Max Maths - Making Real-World Connections SB6 FULL

MAKING REAL-WORLD CONNECTIONS

maths a ×

MAKING REAL-WORLD CONNECTIONS M

CAROLINE FAWCUS

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Contents

Adding and subtracting large numbers

How to Use This Book

77 78 80 82

Numbers to 100 Primes and composites Factors and multiples Square numbers Number problems

2 4 8

Money

Decimals

Tenths and hundredths Adding and subtracting decimals mentally Multiplying and dividing by 10 and 100 Recording money with decimals and word money problems

11 14 16 18 19 20 22 28

2D shapes Polygons

Review of 2D polygons

Triangles

Quadrilaterals

Circles

Length, mass and capacity94 Metric and imperial units 96 Length 98 Mass 100 Capacity 102 Solving word problems 105

Calculating Estimating 32 Mixed operations with a pair of brackets 34 Mental addition and subtraction 36 Written multiplication 39 Division 42 30

Coordinate grids Coordinate grids

108 110 112 118 122 124 126 133 135 136 138 140 142

Parallel and perpendicular lines

Average and spread

44 46 47 49 52

Average Median The mean

Reflections

10 Area and perimeter

Perimeter

Range

Area

Fractions and percentages56 Equivalent fractions 58 Finding a fraction of an amount 60 Fractions and percentages 62 Calculating with percentages 65 Numbers to one million 68 Place value 70 Rounding 74

Area of a triangle Estimating areas

11 Data handling Frequency table

Review of pictographs and bar charts

Two-way tables

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Contents

Sharing a part into a ratio 188 Graphing proportion 190 192 Multiplying 2-digit numbers195 Dividing 3-digit numbers by 2-digit numbers 198 Solving word problems 200 17 Functions and variables 202 Function machines 204 Using symbols 206 Variables and equations 208 18 Time 212 Review of telling the time 214 Measuring time 215 Fractions of an hour 218 Timetables 220 Changes over time 222 19 Solid shapes 226 Faces, edges and vertices 228 Pyramids and prisms 232 Nets 234 16 Further calculations

12 Length, mass and capacity 2

144

Using decimals in measurements Converting measures

146 148

Solving word problems with mixed units

151

13 Angles

154 156

Fractions of whole turn Angles in a triangle

161 Angles in a quadrilateral 164

14 Calculating with fractions and decimals

166

Counting in decimals 168 Calculating with decimals 169 Written calculation 172 Adding and subtracting fractions 175

15 Ratio

178 180 185 186

Ratio and proportion Simplifying ratio Calculating with ratio

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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How To Use This 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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Numbers to 100

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Numbers to 100

KEY questions

What does this diagram show? What do the numbers on the bottom row have in common? Choose another 2-digit number – can you make a similar diagram?

KEY words

array

product

prime

composite

factor

multiply

prime

square

divisible

consecutive

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Primes and composites

Jade is using a one hundred square grid to record the times tables facts. She writes each multiplication fact in the square that corresponds to the answer.

1 11 21 3 × 7

2 12 22 2 × 6 3 × 4 2 × 11

3 13 23

4 14 24 2 × 2 2 × 7 2 × 12 3 × 8 4 × 6

5 15 25 3 × 5

6 16 26 2 × 8 4 × 4 2 × 3

7 17 27 3 × 9

8 18 28 2 × 9 3 × 6 2 × 14 4 × 7

9 19 29

10 20 30 2 × 10 4 × 5 2 × 15 3 × 10 6 × 5

2×4 3×3 2×5

5×5 2×13

The numbers that she can make from a multiplication fact are called composite numbers.

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Numbers to 100

2 When Jade has written all the multiplication facts into her grid, she colours the numbers that can’t be made by a multiplication fact.

2 3

These numbers are called prime numbers . The only times tables they appear are their own and the 1 times table.

