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

About Algebra Book B: Exploring Graphing 596B 978-1-86968-473-0 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

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COPYRIGHT User Friendly Resources, 2007

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Book B

Contents To the Teacher ........................................................................................................................4 ACTIVITIES Activity 1:

Organisation ............................................................................................5 A - Matching Pairs..................................................................................6 B - Finding Relationships ....................................................................7 C - Jumbled Numbers ..........................................................................8 D - Showing Relationships ........................................................ 9 - 10 E - Completing a Set of Ordered Pairs ......................................... 11 F - Finding a Rule ................................................................................ 12

Activity 2:

Organisation ......................................................................................... 13 A - Using Graphs ........................................................................14 - 15

Activity 3:

Organisation ......................................................................................... 16 A - Showing Ordered Pairs .............................................................. 17 B - Graphing Bingo!............................................................................ 18 C - Identifying Ordered Pairs ..................................................19 - 20

Activity 4:

Organisation ......................................................................................... 21 A - Tell a Story ...............................................................................22 - 23

Activity 5:

Organisation ......................................................................................... 24 A - Sketching Graphs ......................................................................... 25 B - Interpreting Graphs ..................................................................... 26 C - More Graphs .................................................................................. 27

Activity 6:

Organisation ......................................................................................... 28 A - Planting Bulbs ............................................................................... 29 B - Graphing Linear Rules ................................................................ 30

Graph Template .............................................................................................................231 Activity Answers ......................................................................................................32 - 33 TESTS AND ASSESSMENT Tests and Related Concepts......................................................................................... 34 Level 1 Test 33 Level 2 Test 34 Level 3 Test 35 Test Answers 36 - 37 Pre/Post Evaluation ................................................................................................40 - 41 Self-Assessment .......................................................................................................42 - 44 ÂŠ Copyright User Friendly Resources.

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Book B

To the Teacher The About Algebra series provides teachers with an accessible resource for working with students aged from 11-14 years in the algebra strand of mathematics. The resource includes 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, 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 B

1

Organisation

CO N C E P T Identifying ifying ordered pairs from a set and finding a rule (To introduce or reintroduce the concept of ordered pairs refer to Book A, page 5.)

EXPLORING GRAPHING

Introduction •

Go back over work on ordered orde sets and reinforce students’ conc understanding of the concept. Look for, “In an ordered set the first element on the left side an and the second element on the right side are always connected by a common operation.”

Teaching Ideas •

Introduce the norm of using (x, y) to show an ordered pair.

•

Ask students how they can show a pattern in the example as a rule. Look for: y = x + 3 or x + 3 = y or y - 3 = x

Reinforcement •

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Use diagrams to show relationships in real world contexts, e.g. arrow diagrams.

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Book B

1A

Matching Pairs

T O P I C

What you need to do: 1. Match up the national flags with the plants that are associated with each of the countries. Draw arrows to connect flags to plants. clover

FIJI

AUSTRALIA

Identifying ng ordered pairs from a set

EXPLORING GRAPHING

eucalyptus

rose

ENGLAND

SOUTH AFRICA

NEW ZEALAND

IRELAND

fern

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protea

palm

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Book B

1B

Finding Relationships

T O P I C Using arrow rrow w diagrams to show a relationship

EXPLORING GRAPHING

What you need to know: The following two examples show how arrow diagrams can represent a relationship.

Examples: •

•

The relationship between members of these pairs is +2

2

4

2+2=4

4

6

4+2=6

6

8

6+2=8

The relationship between members of these pairs is x3

3

12

4

9

5

15

What you need to do: 1. Find the relationship and describe it in words in each of the following examples: a)

c)

b)

10

12

5

20

4

8

2

60

2

2

20

6

12

7

18

8

14

0

25

12

18

5

3

6

d)

2. Now draw and describe two of your own.

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Book B

1C

Jumbled Numbers

T O P I C Identifying tifying g ordered om a set to show pairs from a rule

EXPLORING GRAPHING

What you need to do: A table can represent an arrow diagram. The top line represents one set of numbers, the bottom represents the other set of numbers. 1. Which numbers form pairs to make the relationship +3? You must use all of the numbers! x y

5.25

5 -1

14

8

8.25 17

23 7

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8

10

2 20

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Book B

1D

Showing Relationships

T O P I C

EXPLORING GRAPHING

Example: •

Showing ng relationships relationship on a table

In an ordered set the first element and the second element of an ordered pair are always connected by a common operation.

1

4

2

5

3

6

•

All of the ordered pairs above are formed by adding 3 to the first number of the pair to give the second number.

