Book C TITLE Series Name: Book Name: Book Number: ISBN Number: Published:

About Algebra Book C: Exploring Equations and Expressions 596C 978-1-86968-474-7 2007

AUTHOR Viv McGowan

ACKNOWLEDGEMENTS The publishers wish to acknowledge the work of the following people in the various stages of publishing this resource. Design: Editor:

Glen Honeybone Murray Quartly

PUBLISHERS User Friendly Resources

United Kingdom Office New Zealand Office Freepost RRCD-ALHY-YAAA PO Box 1820 User Friendly Resources Christchurch Parkside Farm Tel: 0508-500-393 Shortgate Lane Fax: 0508-500-399 Lewes BN8 6DG Tel: 0845-450-7502 Fax: 0845-688-0199

Australian Office PO Box 914 Mascot, NSW 2020 Tel: 1800-553-890 Fax: 1800-553-891

WEBSITE www.userfr.com

E-MAIL info@userfr.com

COPYING NOTICE This is a photocopiable book and permission is given to schools or teachers who buy this resource to make photocopies or transparencies of all pages. The copies must be for internal school use only, and may not be given or sold to other educational institutions or teachers from other institutions.

COPYRIGHT User Friendly Resources, 2007

User Friendly Resources specialises in publishing educational resources for teachers and students across a wide range of curriculum areas, at both primary and secondary levels. If you wish to know more about our resources, or if you think your resource ideas have publishing potential, please contact us at the above address.

Book C

Contents To the Teacher 5 ACTIVITIES Section 1:

Solving Simple Equations ...................................................................6 A - Baking a Cake ...................................................................................7 B - Easy Equations............................................................................8 - 9 C - Playing Algebra Mathex.....................................................10 - 13

Section 2:

Using Solution Sets ............................................................................ 14 A - Finding a Route............................................................................. 15 B - Solving using Solution Sets ...................................................... 16

Section 3:

Operations and Brackets .................................................................. 17 A - Changing the Operations ......................................................... 18 B - Moving the Brackets.................................................................... 19 C - Getting your Priorities Right .................................................... 20 D - Getting Tougher ........................................................................... 21

Section 4:

Opposite Operations ......................................................................... 22 A - Doing the Opposite! ................................................................... 23

Section 5:

Generalising Patterns ........................................................................ 24 A - Buying your Lunch....................................................................... 25 B - Going to the Movies .................................................................... 26

Section 6 :

Compacting Terms ............................................................................. 27 A - Squashing Down Equations ..................................................... 28

Section 7:

Practical Equations............................................................................. 29 A - Class Trips and Dances ............................................................... 30 B - Free Fall! ........................................................................................... 31

Section 8:

Relationship Tables ............................................................................ 32 A - Useful Formula .............................................................................. 33 B - More Useful Formula ................................................................... 34

Section 9:

Identifying Like Terms ....................................................................... 35 A - Spot the Coins ............................................................................... 36 B - Spotting Likeness! ........................................................................ 37 C - Establishing Like Terms .............................................................. 38

Section 10:

Simplifying Terms ............................................................................... 39 A - Simplifying Terms (+, –).............................................................. 40 B - Simplifying Terms (x) ................................................................... 41

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Book C

Contents Section 11:

Expanding Expressions..................................................................... 42 A - Expanding Expressions .............................................................. 43 B - Expanding the Brackets Mathex Game ............................44 - 45

Section 12:

Factorising Expressions .................................................................... 46 A - Factorising ...................................................................................... 47 B - Expanding Factors........................................................................ 48

Activity Answers ......................................................................................................49 - 51 TESTS AND ASSESSMENT Level 1 Test

................................................................................................................... 52

Level 2 Test

................................................................................................................... 53

Level 3 Test

................................................................................................................... 54

Test Answers ............................................................................................................55 -56 Self-Assessment .......................................................................................................57 - 60

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Book C

TG

s ' r e h c a e T Guide

To the Teacher The About Algebra series provides teachers with an accessible resource for working with students and uses methods of teaching and learning that will generate not only tangible success, but interest and enthusiasm for algebra and mathematics as a whole. There are three books in the series: • Book A: Exploring Spatial Relationships and Number Patterns (with Teachers’ Guide) • Book B: Exploring Graphing • Book C: Exploring Equations and Expressions Each activity in the books comprises a teacher’s page which includes a plan for organisation and suggestions for implementation. This page is followed by student pages which set up the examples and problems in an accessible way. The activities have been placed in order of complexity from easiest to hardest, and are developed in such a way that they can be used in a consecutive manner or as independent lessons. Each book focuses on a student-centred, problem-solving approach to Mathematics using a variety of established strategies (for example, thinking skills, co-operative group skills, Bloom’s taxonomy) and caters for a variety of learning styles. While there is a major focus on co-operative group work, the resource lends itself to a programme of independent learning as well.

