australian curriculum mathematics year 7

AUSTRALIAN CURRICULUM

MATHEMATICS YEAR 7

Cartesian coordinates

MATHEMATICS YEAR 7

Cartesian coordinates

Student’s name: ________________________________

Teacher’s name: ________________________________

First published 2012

ISBN 9780730744429

SCIS 1564085

© Department of Education WA 2012 (Revised 2020)

Requests and enquiries concerning copyright should be addressed to:

Manager Intellectual Property and Copyright Department of Education 151 Royal Street EAST PERTH WA 6004

Email: [email protected]

This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:

the content descriptions are solely for a particular year and subject

all the content descriptions for that year and subject have been used

the author’s material aligns with the Australian Curriculum content descriptions for the relevant year andsubject.

You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.

creativecommons.org/licenses/by-nc-sa/3.0/au/

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Year 7 Mathematics Cartesian coordinates

© Department of Education WA 2012 – MATHSAC022 Page 1

Contents

Signposts ....................................................................................................................................2

Introduction...............................................................................................................................3

Curriculum details....................................................................................................................4

1. Reviewing the Cartesian plane ..........................................................................................7

2. Plotting points .....................................................................................................................9

3. Tables of values .................................................................................................................15

4. Lines ...................................................................................................................................21

5. Bivariate data ....................................................................................................................29

6. Graphs................................................................................................................................37

7. Travel graphs ....................................................................................................................43

8. Summary.......................................................................................................................49

Review tasks ............................................................................................................................51

Solutions...................................................................................................................................59

Cartesian coordinates Year 7 Mathematics

Page 2 © Department of Education WA 2012 – MATHSAC022

Signposts Each symbol is a sign to help you.

Here is what each one means:

The recommended time you should take to complete this section.

An explanation of key terms, concepts or processes.

A written response. Write your answer or response in your journal.

Correct this task using the answers at the end of the resource.

Calculators may not be used here.

Make notes describing how you attempted to solve the problem. Keep these notes to refer to when completing the Self-evaluation task. Your teacher may wish you to forward these notes.



Introduction This resource should take you approximately two weeks to complete. It comprises seven learning sections, a summary section and a review task section.

The learning sections have the following headings:

Key wordsThese are the main words that you need to understand and use fluently to explain yourthinking.

Warm-upWarm-up tasks should take you no longer than 10 minutes to complete. These are skillsfrom previous work you are expected to recall from memory, or mental calculations thatyou are expected to perform quickly and accurately. If you have any difficulties inanswering these questions, please discuss them with your teacher.

ReviewSome sections have reviews immediately after the warm-up. The skills in these reviewsare from previous work and are essential for that section. You will use these to developnew skills in mathematics. Please speak to your teacher immediately if you are havingany trouble in completing these activities.

Focus problemFocus problems are designed to introduce new concepts. They provide examples of thetypes of problems you will be able to solve by learning the new concepts in this resource.Do not spend too long on these but do check and read the solutions thoroughly.

Skills developmentThese help you consolidate new work and concepts. Most sections include skillsdevelopment activities which provide opportunities for you to become skilled at usingnew procedures, apply your learning to solve problems and justify your ideas. Pleasemark your work after completing each part.

Correcting your work

Please mark and correct your work as you go. Worked solutions are provided to show how you should set out your work. If you are having any difficulty in understanding them, or are getting the majority of the questions wrong, please speak to your teacher immediately.

Journal

Please keep an exercise book to record your notes and to summarise your learning. At the end of each section, write definitions for the key words that were introduced for that section.



Curriculum details Content Descriptions This resource provides learning and teaching to deliver the Australian Curriculum: Mathematics for the following Year 7 Content Descriptions.

Given coordinates, plot points on the Cartesian plane, and find coordinates for a given point (ACMNA178)

Investigate, interpret and analyse graphs from authentic data (ACMNA180)

Content Descriptions 1 2 3 4 5 6 7 R

ACMNA 178

ACMNA 180

Indicates the content description is explicitly covered in that section of the resource.

Previous relevant Content Descriptions

The following Content Descriptions should be considered as prior learning for students using this resource.