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3 Tania is writing the number 70 using multiplication facts. 70 = 7 × 10 She then notices that 10 can also be written as a multiplication fact. 70 = 7 × 5 × 2 composite prime The numbers 2, 5 and 7 are prime. So all composite numbers are made up of prime numbers.

1 Write a multiplication fact to show that the number 18 is a composite number.

2 Write all of the prime numbers between 10 and 20.

Find the next prime number after: (a) 8 (b) 20

4 Write down the one number from each group that is not prime . (a) 1 3 5 7 (b) 11 21 31 41 5 Write down the number from each group that is prime. (a) 2 4 6 8 (b) 5 10 15 20

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Numbers to 100

6 Can you complete this number sentence using only prime numbers?

–

7 Copy and complete the following multiplications using prime numbers.

42 = 2 × 3 ×

110 =

× 5 × 11

75 = 3 ×

× 5

18 =

Consecutive means in order. 14, 15, and 16 are three consecutive numbers that are not prime. Can you find five consecutive numbers that are not prime?

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Factors and multiples

A diagram can help us find the factors of a number. 1 30

2 3

15 10

Draw a diagram to show the factors of 20.

1 Letisha writes down the factors of 12. Jade writes down the factors of 18 .

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

They notice that there are some numbers that are factors of both numbers. These are called common factors.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

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Numbers to 100

2 Carl writes down the first 10 multiples of 3 and 5.

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50

The first two common multiples of 3 and 5 are 15 and 30. The lowest common multiple is 15.

Find the first 6 multiples of (a) 5 (b) 10

The number 20 has 6 factors. Copy and complete the list of factors. 1, 2 10,

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3 Decide if each of these sentences is true or false. (a) The number 56 is a multiple of 6. (b) The number 4 is a factor of 30 and 40. (c) The number 81 is a multiple of 3 and 9.

List the factors of 24 and 16. (a) Circle the common factors in each list. (b) Which is the Highest Common Factor?

List the first ten multiples of 5 and 8. Circle the lowest common multiple.

5 Winston has written the first 8 multiples of 3 and 4 3: 3, 6, 9, 12, 15, 18, 21, 24 4: 4, 8, 12, 16, 20, 24, 28 He notices that 12 and 24 are both common factors. What will be the next common factor after 24?

6 Explain how you know that 25 is a factor of 1775. Write down three more 4-digit numbers that are multiples of 25.

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Numbers to 100

Square numbers

Look at the sequence of dots. What pattern do you see?

Pattern 1 Pattern 2 Pattern 3 Pattern 4

Pattern 5

Let’s count the number of dots in each pattern. Total dots: 1 4 9 16 25 These are the first five square numbers .

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Riana has 15 beans.

She wants to arrange the beans in a square.

1 Use the diagram to help you find the 7th square number.

7 × 7 = 49

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Numbers to 100

2 Draw a diagram to show that the number 36 is a square number.

3 The numbers 64 and 100 are square numbers. What is the square number between 64 and 100?

4 Complete these calculations using square numbers only:

+ 4 = 20

35 = 25 +

+ 1

9 +

29 =

+ 9 +

Letisha has been finding prime numbers that add together to make square numbers. 7 + 2 = 9 17 + 19 = 36 Can you find two more examples where two prime number can add together to give a square number?

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Number problems

7 + 3 = 10

7 x 3 = 21

7 – 3 = 4

Use the numbers in the box to write down: (a) Write down two numbers that have a sum of 60. (b) Write down two numbers that have a difference of 14. (c) Write down two numbers that will give the smallest product.

2 Find two numbers with a total of 100 and a difference of 50.

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Numbers to 100

An arithmagon is a polygon with circles on each edge and a box on each side. The number on each side is the sum of the two circles either side.

(a)

(b)

(c)

(d)

3 12

Now make up some arithmagon puzzles for your friend to solve.

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2D shapes

KEY questions

What geometric shapes can you see these pictures?