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Book C

1

Solving Simple Equations

CO N C E P T Solving olving simple tions using basic equations t ((+, –, x, ÷)) facts

ACTIVITY GUIDELINES FOR TEACHERS

Introduction •

Hand students a copy of tthe Peach Sponge recipe on the next page. In small groups get them to discuss whether it is possible to follow this recipe or not. not They need tto focus on what is wrong with the recipe.

•

As a whole class get students to share their ideas – the key idea to emerge should be that there are facts missing so the recipe can’t really be used.

Teaching Ideas •

Use the following sequence to develop the idea of missing facts, e.g. 3 apples + some pears = number of fruit in a bowl missing facts

•

Ask students whether this equation is solvable - look for the response “only if there is only one missing fact, but not if there are two missing facts.”

•

Re-write the equation as: 3 apples + some pears = 8 pieces of fruit in a bowl

•

Then write it as a number equation: 3+?=8

•

Substitute pronumerals in the equation: 3+p=8 and p = number of pears = 5.

Reinforcement •

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Get students to design a poster entitled “Equations are Easy” which tells reluctant students why they are.

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Book C

1A

Baking a Cake

T O P I C

What you need to do: 1. Work in groups group to identify what is wrong with this recipe.

Solving ng simple equations using basic facts (+, –, x, ÷)

PEACH SPONGE Tin of Peaches Egg Whites (at room temperature) 4 amounts of Sugar Salt Yolks of Eggs 1/2 cup plain Flour 1/2 of Baking Powder Turn on oven. Tip peaches into a dish. Put in oven to heat while making the sponge. Meanwhile beat egg whites until stiff. Beat in sugar, salt and egg yolks. Sift in flour and baking powder. Mix well. Take dish out. Spread sponge mixture over peaches. Bake and serve warm. 2. What facts are missing from this recipe and why would it be difficult to make?

3. Discuss the results with the entire class.

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Book C

1B

Easy Equations

T O P I C

EXPLORING EQUATIONS AND EXPRESSIONS

Examples: •

Solving olving g simple tions using basic equations t ((+, –, x, ÷)) facts

In the problem m below you have to decide what you need to add to 15 to make 39. + 15 = 39 24 + 15 = 39 So the missing number is 24.

•

Sometimes instead of a box you might find a letter like x or b or p. For example: b + 15 = 39 Here you are being asked to solve the unknown value b. In other words, what value of b will make the statement true. Like before, b must equal 24. Easy isn’t it! Addition/subtraction so

24 + 15 = 39 b = 39 – 15 = 24

Multiplication 3xa = 6 a = 6÷3 a = 2

Division c÷2 = 4 c = 4x2 c = 8

Now try solving the problems below.

What you need to do: 1. Find the missing numbers.

HINT: In the problems in part 3 just ask yourself "what would I need to multiply by to make the equation true?" Also 3n is just the same as 3 x n except the ‘times’ sign has been left out. © Copyright User Friendly Resources.

a)

+ 5 = 10

b)

+ 23 = 30

c)

+ 12 =18

d)

+ 18 = 30

e)

– 5 = 12

f)

–8=2

g)

– 24 = 30

h)

– 60 = 100

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Book C

1B

Easy Equations (cont.)

T O P I C Solving olving g simple tions using basic equations t ((+, –, x, ÷)) facts

EXPLORING EQUATIONS AND EXPRESSIONS 2. Now find the value of the unkn unknown letters below. a)

r + 6 = 14 4

b) t – 8 = 12 c)

s – 14 = 22

d) u + 13 = 25 e)

v – 3 = 21

f)

w + 13 = 13

g) x – 13 = 0 h) y ÷ 2 = 12 3. Finally, try some harder problems. a)

6 x l = 18

b) 4 x m = 24 c)

3n = 9

d) 4p = 16 e)

7q = 14

f)

3r = 15

g) 8s = 64 h) 9t = 72

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Book C

1C

Playing Algebra Mathex

T O P I C

EXPLORING EQUATIONS AND EXPRESSIONS

Rules for tthe Game Game: 1. Play in groups o of 3 or 4 student students.