At Year 6 level

Introduce the Cartesian coordinate system using all four quadrants (ACMMG143)

Investigate everyday situations that use integers. Locate and represent these numbers on a number line (ACMNA124)

Proficiency strand statements at Year 7 level At this year level:

Understanding includes describing patterns in uses of indices with whole numbers, recognising equivalences between fractions, decimals, percentages and ratios, plotting points on the Cartesian plane, identifying angles formed by a transversal crossing a pair of lines, and connecting the laws and properties of numbers to algebraic terms and expressions

Fluency includes calculating accurately with integers, representing fractions and decimals in various ways, investigating best buys, finding measures of central tendency and calculating areas of shapes and volumes of prisms

Problem Solving includes formulating and solving authentic problems using numbers and measurements, working with transformations and identifying symmetry, calculating angles and interpreting sets of data collected through chance experiments

Reasoning includes applying the number laws to calculations, applying known geometric facts to draw conclusions about shapes, applying an understanding of ratio and interpreting data displays



General capabilities General capabilities 1 2 3 4 5 6 7 R

Literacy

Numeracy

Information and communication technology (ICT) capability

Critical and creative thinking

Personal and social capability

Ethical behaviour

Intercultural understanding

Indicates general capabilities are explicitly covered in that section of the resource.

Cross-curriculum priorities Cross-curriculum priorities 1 2 3 4 5 6 7 R

Aboriginal and Torres Strait Islander histories and cultures

Asia and Australia’s engagement with Asia

Sustainability

Indicates cross-curriculum priorities are explicitly covered in that section of the resource.

This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:

the content descriptions are solely for a particular year and subject

all the content descriptions for that year and subject have been used

the author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject.

You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.

creativecommons.org/licenses/by-nc-sa/3.0/au/



1. Reviewing the Cartesian planeWhen you complete this section you should be able to:

draw a Cartesian plane identify the parts of a Cartesian plane identify points on a Cartesian plane.

Key words

x-axis y-axis Cartesian plane horizontal axis

vertical axis quadrant ordered pair

Warm-up 1

1. Circle the factors of 7. 1, 5, 7, 9, 14

2. 8 + 9 = _________

3. What is the missing number on this number line?

a = __________

4. Circle the greater fraction. 2

3or

2

4

5. How much is a half of 25? _______

6. 9.3 − 4 = _______

7. 6.1 × 3 = _______

8. Write 0.666… as a fraction. _______

9. Find the next number: 77, 84, 91, 98, ________

10. Determine the size of the missing angle.

_______

?



Review 1

For the Cartesian plane drawn below complete the following.

1. Label the horizontal axis as the x-axis.

2. Label the vertical axis as the y-axis.

3. Complete the scales on the two axes.

4. Label the first, third and fourth quadrants.

5. Match the points A to E on the diagram here with these ordered pairs:

(-9, -6), (-5, 0), (-3, 5), (4, 6), (7, -1).

A ________ B ________ C ________ D ________ E ________

Check your work before continuing.

5

-3

-4 8D

C

BA

E

2nd quadrant



2. Plotting pointsWhen you complete this section you should be able to:

plot points represented by ordered pairs of integers on a Cartesian plane identify points on a Cartesian plane by writing coordinates.

Key words

coordinate

Warm-up 2

1. Circle the common factor of 10 and 20. 4, 5, 6, 7, 8

2. 15 – 7 = _________

3. The temperature was 5 degrees but it dropped 8 degrees.

What is the new temperature? _________

4. Insert <, > or = to make the following sentence true.1 2

4 5

5. 1

250 = _________

6. Round 6.6 to a whole number. __________

7. 6.42 2 = _________

8. Write 70% as a decimal. _________

9. Find the next number: 0.1, 1.2, 2.3, 3.4, ________

10. At what point is the truck?

_________

x1 2 3 4 5

y

12345



Review 2

1. Draw up a Cartesian plane with the x-axis scale from (-10) to 10 and the y-axis scalefrom (-5) to 5.

2. Plot the following points on the Cartesian plane.

A (6, 4) B (-3, 5) C (8, -4) D (-7, -2) E (0, -4)




Focus problem 2

Using the Cartesian plane on the following page, complete the following.

