Max Maths - Making Real-World Connections SB5 FULL

MAKING REAL-WORLD CONNECTIONS

maths a ×

MAKING REAL-WORLD CONNECTIONS M

CAROLINE FAWCUS

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Contents

Rounding decimals to the nearest tenth Decimals with a total of 1 and 10

How to UseThis Book

The four operations 1 Creating number sentences 3 Adding and subtracting mentally 8 Written method for multiplication 9 Written method for division 11 Word problems with the four operations 14

Length, mass and capacity Reading scales

93 95

Using decimals to record measurements Ordering measurements in mixed units

105 Calculating with measures 107

Factors and multiples Factors and multiples

15 17 23 25 29 31 36 41 43 45 49 53 55

Data handling

109 111 114

Statistical investigations

Common factors

Averages The range

Using factors to multiply

117 10 Fractions and decimals 119 Doubling and halving decimals 121 Adding and subtracting decimals 123 Converting between fractions and decimals 125 Adding fractions 128 11 2D shapes 131 Triangles 133 Quadrilaterals 135 Circles 137 2D shapes on a co-ordinate grid 139 Translations 141 12 Patterns and sequences 143 Number patterns 145 Using tables to record results 149 Spatial patterns 153

The three dimensions Points, lines and planes

Coordinates

Numbers to 100000 Five-digit numbers Comparing and ordering numbers

Rounding

Fractions

Count in fractions

Comparing and ordering fractions

Angles

65 67 72 77 79

Angles on a straight line Angles on a triangle

Decimals

Tenths and hundredths Compare and order decimals Multiplying and dividing by 100 and 1000

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Contents

Adding and subtracting money Multiplying with money

13 Proportion

155 157

193 196 198 201 203 207 210 211

Proportion

Profit and loss

Using fractions to describe proportions Finding a fraction of an amount

161

17 Time and timetables

Time

163 165 167 169 172 173 179 181 184 186

Time intervals

Percentages

Calendars

14 Handling data

Solving time problems

Pictographs and charts Changing the scale on the vertical axis

18 More multiplication Multiplying by 19 or 21 mentally

213

215

Line graphs

Multiplying by 50 and 25 mentally Column method for long multiplication

15 Area and perimeter Area and perimeter

218

219

Perimeter

Area

19 3D shapes

225 227 231 233

Using multiplication to find area

Describing 3D shapes Drawing 3D shapes Review of nets of solids

187

16 Money

189 191

Dollars and cents

iii

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How to Use This Book Welcome to Max Maths !

Positivity towards mathematics – the activities provide opportunities to explore mathematics through pattern, reasoning and in meaningful contexts. While developing proficiency, students experience an enjoyment in mathematics, and an appreciation of the beauty and power woven into its fabric. Max Maths is designed specifically to support the new OECS Learning Standards for Mathematics, and to provide regular engagement in all important mathematical processes. Look out for the icons that show: problem solving

The philosophy behind the design of Max Maths is to develop confident , curious , resourceful and proficient mathematicians. The Max Maths approach comprises a comprehensive set of resources that provide: Rich mathematical experiences – each activity uses a Concrete-Pictorial- Abstract (CPA) approach to enrich learning, and develop a visual and physical intuition for mathematical concepts. This means that abstract concepts and calculations are supported with visual representations and concrete materials such as cubes or counters to bring the mathematics to life through practical activities and visual prompts for discussion and understanding. Depth of understanding – the activities use a range of questioning techniques to enable students to develop skill and accuracy alongside a deep mathematical understanding of the concepts. Each year is organised to provide progression over time, with each revisit digging deeper into the skills and mathematical thinking.

reasoning

communicating

connecting

representing Opportunities to develop each

mathematical process are built into the main activities in the Student’s Books and are clearly signposted within the challenges provided in the Teacher’s Guides.

Winston

Jade

Letisha

Leon

Carl

Riana

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HowTo UseThis Book

This student’s book provides a teaching resource to support whole-class teaching and is designed to be accompanied by a workbook for students to write and draw in, to create a record of their achievements. Each unit provides a double-page spread designed to be talked about, prompting students’ curiosity, communication, conjecture and challenge.

Unit 1 – Numbers to 10

1 Numbers to 10

KEY questions

KEY words

Which of these words can you use to describe what you see in the picture? Equal Same as Count Number

How many leaves? How many shells? How many cubes? How many counters? How do you know you have counted them all?

There are many different activity types, carefully ordered, to help students through their learning journey.

Unit 2 – Money and measures

Analogue clocks Let’s Recap

Let’s Practise

1 Estimate and measure. Which units of measurement will you use?

The clock shows that the time is half past three.