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2D shapes

KEY words

squares

rectangles

circles

straight line

curved line

right angle

obtuse

acute

rhombus

parallelogram

isosceles

scalene

equilateral

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Polygons

This shape is a polygon. It has straight sides and it is closed. This means there are no gaps in the perimeter.

This shape is not a polygon because it has curved sides.

1 Sort these shapes into polygons and not polygons.

Polygons

Not polygons

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2D shapes

Review of 2D polygons

We can name shapes by the number of sides.

Three sides

Triangle

Four sides

Quadrilateral

Five sides

Pentagon

Six sides

Hexagon

Shapes with sides of equal length are called regular polygons. Shapes with sides that are different lengths are called irregular polygons.

Each of these words has tri- or quad- in it. Talk to a partner about each object and how its features link to its name. Cross - curricular

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Triangles

We can classify triangles based on the lengths of their sides:

Equilateral

Isosceles

Scalene

Or by the angles inside the triangle:

Right angled

Acute angled

Obtuse angled

Choose one word from each list to categorise each triangle. A B Equilateral Right angled Isosceles Obtuse angled Scalene Acute angled

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2D shapes

1 Copy and complete the table in your books. Add the 5 triangles to the correct place in the table.

Right angled Acute angled Obtuse angled

Equilateral Isosceles Scalene

2 Match the type of triangle to the correct description triangle

a triangle with 2 sides of the same length

a triangle with all sides of the same length

equilateral triangle

any enclosed shape with 3 straight lines

right-angled triangle

a triangle that has a 90° angle

isosceles triangle

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Quadrilaterals

How many different quadrilaterals can you draw on this 9-pin geoboard?

A quadrilateral is a polygon with 4 sides. Some quadrilaterals have special names. A regular polygon is called a square . A quadrilateral is called a rectangle if: Opposite sides are equal length Opposite sides are parallel All angles are right angles A quadrilateral is called a parallelogram if: Opposite sides are parallel Opposite sides are equal length Opposite angles are equal

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2D shapes

1 Look at the quadrilaterals below. They both have the same properties. Can you identify them?

Quadrilaterals that only have 1 pair of parallel sides are called trapeziums . QR // TS QR is parallel to TS 23

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2 Riana studies the shape of her kite. She notices the sides DA and BA are of the same length. She also notices that the sides DC and BC are of the same length. Sides that are next to each other are called adjacent sides. So, we can say that the kite has 2 pairs of adjacent sides that are of equal length. A B

A quadrilateral that has the following properties is called a kite. Pairs of adjacent sides are of equal length. A pair of opposite angles are equal. M

∠ P = ∠ N

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2D shapes

3 Carl makes a square using ice-cream sticks and pins.

He then pushes in the opposite corners of the square to change its shape.

The squeezed square shape still has sides that are of equal length. However, we can see that the internal angles are no longer right angles. The opposite angles are equal. A quadrilateral that has the following properties is called a rhombus.

All sides are of equal length. Opposite sides are parallel. Opposite angles are equal.

AB // DC and DA // CB ∠ A = ∠ C and ∠ D = ∠ B

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Name each of the shapes (a)

(b)

(c)

(d)

1 Use a pencil and ruler to make an accurate drawing of each shape. (a) 130 70 (b)

5 cm

3 cm

130

120 120

6 cm

8 cm

2 Riana is describing a shape. “My shape has 4 sides that are the same length, but the angles are not all the same.” What shape is she describing?

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2D shapes

Draw some different triangles and quadrilaterals using dynamic geometry software.

Use the measure tools to draw your polygons accurately.

74°

AC = 7.1

65.8°

40.2°

Can you draw an example of each these shapes? A right-angled isosceles triangle. A trapezium. A rectangle with a length that is double the width.

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Circles

1 Jenny tells all her friends to stand so they are exactly 2 metres from her. She notices that they made the shape of a circle A circle is a shape where all points on the circle are the same distance from the centre. This is called the radius.

radius

2 The distance across the centre of a circle is called a diameter. It is equal to 2 radii.

4 cm

8 cm

The radius of my circle is 4 cm. Therefore, it has a diameter of 8 cm.

