = 4 Solve these problems. a Boxes of oranges hold 8 oranges each. If I have 7 boxes, how many oranges do I have altogether? b Our hockey team scored goals in each of our 7 games. How many goals did we score in total? c There are 5 frogs in the glass cases at the zoo. Each case hold 7 frogs. How many cases are there? E 9

14 Multiplication facts 7 times table If you get stuck on a 7 times table fact, remember the 8 times table fact and build down. 5 Think of the 8 table fact to get the 7 table fact. 8 table Number to subtract 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 = = 8 = 88 7 = 8 = 96 7 = 6 Add the missing numbers to each fact: a 7 = 8 d 7 = 4 b 7 = 5 e 7 = 49 c 7 = f 7 = 4 7 Use the 8 to complete the 7: E

15 Multiplication facts 9 times table Practise your 9 times table. Use this array to complete the 9 times table: 9 = 9 = 9 = 4 9 = 5 9 = 6 9 = 7 9 = 8 9 = 9 9 = 0 9 = 9 = 9 = Complete these 9 facts. Look out for turnarounds. a 9 = d 9 = b 9 4 = e 9 5 = c 6 9 = f 9 = Find the cost of these items: a 6 fruit salads = c mango juices = e banana splits = b 4 banana splits = d 5 fruit salads = f 7 mango juices = Mango juice Banana split 6 Fruit salad 9 E

16 Multiplication 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 If you want to check whether a number is in the 9 times table add its digits together. If the answer is 9, then it is! For example, if you add the digits of 7 together, you get 9 ( + 7 = 9), so you know that 7 is in the 9 times table. 4 Think of the 0 facts and build down to get the 9 facts. The first one is done for you. 0 table Number to subtract 9 table 0 = 0 9 = 9 0 = 0 0 = = = = = 70 Can you see a pattern in the numbers in the 9 times table? As the numbers get larger the tens digit goes up one and the ones digit goes down one. 8 0 = = = 00 0 = 0 0 = 0 5 Complete the 9: E

17 Multiplication facts times table Practise your times table. Can you see the pattern? Use this array to complete the times table: = = = 4 = 5 = 6 = 7 = 8 = 9 = 0 = = = Complete these facts. Look out for turnarounds. a = d 4 = b 5 = e 9 = c 7 = f 8 = Solve these problems. a There are players in a football team and 0 teams in the league. How many players are there in total? b On each of our 6 class tables is a pot containing pencils. How many pencils are there altogether? c On our school trip we split our class of into groups of. How many children were in each group? E

18 Multiplication facts times table Practise your times table. Use this array to complete the times table: = = = 4 = 5 = 6 = 7 = 8 = 9 = 0 = = = Complete these facts. Look out for turnarounds. a = d 4 = b 5 = e = c 7 = f 9 = Solve these problems. a I make batches of cookies. How many cookies is this altogether? b A florist is selling bunches of roses. She sells 6 bunches. How many roses is this? c Eggs cost for a dozen? If I spend 5 on eggs, how many eggs have I bought in total? 4 E

19 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 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 5

20 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? 4 Complete the number sentence for each set of arrays and then list the factors. a b c d The factors of are: 5 Complete each diagram to show the factors of the number in the middle circle: a 4 c 0 b 6 6 E

21 Mental multiplication strategies multiplying by 0 and 00 When we multiply any whole number by 0, the number is getting 0 times bigger. This means that each digit moves one place value column to the left and we use 0 as a place holder in the ones column. When we multiply any whole number by 00 the number gets 00 times bigger. This means that each digit moves two place value columns to the left and we use 0 as a place holder in the ones and tens columns. Thousands Hundreds Tens Units Use the place value tables to multiply these numbers by 0 and 00: a Th H T O b Th H T O c Th H T O 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 7

22 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 4 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 = 5 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 = 6 Complete the number wheels: a b E

23 Mental multiplication strategies multiplying/dividing by 0 and If you multiply by 0 the answer will always be means 5 lots of 0, which is nothing. The answer is not going to change, whether you have 5 or 5 or,005 lots of nothing. The answer will always be zero. Multiplying by is also very simple. 8 means 8 lots of. 7 means 7 lots of, which is 7. So if you multiply any number by the answer will always be the number with which you started. Dividing by is straightforward too. If we divide a number, we are working out how many equal groups can be made from that number. So, 0 means we have 0 and we want to make one group with it. How many will be in that one group? The answer is 0. So, as with multiplying by, you always end up with your starting number when you divide by. = = = (In case you are wondering, you can t divide by 0. We can t split, say, a bag of sweets into groups of nothing it doesn t make any sense to divide a number by zero. It can t be done. We say that dividing by 0 is undefined.) Solve these calculations: a 6 = b 9 = c 0 = d 59 0 = f 4 = e 7 = g 848 = = = 0 = 0 0 = impossible! h = i 44 = j 999 = k 0 0 = l 44 = m 74 0 = E 9