The mass of:

Estimate: Heavier than, lighter than or the same as one kilogram?

Measure

(a) 8 exercise books (b) Your shoe

The clock shows that the time is quarter past six.

(c)

A full pencil pot

Who do you agree with? Why? What do we need to know about when reading the time on an analogue clock? Let’s Learn Together 1 It takes the minute hand 5 minutes to move from one number to the next. It takes the minute hand 60 minutes to move all the way around the clock. There are 60 minutes in 1 hour.

(d) Your games bag

2 Work together in a small group to carry out this investigation

Does the tallest person have the longest stride?

Make estimates and think about the unit of measurement you will use. Record your measurements to the nearest whole unit. Make a table of your results. What does the information tell you? Can you answer the original question?

We can use what we know about counting in multiples of five to help us work with time.

60 minutes 0 minutes

55 minutes

5 minutes

50 minutes

10 minutes

Leon has only part of an old tape measure. Some of the numbers have rubbed off. He uses it to measure these objects in centimetres. What is the actual measurement of each object? Explain why. 28 40 42

15 minutes

45 minutes

40 minutes

20 minutes

35 minutes

25 minutes

30 minutes

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The four operations

KEY questions

How many different calculations can you make that will give the answer 186? Can you make 186 by combining two or more operations? What are the four operations? What strategies do you know to calculate with each operation? What does the equals sign indicate? Can you find some more calculations that give the same answer?

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The four operations

KEY words

product

sum

difference

add

subtract

multiply

divide

place value equal equivalent

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Creating number sentences Start Thinking

The number of dots is one added to the answer of five times four.

The number of dots is five times four, and then add on one more.

Talk to a partner about what is the same and what is different about the two methods. How could you write each of these methods as number sentences?

Let’s Learn Together

1 There are 3 boxes of 8 chocolates, plus 4 extra chocolates.

We can write a number sentence to find the total number of chocolates. 4 + 3 × 8 3 × 8 + 4 They are both correct ways to write the number sentence.

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The four operations

When we solve the number sentence, we don’t always solve from left to right. There is a correct order of operations. 4 + 3 × 8 3 × 8 + 4 = 4 + 24 = 24 + 4 = 28 = 28 When a number sentence includes a mix of operations, always do multiplication first. Multiplication Addition

2 There are 15 marbles. Winston takes away 7 marbles and then Jade takes away 4.

We can write a number sentence to find the number of marbles left. Marbles = 15 − 7 − 4 = 8 − 4 = 4 Write each line of working out on a new line.

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Let’s Try It

1 Copy and complete the two number sentences to find the number of pencils.

There are 58 counting blocks. Winston takes 35 blocks away and Jade takes 11 blocks away. Jade writes and solve a number sentence to find out how many are left. Blocks left = 58 − 35 − 11 = 58 − 24 = 34 Explain the mistake that Jade has made.

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The four operations

Let’s Practise

1 Letisha is 162 cm tall. Leon is 5 cm shorter than Letisha. Riana is 4 cm shorter than Leon. Copy and complete the calculation to find the height of Riana. Riana’s height = − −

−

2 Solve each of these calculations. Show your working.

(a) 38 − 10 − 4 (b) 80 − 22 − 10 (c) 45 − 15 − 9 (d) 103 − 31 − 8

3 Arrange the digits in 2 different ways to make a correct calculation.

5 6 8

= 46

4 Solve each of these calculations. Show your working.

(a) 8 × 4 + 7 (b) 1 + 3 × 6 (c) 10 + 4 × 3 (d) 7 × 3 − 1

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Calculate these mentally. (a) 76 − 15 − 8

(b) 19 + 43 + 7

(d) 15 − 3 × 4

(e) 8 ÷ 2 + 3

(f) 42 − 9 − 5

6 Write a number sentence for these word problems.

(a) I buy 4 pens that cost $2 each. I pay with a $10 note. What is my change? (b) I invite 8 friends to my party. I buy 4 sweets for each friend. I also buy an extra 5 sweets to give as prizes. How many sweets do I buy?

Target This is a game for 2–4 players. You will practise your mental calculation. You will need a die. Each player rolls the die 4 times. Record your digits in your book. Your teacher will give you a two-digit target number.

7 6

Each player uses their 4 digits to get as close to the target as possible in one minute. You can use all 4 operations.

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The four operations

Adding and subtracting mentally Let’s Learn Together 1 We can use our knowledge of counting in 100s to add 99. What is 367 + 99?

I know that 367 + 100 = 467.

+100

−1

367

467

350 500 420 430 440 450 460 470 480 490 360 370 380 390 400 410

466

+99

Adding on 100 is one too many. We can take away 1 to find the answer: 367 + 99 = 466. 2 We can use a number line to find the difference between two numbers. How could we find the difference between 789 and 1002?

