Number Strategies Series: Working on Algebra by Teacher Superstore

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Acknowledgements i. Clip art images have been obtained from Microsoft Design Gallery Live and are used under the terms of the End User License Agreement for Microsoft Word 2000. Please refer to www.microsoft.com/permission.

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The purchasing educational institution and its staff have the right to make copies of the whole or part of this book, beyond their rights under the Australian Copyright Act 1968 (the Act), provided that: 1.

The number of copies does not exceed the number reasonably required by the educational institution to satisfy its teaching purposes;

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Copies are not sold or lent;

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The Act allows a maximum of one chapter or 10% of the pages of this book, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that

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Except as otherwise permitted by this blackline master licence or under the Act (for example, any fair dealing for the purposes of study, research, criticism or review) no part of this book may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address below.

o c . che e r o t r s super Published by: Ready-Ed Publications PO Box 276 Greenwood WA 6024 www.readyed.com.au info@readyed.com.au

ISBN: 978 1 86397 828 6 2

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Any copying of this book by an educational institution or its staff outside of this blackline master licence may fall within the educational statutory licence under the Act.

Reproduction and Communication by others

Contents Teachers’ Notes Curriculum Links

4 5

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Number Properties

(Teachers’ Notes)

Groups of Numbers

Plotting Points

Multiples

Join the Dots

Tables and Coordinates

Prime Factor Trees

Plotting Tables of Values

Can it be divided by this number?

Graphs and Coordinates

Divisibility Tests

Match Them Up

More Divisibility Tests

Marble Clusters

Factors

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(Teachers’ Notes)

Tables and Graphs

29 30

31 32 33 34

14 Farmer Tom’s Fence 35 © R e a d y E d P u b l i c a t i o n s Conjectures 2 15 Writing a Short Story 36 •f orr evi ew puWriting r po se sonl y•37 a Short Story Conjectures 1

Number Patterns

Roast Dinner

Social Media

Complete The Pattern 1

Handshakes and Kisses

Complete The Pattern 2

Super Savings

Complete The Pattern 3

Square Numbers

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Cube Numbers

Triangular Numbers Fibonacci Numbers Pascal’s Triangle 1 Pascal’s Triangle 2 Pascal’s Triangle 3

Answers

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(Teachers’ Notes)

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44 46

Teachers’ Notes This resource is focused on the Number and Algebra Strand of the Australian Curriculum for students in Year 5 ,Year 6 and Year 7, aged between 10 and 12 years old. Each section provides students with the opportunity to explore a key area of their algebraic understanding, often with the opportunity to explore their real life contexts or extend their exploration further.

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While students are to be encouraged to use their mental arithmetic skills at every opportunity, many tasks will be made more efficient with the use of a calculator. When it is advisable to use a calculator you will see this symbol.

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The section entitled “Groups of Numbers” exposes students to various categories of numbers, their uses in calculations and their real life applications. The section on Conjectures allows students to really exercise their mathematical justification.

The section entitled “Number Patterns” exposes students to a large variety of different patterns, from the basic to the famous. Students will learn about some famous Mathematicians and explore some of the fascinating patterns that exist within these famous number sequences.

The section entitled “Rules, Table and Graphs” is a more advanced section which builds towards students being able to move fluidly between the three different representations of functions. Each task is set within a real life context, many of which would be familiar to students. Once mastered, students will have an excellent basis for future function work.

© ReadyEdPubl i cat i ons •f o rprefaced r evbyi e wp ur pexplaining oses l y • Each section is also a Teacher Notes page, the o idean and purpose behind each activity. Included here are methods to extend the activities or modify the activities based on the level of individual student ability.

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The majority of activities are scaffolded into two sections; Task A builds up the general skills to be mastered, usually enabling students competence in a given skill or an understanding of the basic number sequence. Task B explores the skill further with a more in-depth investigation or consideration and often extends the concept further.

Most activities contain a Challenge at the bottom of the page. These challenges range from Individual Challenges, through to Research and Small Group Challenges. Each of these are designed to complement the activity page, yet extend beyond the material. They are designed to engage student interest and appreciation for Mathematics as well as exposing students to the idea that Mathematics can be a creative and investigative pursuit. Challenges can be included in the lesson of the day, or used as a stand-alone lesson when time permits. Many can be set as homework or assignment tasks over a longer period of time. Research tasks do tend to include the use of internet resources and it is advisable that computer resources are organized in advance.

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It is hoped that Working On Algebra will be used to help guide teachers in their teaching strategies and methods of presentation. While some activities are designed to be extra practice for students, many others can be used to present and teach students new concepts.

National Curriculum Links Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction (ACMNA107) Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence (ACMNA133)

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Identify and describe properties of prime, composite, square and triangular numbers (ACMNA122)

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Identify and describe factors and multiples of whole numbers and use them to solve problems (ACMNA098) Given coordinates, plot points on the Cartesian plane, and find coordinates for a given point (ACMNA178)

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Teachers’ Notes

Number Properties Groups of Numbers

This activity exposes students to different types of numbers and their definitions. Task C can be extended into an assignment or a small group task over a few lessons.

Multiples

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Factors

Factors and Their Prime Factors

This activity is best attempted after “Prime Factor Trees”. This activity helps reinforce the idea that once two factors are known, all factors can be known for any number. Task C provides those more curious and capable students with an opportunity to research the real life and important applications of prime numbers.

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This activity revises the concept of multiples and provides students with a method to determine a Lowest Common Multiple. The mini quiz task is a useful strategy to employ to determine whether students have understood a topic and can used often to engage student participation and knowledge.

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Divisibility Tests

This task should be attempted after “Can it be divided by this number?”. This task encourages students to explore other divisibility tests with the more difficult tests left for Task C and those more capable students.

© ReadyEdP l i cat i onPrimes s Eu asyb Calculation Using This activity is best completed after “Prime Factor Trees”. Students are encourage to •f orr evi ew pu r p os eso nl y• multiply number by first determining their

Prime Factor Trees

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Understanding that all numbers can be broken down into the product of their prime factors is an important concept and this method provides students with an easy and visual approach to find these prime factors.

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prime factors. The early concepts of indices and manipulation for multiplication come into play here.

Conjectures 1

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This activity revises the concept of factors and provides students with a method to determine the Highest Common Factor. Task C is suitable as an extension activity and is probably best attempted by your more able students.

Students are asked to consider various mathematical statements, called conjectures, and determine whether they are either true or false. This provides students with an opportunity to think about how they know what they know. Task C can extend this concept further by considering geometrical statements.

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Task A encourages students to think about how they know if whether a number can be divided (without a remainder) by a particular one digit number. Task B then introduces students to a simple divisibility test to determine whether a number can be divided by 3 without leaving a remainder. Task C can be treated as a research project or for the more able students, as an investigative task. 6

Conjectures 2

This activity is best completed after “Conjectures 1”. These conjectures are more advanced and tie in many of the concepts covered in the previous 9 activities.

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Groups of Numbers

* Task a

Choose a word from the list to make each of the following definitions true. Some words can be used more than once.

Itself

Facto

tion

Multiplica

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Evenly

Integ

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a. Prime Number: A number that is only divisible by _____ and ______. b. Composite Number: A number that has _______ other than itself and one. c. Factors: The numbers that can be divided _______ into another number. d. Multiples: The numbers that are created when we multiply one integer (whole number) by another _______. These can be thought of as the _____________ tables of a number.

e. Square Numbers: The number created when we multiply a number by _______.

Write down the next ten numbers for each set of numbers:

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* Task b

2, 3, 5, ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ .

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4, 6, 8, ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ .

1, 4, 9, ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ .

e. Multiples of 5:

5, 10 , 15, ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ .

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a. Prime Numbers: b. Composite Numbers:

o c . c e h r d. Cube Numbers: 1,e 8, 27, ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ . o t r s super

* Task c: research Challenge

In small groups of 3 or 4 students research the Sieve of Eratosthenes. Who invented it? What is it used for? Display your findings on a poster with clear explanations and examples of how the sieve works. 7

Multiples Task a * List the first 10 multiples for each of these numbers. The first one has been done for you. a. 3:

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b. 2: c. 7:

e. 25:

175 60

f. 15:

g. 12:

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d. 10:

105 72

120

What is a Lowest Common Multiple? A Lowest Common Multiple (LCM) is the lowest multiple that two numbers have in common. To find the LCM for 3 and 4 we list some of their multiples and then look for the lowest one they have in common. 3: 3, 6, 9, 12, 15, 18 …

4: 4, 8, 12, 16 …

Find the lowest common multiple for each of the following:

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* Task b

a. 3 and 5 b. 6 and 7

c. 10 and 15 d. 6 and 8

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* Task c: Partner Challenge

Create a one page mini quiz for your partner on multiples and LCMs. Before giving it to your partner to try, make sure you have already written out a detailed marking key with all the answers on a separate piece of paper.

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As we can see, the lowest number they have in common is 12.

Quiz !

Factors

* Task a

List all of the factors for each of the following numbers. The first one has been done for you. 1

a. 8:

b. 10: c. 24:

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e. 36:

f. 48:

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d. 100:

What is a Highest Common Factor? A Highest Common Factor (HCF) is the largest factor that two numbers have in common. To find the HCF for 12 and 18, we firstly list each of their factors and then look for the largest factor they have in common.

© ReadyEdPubl i cat i ons Find ther highest common factor for each ofo these numbers. Then firstl one has been done for o r e v i e w p u r p s e s o y • Task• b f you. As we can see, the largest number they have in common is 6. This is the HCF. * 12: 1, 2, 3, 4, 6, 12

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b. 15 and 60

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c. 27 and 36 d. 36 and 90 e. 20 and 24

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

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a. 12 and 18

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Personal Challenge * ToTaskfindc:a faster way to find the HCF, research Euclid’s Algorithm. An algorithm is a set of rules to follow to complete a calculation. It’s a lot like following a recipe. Once you have understood how the algorithm works, use it find the HCF for these numbers: a) 252 and 105 b) 2322 and 654 9

Prime Factor Trees Any number can be expressed as a product of its prime factors. Look at the two examples below. Example 1:

24 x

Choose any two numbers that multiply to give you 24

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2x3

These are all prime numbers so the tree finishes here

2x2

which are all prime numbers

2x3

x2 © ReadyEdPubl i cat i ons •f orr evi ew pu posesonl y• d.r 80

120 x

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12 x4

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x4 x2

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16 x

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Complete each of these prime factor trees.

2x 2x 4x x

Can it be divided by this number?

* Task a

Explain how you know whether a number can be divided these numbers. Use a few examples to explain your answers.

a. How do you know whether a number can be divided by 2? _______________________________________________________________________

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b. How do you know whether a number can be divided by 10?

_______________________________________________________________________ c. How do you know whether a number can be divided by 5?

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_______________________________________________________________________

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d. If you know a number can be divided by 4 , can it also be divided by 2? How do you know? _______________________________________________________________________ e. If you know a number can be divided by 15, can it also be divided by 5? How do you know? _______________________________________________________________________

f. If you know a number can be divided by 6, can it also be divided by 3? How do you know? _______________________________________________________________________

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* Task b

Circle any of the following numbers that are divisible by 3.

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a. 312 e.g. 3+1+2=6

g. 2 652

b. 450

h. 3 216

c. 617

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A number is divisible by 3 if when we add the digits of a number we see that it is a multiple of 3. We can see that 312 can be divided by 3 because 3 + 1 + 2 = 6 and 6 is a multiple of 3.

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d. 723

j. 5 355

e. 982

k. 12 693

f. 1 012

l. 125

*Check your answers with a calculator.

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Divisibility Tests

* Task a

Circle all the numbers that are divisible by 2.

a. 15

d. 36

g. 7

j. 234

b. 22

e. 48

h. 95

k. 511

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l. 64

i. 102

f. 8

c. 41

Explain how you know if a number can be divided evenly by 2.

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_______________________________________________________________________ For a number to be divisible by 4 then the last 2 digits of the number must be divisible by 4.

Task B

b. 24

c. 218

12 is divisible by 4 so 112 is divisible by 4

d. 348

g. 630

j. 2 522

e. 420 h. 1 016 k. 4 136 © Re adyEdPu bl i cat i ons f. 514 i. 1 764 5 210 •f orr evi ew pur p osesonl yl. •

* Task C

a. 10

Circle all the numbers that are divisible by 5. c. 55

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b. 24

d. 32

e. 48

g. 220

f. 100

h. 315

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a. 112

Look at the last two digits of the numbers below and decide whether they are divisible by 4. Circle these numbers.

. t e o _______________________________________________________________________ c . c e For a number to be divisibleh by 6e then the number must first be divisible byr both 2 and 3. Remember, a o t r s s rwe can see that it is a multiple of 3. u number is divisible by 3 if when we add the digits ofe a number p Explain how you know if a number can be divided evenly by 5.

* Task D

a. 312

Circle all the numbers that are divisible by 6.

312 is divisible by 2 and 3 therefore divisible by 6

d. 48

g. 212

j. 738

b. 423

e. 92

h. 525

k. 873

c. 642

f. 990

i. 552

l. 252

More Divisibility Tests For a number to be divisible by 8 then the last 3 digits of the number must be divisible by 8. For example the number 5 816 is divisible by 8 because 816 is divisible by 8.

Task a

Without using a calculator, write down 10 numbers, greater than 1000, that are divisible by 8. Have your partner check your answers using a calculator.

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For a number to be divisible by 12 then the number must first be divisible by both 3 and 4.

* Task B

b. How do you know whether a number can be divided by 4?

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_______________________________________________________________________ c. Without a calculator, write down 10 numbers, greater than 10 000, that are divisible by 12. Have your partner check your answers using a calculator. E.g. 3 0 6 2 4

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Task c: Personal Challenge * Find out how you know whether a number can be divided by 7 and whether a number can be divided by 11. Write a half page report, with examples, explaining the techniques you need to use.

Conjectures 1 A conjecture is something we think is probably true after we’ve looked at a few examples. For example, we might conjecture or guess that if a number can be divided by 3 then it can also be divided by 6. If this conjecture is false then we need to find an example where it is false.

Trure o e Fals

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b. Two more than an odd number is always another odd number. q True q False

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For each of the following tick whether the conjecture is true or false. If it is true provide 5 examples of where it is true. If it is false, provide just one example that shows it is false. d. All multiples of 10 are even numbers. a. If a number can be divided by 4 then q True q False it can also be divided by 2. q True q False

Task a

e. An even number plus an odd number is always an odd number. q True q False

For each of the following conjectures explain clearly why they are true.

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* Task b

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c. If a number can be divided by 5 then it can also be divided by 10. q True q False

a. When two even numbers are added together the result is an even number.

. tbe divided by 10 then it can also be divided by 5. o b. If a number cane c . c e _______________________________________________________________________ her r o t s su er palso c. If a number can be divided by 6 then it can be divided by 3. _______________________________________________________________________

_______________________________________________________________________ Task c: Personal Challenge *Think about each of these geometric conjectures, that is, conjecture that have to do with shapes. Write a sentence on why each conjecture is either true or false. 14

Conjecture 1: All squares are rectangles. Conjecture 2: All triangles have a right angle. Conjecture 3: All quadrilaterals have a pair of parallel sides.