24 Mental multiplication strategies multiplying numbers There is a law in maths called the Commutative Law. This states that for certain types of calculation, the order of the numbers doesn t matter. The answer will be the same. It is true for addition. + 4 = = 7 The same is true for multiplication = = 7 5 = 0 5 = = = 56 If you are multiplying more than two numbers, the Commutative Law still applies. 6 = 6 6 = 6 6 = 6 6 = 6 6 = 6 6 = 6 Solve these multiplications: a 4 4 = b 0 5 = c 7 = d 5 8 = e 6 = f = Using the Commutative Law, create two different correct multiplications using the same numbers: Does the Commutative Law work for subtraction and division too? a 8 = b = E

25 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 Double IN OUT 5 0 b Double-double IN OUT 5 60 Can you see what double-double is the same as? Yes, that s right, it s the same as Complete these doubling wheels: 7 8 a Double 9 b 4 Double E

26 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 Double 0 once 0 Double 0 twice 40 8 = 88 Double once Double twice 44 Double 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 = Double once 4 Double twice 48 Double 5 once Double 5 twice c 8 4 = Double 8 once Double 8 twice d 4 = Double once Double twice e 6 8 = Double 6 once Double 6 twice Double 6 three times f 5 8 = Double 5 once Double 5 twice Double 5 three times g 8 = In this last table choose a -digit number to multiply by 8 and double it three times. Double Double Double once twice three times E

27 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

28 Mental multiplication strategies compensation strategy 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 = We have rounded up Use the compensation strategy to answer these questions. to 0. So instead of This time you need to look for more than one extra group 4 8 we have 4 0. to subtract: This is more groups of 4. So we subtract 8. a So, 4 8 = b 7 So, 7 = 4 E

29 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 5

30 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 larger 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 Double 8 6 = 48 Make these problems easier by using doubling and halving. Shade an array for each: a 8 Halve Double b 4 4 Halve Double 6 E

31 Mental multiplication strategies doubling and halving Use the doubling and halving strategy to solve these: a 4 b 48 5 Halve Double Halve Double c 6 5 d 64 5 Halve Double Halve Double Follow this doubling and halving trail through to the bottom: a Halve Double b Halve Double c Halve Double 8 56 =? 8 5 =? 8 45 =? So, 8 56 = So, 8 5 = So, 8 45 = d What do you notice? E 7

32 Mental multiplication strategies word problems When you are faced by a word problem, read it carefully. Ask yourself What are the important numbers? Which key words give clues to the correct operation? Jim makes boxes of biscuits for his 5 friends. There are 6 biscuits in each box. How many biscuit does he make altogether? Important numbers: 5 friends 6 biscuits in each box Key words/operations: altogether suggests multiplication 5 6 Strategy: split 5 6 = 5 0 and = = = 80 If I buy 4 packets of sweets and each packet contain 6 sweets, how many sweets will I have altogether? Every minute I complete one length of the swimming pool. How many lengths will I have swum in one hour? 8 E

33 Mental multiplication strategies word problems Jimmy lines up his soldiers in lines of 9. If he has 8 lines, how many soldiers does he have? 4 There are 5 toys in a tin. Lily has 8 tins. How many toys does she have altogether? 5 Eggs come in boxes of a dozen. Our local shop has 4 boxes on its shelves. How many eggs is this? 6 Mike and I are playing darts. On my first throw I score double 5s. Mike s score is twice mine. How many do we score between us? Read carefully! What are the important numbers? What are the key words? What operations do I need? What is the best strategy? E 9

34 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 0 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? = 0 E 4

35 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

36 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

37 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

38 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? = 4 E 4

39 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 because the number is getting 0 times smaller. When we divide any Thousands Hundreds Tens Ones number by 00, we move the number two place value spaces to the right because the number is getting 00 times smaller Use the place value tables to divide these numbers by 0 and 00. a Th H T O b Th H T O c Th H T O d Th H T O Use patterns to solve these: a 400 = = = b 5600 = = = c 500 = = = Use a calculator to solve these: a = b 49 0 = E 5 5

40 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 halving-halving function machine. The number goes IN and is halved and then halved again and comes OUT. IN OUT RULE: Halve RULE: Halve 6 E 5