789 to 790 is 1 .

1000 to 1002 is 2 .

+10

790 to 800 is 10 .

1000 1002

800

789 790

800 to 1000 is 200 .

+200

Let’s Practise

What is the difference between: (a) 3998 and 5000 (b) 995 and 1032 (c) 18001 and 19003?

There are different ways that you can use the number line. Did you do it the same way?

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Written method for multiplication Let’s Learn Together

Find the product of 327 and 5. First multiply the ones by 5.

3 2 7

5 7 ones × 5 5 = 3 tens and 5 ones

Next multiply the tens.

3 2 7

5 2 tens × 5 5 = 1 hundred and 0 tens

Finally multiply the hundreds.

3 2 7

5 3 hundreds × 5 1 6 3 5 = 1 thousand and 5 hundreds 1 3

So the product of 327 and 5 is 1635.

Let’s Try It

Help Leon finish his written calculation.

8 1 7 3

I’m calculating 817 × 3. I’ve estimated my answer will be a bit more than 2400.

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The four operations

Let’s Practise

Find the product of these numbers

(a) 2 and 163

(b) 204 and 5

(d) 612 and 3

2 Work out the following. Find an estimate first by rounding the greater number to the nearest hundred. (a) 358 × 2 Estimate: (b) 5 × 1080 Estimate:

Answer:

(d) 3329 × 6 Estimate:

Answer:

The letters A, B, C, D and E represent digits. Here are some clues about the numbers. ABC × D = EBB

A B C

A × A = B A × B = C C − 1 = E C = 8 Can you work out what numbers the letters represent?

E B B

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Written method for division Start Thinking

How can these counters be divided equally between three people?

is equal to

Let’s Learn Together

I have 4254 sheets of paper. I divide them between 3 printers. How much do I put into each printer? Let’s use a short division method.

Divide the thousands: Four thousands divided by 3 is one thousand with a remainder of 1000. Regroup the remaining 1000 for 10 lots of 100 . Record them in the hundreds column.

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The four operations

Divide the hundreds by three: Twelve hundreds can be divided equally by 3.

Divide the tens by three: Five tens divided by 3 is 1 ten with a remainder of 2 tens. Regroup the two 10 for twenty 1 . Record them in the ones column.

Divide the ones by three: Twenty-four ones can be divided equally by 3.

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Let’s Practise

(b)

(a)

6 1 7 4

7 3 2 6 9

(d)

(c)

8 5 4 9 6

5 7 1 7 5

(f)

(e)

3 2 9 3 4

6 7 3 6 2 6

Work out these division sums. Give remainders with your answers. (a) 8547 ÷ 2 (b) 1634 ÷ 4 (c) 3265 ÷ 7 (d) 3317 ÷ 9 (e) 32633 ÷ 5 (f) 91608 ÷ 6

Books are packed into boxes of 6. How many boxes are needed for 1854 books?

4 Winston has calculated that 8037 ÷ 3 = 2645 remainder 2. Look at his working and work out what mistake has he made.

2 6 4 5

2 0

1 3

1 7

3 8

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The four operations

Word problems with the four operations Let’s Learn Together

Carl’s book has 140 pages. He reads 1 __ 2 of the book. He then reads another 18 pages. How many pages has he read?

Let’s show the word problem in a diagram. This bar shows Carl’s book.

140 pages He reads half of it.

70 pages He then reads an extra 18 pages.

18 pages

He reads a total of 88 pages.

70 pages

Let’s Practise

1 Riana has 20 sweets. She eats 8 sweets and then shares the rest with a friend. How many does she give her friend? 2 Letisha wins a prize of $4128. She gives $1500 away. She shares the remaining money between 4 people. How much do they each get? 3 A holiday costs $399 for adults and $120 for children. How much does it cost for 5 adults and 1 child?

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Factors and multiples

KEY questions

What is the difference between a factor and a multiple? How can factors and multiples help her work out how many pens to put in each group?

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Factors and multiples

KEY words

factors

multiples

common factors

common multiples

times tables

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Factors and multiples Start Thinking

Here are 12 tiles. How many different ways can you arrange them to make a rectangle? Can you arrange the 12 tiles into a square?

Let’s Learn Together

1 Carl has 10 jelly beans. He arranges them in an array in different ways.

1 × 10 = 10

He then arranges them to make 2 rows of 5 jelly beans.

2 × 5 = 10

The numbers 1, 2, 5 and 10 all divide into 10. 1, 2, 5 and 10 are the factors of 10. 10 is a multiple of 1, 2 and 5.