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2D shapes

1 A circle is used to draw a triangle. Two points are on the perimeter and one point is on the centre.

What is the name for this type of triangle?

Start with a circle. Draw three lines from the centre to the circumference of the circle. What shape will it make?

When all the vertices of a polygon lie on the edge of a circle then we say the polygon is inscribed in the circle. Can you use this method to draw a square inscribed in a circle?

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Calculating

KEY questions

Estimate the number of sheets of paper. What multiplication strategies could you use to work out the exact number?

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Calculating

KEY words

multiplication

division

addition subtraction

approximation

estimation

rounding

mental strategies

written strategies

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Estimating

How many chillis can you see in this picture?

Do I have enough money to buy 28 of these books?

$4.75 per book

Winston wants to buy 28 books that are priced at $4.75. He has $185. To calculate how much he must pay he needs to work out 28 × $4.75. He can answer his question quickly by making an estimate. 28 books is a little less than 30 books. $4.75 is a little less than $5 30 × $5 = $150 By estimating, Winston knows that he will have enough money.

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Calculating

Estimate the answer to 4482 ÷ 88 4482 is close to 4500

88 is close to 90. 4500 ÷ 90 = 50 The answer will be about 50.

1 Estimate the cost of 51 televisions priced at $299 each.

2 A restaurant bill for 9 people is $178.50. Estimate how much each person must pay.

3 A cinema has 28 rows. There are 22 seats in each row. A cinema ticket costs $9.80. How much money will the cinema receive if the cinema is full? 1 The answer to three of these questions is wrong. Use estimating to check whether each answer is reasonable. A 892 281 + 1893 = 894 174 B 9132 – 1922 = 5980 C 956 × 18 = 20 076 D 814 ÷ 11 = 31

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Mixed operations with a pair of brackets

1 There are 3 plates of fruit. Each plate has 5 bananas and 3 apples. How many pieces of fruit are there altogether?

Each plate has a total of 8 pieces of fruit. To find the total number of pieces, we multiply 8 pieces by 3. 8 × 3 = 24 There is a total of 24 pieces of fruit. However, Carl and Letisha got different answers. Their workings are shown below.

3 × 3 + 5 = 9 + 5 = 14

3 × 5 + 3 = 15 + 3 = 18

Carl and Letisha need to add brackets as shown below. 3 × (5 + 3) or 3 × (3 + 5)

When there are brackets, always complete the operation inside the brackets first. In this case, we complete the addition first. Then we multiply the sum. 3 × (5 + 3) or 3 × (3 + 5) = 3 × (8) = 3 × (8) = 24 = 24

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Calculating

2 Let’s find the total number of items using brackets. (a)

3 × (4 + 2) = 3 × (6) = 18

(b)

(4 − 2) × 5 = (2) × 5 = 10

Carry out the following. (a) 3 × (3 + 3)

(b)

(6 × 5) ÷ 10

= 3 × (

)

= (

) ÷

(d)

)

(8 + 1) × 6

− (24)

= (

) × 6

(e) 40 − (5 × 5)

(f )

12 + (

÷ 6)

− (

)

+ (4)

Find the solutions to the following.

(b)

(a) (2 × 6) + 13 =

50 ÷ (2 + 8) =

(d)

4 + (3 × 4) =

(e) 4 × (30 ÷ 6) =

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Mental addition and subtraction

Look at the calculations in the box. Talk to a partner about the mental strategies you could use to calculate the answer. 520 + 370 645 – 213 4003 – 3998 Compare your strategies with Leon and Letisha in the next section.

$520 $370

Leon notices that both of these are multiples of tens. 52 + 37 = 89 So the prices add up to 89 tens. This is equal to 890.

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Calculating

2 Use a mental strategy to find the difference between 645 and 213

Letisha uses a strategy that uses addition. 213 +

613 Next, she adds tens to reach 643. +

643 Finally, she adds ones to reach 645.

645

Letisha adds up the hundreds, tens and ones to find the difference.