41 Mental division strategies halving strategy We also use halving-halving to divide by 4. Look at these diagrams: Halve 6 once Halve 6 twice Use the tables for halving-halving 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 7

42 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 + = + = 8 E 5

43 Mental division strategies word problem 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 9

44 Mental division strategies word problem Remember the steps and questions to ask yourself when you are trying to solve a word problem. Four friends have a party. They share out all the food equally. There are 64 blueberries in total. How many do they get each? Important numbers: 4 friends 64 blueberries Key words/operations: share = multiplication 64 4 Strategy: halving 64 = 8 8 = 4 Tom, Milo and Xav have been trick and treating. They agree to share their sweets out equally between them. They have sweets in total. How many do they get each? 4 Lillies are sold in bunches of 7. A florist has 4 lillies. How many bunches can she make? Read carefully! What are the important numbers? What are the key words? What operations do I need? What is the best strategy? 40 E 5

45 Mental division strategies word problem 5 Jon needs to buy some files. They cost 9 each. He has 7. How many files can he buy? 6 Andy loves astronomy. He s worked out that he can see about 000 stars with his new telescope. If there are about 00 stars visible in any one galaxy. How many galaxies can he see? 7 Kate has been planting trees. She has planted a total of 55 trees in rows of 5. How many rows has she planted? 8 Charles is saving up to buy a new bike. The bike costs 70. He gets 74 for his birthday, and 4 pocket money a week. How many weeks will he have to save until he can get the bike? E 5 4

46 Written methods short multiplication H T O Start with the ones. 4 = ones. Rename this as ten and ones. Put the in the ones 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 O b H T O c H T O d H T O e H T O f H T O g H T O h H T O i H T O E 6

47 Written methods short multiplication Solve these multiplications: a Th H T O b Th H T O c Th H T O d Th H T O e Th H T O f Th H T O Use short 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 O 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 O E 6 4

48 Written methods short division Another way to represent division is with the division symbol. T O This is the same as 6 6 = 6 If the answer is a single digit, it should go in the ones column. Solve these division problems using the division symbol: a 5 5 b 4 8 c 9 8 d e 4 f 4 6 g 5 5 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? 44 E 6

49 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 ones place. H T O 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 E 6 45

50 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 ones 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 O 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

51 Patterns and algebra skip counting Using a 00 square can help us to identify skip counting patterns. Colour the counting pattern on each 00 square: a Count in 6s. b Count in 7s c Count in 9s d Count in s and 6s. Shade the s and circle the 6s e Look at the completed number square in question d. Describe the pattern that you see. What is the relationship between counting in s and 6s? Explain your answer. E 7 47

52 Patterns and algebra skip counting Complete these number patterns by looking for skip counting patterns. a b c Colour the skip counting pattern for s up to 0. If you kept going on a complete hundred grid, would 5 be coloured in? How can you tell without using a whole hundred grid? 4 Only numbers are shaded in each of the skip counting patterns below. Work out the pattern and complete the shading: a b This shows a skip counting pattern of: This shows a skip counting pattern of: 48 E 7

53 Patterns and algebra completing and describing patterns This is a pattern involving multiplication. The pattern begins at. The rule is: multiply by , Figure out the missing numbers in each pattern and write the rule. a b Rule: Rule: Some number patterns can be formed with two operations each time. For example: The rule is to multiply by and add each time. Complete these number patterns, by following the rules written in the diamond shapes. Describe the rule underneath. + 5 x + 5 x + 5 x The rule is Roll a die to make the starting number. Continue the sequence by following the rule: a Rule: + 4 b Rule: + c Rule: + E 7 49

54 Patterns and algebra predicting repeating patterns When we use number patterns in tables, it can help us to predict what comes next. Look at the table below and how we can use it to predict the total number of sweets needed for any number of children at a party. This table shows us that sweet bag contains 8 sweets and bags contain 6 sweets. We can see that the rule for the pattern is to multiply the top row by 8 to get the bottom row each time. Number of sweet bags Number of sweets To find out how many sweets are in 0 bags, we don t need to extend the table, we can just apply the rule. 0 8 = 80. So, 0 bags contain 80 sweets. This helps us plan how many sweets are needed for a party. Complete the table for each problem: a Tom receives 5 a week pocket money as long as he does all his chores. How much pocket money does Tom get after 0 weeks? Weeks Pocket money 5 0 b A flower has 7 petals. How many petals are there in a bunch of 0 flowers? Flowers Number of petals 7 4 c A flag has 6 stars. How many stars are there on 0 flags? Flags Number of stars 6 d At a pizza party, each person eats pieces of pizza. How many pieces of pizza do 0 people eat? Guests Pizza pieces 9 50 E 7