We need to know our multiplication facts to find factors and multiples.

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Factors and multiples

Find all the factors of the number 12.

1 × 12 = 12

2 × 6 = 12

3 × 4 = 12

The factors of 12, in order, are 1, 2, 3, 4, 6 and 12

Find the first 3 multiples of 12. The first multiple of 12 is 12:

12 × 1 = 12

The second multiple of 12 is 24:

12 × 2 = 24

The third multiple of 12 is 36:

12 × 3 = 36

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Let’s Practise

1 Carl, Jade and Winston have found the factors of 42. Carl’s work:

Factors of 42: 21 2 7 1 42 3 14

Jade’s work: 1

Winston’s work:

1 x 42 = 42 6 x 7 = 42 2 x 21 = 42 The factors are 1, 2, 6, 7, 21, 42

Check Carl, Jade and Winston’s work. Who has found all of the factors? Talk with a partner about which method you think is better. Explain your reasons.

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Factors and multiples

2 Sort these numbers into factors and multiples of the number 20: 100, 2, 4, 400, 80, 5, 1, 32, 10

Factors Multiples

Use the words

and

to copy and

complete these sentences. A factor of a number is always

the number.

A multiple of a number is always

(a) Find the first 3 multiples of 7. (b) Find the first 3 multiples of 20.

5 Copy and complete these factor diagrams:

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6 Put these numbers together to make three factor pairs of the same number: 10 2 14 35 5 7

The first four multiples of 15 are listed. 15 30 45 60 Will the 100th multiple be odd or even? Explain your answer.

(a) List all the factors for the first five square numbers. Count the number of factors for each square number. (b) Choose 5 non-square numbers. Find the number of factors for each number.

Number of factors

Factors

4 9

16 25

You could record your work in tables like this. What patterns do you notice? Use your results to make a conjecture about the number of factors for square and non-square numbers.

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Factors and multiples

Find the factors Instructions: This is a game for two players. You need a set of digit cards from 0–9. Shuffle the cards and lie them face down on the table. Take it in turns to select two digit cards. The first player chooses any two cards and makes a 2-digit number. They score 1 point for every factor that they can name. When player 1 has named as many factors as they can, player 2 can attempt to name any remaining factors, scoring 2 points for each factor. At the end of the round, return the cards to their place on the table. Repeat, with player 2 starting the round by selecting 2 cards. The winner is the first player to score 25 points.

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Common factors Let’s Learn Together

1 The factor diagram shows the factors of 45 and 54.

54 18 27 9

1 3 2 6

There are some factors that are common to both number. The numbers 3 and 9 are common factors of 45 and 54. The highest common factor is 9.

2 Look at the first 9 multiples of the numbers 2 and 3.

2: 2, 4, 6, 8, 10, 12, 14, 16, 18 3: 3, 6, 9, 12, 15, 18, 21, 24, 27 There are some multiples that are common to both numbers. The numbers 6 and 18 are common multiples of 2 and 3. The lowest common multiple is 6.

I predict that the next common multiple is 24.

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Factors and multiples

Let’s Practise

1 Find two numbers that both have 6 as a factor.

(a) Find the factors of 28 and 32. (b) Underline the numbers that are common factors of 28 and 32.

3 The numbers in the red circle are multiples of 25. The numbers in the blue circle are multiples of 10.

30 60 10 40

100

(a) Use the diagram to write down some common multiples of 10 and 25. (b) Leon says the number 120 should go in the overlapping section. Explain what mistake Leon has made.

Find the highest common factor of: (a) 21 and 35 (b) 56 and 77 Find the lowest common multiple of: (a) 8 and 10 (b) 7 and 12

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Using factors to multiply Start Thinking

1 × 7 = 7 2 × 7 = 14 3 × 7 = 21 4 × 7 = 28 5 × 7 = 35 6 × 7 = 42

1 × 14 = … 2 × 14 = … 3 × 14 = … 4 × 14 = … 5 × 14 = … 6 × 14 = …

Jade doesn’t know her 14 times tables. How could Jade use her 7 times tables to complete the 14 times tables?

Let’s Learn Together

Carl wants to multiply 7 by 15. He starts by multiplying 7 by 3.

7 × 3 = 21 He then multiplies his total by 5.

21 × 5 = 105

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Factors and multiples

Now I have 15 lots of 7, so 15 × 7 = 105. 7 × 15 = 7 × 3 × 5 = 21 × 5 = 105

15 is 5 × 3. I can multiply by 15 by multiplying by 3 and then by 5.

Let’s Try It

1 Copy and complete the calculation to 12 × 18. 12 × 18 = 12 × 9 ×

Multiplying by 18 is tricky! We can use the multiplication fact 9 × 2 = 18 to help us.