Conjectures 2 For each of the following conjectures, decide if they are true or false. If they are true write down five examples that show they are true. If they are false, find just one example that shows they are false. a. All prime numbers are odd numbers. q True q False

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f. The product of two consecutive counting numbers is always a multiple of 4. q True q False

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b. The product of an even number and an odd number is an even number. q True q False

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e. When you multiply two consecutive odd numbers you always end up with an odd number. q True q False

c. The sum of three consecutive numbers (numbers that are one after the other) is a multiple of 3. q True q False

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d. When you square an odd number and then subtract one, the answer can always be divided by 8 with no remainders. q True q False

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q True q False

h. All composite numbers less than ten are even numbers. q True q False

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Teachers’ Notes

Number Patterns Complete the Pattern 1

This activity is best completed after “Square Numbers” and explores the same concepts, this time about cubic numbers. Task C should keep students busy with their investigations and is the beginnings of one of the most puzzling Mathematical problems of all time, Fermat’s Last Theorem.

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Complete the Pattern 2

Similar to “Complete the Pattern 1”, except here students will find that each pattern involves the multiplication or division of numbers. Task C is a fun pattern which is as it’s name suggests. You “look” and see one 1, so the next number is 11, then you look and seen two 1s, so the next number is 21 and so on. Calculators can be used for this activity.

Triangular Numbers

Triangular numbers are a great number sequence that can be easily visualized. This activity exposes students to this sequence with further explorations.

Fibonacci Numbers

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Task A encourages students to look for patterns within each number sequence. Each pattern involves the addition or subtraction of numbers only. Task B enables students to describe the patterns they see and to extend the patterns further. Task C provides students with an opportunity to explore an interesting number pattern. Calculators can be used for this activity.

Cube Numbers

This activity is a rich task, inviting students to explore the Fibonacci sequence and a few of the many patterns to be found within the sequence itself. In Task B students have the opportunity to explore the Golden Ratio and a class discussion of where the Golden Ratio can be found in “real life’ might spark an extra interest and enthusiasm from students. Task C is a great opportunity for students to extend their understanding of the history of Mathematics.

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opportunity for students to test their knowledge and skills. Pascal’s Triangle 1 This task exposes students to this famous triangle and number sequence. Students are encouraged to explore further patterns within the triangle and to research Blaise Pascal’s place in history. This activity is more difficult because students need to investigate each sequence very carefully. You may like to have students work in small groups to discuss the patterns they see.

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Pascal’s Triangle 1

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This task exposes students to this famous triangle and number sequence. Students are encouraged to explore further patterns within the triangle and to research Blaise Pascal’s place in history.

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This activity enables students to explore square numbers and gain a visual understanding for their name. Task B combines square numbers with other number patterns and some may be a little tricky. Task C provides the students with an opportunity to explore where square numbers are used and a chance to learn about this important Mathematician.

Pascal’s Triangle 2

This task is best completed after “Pascal’s Triangle 1” and “Fibonacci Numbers”. Exploring some of the patterns to be found by adding numbers in Pascal’s Triangle allows students to explore the triangle further.

Pascal’s Triangle 3

A fun and important application of Pascal’s Triangle is in the use of combination theory. Here students can explore the connection between choosing groups of objects and what each row of the triangle represents. Task C allows for further practice of these concepts.

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Complete The Pattern 1

* Task a

Look carefully at each of these number patterns and find the next three numbers in the sequence. The first one has been done for you.

23 27 31 a. 7, 11, 15, 19, ___ , ___ , ___ .

We add 4 each time.

b. 200, 190, 180, 170, ____, ____, _____.

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c. 30, 37, 44, 51, ____, _____, _____.

e. 1 , 1 , 1 , ____, ____, _____. 12 6 4

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d. 30 000, 28 200, 26 400, ______, ________, _______.

Hint: Make all the fractions have the same denominator first before completing the pattern.

f. 40, 43, 41, 44, 42, _____, _______, ______

Look closely, this one is tricky!

the numbers in the sequences below and describe the pattern you notice . * Task b©Complete ReadyEdPubl i cat i ons

a. 80, 75, 70, 65, ___ , ___ , ___ , ___ ,

•f orr evi ew pur posesonl y•

Describe the pattern:___________________________________________________

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b. 1, 3, 6, 15, ___ , ___ , ___ , ___ ,

c. 320, 370, 420, 470, ___ , ___ , ___ , ___ ,

. te o c d. 1 , 1 , 3 , ___ , ___ , ___ , 7 . e 20 10 20 c 20 her r o t s super Describe the pattern:___________________________________________________

Describe the pattern:___________________________________________________ Hint: Make all the fractions have the same denominator first before completing the pattern.

* Task c: Personal Challenge

Explain what’s happening in this number sequence: 1 + 3 = 4 4 + 5 = 9 9 + 7 = 16 16 + 9 = 25 What are the next 10 calculations in this sequence? Do you think this sequence continues to work forever? 17

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Complete The Pattern 2

Task a

Look carefully at each of these number patterns and find the next three numbers in the sequence. The first one has been done for you.

3 2 _____, 6 4 _____ 128 a. 2, 4, 8, 16, ____,

We are multiplying by 2 each time.

b. 1, 10, 100, 1000, _____, _____, _______

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c. 1000, 500, 250, _____, ______, ______

e. 9000, 6000, 4000, ____, ______, _______ f. 1 , 1 , 1 , ____, ____, _____. 2 6 18

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d. 1, 3, 9, 27, _____, _____, _____

Complete the numbers in the sequences below and describe the pattern you notice . * Task b © ReadyEdPubl i cat i ons a. 2, 10, 50, ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , f orr evi ew pur posesonl y• Describe• the pattern:___________________________________________________ b. 300, 30, 3, ___, ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ ,

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Describe the pattern:___________________________________________________

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c. 1, 1, 2, 6, 24 , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ , ___ ,

. t e1 o 1 1 c d. 1000 , 100 , 10 , ___ , ___ , ___ , ___ , ___ , ___ , ___ ,. ___ , ___ , ___ , che e r o t r s super Describe the pattern:___________________________________________________

Describe the pattern:___________________________________________________

Task c: Personal Challenge * The following number sequence is called the “Look and Say” sequence. 1, 11, 21, 1211, 111221 …. Why is it called the “Look and Say” sequence? What are the next 5 numbers in the sequence? Create your own “Look and Say” sequence. 18

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Complete The Pattern 3 Find the next three terms in each of the following number patterns. The first one has been done for you. a. 7, 10, 13, 16, b. 4, 20, 100,

, ,

d. 45, 39, 33, 27,

e. 1, 2, 5, 10, 17,

f. 0.00006, 0.0006, 0.006,

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c. 8000, 2000, 500,

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You need to look at each sequence carefully to determine whether you need to add, subtract, multiply or divide by a certain number. Sometimes you might need to do a bit of both! Try working in a small group so that you can help each other and share your ideas.

, d ,© Rea y EdPubl i cat i ons • rr ev ew pur posesonl y• h. 1, 2, 2, f 4,o 8, 32, , i , g. 5, 11, 23, 47,

i. 2, 4, 6, 10, 16, 26,

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What are the next three numbers …

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j. 2, 8, 38, 188,

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k. 1 , 2 , 4 , 2 6 18

m. 18, 20, 24, 30, 38,

n. 5, 0, 10, 5, 15, 10,

o. 1000, 200, 30, 4,

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Square Numbers A square number is a number that you get when you multiply a number by itself. For example 2 ×2 = 4, so 4 is a square number. You can also think of a square number as the area of a square. The area of this square is 25 cm2. So 25 is a square number.