1. Write in the scale on the axes so that it goes from (-12) to 12 on both axes.

2. Connect, in order, the points (7, -1), (7, 1), (6, 1), (6, 0) with lines.

3. Connect, in order, the points (4, 2), (4, 4), (3, 4), (3, 3) with lines.

4. Connect, in order, the points (-3, 3), (-3, 4), (-4, 4), (-4, 2) with lines.

5. Connect, in order, the points (-6, 0), (-6, 1), (-7, 1), (-7, -1) with lines.

6. Connect, in order, the points (4, -12), (3, -12), (0, -11), (-3, -12), (-4, -12),

(-0.5, -9), (-1, 1), (-7, -3), (-8, -3), (-8, -2), (-1, 5), (-1, 10), (0, 11), (1, 10),

(1, 5), (8, -2), (8, -3), (7, -3), (1, 1), (0.5, -9), (4, -12) with lines.

If you have plotted these five sets of points correctly and connected each set with lines, you should be able to recognise the outline of an object that uses coordinates in its navigation.


René Descartes The Cartesian plane is named after 17th century French mathematician René Descartes.

Descartes developed the system of Cartesian coordinates as a way of providing a link between the algebraic representations of equations, and geometry. In other words, he provided a system of representing equations and functions with diagrams.

The diagram shown here is a representation of the

function 3 4y x x .

x-4 -2 2 4

y

-10

-5

5

10



Cartesian plane for Focus problem 2.



Skills development 2

1. After first writing in the scales so that each square is equal to one, write down thecoordinates of the points shown on this Cartesian plane.

A ______ B ______ C ______ D ______ E ______ F ______ G ______

H ______ I ______ J ______ K ______ L ______ M ______ N ______


AB

C

D

E

F

G

H

I

J

K

L

M

N



3. Tables of valuesWhen you complete this section you should be able to:

plot points on a Cartesian plane from a table of values.

Key word

ordered pair

Warm-up 3

1. Circle the square number. 5, 10, 15, 25

2. 6 7 = ___________

3. What is the missing number?

a = ___________

4. Locate 2

10 on the number line.

5. Find two-quarters of 20. ___________

6. Estimate the sum by first rounding to whole numbers

1.8 + 5.8 ___________

7. 4.7 2 = ___________

8. Write 3

4 as a percentage. ___________

9. Find the next number:1 1 5

6 2 6, , , ___________

10. Determine the probability this spinner will land on a 2.

Express your answer as a fraction.

_________

1

2

3

0 1

-6-9 0 3a



Review 3

Example Identify the pattern to enable completion of this table of values.

Solution The y values are increasing by 4 for every increase of 1 in the x values. Hence the completed table is as follows.

1. Complete the following tables of values by following the patterns.

(a)

x 1 2 3 4 5 6 7 8

y 1 4 7 10 13

(b)

x 1 2 3 4 5 6 7 8

y 19 17 15 13 11

(c)

x 1 2 3 4 5 6 7 8

y 14 11 8 5 2

(d)

x 1 2 3 4 5 6 7 8

y 1 5 9 13 17

(e)

x 1 2 3 4 5 6 7 8

y 1 4 9 16 25


x 1 2 3 4 5 6 7 8

y 3 7 11 15 19

x 1 2 3 4 5 6 7 8

y 3 7 11 15 19 23 27 31




Example Complete this table of values then plot the ordered pairs as points on a Cartesian plane.

Solution

x -3 -2 -1 0 1 2 3 4

y -2 -1 0 1

x-3 -2 -1 1 2 3 4

y

-2

-1

1

2

3

4

5

x -3 -2 -1 0 1 2 3 4

y -2 -1 0 1 2 3 4 5



1. Complete each table then plot the ordered pairs as points on the given Cartesian plane.

(a)

x 1 2 3 4 5 6 7 8

y 3 3.5 4 4.5 5

(b)

x -3 -2 -1 0 1 2 3 4

y 1 3 5 7 9

x-1 1 2 3 4 5 6 7 8 9

y

-1

1

2

3

4

5

6

7

x-3 -2 -1 1 2 3 4

y

-2

2

4

6

8

10

12

14

16



(c) You will need to draw up an appropriate scale for this Cartesian plane.

x -3 -2 -1 0 1 2 3 4

y 8 5 2 -3 -6


x

y



4. LinesWhen you complete this section you should be able to:

recognise points on a straight line.