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Let’s Practise

1 Find the missing number in the calculation.

×3

= 210

×7

×…

2 Complete this calculation in two different ways.

×…

= 4500

×…

×45

3 Winston multiplies a number by 9 and then multiplies by 7. How much bigger than his original number is his answer?

Complete the multiplication fact for 25. × = 25 (b) Use the multiplication fact to calculate 32 × 25.

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Factors and multiples

(a) Write some factor pairs of 28. 1 × 28 (b) Talk to a partner about which factor pair is the most helpful to use to multiply by 28.

Calculate 45 × 18.

Here is a multiplication fact: 5 × 7 × 9 = 315 Use this fact to calculate 22 × 315

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The three dimensions

KEY questions

Can you find an example of a point on the map? Can you find an example of a line on the map?

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The three dimensions

KEY words

points

lines

planes

parallel

perpendicular

dimensions

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Points, lines and planes

One of the pictures is the odd one out. Talk with your partner about what is the same and what is different in each picture. Can you spot the odd one out and explain why? Cross - curricular

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The three dimensions

Let’s Learn Together

Letisha has marked the point A. Put your finger on the point A. Can you move your finger and still be on the point A? A

Letisha has drawn a line segment AB. Put your finger on the line segment AB. Can you move your finger and still be on the line segment AB? A B

A line is a set of points with no thickness or curves. You can move along a line in either direction. It is one-dimensional .

A line goes on and on for ever. If only part of a line is shown, it is called a line segment.

Letisha has drawn a plane shape ABCD. Put your finger in the plane shape. Can you move your finger and still be in the plane?

A plane is a flat surface. You can move left and right, and up and down. It is two-dimensional .

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Let’s Try It

1 (a) Draw a line that is 4 cm long. Label your line AB. How many dimensions does your line have? (b) Draw a rectangle that is 5 cm by 3 cm. How many dimensions does your shape have?

2 Explain to your partner why the lines a and c are not parallel .

d e

(b) Find the pair of lines that are parallel.

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The three dimensions

Let’s Practise

Match the word to the diagram. Point

Line

Plane

Part of the line a is shown on the grid.

A B C

line a

Which point belongs to the line a?

(a) Mark a point D in your book. Draw a 4 cm vertical line that passes through your point. Explain why your line has one dimension. (b) Explain why a point has zero dimensions.

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Draw three lines that intersect at three different points. Draw three lines that intersect at two different points. Draw three lines that intersect at one point. Draw three line segments that do not intersect.

Points, lines and planes This is a game for 2 players. Start with a set of points marked in an array. Take it in turns to connect two adjacent points with a line segment. Points can be connected horizontally or vertically.

If you close a box, write your initial in the box. The game is finished when all the boxes are complete. The winner is the player with the most boxes.

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The three dimensions

Coordinates Start Thinking

Imagine you are walking in a big park and trying to meet up with a friend. What are the different ways you could describe your location.

1 2 3 4 5 6 7 8

Let’s Learn Together

A plane is a 2-dimensional space. Coordinate grids are used to describe the position of a point or a line on a plane. The two dimensions are shown by an x-axis and a y-axis.

10 9 8 7 6 5 4 3 2 1 0 0 12345678910 x -axis y -axis

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Let’s Try It

(a) Draw a set of axes in your book.

0 1 2 3 4 5 6

(b) Draw a vertical line that passes through the x-axis at 5. (c) Draw a horizontal line that passes through the y-axis at 2. (d) Write down the coordinates of the point where they intersect. Let’s Practise

1 (a) Draw a set of axes in your books from 0 to 6.

Draw a horizontal line that passes through the x-axis at 5. Draw a vertical line that passes through the x-axis at 1. Draw a horizontal line that passes through the y-axis at 2. Draw a horizontal line that passes through the y-axis at 4. (b) Write down the coordinates of two points that lie inside the rectangle.

Choose four objects from the grid. Write their coordinates. For example, Apple = (4, 6)

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The three dimensions

2 3 4 5 6 7 8

9 10

A co-ordinate grid is sometimes called a Cartesian grid.

That’s because it was named after a mathematician called Descartes.

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3 Riana says that the co-ordinates of the point are (4, 3).

5 4 3 2 1 0 0 1 2 3 4 5

Explain the mistake that Riana has made.

4 What shape do these coordinates make? (3, 4), (7, 4), (7, 8), (3, 8).

10 9 8 7 6 5 4 3 2 1 0 0 12345678910 x -axis y -axis

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The three dimensions

Create some points, lines and planes using a dynamic geometry software package. Look for these options in the menu to help you:

Point

Segment

Line

Polygon

Circle with Centre

Regular Polygon

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Numbers to 100 000

KEY questions

What is the difference between a digit and a number? What is the value of each digit in a 5-digit number?