= 432

645 − 213 = 432

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Subtract 3998 from 4003 mentally.

Toby notices these numbers are close to each other. He counts up from the smaller number to the larger number.

+ 1

3 998 3 999 4 000 4 001 4 002 4 003

Complete the following mentally. (a) 780 − 180 =

(b) 3000 + 2500 =

(d) 893 − 543 =

(e) 5700 + 1200 =

(f) 660 + 220 =

(g) 726 − 321 =

(h) 6400 + 4400 =

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Calculating

Written multiplication

When we want to multiply, we can use a formal written method.

1 4 6 2 4

Multiply each digit in turn.

1 4 6 2 4 8

6 tens multiplied by 4 is 24 tens.

1 4 6 2 4 24 8

1 4 6 2 4

4 8

We can regroup the 20 tens to 2 hundreds.

1 4 6 2 4 24 4 8

4 hundreds multiplied by 4 is 16 hundreds.

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1 4 6 2 4 8 4 8

1 thousand multiplied by 4 is 4 thousands.

1 4 6 2 4 5 8 4 8

1462 x 4 = 5848

1 Use a formal written method to calculate: (a) 3812 x 7 (b) 1808 x 9 (c)

81762 x 3

2 Explain the mistake that Peter has made:

6 3 0 7 5 4 5 3 5

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Calculating

Solve these word problems. (a) There are 9 biscuits in a pack. How many biscuits are in 135 packs? (b) John has 5 pieces of ribbon. Each ribbon is 287 cm long. How many centimetres of ribbon does he have in total? (c) There are 638 people at a concert. They each pay $7 for a ticket. How much has been paid? (d) A cruise liner travels 1921 km in one trip. How many kilometers does it travel in 4 of these trips.

Arrange these digit cards to make a multiplication calculation.

1 2 3 4 5 6

How close can you get to a quarter of a million?

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Division

A box contains 94 tomatoes. They are divided into 4 paper bags equally. How many tomatoes will each bag contain?

We need to divide 94 by the number of bags.

9 1 4 _ 4 ) 2 3

remainder 2

Each bag will contain 23 tomatoes. There will be two tomatoes left over.

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Calculating

1 There are 208 students in a school. They are divided into teams of 5. How many will be in each team? How many will be left over?

2 Peter needs 187 cards. Cards are sold in packs of 8. How many packs of cards should he buy?

3 How many complete weeks are in 162 days?

Divide 3-digit numbers by single-digit numbers: Dice divisions You need: pencil, paper, a die. Roll the die 3 times to create a 3-digit number. Every correct division you can make without a remainder wins you 10 points. Every correct division with a remainder wins you 5 points. For example:

Points

÷2 ÷3 ÷4 ÷5 ÷6 ÷7 ÷8 ÷9

216



The winner is the first player to achieve 200 points.

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Average and spread

KEY questions

What is the same about the height of the people in the two pictures? What is different about the height of the people in the two pictures? What statistics would you collect to find out which group was taller?

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Average and spread

KEY words

representative

summary statistic

average

mean

median

mode

data set

frequency

range

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Average

Jade’s class had a maths test. It is scored out of 20. These are their scores.

17 15

Which one score do you think best summarises how well the class did?

An average is a single number that is a good summary of the whole data set.

There are three different ways to calculate an average. These are the Mean, the Median and the Mode.

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Average and spread

Median The median is the middle value in a data set. Jade’s teacher has written the test scores in order from smallest to largest.

11 12 13 13 13 14 15 15 15 16 17 17 17 17 18 18 18 19 20 20

In this case there are two middle values. The median is midway between these two values. 16 .5 16 17 The median is 16.5 Exactly half the class scored more than 16.5 and exactly half the class scored less than 16.5 Mode The mode is the value that has the highest frequency.