Key word

linear function

Warm-up 4

1. A prime number has ______ factors.

2. 45 5 = _________

3. The temperature was minus 10 degrees but it went up 6 degrees.

What is the new temperature? _________

4. Express the value of w as a fraction.

5. 1

48 = _________

6. 726.3 1000 = _________

7. 2 14.8 _________

8. Write 90% as a fraction. _________

9. Find the next number: 74, 71, 68, 65, _________

10. Determine the size of the missing angle.

_________

10

w

? 119°



Review 4

Example Complete the table of values here for the linear function 3 2y x by substituting in the x values to find the y values.

Solution

1. Complete these tables of values for each of the functions.

(a) 5y x

x -3 -2 -1 0 1 2 3 4

y 2 9

(b) 2 2y x

x 0 1 2 3 4 5 6 7

y 2 16

(c) 3 3y x

x 1 2 3 4 5 6 7 8

y 0 21

(d) 4y x

x -3 -2 -1 0 1 2 3 4

y -7 0


x -3 -2 -1 0 1 2 3 4

y -11 -8 -5 -2 1 4 7 10

x -3 -2 -1 0 1 2 3 4

y -11 -8 -5



Focus problem 4

The diagram below shows a conference table made up of three square tables with chairs at each place.

1. Complete the following table of values that models various numbers of tables. Note thatx represents the number of tables and y represents the number of chairs needed.

x 1 2 3 4 5 6 7 8

y 4 6 8

2. Label the axes and complete the scales on the following Cartesian plane so that theordered pairs in the table above can be plotted.

5

5



3. Plot the first three ordered pairs from the table, (1, 4), (2, 6) and (3, 8), on the Cartesianplane.

4. Plot the remaining five ordered pairs from the table on the Cartesian plane.

5. What do you notice about the eight ordered pairs you have plotted?

_______________________________________________________________________

6. Use a ruler to draw a line through the eight points extending to both edges of the grid. Beas accurate as possible when you draw the line.

7. Using the line to help you, read off how many chairs would be required for 11 tables.

________________

8. The algebraic rule or function that gives the number of chairs for a given number oftables can be written as the linear function 2 2.y x Use this rule to calculate the

number of chairs required for:

(a) 20 tables ________ (b) 100 tables. ________

9. This function is known as a linear function. Why do you think the term linear is used todescribe the function?

_______________________________________________________________________


Linear functions Algebraic rules that calculate a value from a given starting value are known as functions. In the problem above the function 2 2y x is known as a linear function. The word linear comes from the fact that the values generated by the function will all fall on a line when plotted.

The diagram here shows a graph of the linear function 2 1.y x

x-4 -2 2 4

y

-10

-5

5

10




A function can be represented in three formats: using algebra, as a table of values, or as a graph on a Cartesian plane.

Example Complete a table of values and draw a graph on a Cartesian plane of the linear function

2 1.y x Draw in a line to show that any x value is possible. Use the line to read off the y

value when x is 2.5.

Solution

When x = 2.5, reading off the line gives y = 4.

x -3 -2 -1 0 1 2 3 4

y -7 -5 -3 -1 1 3 5 7

x-3 -2 -1 1 2 3 4

y

-9-8-7-6-5-4-3-2-1

12345678



1. (a) For the linear function 2y x complete this table of values.

x -3 -2 -1 0 1 2 3 4

y 1

(b) Plot the eight points, determined by the ordered pairs in the table, on the Cartesianplane below.

(c) Draw in a line on the graph to represent the linear function.

(d) Use the line to determine the value of y when x = 0.5. _______

x-3 -2 -1 1 2 3 4

y

-2

-1

1

2

3

4

5

6



2. Two ordered pairs for a linear function are represented by points on the Cartesian planebelow.

(a) What are the coordinates of these two points? ______________________________

(b) Draw a line through the points extending to the edges of the Cartesian plane shown.

(c) Use the graph of the linear function to complete this table of ordered pairs.

x -4 -3 -2 -1 0 1 2 3 4

y 3 5

(d) Which of these linear functions is the correct one for the graph and table?