Solving olving simple tions using basic equations t ((+, –, x, ÷)) facts

2 nee one person tto be their runner. 2. Each group needs 3. Four students act as markers sitting at the front of the class with their answers hidden. 4. The markers hand out questions in order, one at a time, to each runner. 5. When a group has solved a problem the runner takes the answer to the marker who replies ‘correct’ or ‘incorrect’. 6. A group can have as many tries at a problem as it likes, but if it passes then the group can not go back to that question. 7. Your teacher may set a time limit on the game or finish when the first group completes all questions correctly.

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Book C

1C

Mathex Playing Cards Level A

T O P I C Solving olving simple tions using basic equations t ((+, –, x, ÷)) facts

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EXPLORING EQUATIONS AND EXPRESSIONS

What you need to d do: 1. The following three t pages need to be photocopied so that there is sufficient copies for the group g work in this th activity.

1

4r = 16

; r=

2

10p = 30

; p=

3

x ÷ 2 = 12

; x=

4

n ÷ 12 = 10 ; n =

5

x + 6 = 12

; x=

6

y–3=8

; y=

7

n + 32 = 40 ; n =

8

13 + t = 21

; t=

9

17q = 34

; q=

10

t ÷ 150 = 2

; t=

11

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Book C

1C

Mathex Playing Cards Level B EXPLORING EQUATIONS AND EXPRESSIONS

T O P I C Solving olving simple tions using basic equations t ((+, –, x, ÷)) facts

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1

3x + 3 = 12

; x=

2

10p – 4 = 16

; p=

3

11y + 5 = 27

; y=

4

4n – 3 = 9

; n=

5

17x – 2 = 32

; x=

6

12p – 27 = 9

; p=

7

(q ÷ 3) + 4 = 16 ; q =

8

(n ÷ 2) – 3 = 4

; n=

9

8÷y=4

; y=

10

4x ÷ 2 = 3

; x=

12

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1C

Book C

Playing Algebra Mathex Answers

T O P I C Solving olving simple tions using basic equations t ((+, –, x, ÷)) facts

EXPLORING EQUATIONS AND EXPRESSIONS

Level A An Answers nswers

1

r=4

6

y = 11

2

p=3

7

n=8

3

x = 24

8

t=8

4

n = 120

9

q=2

5

x=6

10

t = 300

Level B Answers

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1

x=3

6

p=3

2

p=2

7

q = 36

3

y=2

8

n = 14

4

n=3

9

y=2

5

x=2

10

x = 1.5

13

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Book C

2

Using Solution Sets

CO N C E P T Solving olving expressions using solution sets

ACTIVITY GUIDELINES FOR TEACHERS

Introduction •

For any problem there can be more than one solution. Students need some solutions can be longer than others but this to be aware that som does not make them incorrect.

Teaching Ideas •

Using Finding a Route (Activity 2A), ask students to identify the most direct route (Howe Street - Penelope Drive - Grey Street).

•

Using lengths of string have students compare the length of their route with the most direct route.

•

Introduce three concepts: = equal to > greater than < less than How does their route compare with the direct route? Which symbol should they use? E.g. my route > direct route

•

Extend the above symbols to include ≥ greater than or equal to ≤ less than or equal to

Reinforcement •

Get students to develop an easy way of remembering what these symbols mean.

< •

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≤

=

≥

>

Students could work in pairs or groups and produce a chart to display on the wall.

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Book C

2A

Finding a Route

T O P I C Using th the he symbols ≤, <, =, >, ≥

EXPLORING EQUATIONS AND EXPRESSIONS

What you u need to do: 1. Identify one w way of getting from point A to point B (it does not need to be route). the shortest ro oute).

1cm = 10m

Ma

Ma

ry

Penelope Drive

John Street

ns

ion

Av en u

e

A Ho we Str eet

Ro

ad

Reserve

Gre yS tre et

B

Street Green

lo Pene

ive pe Dr

Brown Drive

2. Describe your route in words, e.g. “Starting from point A turn … … which finally gets you to point B.”

3. Use a length of string, laid out on the map following your route, to measure the ‘length’ of your route.

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