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Numbers to 100000

KEY words

place value

hundred thousands

place value

digits

number

numerals

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Five-digit numbers Let’s Learn Together

Count together in 10 000.

10 000

Eighty thousand, ninety thousand, one hundred thousand.

Ten thousand, twenty thousand…

There are ten lots of ten thousand in a hundred thousand.

Hundred Thousands

Ten Thousands

Hundreds

Tens

Ones

Thousands

5 0 0 0 2 0 0 0 0 1 0 0 4 0 3 2 5 1 4 3

Twenty-five thousand, one hundred and forty-three.

20 000 + 5 000 + 100 + 40 + 3 = 25 143

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Numbers to 100000

Let’s Try It

1 Write the number shown by the abacus in words and numerals.

Ten Thousands Tens

Thousands Ones

Hundreds Hu ndred s

Tens Tens

Ones Ones

1 A bead is added to the thousands column. Write the new number. Let’s Practise

Write these numbers in words: (a) 49 281 (b) 19 080

(d) 80 214

Write these numbers in numerals: (a) twenty thousand, nine hundred and eleven (b) eighty three thousand, four hundred and ninety nine

Add three thousand to the number.

Hundred Thousands

Ten Thousands

Hundreds

Tens

Ones

Thousands

4 The place value chart shows one hundred thousand.

Hundred Thousands

Ten Thousands

Thousands

Hundreds

Tens

Ones

Write the number that is half of one hundred thousand.

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Comparing and ordering numbers Start Thinking

Add ten beads to the abacus to make the largest possible number.

Ten Thousands

Thousands

Hundreds

Tens

Ones

Can you rearrange the beads to make the smallest number?

Let’s Learn Together

1 Jade and Leon have each made a number using the digits 1–6.

My number is smaller because I have put a smaller digit in the ten thousands column.

We’ve used the same digits but our numbers have different values.

25634

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Numbers to 100000

2 Jade and Leon estimate the position of their number on a number line.

0 10 000 20 000 30 000 40 000 50 000 60 000 70 000 80 000 90 000 100 000

My number line goes up in ten thousands. My number is a bit more than fifty thousands. thousands.

25000

50000

75000

100000

My number line goes up in twenty five twenty five thousands. My number is a bit more than . My number is a bit more than twenty five thousand.

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Let’s Practise

1 The Max Maths team bought tickets to the fair. Each ticket has a 5-digit number. Arrange the numbers from smallest to greatest.

28600

2 Write these statements using > or < to show the greater or lesser number:

(a) 29 382

(b) 30181

2938

31091

(d) 81817

18 170

19118

3 (a) Write these numbers in order from smallest to largest: 99203 93203 95203 91203 97203 (b) How much do the numbers increase by each time?

4 Look at the number line. What are the values of the numbers marked with arrows? a b c

100 000

5 Estimate the value of each number marked with an arrow.

60 000

70 000

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Numbers to 100000

Five-digit numbers You will need: a blank place value chart; a dice. The first player throws the dice. Place the digit into one of the empty spaces in your place value chart.

Ten Thousands

Hundreds Thousands

Tens

Ones

Take it in turns to throw the dice and record the digit in your place value chart. Keep going until all players have created a 6-digit number. Compare your 6-digit numbers. Say each number out loud.

Who has the largest number? Who has the smallest number? Play 3 more rounds.

In round 4, the winner is the player with the number closest to five hundred thousand.

In round 3, the winner is the player with the smallest number.

In round 2, the winner is the player with the largest number

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Rounding Start Thinking

Use the digit cards to make three 3-digit numbers. Use each card only once.

Which of your numbers is closest to 500? Which of your numbers is closest to 250?

Let’s Learn Together

The number line shows the number 16200 on the number line.

16 000

17 000

The number 16 200 is between 16 000 and 17 000. It is closer to 16 000. 16 200 is approximately 16 000, when to the nearest thousand.

We don’t always need to know the exact number of something. Sometimes it’s OK to talk about approximate numbers.

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Numbers to 100000

1 Let’s round the number 3460 to the nearest hundred.

The multiple of hundred before is 3400. The multiple of hundred after is 3500.

3460

multiple of one hundred after

multiple of one hundred before

3400

3500

The number 3460 is closer to 3500. So when rounded to the nearest hundred, 3460 is approximately 3500. Let’s Try It

Winston and Jade are talking about the approximate number of spaces in a car park.