Hint: Substitute values to see if the function fits.

I: 2 2y x

II: 0.5 2y x

III: 0.5 4y x

IV: 2 4y x


x-4 -3 -2 -1 1 2 3 4

y

-1

1

2

3

4

5

6



5. Bivariate dataWhen you complete this section you should be able to:

plot bivariate data on a graph read and interpret bivariate data from a graph.

Key word

bivariate data

Warm-up 5

1. Circle the prime factors of 6. 1, 2, 3, 4, 6, 12

2. 139 + 47 = _________

3. What is the missing number on this number line?

h = __________

4.3 2

8 8 __________

5. Find a fifth of 35. __________

6. 1.3 cm = __________ mm

7. 6 – 5 1 = _________

8. Write 3

8 as decimal. _________

9. Find the next number: 3.1, 2.8, 2.5, 2.2, ________

10. The truck is at (0, 3).

If the truck moves 4 units right, where will it then be?

__________

x1 2 3 4 5

y

12345

-1-7 -3 1h



Review 5

1. (a) Plot the following points on this Cartesian plane.

A(3, 7), B(-4, 5), C(2, -9), D(-7, -2), E(-5, 0), F(3, 0)

(b) Write down the coordinates of these points from the Cartesian plane.

G ________ H ________ I ________ J ________ K ________


x-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

y

-11-10

-9-8-7-6-5-4-3-2-1

123456789

10

G

H

I

J

K



Focus problem 5

The graph below shows the risk of shortages of elements important to clean energy in a visual and easy to follow format.

It is a graph of bivariate data. That is, data with two measurements. These are:

Supply risk – on the horizontal axis Importance to clean energy – on the vertical axis.

The graph is shown as an example of a real representation of bivariate data. However, it does not strictly follow graphing principles as the scales are not evenly spaced from zero.

1. What is the supply risk value for Cerium? ________________

2. What is the importance to clean energy value of Yttrium? ________________

3. Which elements have the highest supply risk?

________________________________

Supply risk

Impo

rtan

ce to

cle

an e

nerg

y

DyNd

Y TbGa Li

Ce

Sm

Y - Yttrium

Ce - Cerium

Nd - Neodymium

Sm - Samarium

Tb - Terbium

Dy - Dysprosium

Li - Lithium

Ga - Gallium

Key to elements

Rare earth elements supply/demand

1 2

1

2

3

3

4

4



4. Which elements have the highest importance to clean energy?

________________________________

5. Which element has the lowest supply risk?

________________________________

6. Write a comment that compares the supply risk of Cerium to Lithium.

_______________________________________________________________________

7. Write a comment that compares the importance to clean energy of Gallium to Cerium.

_______________________________________________________________________

8. Which has the highest supply risk, Terbium or Neodymium?

_______________________________________________________________________


Rare earths A series of elements that are rare in our environment or difficult to mine hold the key to a lot of clean fuel and green energy technologies.

Dysprosium and Neodymium are used in wind turbines and electric vehicles motors.

Gallium is used in solar panels.

Cerium and Lithium are used in batteries for electric vehicles.

The shortage of these elements due to their rarity or the difficulty of mining them is a concern for the future progress of these technologies.




Example

The graph of bivariate data here shows the age and height of five children.

1. Who is the tallest child?

2. Which two children are the same age?

3. Who is older, Ava or Ella?

4. Which child is most likely to be considered short for their age?

Solution

1. Ryan

2. Cooper and Ella

3. Ella

4. Ella

Age

Height

Ella

Cooper

Ava

Ryan

Lily



1. Answer the following questions for the graph below that shows the number of books readper year for children of varying ages.

(a) Who reads the most books? _______________

(b) Who is the youngest? _________________

(c) Which two children read the same number of books? _________________________

(d) What do you notice about the ages of Baden and Jennifer? ____________________

(e) What seems to happen, as children get older? _______________________________

Age

Number of books read per year

Aasha

Baden

Caeleah

Dacey

Earlyn

Jennifer



2. The bivariate data graph here shows information about students’ part time jobs over oneweek.

(a) Who works the most hours? ____________________

(b) Who is paid the highest rate of pay and what is that rate of pay? ________________

(c) Which two students work the same number of hours? ____________________

(d) Who has the lowest pay rate? _________________________

(e) How much does Emile earn for the week? _______________________

(f) Add Fraser to the graph. He worked 16 hours at a pay rate of $10 per hour.