* Task a b. 2 x 2 =

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i. 9 x 9 =

= 4 = 32 = 9

= 64

= 102 =

= 112 =

e. 5 x 5 =

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d. 4 x 4 =

= 144

=t m. 13 xu 13b = l © ReadyEd P i ca i ons •=f or=r e i ew p r po=s es nl y• 49v n. u 14 = o

= 62 =

Each of the patterns below involve square numbers. Fill in the missing spaces for each of the number patterns.

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a. 1, 9,

, 49,

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b. 2, 5, 10, 17, c. 2, 8, d. 0, 3, 8,

, 32,

m . u

Task b

o c . che e r , o t r s super ,

, 24,

* Task c: Research Challenge

In small groups of 3 or 4 students, research the importance and role of square numbers to Pythagoras’ theorem. In particular, write a report about Pythagorean Triples and provide at least 15 Pythagorean Triples in your report.

5 cm

Using your calculator, and the boxes below, find the first 14 square numbers.

a. 1 x 1 = 12 =

5 cm



Cube Numbers A cube number is a number that you get when you multiply a number by itself twice. For example 2×2×2 = 8, so 8 is a cube number. Another way to think about a cube number is as the volume of a cube. In this cube the volume is 5 × 5 × 5 = 125 cm3. So 125 is a cube number.

* Task a

5 cm

r o e t s Bo r e p ok u S

Using your calculator, and the boxes below, find the first 14 cube numbers. h. 8 x 8 x 8 =

b. 2 x 2 x 2 = 23 =

= 93 = =

c. 3 x 3 x 3 =

= 27

d. 4 x 4 x 4 =

k. 11 x 11 x 11 =

= 53 =

ew i ev Pr

Teac he r

a. 1 x 1 x 1 = 13 =

= 1000

= 123 =

n. 14 x 14 x 14 =

= 73 =

Each of the patterns below involve cube numbers. Fill in the missing spaces for each of the number patterns.

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Task b

a. 0, 7, b. 0.5, c. 2,

d. 1, 27,

, 63, . te

m . u

o c . , 13.5, c , 62.5, e her r o t s s r u e p , 54, , ,

Task c: Personal Challenge * Can you find three numbers, a, b and c, so that a3 + b3 = c3 ? Research and discover how many such numbers are possible. Take note of any famous theorems you find in your research. 21



Triangular Numbers

* Task a 1st triangular number

Draw the next three diagrams in the pattern below and answer each of the questions.

2nd triangular number

3rd triangular number

4th triangular number

5th triangular number

r o e t s Bo r e p ok u S

6th triangular number

Teac he r

Why do you think they are called triangular numbers?

ew i ev Pr

_____________________________________________________________________ Explain how the triangular number pattern works.

_____________________________________________________________________ Using your calculator to help you, write down the first 10 triangular numbers.

_____________________________________________________________________

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2 3 4 5 6 7

. te

Sum of Triangular numbers (S)

1 1+3=4 1+ 3 + 6 = 10

m . u

Sequence Number (n)

o c . che e r o t r s super 1 + 3 + 6 + 10 = 20

8 9 10 Explain how you calculate the numbers in the third column. _______________________________________________________________________ 22



Fibonacci Numbers

* Task A

The first six numbers of the Fibonacci pattern are written below. Find the next 10 numbers in the Fibonacci pattern.

1, 1, 2, 3, 5, 8, ___, ___, ___, ____, ____, _____, _____, _____, _____, ______ Explain how you find the next numbers in the Fibonacci sequence.

r o e t s Bo r e p ok u S

______________________________________________________________________

There are many interesting patterns that can be found within the numbers of the Fibonacci sequence. Complete the table: Equation

Answer

Number in the Fibonacci pattern 3rd

12 + 12 12 + 22 22 + 32 32 + 52

ew i ev Pr

Teac he r

Task b

+ 21 © 13Re adyEdPubl i cat i ons 34 r +e 55v •f or i ew pur posesonl y• 2

w ww

m . u

The Golden Ratio is a special number that was particularly important to Ancient Greek architecture. The number is 1.61803399 …

* Task c

Complete each of the following:

2 ÷1 = 3÷2= 5÷3= 8÷5= . te calculations for the next 10 pairs of Fibonacci numbers. o *Continue these division c . c e __÷__ = __ ÷__ =h __ ÷ __ = __ ÷ __ =r __ ÷ __ = er o t s super 1÷1 =

__÷__ =

__ ÷__ =

__ ÷ __ =

What do you notice about the division of these numbers and the Golden Ratio? _____________________________________________________________________ Research Challenge *Who TaskwasD:Leonardo Fibonacci? Write half a page on Leonardo Fibonacci’s life and his importance to the history of Mathematics. 23

Pascal’s Triangle 1

* Task a

Complete rows five, six, seven and eight of Pascal’s Triangle below.

Row 0

Row 1

Row 2

Row 3

Row 4

Teac he r

r o e t s Bo r e p ok u S 1

Row 6 Row 7 Row 8

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Row 5

© ReadyEdPubl i cat i ons ______________________________________________________________________ •f orr evi ew pur posesonl y• * Task b Let’s look at some of the patterns within the diagonals of Pascal’s Triangle. Explain how you found each of the numbers in the next four rows of the triangle.

The second diagonal is 1, 2, 3, 4, 5 …

m . u

w ww

The first diagonal is 1, 1, 1, 1, 1 …

1. What pattern do you notice in the third diagonal?____________________________

. te

o c . 2. What pattern do you c notice in the fourth diagonal?___________________________ e h r e o t r What are these numbers called?____________________________________________ s super What are these numbers called?____________________________________________

3. What pattern do you notice in the fifth diagonal?_ ___________________________ How is this pattern related to the fourth diagonal of numbers?____________________

* Task c: Research Challenge

Who was Blaise Pascal? Write half a page on Blaise Pascal’s life and his importance to the history of Mathematics.



Pascal’s Triangle 2 Let’s explore the sums of the numbers in the rows of Pascal’s Triangle.

Task a First we need to complete the missing spaces in Pascal’s Triangle below.

Sum = 1

=4 =

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Teac he r

1 1 Sum = 1 + 1 = r o e t s B r o=o 1 2 p1e Sum u k 1 S 3 1 Sum = 1 6 1 Sum = 1 1 Sum = 1 1 Sum =

= 16

= 32 =

Sum of the Numbers (S)

w ww

1. What pattern do you notice in the second row of the table?

m . u

Row (n)

. ta rule we can use to help us find the number in the second rowo There ise of the table. c Sum of the Numbers = 2 . c e If we want to know the sum the numbers in Row 4 we would calculate 2 = 2 x 2 x 2 x 2 = 16 hofe r o t r s s r u e p 2. Explain how you can use the rule to find the sum of the numbers in row:

_______________________________________________________________________ Row Number

a) 6 ___________________________

b) 10 ___________________________

3. Use your rule, and your calculator, to find the sum of the numbers in the:

a) 12th row of the triangle____________________________________________

b) 25th row of the triangle____________________________________________ 25

Pascal’s Triangle 3

* Task a

Lucy is playing with her alphabet blocks. She has four alphabet blocks in front of her, the letters A, B, C and D.

1. How many ways can Lucy choose no blocks from this group? Only one way, by just sitting there!

r o e t s Bo r e p ok u S

2. How many different ways can Lucy choose one block from this group?

Teac he r

ew i ev Pr

3. How many different ways can Lucy choose two blocks from this group? She can choose AB, AC, AD, BC, BD or CD. So there are 6 ways. 4. How many different ways can Lucy choose three blocks from this group? 5. How many different ways can Lucy choose four blocks from this group?

w ww

Number of Different Groupings

2 15

Do you notice anything special about the numbers in this table and one of the rows in Pascal’s Triangle?