11 12 13 13 13 14 15 15 15 16 17 17 17 17 18 18 18 19 20 20 3 3 4 3 2

4 people scored 17. It has the highest frequency. The mode is 17.

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(a) Which colour car is the mode?

(b) Which sport is the mode?

Which set of data does not have a mode?

3 Letisha recorded the number of minutes she waited for the bus for one week. 4 15 7 8 2 5 7 (a) Write the data in order from smallest to largest. (b) Find the median.

4 Find the mode and median for each set of data. (a) 8, 2, 3, 1, 8, 1, 4, 8, 5 (b) 10, 70, 40, 40, 30, 20, 50 48

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Average and spread

The mean Riana, Jade and Leon each have some sweets.

They have 15 sweets altogether between 3 people. 15 ÷ 3 = 5

The mean average number of sweets is 5.

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Some rectangles are made from squares.

(a) How many squares are there in total? (b) How many rectangles are there? (c) What is the mean average number of squares per rectangle?

1 Five children have 30 pencils between them. How many pencils do they have on average?

2 Find the mean average number of buttons.

3 Match each data set to its mean average. Data set

Average

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Average and spread

Cross - curricular

A batting average is calculated by the number of runs divided by the number of innings where they were ‘out’.

Batting average = Runs scored __________________ Number of times out This table shows some statistics from the St Lucia Zouks in 2020. Most runs Player Mat Inns

NO Runs

225

RL Chase

224

Najibullah Zadran

211

ADS Fletcher

182

RRS Cornwall

166

M Deyal

156

Mohammad Nabi

J Glen

DJG Sammy

SC Kuggeleijn

KS Melius

For example, RL Chase played 9 innings but was not out 3 times. He was out 6 times. His batting average is: 225 ÷ 6 = 37.5 Calculate the batting average for the rest of these players.

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Range

Letisha owns a shoe shop. She records the sizes of the last 100 pairs of shoes she sold.

Size

30 31 32 34 35 36 37 38

Frequency

11 14 18 24 12 10

The range is a measure of how spread out the data is. It is calculated by finding the difference between the smallest and the largest value. The smallest shoe size was 30. The largest shoe size was 38. 38 − 30 = 8 The range is 8.

Find the range in height of these mountains.

3214 m

2796 m

2395 m

1847 m

1203 m

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Average and spread

Find the range of these sets of data. (a) 4, 6, 6, 9 ,5, 8 (b) 19, 19, 31, 42, 24, 29, 25

2 The youngest member of a club is 18. The range of ages is 25. (a) What is the age of the oldest member? (b) A new member joins who is 16. Will the range increase, decrease or stay the same?

Conduct a statistical investigation to answer this question. How has the length of pop songs changed over time? Research length of the top 20 songs sold in 1980 and in 2020. Collect your results in a table. Calculate the average length of song in 1980s and in 2020. It might help to convert the length of songs to seconds. What does your data show?

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Form small groups and measure the height of each group member. Record the height of each person in a table. Use a spreadsheet application to find the average height of the members in your group. Step 1 Start the application and create a new document.

Type the names of the members in your group in column A.

Step 2

Leon Riana Jade Carl

Type the heights of the members of the group in column B. Note all the heights must be in the same units.

Step 3

Leon Riana Jade Carl

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Average and spread

In a cell below the names, type ‘Sum’. In the cell below the heights, type ‘=SUM(cell of first height : cell of last height)’ Then press ‘ENTER’. The sum will appear in the cell.

Step 4

Leon Riana Jade Carl

In a cell below the sum, type ‘Average’. In the next cell across, type ‘= cell of the sum / the number of members in your group)’ Then press ‘ENTER’. The average will appear in the cell.

Step 5

Leon Riana Jade Carl

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Fractions and percentages

KEY questions

What fractions can you see in the diagram? What percentages can you see in the diagram?

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Fractions and percentages

KEY words

equivalent

proper fraction

improper fraction simplify proportion

mixed fraction

percent

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Equivalent fractions

Leon uses this fraction wall to find equivalent fractions.