55 Patterns and algebra predicting repeating patterns Each of these kids wrote the first numbers of a skip counting pattern of 6, starting at different numbers. Each kid s sequence goes down the column. Imagine the sequence continues. Mel Brianna Brad Gen Jo Kate a Who had the number 4 in their column? b Who had the number 50 in their column? Look at each pattern of shapes and complete the table below: Repeat section Number of circles Number of triangles Show what this entire sequence would look like with 0 repeat sections: Look for the section that repeats. What is it made up of? This is the rule. E 7 5

56 Patterns and algebra predicting growing patterns Number patterns in tables can help us with problems like this. Mia is making this sequence of shapes with matchsticks and wants to know how many she will need for 0 shapes. Shape Shape Shape Shape number Number of matchsticks To find out how many matchsticks are needed for 0 triangles, we don t need to extend the table, we can just apply the function rule: Number of matchsticks = Shape number Complete the table for each sequence of matchstick shapes and find the number of matchsticks needed for the 0th shape. a Shape Shape Shape Shape number Number of matchsticks 4 b Shape Shape Shape Shape number Number of matchsticks 6 c Shape Shape Shape Shape number Number of matchsticks 7 5 E 7

57 Patterns and algebra predicting growing patterns Look at these growing patterns. Complete the table and follow the rule to draw Picture 5: a Picture Picture Picture Picture 4 Picture 5 l l l l l l l l l l l l l l l l Picture number 4 5 Number of dots 5 7 Rule Picture number = Number of dots b Picture Picture Picture Picture 4 Picture 5 Picture number Number of squares Rule Picture number + = Number of squares How many squares will Picture 8 have? E 7 5

58 Patterns and algebra function machines This is a function machine. Numbers go in, have the rule applied, and come out again. IN 6 RULE: 4 OUT Look carefully at the numbers going in these function machines and the numbers coming out. What is the rule? a b IN 8 RULE: OUT 48 IN 8 RULE: OUT What numbers will come out of these function machines? a b IN 8 08 RULE: 9 OUT IN 8 5 RULE: 7 OUT 45 What numbers go in to these number function machines? a b IN RULE: 9 OUT 90 6 IN RULE: 6 OUT E 7

59 Patterns and algebra understanding equivalence Look at these balanced scales. On one side there is the sum 4 and on the other side there is a total of triangles. This makes sense because it shows the equation 4 =. Equation is another word for a sum. With equations, both sides must be equal. 4 4 = Balance each set of scales by writing a number in the box that is equivalent to the total number of shapes. Then write the matching equation. 6 a 4 b Balance each set of scales by writing a number in the box. Then write the matching equation a 6 54 b E 7 55

60 Patterns and algebra balanced equations using + and There are different equations we could write for one set of balanced scales = 4 8 = 4 Work out the values of the symbols in each problem: a = b c d E 7

61 Patterns and algebra balanced equations using + and Find the values of these symbols: a If is 5, what is the value of? 5 = 5 = b If is 8, what is the value of? 8 = 6 = Find the values of both symbols from the clues: a If both sides are equal to 6, what is the value of each symbol? = = b If both sides are equal to 0, what is the value of each symbol? 5 = 5 = = E 7 57

62 Patterns and algebra using symbols for unknowns Write an equation for these word problems. Write an equation using a s for the unknown number. a Bec collects stickers. She has 48 bumper stickers, glitter stickers and 5 smiley face stickers. How many stickers does Bec have in her collection? = s s = b Charlie saved 5 a week of his pocket money over 8 weeks but then spent 5. How much did Charlie have at the end of 8 weeks? s = c 5,000 people are spectators at a football match.,700 are there to support Team A while the rest are there to support Team B. How many spectators support Team B? s = In this triangle, the numbers on the sides are the totals. So 0 + = Work out the value of the other symbols: = 0 = 58 E 7

63 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 8 59

64 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 8

65 Doubling strategy to 0 apply Getting ready This is a game for two players. You will need a copy of page 6, a die and a pencil to write down your scores. You may like to make extra copies of page 6 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 double-double or for double-double-double. For example, Player has rolled a 5 and has chosen strategy double-double-double. 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 Double Double Double Double Double Double Sample game Player Die Strategy Score Total 56 Player Die Strategy Score Total 5 E 8 6

66 Doubling strategy to 0 apply Strategy Strategy Strategy Double Double Double Double Double Double Player Die Strategy Score Total Player Die Strategy Score Total 6 E 8

67 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. = = = = = = = E 8 6

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