6 1

m . u

Groups of Blocks

. te o ___________________________________________ c . che e ___________________________________________ r o t r s super ___________________________________________ c: Pascal’s Triangle Challenge * TUseaskPascal’s Triangle to answer the following questions. • If Lucy had 8 blocks, how many different ways can she choose a group of 3 blocks? • If Lucy had 12 blocks, how many different ways can she choose a group of 8 blocks? • If Lucy had 15 blocks, how many different ways can she choose a group of 7 blocks? 26

Teachers’ Notes

Tables and Graphs Plotting Points

This activity serves as an introduction to the first quadrant of the Cartesian Plane. Students are encouraged to view plotting coordinates as performing a translation.

A task which explores another linear pattern and enables students the opportunity to compare and contrast and explain how they have determined their answers.

r o e t s Bo r e p ok u S

Join the Dots

Tables and Coordinates

Crucial to future understanding of function work, students are made aware through this activity that a table of values is a more efficient method of recording a series of coordinates.

Writing a Short Story

Students are encouraged to compare and contrast different linear patterns and to extend the pattern beyond the available table of values. Describing linear patterns in terms of slope is an important foundational concept.

Roast Dinner

ew i ev Pr

Building on the Plotting Points activity, this activity allows students to follow a series of translation instructions while improving their understanding of plotting points.

Teac he r

Farmer Tom’s Fence

Another familiar problem and linear relationship, students can explore and compare various graphs, exposing them to concepts of slope. Students are encouraged to create their own table of values and you might like to hold a class discussion before students embark on Task C.

Graphs and Coordinates

w ww

Students are encouraged to fluidly transform a graph of coordinates into a table of values. The emphasis is on ensuring that the independent, or horizontal variable, is arranged in the table in ascending orders.

A very relevant linear relationship, students can explore how mathematical relationships exist even in their spare time! In a familiar context, students also have the opportunity to add their own experiences to the task.

m . u

Plotting Tables of Values

In this activity students practice the skill of turning a table of values into a graph of plotted coordinates. The scale on each axis is already established and it is important to discuss with students the reason for using different scales.

Handshakes and Kisses

. t eUp! o Match Them c . che e r o t r s super

A great task to test student understanding after completing the previous tasks. Students need to carefully analyse the graphs, tables and rules and, as the title suggests, match them up.

Marble Clusters

This task begins with an easy visual and requires students to think about the connections between a table of values and the shape of a graph.

The familiar handshakes problem is explored here in more detail and extended by considering the related problem of exchanging kisses! Students are encouraged to explore the connection between the two problems and to consider the shape of curved graphs.

Super Savings

A great task to get students thinking about how even small amounts of savings can go a long way in the long term. Another opportunity to explore curved graphs and to consider a financial application. Students also have the opportunity to add their own experiences and create their own pattern. 27

Plotting Points To plot a point on a grid, or the Cartesian Plane, we consider the Left/Right point (or coordinate) and the Up/Down point (or coordinate). For example, if we are given the point (3, 5) this means we plot the coordinate 3 right, and 5 up, from the origin, or the (0, 0) point.

Task a

Plot each of the following coordinates on the grid below and be sure to write each letter next to its coordinate. The first one has been done for you. y

r o e t s Bo r e p ok u S 10

A: (2, 6) B: (5, 1)

C: (8, 7)

D: (1, 5) E: (3, 9)

ew i ev Pr

Teac he r

9 7

6 5 4 3

2 © R eadyEdPubl i cat i ons F: (4, 3) 1 •f orr evi ew pur posesonl y•

w ww

A: (2, 7) B: C: D: E: F:

. te98

o c . 7 c e h r e o t r 6 s super G

5 4 3

1 28

m . u

* Task b

1 2 3 4 5 6 7 8 9 10 For each of the points on the graph below, write their coordinates. Remember you write the horizontal, or x coordinate first, and the vertical, or y coordinate second. The first one has been done for you. y

Join the Dots When you were little you might have enjoyed dot-to-dot books. You might like to ask your parent or guardian about them! Follow the steps below to plot the points and join each point up with a line. The first few have been done for you.

Task a

Start at (2, 3)

move 3 right and 4 up

Teac he r

move 5 right and 2 down move 2 left and 1 down move 3 left and 7 up move 1 right and 4 down move 5 left and 8 up

move 3 right and 6 down.

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r o e t s Bo r e p ok u S 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Start

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__________________________ __________________________

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m . u

* Task B

Start

o c . che e r o t r s s r u e p 4

__________________________ 8 7 __________________________ 6 __________________________ 5 __________________________ __________________________

3 2

__________________________ 1 __________________________

10 29

Tables and Coordinates When we see a table of values, we can turn it into a set of coordinates. The first row of the table represents the horizontal coordinate and the second row represents the vertical coordinate. Turn each of the following tables of values into coordinates. The first one has been started for you.

Task a

(2,4)

(3,6)

(4,8)

Teac he r

. te (2,5) (3, 7) x y

(5, 13) (3, 9) x y

Create a table of values for each of the following sets of coordinates. Make sure the first row, the x coordinate row, is in ascending (increasing) order.

Task B

w ww

m . u

ew i ev Pr

(1,2)

r o e t s Bo r e p ok u S

o c (11, 19) . che e r o t r s super (4, 9)

(5, 11) (6, 13)

(7, 15)

(9, 17)

(1, 10) (8, 21) (7, 12)

(2, 11)

(6, 15)

(4, 8)

Plotting Tables of Values We know that a table of values is just another more efficient way of writing coordinates. We can therefore graph a table of values just like we would graph coordinates.

Graph each of the following tables of values on the Cartesian Plane (grid) provided.

x y

r o e t s Bo r e p ok u S y

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Teac he r

w ww

. te 20

m . u

o c . che e r o t r s super

x 31

Graphs and Coordinates Turn each of the following graphs of coordinates into a table of values. Remember each table needs two rows, one for the horizontal or x coordinate, and one for the vertical or y coordinate.

r o e t s Bo r e p ok u S y

ew i ev Pr

Teac he r

15 10 5

w ww

m . u

2) y

. te 35 o c . 30 che e r o 25 t r s super 40

20 15 10 5

1 32

Match Them Up Below are four linear (straight line) graphs, six rules and four tables of values. Match each graph with a table of values and an equation. A great way to do this might be to use a different colour highlighter for each graph. Graph A 20

Graph B 20

r o e t s Bo r e p ok u S

14 12

Graph C

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Teac he r

Graph D

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2

w ww

. te 1

8.5

o c . c e 3h 4e 5 6 x 1 o 2 r 3 4 5 t r s s upe 8 7.5 7 6.5 yr 3 6 9 12 15

Table 1

20 16 12

Table 2

Table 3

m . u

6 18

Table 4

12 17 22 27

Marble Clusters

* Task a

Michael is arranging his marbles in clusters. Examine the clusters of marbles below.

Cluster 1

Cluster 4

Cluster 2

Cluster 3

r o e t s Bo r e p ok u S Cluster 5

Cluster 6

b. Use the information from the pattern to fill in this table. Cluster (C) Number of Marbles (M)

1 3

3 7

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Teac he r

a. Draw the next three marble clusters in the pattern.

c. Describe the pattern in the numbers from the second row of the table.