Hours worked (h)8 9 10 11 12 13 14 15 16 17 18 19 20

Pay rate ($/h)

2

4

6

8

10

12

14Alex

Bailey

Cadell

DakotaEmile



6. GraphsWhen you complete this section you should be able to:

interpret and analyse line graphs from data.

Key words

line graph bivariate data

Warm-up 6

1. 2.03 10 = _________

2. 35 – 27 = _________

3. The temperature is minus 7 degrees.

How much will it need to increase to get to 2 degrees? _________

4.6 1

7 7 _________

5. 1

630 = _________

6. 7.9 g = _________ mg

7. 10 5 2 = _________

8. Write 0.9 as a percentage. __________

9. Find the next number here: 8 5 2

10 10 101 , 1 ,1 , ________

10. A six-sided die is rolled.

Express, as a fraction, the probability that it lands on a 5.

_________



Review 6

Answer the following questions for the graph of bivariate data shown above.

1. Write down the ordered pairs for the following four points on the graph.

A ________ B ________ C ________ D ________

2. Complete the following ordered pairs from the points plotted on the graph.

(1, ___) (6, ___) (___, 20.8) (___, 19.2)

3. At which ordered pair is the value on the y-axis the lowest?

____________

4. At which ordered pair is the value on the y-axis highest?

____________


Daily maximum temperatures

Day in June1 2 3 4 5 6 7 8 9 10 11 12 13 14

Max

imum

tem

pera

ture

16

17

18

19

20

21

22

23

24

25

A B

C

D



Focus problem 6

The line graph here shows the level of water in a dam over a period of nineteen March days.

Dam Water Levels

1780

1790

1800

1810

1820

1830

1840

1850

18605-

Mar

6-M

ar

7-M

ar

8-M

ar

9-M

ar

10-M

ar

11-M

ar

12-M

ar

13-M

ar

14-M

ar

15-M

ar

16-M

ar

17-M

ar

18-M

ar

19-M

ar

20-M

ar

21-M

ar

22-M

ar

23-M

ar

Date

Wat

er le

vel (

mm

)

A line graph is a graph suitable for bivariate data. It is most often used to plot a value over a period of time. Data points are first plotted, and then these points are connected by lines as an estimate of the values between the data points.

1. What are the two measurements represented on this line graph?

__________________________________________________

2. Complete these two ordered pairs from the graph.

(9 March, ______ mm)

( _________, 1800 mm)

3. What was the water level on 12 March ____________________

4. On what day/s was the water level 1815 mm? _______________________

5. How much did the water level fall over the nineteen days? _____________

6. A trend is a pattern in data. Comment on the overall trend in the graph.

_______________________________________________________________________



7. The change in water level is due to evaporation. Estimate how much is evaporating perday.

__________________

8. Comment on what might have happened between the readings on 14 March and 15March and between the readings of 17 March and 18 March.

_______________________________________________________________________


Water storage In Australia there are about 500 large dams storing about 80 000 GL of water.

There are also over two million farm dams storing about 7000 GL of water.

Evaporation from these dams averages about two metres per year, and this evaporation results in the loss of large quantities of collected water from our water storage.

In recent years, many techniques have been researched to reduce this evaporation. Using trees as windbreaks, covering the water with floating objects including old car tyres, and spreading chemical barriers on the water, are some of the techniques.




Example Explain the following terms.

trend, line graph, bivariate data, ordered pair, coordinate, Cartesian plane, axes

Solution Trend – a pattern that occurs in data (Line graphs make it possible to see a trend.)

Line graph – a graph of bivariate data drawn on a Cartesian plane by connecting up the data points with line segments

Bivariate data – data with two measurements or variables for each recording

Ordered pair – a pair of values that determine a point on the Cartesian plane

Coordinate – a value that fixes a point according to a distance along a horizontal or vertical axis

Cartesian plane – points are determined in two dimensions by referencing the x and y axes

Axes – the plural of axis eg The Cartesian plane has two perpendicular axes.