1 24

1 12

1 8

1 6

1 4

1 3

1 2

The diagram shows 6 __ 9

shaded.

6 and 9 are both divisible by 3.

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Fractions and percentages

÷3

6 9

2 3

÷3

The fraction 6 __ 9

2 __ 3 when written in its simplest form.

1 Use the fraction wall to write six more pairs of equivalent fractions.

Complete these equivalent fractions.

8 ______

3 __ 4

1 __ 3

4 __ 5

= ______ 9

= ______ 12

(a)

(b)

(c)

7 ___ 14

3 ___ 10

15 ______

1 ______

4 __ 6

2 ______

(d)

(f)

(e)

3 What fraction of the grid has been shaded? Write your fraction as simply as possible.

4 Change one fraction in each pair so they have a common denominator.

5 __ 6

3 __ 8

6 ___ 10

1 __ 4

14 ___ 20

2 __ 3

(b)

(c)

(a)

Circle the greater fraction in each pair.

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Finding a fraction of an amount

Recall that we can find fractions of quantities by dividing. What is 1 __ 3 of 21? 21 7 7 7

1 __ 3

of 21

= 21 ÷ 3 = 7

2 __ 5

Find

of these balls

Represent the problem using a bar model. 1 __ 5 of 15 = 15÷ 5 = 3

1 __ 5 2 __ 5

of15=3

of15=6

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Fractions and percentages

1 The model shows 48 divided into 6 equal parts.

Use the model to write down

5 __ 6

1 __ 6

2 __ 6

(a)

(b)

(c)

of 48

1 __ 7

of35=5

Use this fact to find

3 __ 7

5 __ 7

2 __ 7

(a)

(b)

(c)

of 35

(a) Find 3 __ 5

of 40 (b) Find 3 __ 4

3 ___ 10

of 20 (c) Find

of 70

Leon opens a box of 48 biscuits.

Winston takes 1 __ 4

of the biscuits.

Jade then takes 6 biscuits Letisha counts how many are left and then takes 1 __ 5 What fraction of the box remains?

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Fractions and percentages

Which of the fractions in each pair is greater?

5 __ 6

34 ____ 100

45 ____ 100

2 __ 3

and

Talk to a partner about which pairs you found easier to compare.

A percentage is a fraction with a denominator of 100.

34 ____ 100

= 34%

1 Look at the coloured squares on the 100-grid.

What percentage of the squares are red? 3 out of 100 squares are red. = 3%

3 ____ 100

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Fractions and percentages

2 We can use equivalent fractions to help us write a quantity as a percentage. What percentage of this grid is shaded?

The fraction that is shaded is 6 ___ 10 . =

6 ___ 10

60 ____ 100

The percentage of the grid that is shaded is 60%

Write 32% as a fraction. 32% =

32 ____ 100

÷2

32 100

16 50 = =

8 25

÷2

8 ___ 25

1 What percentage of the grid opposite is: (a) Orange? (b) Blue? (c) Green?

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2 Carl says that 20% of this grid is orange. Explain what mistake he has made.

3 (a) What fraction of the grid is shaded?

(b) Write this fraction as a fraction out of 100. (c) What percentage of the grid is shaded? 4 Write each of these percentages as a fraction. Simplify the fraction where possible.

(a) 30% (b) 25% (c) 48% (d) 17%

5 Write these fractions and percentages in ascending order.

17 ___ 50

6 ___ 10

35%

50%

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Fractions and percentages

Calculating with percentages

1 These are some common fraction-percentage equivalents you should learn by heart.

2 We can use the fractional equivalents of percentages to calculate quantities. (a) Find 75% of 40 kg. Recall 75% = 3 __ 4 75% of 40kg = 3 __ 4 of 40kg

1 __ 4

of 40 = 10

3 __ 4 of40=10×3 = 30 75% of 40kg = 30kg

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(a) Find 20% of 50.

(b) What is 35% of 80?