_____________________________________________________________________

. te

– Curved

o c . che e – Sloping Up r o t r s– Sloping Down super – Straight

e. Use the graph to help you find out how many marbles there will be in cluster 7. _______________________________ 1

d. Circle all the phrases that you can use to describe the pattern you see in the points on the graph.

m . u

19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

w ww

Number of Marbles

Michael’s Marbles

Cluster

Farmer Tom’s Fence

* Task a

Farmer Tom is building a fence for his farm. Each day he adds on another section of his fence. This is what he has been doing so far. Draw, in the space below, what Farmer Tom’s fence will look like on day 4 and on day 5.

r o e t s Bo r e p ok u S Day 2

Day 3

Day 4

Day 5

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Teac he r

Day 1

* Use the diagrams above to fill in the table. Some values has been completed for you.

Day

Number of Fence Pieces

© ReadyEdPubl i cat i ons _______________________________________________________________________ •f orr evi ew pur posesonl y• 1. How many fence pieces does Farmer Tom add each day? Explain how you found your answer.

2. What is the total number of pieces in the fence by the end of Day 6?

w ww

m . u

3. Describe in words the pattern you see in the second row of the table.

Farmer Tom’s Progress

. te _________________________ o 40 c . che _________________________ e r o t r 30 p s su er *Plot the table of values for Farmer Tom’s Progress on the graph below.

4. Use your graph to determine on which day Farmer Tom will have completed 41 pieces of his fence. _________________________

Number of Fence Pieces

_________________________

20 10 1

Day

6 35

Writing a Short Story

* Task a

Part 1

Martha is writing a short story. Each day she decides to write 200 words of her short story, except the first day where she writes 500 words. * Use the information above to fill in the table below.

End of Day (D) Total Number of Words Written (W)

Day 1 500

Day 2

Day 3

Day 4

Day 5

Day 6

Day 7

Day 8

Day 9 Day 10

r o e t s Bo r e p ok u S 1100

1900

1. Describe in words the pattern you see in the second row of the table.

Teac he r

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_______________________________________________________________________ 2. Imagine plotting this table of values. Circle which phrases you think would match the shape of the graph. Curved Straight Sloping Up Sloping Down

PLot the points

2500

m . u

2000

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Total Number of Words Written (W)

3000

*Plot the table of values on the graph below and check whether your answer to the question above is correct.

1500

. te

1000 500

o c . che e r o t r s super

End of Day (D) 3. Using the pattern you have found, how many words of her short story will Martha have written by the end of day 15? 4. If she continues to write this many words every day, how much will Martha have written by the end of October if she started writing on the 1st of October? 36

Writing a Short Story

* Task b

Part 2

Unfortunately Martha doesn’t like to write essays as much as she likes to write short stories. She has a 2000 word History essay to write. On the first day she writes 200 words, the next day she writes 180 words, then 160 words the next day.

1. What will the graph of this situation look like? Explain in as much detail as possible.

r o e t s Bo r e p ok u S

_______________________________________________________________________ _______________________________________________________________________

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Teac he r

2. How many words will Martha have written on the 6th day?

3. How many words will Martha have written, in total, by the end of day 6?

End of Day (D) Total Number of Words Written (W)

Use the above pattern to fill in the table of values below. Day 1

Day 2

Day 3

Day 4

Day 5

Day 6

Day 7

Day 8

m . u

4. If this pattern continues, Martha will soon stop writing her essay altogether! On which day does Martha write nothing for her essay?

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*You might like to extend the table above to help you work out the answer.

. te

o c . che e r o t r s super

5. Martha’s essay has to be 2000 words long. Will Martha ever finish writing her essay? _______________________________________________________________________ 37

Roast Dinner

* Task a

Part 1

The two graphs below represent the number of minutes it takes in a 180˚C oven to cook a roast dinner. Graph 1

300

250

r o e t s Bo r e p ok u S 200

Minutes

200 150 100

100

1kg

150

2kg

3kg

4kg

1kg

2kg

ew i ev Pr

Teac he r

Minutes

Graph 2

3kg

4kg

1. One of the graphs represents cooking roast chicken and the other cooking roast lamb. Explain which graph represents which roast dinner and why.

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y• ______________________________________________________________________ ______________________________________________________________________ 2. Explain all the similarities you see between the two graphs above.

m . u

______________________________________________________________________

w ww

______________________________________________________________________ 3. Explain all the differences you see between the two graphs above.

. te o ______________________________________________________________________ c . cofh e *Create a table values for each of the above graphs in the space below. r er o st super ______________________________________________________________________

Graph 1

Minutes

Kilograms Graph 2

Minutes Kilograms 38

Roast Dinner

Part 2

4. Explain the pattern you see in the table for Graph 1. ______________________________________________________________________ 5. Explain the pattern you see in the table for Graph 2. ______________________________________________________________________ 6. Estimate, using your graphs, how long it would take to cook a 1.7 kg roast chicken and how long it would take to cook a 1.7 kg roast lamb.

r o e t s Bo r e p ok u S

____________________________________________ ____________________________________________ ____________________________________________

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Teac he r

______________________________________________________________________ 7. Do you think that these graphs can go on forever? Explain your answer.

Now let’s consider cooking a roast turkey. * Task B © ReadyEdPubl i cat i ons 1. Create a table of values that you think would best represent cooking a roast turkey. You might like to research the weight and cooking times on the internet •f orr evi ew pur posesonl y• first before doing so.

Minutes

w ww Kilograms

. te

*Plot your table of values on the axes below.

m . u

Cooking a Roast Turkey

___________________________________ ___________________________________

Minutes

o c 2. Is your graph similar or different to the . c e he r graphs for cooking roast lamb and roast o t r s chicken? Explain your answer. s uper ___________________________________ ___________________________________ KG

Social Media

* Task a

Part 1

Justine has just joined the latest social media craze, FriendFace. The graph below shows the number of new people she invites to be her “friend” each day for the first 10 days.

55 50 45 40

Teac he r

30 25

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Friend s (F)

r o e t s Bo r e p ok u S

20 15 10 5

Day 1

Day 2

Day 3

Day 4

Day 5

Day 6

w ww

New Friend Invites (F)

. te

1. If all friends accept Justine’s invite, how many friends in total will she have by the end of day 6?

Day 7

Day 8

Day 9 Day 10

m . u

Day (D)

o c 2. Describe the pattern you see in the second row of the table. . che e r o t r s super _______________________________________________________________________ 3. Describe the shape of the above graph in as much detail as possible.

_______________________________________________________________________ 4. If this pattern continues, how many friends will Justine invite on day 20? 5. If this pattern continues, on which day will Justine invite 64 new friends? 40

Social Media

* Task b

Part 2

Brenton wants to remove some of his “friends” from his FriendFace account. On the first day he removes 40 friends, the next day 34 friends, the next day 28 friends and so on.

Use this pattern to fill in the missing spaces in the table below. Day (D) Friends Removed (F)

Day 1

Day 2

Day 3

Day 4

Day 5

Day 6

r o e t s Bo r e p ok u S 40

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Teac he r

1. Can Brenton keep removing friends in this way everyday for a month? Explain your answer. _______________________________________________________________________

PLot the Values * Plot the points from your table on the graph from Task A (page 38).

2. Describe any similarities you see between Justine’s graph and Brenton’s graph.

_______________________________________________________________________

© ReadyEdPubl i cat i ons _______________________________________________________________________ •f orr evi ew pur posesonl y• 4. Is Justine adding friends faster than Brenton is removing them? 3. Describe any differences you see between Justine’s graph and Brenton’s graph.

Explain how you decided upon your answer.

m . u

w ww

_______________________________________________________________________ 5. Is there a day where both Brenton and Justine will invite/remove the same number of friends?