Spaces: 1812

Winston says: there are approximately 1000 spaces in the car park. Jade says: there are approximately 2000 spaces in the car park. (a) Do you agree with Winston or Jade? Explain your answer.

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(b) Copy and complete the diagram to help you round the number 1812 to the nearest one hundred.

multiple of one hundred after

multiple of one hundred before

1812

1812 rounded to the nearest hundred is …

Let’s Practise

1 Letisha is rounding the number 4281 to the nearest 100. (a) Which of these number lines will help her the most?

2000

3000

4000

5000

6000

7000

4000

4100

4200

4300

4400

4500

4600

C 4230 4240 4250 4260 4270 4280 4290 4300 4310

4277

4278

4279

4280

4281

4282

4283

(b) Use your chosen number line to round 4281 to the nearest 100.

The price of a car is £2819. The salesman says the price is about £2000. Explain whether this is a good approximation.

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Numbers to 100000

The number line goes up in hundreds.

2000

4000

Use the number line to find: (a) two numbers that round up to 3000 (b) two numbers that round down to 3000 when rounded to the nearest thousand.

Round each number to the nearest ten: (a) 281 (b) 1722 (c) 967

5 Round each number to the nearest hundred: (a) 606 (b) 12765 (c) 7278 6 Round each number to the nearest thousand: (a) 6182 (b) 91827 (c) 12999

Roll 4 dice. Arrange the dice to make a 4-digit number. How close to a multiple of a thousand can you get? Rearrange the dice again. How far away from a multiple of a thousand can you get?

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Fractions

KEY questions

What fractions can you see in the picture? How much is 3 __ 4 of a metre? What is the difference between a proper fraction and an improper fraction?

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Fractions

3 __ 4

I would like of a metre please.

KEY words

equivalent

proper

improper

mixed fraction

numerator

denominator

sequence

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Count in fractions Let’s Recap

The sequence is counting in steps of one fifths up to one whole. What is missing from the sequence?

3 __ 5

1 __ 5

2 __ 5

* 1 whole Draw your own sequence of fractions with a different denominator. Start Thinking

4 5

7 8

3 4

6 7

5 6

1 __ 6 of a cup of water into each of these cups?

Can you pour

Which cups will overspill when they are filled with 1 __ 6 of a cup of water?

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Fractions

Let’s Learn Together

Riana and Leon are using a number line to count in sixths. Leon counted in sixths using improper fractions.

1 6

2 6

3 6

4 6

5 6

6 6

7 6

8 6

9 6

10 6

11 6

12 6

13 6

14 6

Six sixths is a whole.

Riana counted in sixths using mixed numbers.

1 6

2 6

3 6

4 6

5 6

6 6

1 6

1 1 2 6

3 6

4 6

5 6

6 6

1 6

2 6

Fractions greater than one can be written as mixed numbers and improper fractions.

is the same as 1 5 __ 6 .

11 ___ 6

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Let’s Try It

1 Copy and complete the number lines to show the whole numbers written as improper fractions.

1 3

2 3

4 3

5 3

7 3

8 3

1 4

2 4

3 4

5 4

6 4

7 4

9 4

10 4

11 4

What do you notice about the numerators?

3 __ 5

2 Jade and Winston use different methods to count on 1 __ 5

six times from

Jade’s method

10 5

3 5 3 5 9 5

6 5 4 5

9 5 5 5

6 5

7 5

8 5

9 5

4 5

Winston’s method 1 __ 5 + 1 __ 5 + 1 __ 5 + 1 __ 5 + 1 __ 5

1 __ 5

= 1

3 __ 5

1 __ 5

4 __ 5

+ 1

= 1

3 __ 8

1 __ 8

eleven times from

Count on

Use both Jade’s method and Winston’s method. Which method did you prefer?

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Fractions

Let’s Practise

1 Put the improper fractions into two groups:

Equivalent to a whole

Not equivalent to a whole

10 ___ 20

15 ___ 10

20 ___ 5

15 ___ 4

12 ___ 4

2 Use the number line to convert between improper and improper fractions. 10 5 12 5 1 5 2 5 5 5 7 5 9 5

3 5

4 5

1 5

3 5

1 5

8 __ 5

10 ___ 5

1 __ 5

(a) 1

(b) 2

(c)

(d)

3 Give your answers to these questions as mixed numbers: (a) Count on 1 __ 2 5 times from 9. (b) Count on eight times from

7 ___ 10.

1 ___ 10

1 __ 3

8 times from 2 1 __ 3 .