•

On a graph these ordered pairs would be read as the co-ordinates (x,y) e.g. (x = 1; y = 4) You could show these ordered pairs on a table:

•

x

1

2

3

4

y

4

5

6

7

The relationship shown on the table is “y is always 3 more than x.” These ordered pairs can be shown as co-ordinates on a graph: 10

y

9 8 (4, 7)

7 6

(3, 6)

5

(2, 5)

4

(1, 4)

3 2 1 0

1

2

3

4

5

6

7

8

9

10

x

What you need to do: 1. Complete the table for each of the relationships. a)

y is 3 times x (y = 3x) x

2

4

6

8

10

12

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Book B

1D

Showing Relationships

T O P I C Showing ng relationships relationship on a table

EXPLORING GRAPHING

What you need to do: (continued) b) y is 1/3 of x (y = x/3) x

6

9

12

15

18

y c)

y is 1 more than x ( y = x + 1) x

5

6

7

8

9

12

15

10

y d)

x is 4 times a (b = 4 x a) x

3

6

9

y 2. Complete each table, and state in words the relationship shown on that table. a)

x

5

6

7

8

9

y

3

4

5

4

5

6

3

4

5

8

10

The relationship is: b)

x

2

3

y

8

12

The relationship is: c)

x

1

2

y

4

5

The relationship is: d)

x

2

4

6

y

1

2

3

The relationship is:

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Book B

1E

Completing a Set of Ordered Pairs

T O P I C Completing pleting ng ordered airs from an pairs ordered set

EXPLORING GRAPHING

Example: •

In the following sequence of you can recognise a consistent relationship: {(10,5,) (8,4) (6,3)} (Remember - in an ordered pair it always goes (a, b)) 5 is half of 10 4 is half of 8 3 is half of 6 so b is always half of a or y = x/2

What you need to do: 1. Find the missing number and describe the relationship in each case. Use the letter x to represent the first member of the pair and the letter y for the second member. a)

{(3,8) (4,9) (5,10) (6,

b) {(7,5) (8,6) (9,7) ( c)

)} ,8)}

{(18,6) (21,7) (24,8) (

,9)}

2. Write and describe two examples of your own. Don’t forget to use the correct format!

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Book B

1F

Finding a Rule

T O P I C Finding ng a rule for a nce of ordered sequence pairs (an ordered set)

EXPLORING SEQUENCES

Example: •

In the following followin ng example: {(3,4) (4,5) (5,6)} {(3,4 The relationship between x and y (the first and second member of each pair) is: y=x+1 e.g. 4 = 3 +1, 5 = 4 +1, and so on.

What you need to do: 1. Describe the rule (using algebraic symbols) for each of the following sets of ordered pairs: a)

{(9,8) (8,7) (7,6) (6,5)}

b) {(10,5) (12,7) (14,9) (16,11)}

c)

{(10,5) (20,10) (30,15) (40,20)}

E VA LUAT I O N • Define an ordered set in words and give an example different from the ones done in class. Do this in pairs or in a group. • Formal examples of relationship diagrams.

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Book B

2

Organisation

CO N C E P T Using graphs to represent number or informal relationships

ACTIVITY GUIDELINES FOR TEACHERS

Introduction •

Have groups brainstorm the term graphing. Use who, what, why, trigger Groups could record their ideas on when, where, how as triggers. large sheets of paper. The aim is to develop a ‘graphing vocabulary’ from this.

•

The whole class could discuss the vocabulary list and then display the sheets on the classroom walls.

Teaching Ideas •

Have a class discussion on the activity: a) What was wrong with the graph? b) Why was it a problem?

•

Devise a rule about graphs which begins: “A graph must…"

•

The point of these activities is to demonstrate to students that a graph must tell the entire story.

Reinforcement •

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In groups students gather data of their own choice, create a data table and then present the graph and the data table on a large sheet of paper for display.

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Book B

2A

Using Graphs

T O P I C

EXPLORING GRAPHING

The graph m must be a able to tell the whole story!

Using graphs to present number or represent informal relationships

Temperature of Our Class 35

Temperature (oC)

30 25 20 15 10

Fri

Thur

Wed

Tue

0

Mon

5

Day of the Week

What you need to do: This can be done as groups or in pairs. 1. Make a list of the different features of this graph.

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Book B

2A

Using Graphs

T O P I C Using graphs to present number or represent informal relationships

EXPLORING GRAPHING

What’s wrong wron ng with this th graph?

7 6 5 4 3 2

Fri

Thur

Wed

Tue

0

Mon

1

2. Finish the graph and then use the "Graphing Peer Checklist" to check out your graph.

GRAPHING PEER CHECKLIST

E VA LUAT I O N • Use the 'Graphing Peer Checklist' • Use individual formal graphing exercises from given or collected data.

The graph has a title. Each axis has a label and, if appropriate, units. Each axis has a scale or categories. Each scale goes up evenly. Signed: Peer checker: ________________________________________ Graph creator: _______________________________________ Teacher: ____________________________________________ ___________________________________________________

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