. te o c Task c Do you use a Social Media application like FriendFace? . * c e he 1. Do you, or someone you know, use something similar tor o t r s FriendFace? super

_______________________________________________________________________

______________________________________________________________________ 2. When you, or someone you know, first joined this application, how many friends were invited each day? Record your results in a table in the space below. Day (D) New Friend Invites (F) 41

Handshakes and Kisses

* Task a



Part 1

When people come together for a meeting they shake hands with each other. If there are two people at a meeting there will be one handshake. If there are three people at a meeting there will be three handshakes.

1. How many handshakes will there be if there are four people at a meeting? *This graph shows the number of people at a meeting and the number of handshakes. Use this graph to fill in the table below. 50

Number of Handshakes (H)

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Teac he r

r o e t s Bo r e p ok u S

35 30 25 20 15 10

w ww

. te

Number of People (P)

9 5

10 6

m . u

o c . 2. The numbers in the second row of the table are a special pattern and have a special c e h r name. What is the name of these special numbers? er o t s s r u e p _______________________________________________________________________ Number of Handshakes (H)

3. Describe in words the pattern you see in the second row of the table. _______________________________________________________________________ 4. How many handshakes would you expect if there were 15 people at the meeting?

Handshakes and Kisses

Task b

Part 2

When Italian families get together they greet each other by giving each other a kiss on each cheek. If two Italian people meet there would be four kisses in total. Fill in the table below for the number of total kisses given depending on the number of Italian family members meeting.

r o e t s Bo r e p ok u S

Number of Italian People (P)

ew i ev Pr

Teac he r

Number of Kisses (K)

1. If you plotted this new table of values, would the graph have the same shape or a different shape? Explain your answer. _______________________________________________________________________

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3. How many kisses would you expect if there were 15 Italian relatives meeting at the same event?

. te

m . u

_______________________________________________________________________

o c . che e r o t r Answer: s super

4. How are the numbers in the second row of the “Kisses” table related to the numbers in the second row of the “Handshakes” table? _______________________________________________________________________ _______________________________________________________________________ 43



Super Savings

* Task a

Part 1

Lillian has decided to start a savings account. The table below shows how much money Lillian will put aside to save each day.

Day (D)

Day 1

Amount Saved (A)

Day 2

Day 3

Day 4

Day 5

Day 6

Day 7

Day 8

$0.01 $0.02 $0.04 $0.08 $0.16 $0.32

r o e t s Bo r e p ok u S

* Complete the remaining spaces in the table above.

2. How much has Lillian saved in total by the end of the 10th day?

ew i ev Pr

Teac he r

1. How much has Lillian saved in total by the end of the 9th day?

3. Do you think it is possible for Lillian to continue saving like this for a whole month? Explain your answer clearly. ___________________________________________________________ 4. Explain the pattern you see in the second row of the table.

___________________________________________________________

saves each day for the first few days.

w ww Day (D)

Day 1

Day 2

Day 3

Day 4

Day 5

Day 6

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© Re ad y Ed P ub l i c a t i o n s *Graph the points from Lillian’s table on Graph A in Black. Plot the points •f orr evi ew pur posesonl y• Task b Troy has his own savings plan. He will start by saving $0.20 on the first day and will * save an extra $0.20 each day. Complete the table below showing the amounts Troy Day 7

Day 8

$0.20 $0.40 $0.60 $0.80 . te o c *Graph the points from Troy’s table on Graph . A in Blue. Plot the points che e r o t r s su 1. What do you notice about the shapes ofp Lillian’s and Troy’s graphs? er Amount Saved (A)

_______________________________________________________________________ 2. Who is saving the fastest? Explain how you found your answer. _______________________________________________________________________ 3. Can Lillian and Troy keep saving like this over the long term? Explain who will have to change their savings plan first and why. _______________________________________________________________________ 44

Super Savings

Part 2

Graph A

r o e t s Bo r e p ok u S amount saved (A)

Teac he r

amount saved (A)

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

© ReadyEdPubl i cat i ons What is your savings plan? Do you receive pocket money or are you given an allowance for •f o rsmall r ejobs vi ewyourp ur po s es n l y • doing around house? How much could youo save each week?

* Task c

Day (D)

week (W)

Create a table of values that shows your savings plan for the next 8 weeks.

w ww

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2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Graph B

o c . Plot the points c *Plot your table of values on Graph B above. e her r o t s 1. Do you think this savings plan s willu bep successful? er Explain why or why not.

___________________________________________ ___________________________________________ 2. Share your savings plans with a few other classmates and compare ideas. 45

Answers Number Properties

d) 2×2×2×2×5 e) 2×2×2×5×5 f) 2×2×2×2×2×2×5

Groups of Numbers p7 Task A a) One, itself b) Factors c) Evenly d) Integer, multiplication e) Itself

Can it be divided by this number? p11

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Multiples p8

Task a b) 2, 6, 8, 10, 12, 14, 16, 20 c) 14, 21, 28, 42, 49, 56, 63, 70 d) 10, 30, 40, 50, 60, 80, 90, 100 e) 25, 50, 100, 125, 150, 200, 225, 250 f) 15, 30, 45, 75, 90, 120, 135, 150 g) 12, 36, 48, 60, 84, 96, 108

Divisibility Tests p12

Task A Circled numbers: 22, 36, 48, 8, 102, 234, 64 If it is an even number.

Task b Circled numbers: 24, 348, 420, 1016, 1764, 4136

Task © ReadyEdCircled Pcu b l i c a t i o n s numbers: 10, 55, 100, 220, 315, 2045 If it ends in 0 or 5. •f orr evi ew pTask ur p o s esonl y• d Circled numbers: 642, 48, 990, 552, 738, 252

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Task a b) 1, 2, 5, 10 c) 1, 2, 3, 4, 6, 8, 12, 24 d) 1, 2, 4, 5, 10, 20, 25, 50, 100 e) 1, 2, 3, 4, 6, 9, 12, 18, 36 f) 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Task b b) 4 c) 15 d) 9 e) 18 Prime Factor Trees p10 a) 2×2×2×2 b) 2×2×2×2×3 c) 2×2×2×3×5

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More Divisibility Tests p13

Factors p9

Task b 312, 450, 723, 2652, 3216, 4122, 5355, 12693

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Task b a) 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 b) 9, 10, 12, 14, 15, 16, 18, 20, 21, 22 c) 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 d) 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197 e) 20, 25, 30, 35, 40, 45, 50, 55, 60, 65

Task b a)15 b)42 c)30 d)24

Task A a) Even numbers b) Ends in 0 c) Ends in 5 or 0 d) Yes because 2 is a factor of 4 e) Yes because 5 is a factor of 15 f) Yes because 3 is a factor of 6

Task A Students to check their chosen numbers with their partner Task b a) The sum of the digits is a multiple of 3 b) The last two digits are divisible by 4 c) Students to check their chosen numbers with their partner

o c . che e r o t r s super Conjectures 1 p14 Task A a) True b) True c) False , 25 d) True e) True Task b a) Answer will be a multiple of 2

b) 5 is a factor of 10 c) 3 is a factor of 6

Complete the Pattern 3 p19 a) 19, 22, 25 b) 500, 2500, 12500 c) 125, 31.25, 7.8125 d) 21, 15, 9 e) 26, 37, 50 f) 0.06, 0.6, 6 g) 95, 191, 383 h) 256, 8192, 2097152 i) 42, 68, 110 j) 938, 4688, 23438 k) 8/54, 16/162, 32/486 l) 5/48, 6/48, 7/48 m) 8, 60, 74 n) 20, 15, 25 o) 0.5, 0.06, 0.007