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Let’s Learn Together

1 Riana, Winston, Leon and Jade each took a piece of paper. They folded it and coloured some of the parts. Riana’s piece of paper was folded into 2 equal parts. She coloured 1 of the equal parts. She coloured one half of the paper. 1 __ 2 1 __ 2 Winston’s piece of paper was folded into 6 equal parts. He coloured 3 of the equal parts. He coloured three-sixths of the paper. 1 __ 6 1 __ 6 1 __ 6 1 __ 6 1 __ 6 1 __ 6 Leon’s piece of paper was folded into 8 equal parts. He coloured 4 of the equal parts. He coloured four-eighths of the paper. 1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8 Jade’s piece of paper was folded into 10 equal parts. She coloured 5 of the equal parts. She coloured five-tenths of the paper.

1 ___ 10

Fractions that have the same value but different numerators and denominators are called equivalent fractions . 1 __ 2 = 3 __ 6 = 4 __ 8 = 5 ___ 10 Equivalent fractions can be found by multiplying the numerator and denominator by the same number.

Multiplying the numerator and the denominator by the same

× 3

× 5

1 2

3 6

1 2

5 10

number results in an equivalent fraction.

× 3

× 5

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Fractions

Let’s Try It

1 What equivalent fractions do these strips of paper show?

2 Copy and complete these equivalent fractions.

3 __ 7

5 __ 8

25 ___

8 ___

9 ___

2 __ 7

2 __ 5

= ___ 14

(a)

(c)

(d)

(b)

Let’s Practise

Write 5 fractions equivalent to 5 __ 6 .

2 Which of these fractions are equivalent to 3 __ 5 :

6 ___ 10

30 ___ 50

18 ___ 40

9 ___ 25

40 ___ 45 so that the numerator is as small as possible.

Find an equivalent fraction for

Find a fraction equivalent to 3 __ 4

where the numerator and denominator have a

difference of 9.

Riana has been counting in equal steps to 5. She can do 1 step of 5. She can do 20 steps of 1 __ 4 . What other steps can she do to get to 5?

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Comparing and ordering fractions Let’s Recap

Fractions can be compared when their numerators or denominators are the same.

1 Write these fractions in order of size by comparing numerators: 5 __ 7 3 __ 7 9 __ 7 2 __ 7 6 __ 7 2 Write these fractions in order from largest to smallest by comparing denominators: 3 __ 7 3 __ 5 3 ___ 10 3 __ 8 3 __ 2

Let’s Learn Together Which is the larger fraction? 3 __ 5 or

7 ___ 10

We can use our knowledge of equivalent fractions!

How can we compare fractions if the denominators or numerators are not the same?

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Fractions

We can find an equivalent fraction so that the denominator is the same.

×2

3 __ 5

6 ___ 10

3 5

6 10

I noticed that if I double 5 I get 10!

×2

7 ___ 10

6 ___ 10

and

Now we can compare

7 ___ 10

So,

is the larger fraction.

Let’s Try It

1 (a) In each pair, change the fraction in bold so that denominators are the same.

1 __ 3

2 __ 7

5 __ 6

3 ___ 14

1 __ 9

2 __ 5

6 ___ 10

4 ___ 27

3 __ 4

3 __ 8

7 ___ 24

13 ___ 16

(b) For each fraction in part a, circle the larger fraction.

2 (a) Write these three fractions with a denominator of 12. 3 __ 4 6 ___ 24 5 __ 6 (b) Now write them in order of size from smallest to largest.

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Let’s Practise

6 ___ 15

= ___ 30 Use your answer to decide which is smaller:

(a)

6 ___ 15

17 ___ 30

(b)

Find the larger fraction in each pair.

6 ___ 10

5 __ 8

15 ___ 24

1 __ 2

2 __ 5

2 __ 3

3 __ 5 of his pocket money. Tim has spent

13 ___ 20

Freddie has spent

of his.

Who has spent the most?

Sort the fractions into groups. Copy the table into your book. 3 __ 9 , 2 __ 6 , 2 __ 7 , 6 __ 9 , 4 ___ 15 ,

5 ___ 12

1 3

less than

equal to

more than

2 6

Find three different fractions that would make this statement correct. 3 __ 7 < ____ < 6 ___ 21 Are your fractions written in their simplest form?

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Fractions

You need: 2 counters; 2 dice On your turn, roll two dice. Use the dice to make a proper fraction. Move forward that fraction on one of your number lines. You complete a line when you move past the number 3. The winner is the first player to complete all three of their lines.

Player 1 game board

Use for denominators of 1, 2 or 4

Use for denominators of 1 or 5

Use for denominators of 1, 2, 3 or 6

Example:

2 4

Use for denominators of 1, 2 or 4

Player 2 game board

Use for denominators of 1 or 5

Use for denominators of 1, 2, 3 or 6

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