(d) Find 80% of 60.

(e) What is 12% of 20?

(f) Find 50% of 84.

(g) Find 30% of 70.

(h) What is 25% of 16?

(i) What is 95% of 200?

(j) Find 75% of 1000.

10% of a number is 24. Use this fact to find (a) 5% of the number (b) 1% of the number (c) 30% of the number

1 There are 25 cars in a parking lot. 4 of the cars are blue. What percentage of the cars are blue? 2 There are 200 learners in Year 5, of which 88 are girls. What percentage of the learners are girls? 3 24 students out of the class of 50 are girls. What is the percentage of female students in the class?

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Fractions and percentages

Look through some newspapers and magazines. What examples of percentages can you find?

Carl has noticed a lot of percentages today.

Researchers say over 20% of people will be affected.

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Numbers to one million

KEY questions

What is the place value of each digit in large numbers? Where else in life might you use large numbers?

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Numbers to one million

KEY words

millions

hundred thousands

place value

digits

estimate

round

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Place value

Look at the number beads below.

200 000

800 000

1000

+ 600

+ 40

+ 3

This is equal to

2 8 1 6 4 3

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Numbers to one million

2 Let’s read the number represented by the beads as a numeral and in words.

This is the number 3 490 526. We read it three million, for hundred and ninety thousand, five hundred and twenty-six. What happens if we add another ten thousand?

The 10 ten thousands are exchanged for one hundred thousand.

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3 Position the number 412 150 on a number line.

+1000

412 000

413 000

The number line has ten intervals in one thousand. Each interval is 100. The number 412 150 is halfway between 412 100 and 412 200.

412 100

412 200

1 (a) Read each number in the sequence out loud. + 100 000 + 100 000

+ 100 000

305 118

405 118

505 118

605 118

205 118

(b) Write the number that comes next in numerals and in words.

2 Arrange the numbers from the smallest to the greatest.

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Numbers to one million

1 Write the number represented by the beads in numerals and in words.

2 Write these number statements using < or > to show which is greater.

(a) 48 382

38 181

(b) 112 291

29 319

401 091

Riana is counting on in fifty thousands.

+ 50 000

316 000

416 000

466 000

516 000

366 000

What will be the first number greater than one million?

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Rounding

What could the numbers at a and b on the number line be?

14 000

15 000

1 The number of people at a football match was 340 287. What is the number to the nearest hundred? The number 340 287 is shown on the number line.

340 200

340 300

340 287 rounded to the nearest 100 is 340 300.

2 What is 43 291 rounded to the nearest thousand. The number 43 291 is between 43 000 and 44 000 43 291 43 000 44 000 It is closer to 43 000 than 44 000. So, 43 291 rounded to the nearest thousand is 43 000.

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Numbers to one million

1 Use the number line to round each number. (a) Round 3690 to the nearest hundred.

3 400

3 500

3 600

3 700

(b) Round 8150 to the nearest hundred.

8 000

8 100

8 200

8 300

68 000

69 000

(d) Round 53520 to the nearest thousand.

53 000

54 000

2 Winston, Leon and Jade want to round the number 183 813 to the nearest thousand.

(a) Which is the correct answer? (b) Can you explain the mistake the other children have made?

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What does one million look like? Visualise 10 Type 10 of the same character on the page.

tttttttttt

Visualise 100 Repeat with different characters until you have ten sets of ten.

tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%%

Visualise 1000 Now ‘copy’ and ‘paste’ each hundred until you have 1000 characters on the page. tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% tttttttttt**********$$$$$$$$$$vvvvvvvvvv))))))))))!!!!!!!!!!gggggggggghhhhhhhhhhllllllllll%%%%%%%%%% Visualise 10 000 ‘copy’ and ‘paste’ your set of thousand to create 10 000 characters!! Can you change the font size and page formatting to fit all the characters on one page? Use the word count to check! Visualise 1 000 000 How many of your pages would you need to show one million characters?

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