Conjectures 2 p15 a) False, 2 b) True c) True d) True e) True f) False, 2×3 g) True h) False, 9

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Square Numbers p20

Complete the Pattern 1

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Task A a) 1 b)22 c)3×3 d)42, 16 e) 52, 25 f) 6×6, 36 g) 7×7,72 h) 8×8, 82 i)92, 81 j)10×10,100 k) 11×11, 121 l) 12×12, 122 m) 132, 169 n) 14×14, 196

Task A b)160, 150, 140 c) 58, 65, 72 d) 24600, 22800, 21000 e) 4/12, 5/12, 6/12 f) 45, 43, 46

Task b a) 25, 81, 121 b) 26, 37, 50 c) 18, 50, 72 d) 15, 35, 48

Complete the Pattern 2 p18

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Task A b) 10000, 100000, 1000000 c) 125, 62.5, 31.25 d) 81, 243, 729 e) 2666.67, 1777.77, 1185, 185 f) 1/54, 1/162, 1/486

Cube Numbers p21 Task A a)1 b)8 c)33 d)43, 64 e)5×5×5, 125 f)63,216 g)7×7×7, 343 h)83, 512 i)9×9×9, 729 j)10×10×10, 103 k)113, 1331 l)12×12×12, 1728 m)13×13×13, 2197 n)143, 2744 Task b a) 26, 124, 215 b) 4, 32, 108 c) 16, 128, 250 d) 125, 343, 729

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Task b a) 60, 55, 50, 45, 40, 35, 30, 25, 20, 15 Subtract 5 each time b) 21, 28, 36, 45, 55, 66, 78, 91, 105, 120 Add consecutive counting numbers c) 520, 570, 620, 670, 720, 770, 820, 870, 920, 970 Add 50 each time d) 4/20, 5/20, 6/20 Add 1/20 each time

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Task b a) 250, 1250, 6250, 31250, multiply by 5 each time b) 0.3, 0.03, 0.003, 0.0003, divide by 10 each time c) 120, 720, 5040, 40320 multiply by consecutive integers d) 1, 10, 100, 1000, multiply by 10 each time

Triangular Numbers p22 Task A

4th triangular number

5th triangular number

6th triangular number 47

You can make triangles out of the number of dots Add consecutive integers 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 Task b 3, 10, 15, 21, 28, 36, 45, 55 35, 56, 84, 120, 165, 220 Fibonacci Numbers p23

Task a

1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

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Task A 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 Add two previous terms Task b

Pascal’s Triangle 2 p25

Sum = 1

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2 5 13 34 610 4181

3rd term 5th term 7th term 9th term 15th term 19th term

Task c 1, 2, 1.5, 1.66666, 1.6 1.625, 1.615, 1.619, 1.618, 1.618 The more divisions you do, the closer you get to the Golden Ratio

Sum = 1+1=2

1+2+1 =4 Sum = 1+3+3+1= 8 Sum = 1+4+6+4+1= 16 Sum = 1+5+10+10+5+1= 32 Sum = 1+6+15+20+15+6+1= 64 Sum =

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12 + 12 12 + 22 22 + 32 32 + 52 132 + 212 342 + 552

Task b 1. Multiply by 2 each time 2. a) 26 b) 210 3. a) 4096 b) 33 554 432

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Task A Row 5 = 1, 5, 10, 10, 5, 1 Row 6 = 1, 6, 15, 20, 15, 6, 1 Row 7 = 1, 21, 35, 35, 21, 7, 1 Row 8 = 1, 8, 28, 56, 70, 56, 28, 8, 1

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Add the two numbers above, immediately to the left and right

Task b

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Pascal’s Triangle 1 p24

There are the 6th row of Pascal’s Triangle

Rules, Table Graphs

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Task b 1. 1, 3, 6, 10 … Triangular numbers 2. 1, 4, 10, 20 … Tetrahedral numbers 3. 1, 5, 15, 35 … The sum of the tetrahedral numbers

Plotting Points p28 Task a y

10 9

8 7

6 5

4 3

2 1

Plotting Tables and Values p31 1)

Task B B=(4,5) C=(6,9) D=(7,2) E=(1,2) F=(8,6) G=(1,9) H=(5,7)

Join the Dots p29 Task a

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

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Start

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x

Task b Start at (8,9) move 2R 3D, 3L 2D, 2L 2D, 3L, 2R 3U, 3L, 3L 3U, 2R 1D, 1D Tables and Coordinates p30

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Task B 1) x 2 y

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1) x

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Task a 1) (5,10) (6,12) (7,14) (8,16) 2) (0,15) (1,20) (2,25) (3,30) (4,35) (5,40) (6,45) (7,50) 3) (2,20) (4,19) (6,18) (8,17) (10,16) (12,15) (14,14) (16,13)

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Match Them Up p33 a) Graph A = Table 2 b) Graph B = Table 3 c) Graph C = Table 4 d) Graph D = Table 1

Marble Clusters p34 Task a b)

c) increase by 2 each time, starting at 3 d) straight, sloping up e) 15 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Writing a Short Story p36-37 Task a Day 1

Day 2

Day 3

Day 4

500

700

900

1100 1300 1500 1700 1900 2100 2300

Day 7

Day 8

Day 9

Day 10

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1500 1000

500

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2000

Farmer Tom’s Fence p35 Task a

Day 5

2 11

3 16

Task b 1) Straight, sloping downwards 2) 100 3) 900

4 21

5 26

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40 30

Day 2

Day 3

Day 4

Day 5

Day 6

Day 7

Day 8

200

380

540

680

800

900

980

1040

4) Day 11 5) No

1) five, goes up by 5 each day 2) 31 3) increase by 5 starting at 6 4) Day 8

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Roast Dinner p39-40

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Day 4

Task a 1) Graph 1: Roast Chicken because it is smaller and cooks quicker Graph 2: Roast Lamb 2) Straight lines, increasing graphs 3) Different slopes, different first values

o c . che e r o t r s super KG 0.5 1 1.5 2 2.5 Graph 1

100 130 160 190

0.5

1.5

130 170 210 250

Graph 2

10 1

Day 6

1) Increase by 200, starting at 500 2) Straight, Sloping Up 3) 3300 4) 6500

1 6

Day 5

Day

4) Increase by 30 each time 5) Increase by 40 each time

2.5

6) Chicken: 110 minutes, Lamb: 140 minutes 7) No, because roast meals will not continue to increase in weight forever.

2) Triangular numbers 3) Adding consecutive integers 4) 105

Task b Check student responses and have students check each other’s answers.

Task b

Social Media p40-41

1) Same shape, but steeper 2) Curved, sloping up 3) 420 4) 4 times bigger

Task a

12 16 20 24 28 32 36 40 44

12 24 40 60 84 112 144 180

Super Savings p44-45 Task a

Day 7 0.64

40 34 28 22 16 10

1) No, soon he will be removing a negative number of friends

Day 8 1.28

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1) 108 2) Increasing by 4 each time 3) Straight, sloping up 4) 84 5) 15 Task b

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1) 5.11 2) 10.23 3) No, by day 30 she needs more than $5 million 4) Each day Lillian saves twice as much as the day before Task b

Day 5 1.00

Day 6 1.20

Day 7 1.40

Day 8 1.60

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2) Both are straight lines 3) One is sloping up, the other sloping down 4) No, Justine’s goes up by 4, Brenton’s goes down by 6 5) No

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

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Task c Have students compare answers and/or share with the class. Handshakes and Kisses p42-43 Task a 1) 6

10 15 21 28 36 45

Task c Students can compare their graphs and tables with each other and you may like them to present to the class. 51

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