1. The line graph here shows the number of social network friends Amy had each month in2011. The values were recorded on the first of every month.

(a) Comment on the overall trend in the data.

____________________________________________________________________

(b) What are the measurements on the two axes?

____________________________________________________________________

Social network friends

Month

Num

ber

of fr

iend

s

120130140150160170180190200210220

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

P



(c) What are the coordinates of point P?

____________________

(d) Complete the following ordered pairs.

(Jan, _____) (Aug, _____ ) ( _____, 205) ( _____, 172)

(e) Explain why the data represented in the graph is bivariate data.

____________________________________________________________________

(f) In what months did the number of friends Amy had decrease?

____________________________________________________________________

(g) In what month did the number of friends Amy had stay the same?

____________________________________________________________________




7. Travel graphsWhen you complete this section you should be able to:

interpret and analyse travel graphs.

Key word

travel graph

Warm-up 7

1. 27.2 10 = _________

2. 64 3 = _________

3. The temperature is 2 degrees.

How much will it need to decrease to get to minus 2 degrees? ____________

4.3 1

6 3 _________

5. 2

525 = _________

6. 9.7 L = ________ mL

7. 2 3 + 8 = _________

8. Find 50% of $120. ____________

9. Describe the rule for the following pattern.

22, 20, 18, 16, 14, …

____________________________________________________________

10. What shape is at (-1, 3)?

__________________

x-5 -4 -3 -2 -1 1 2 3 4 5

y

1

2

3

4

5



Review 7

1. The number line below shows times from 5.10 pm to 5.30 pm.

Write in the times along the scale at each mark.

2. The number line below shows times from 1.00 pm to 5.00 pm.


3. The number line below shows times from 7.00 am to 11.00 am.


4. The number line below shows times from 5.00 am to 8.00 am.



5.205.155.10 5.25 5.30

3.002.001.00 4.00 5.00

9.008.007.00 10.00 11.00

6.005.00 7.00 8.00



Focus problem 7

The line graph of bivariate data below is known as a travel graph.

This is because it shows movement over a period of time.

This travel graph shows the movement of a train from the start to the end of a trip.

Train movement

0

2

4

6

8

10

12

14

16

18

20

7.00 7.05 7.10 7.15 7.20 7.25

Time

Dis

tanc

e fr

om s

tart

(km

)

The train leaves from the depot at 7.00 am.

1. Label the point on the graph that represents its starting point A.

It then travels to a station where it remains for two minutes.

2. Label the point where it arrives at this first station, B and where it departs, C.

It then travels to a second station where it remains for two minutes.

3. Label the point where it arrives at this second station, D and where it departs, E.

It then travels to a third station where it remains for two minutes.

4. Label the point where it arrives at this third station, F and where it departs, G.

It finally travels to the end of the line arriving at 7.25 am.

5. Label the end of the line H.

6. At what time does the train arrive at the second station? _________

7. At what time does it leave the third station? _________



8. How far is the first station from where the train starts? _________

9. How far does the train travel on the whole trip? _________

10. How long does the whole trip take? _________

11. How long does it remain at each station? _________

12. Describe the sections of the graph where the train is at the stations.

_____________________________________________________________________

13. The section of the graph from the third station to the end of the line is steeper than thesections between other stations. What do you think this shows?

_____________________________________________________________________


High speed trains Australia’s fastest train is capable of travelling at about 160 kilometres per hour (kph). However, due to safety issues with the tracks, the average speeds are well below this. High speed trains are not a priority in Australia due to low populations and cheap airfares.

The world’s fastest train, China’s CRH380A, has a fastest speed of 486 kph, although other trains have gone faster under test conditions.

The train shown here, the Japanese Shinkansen, has a top speed of 446 kph.




The slope of a line in a travel graph indicates the speed during that period of travel.

Example For the travel graph of a car trip shown here, answer the questions that follow.

Car trip

0

20

40

60

80

100

120

140

160

7.00 8.00 9.00 10.00 11.00

Time (am)

Dis

tanc

e fr

om s

tart

(km

)

. . .

. .

1. When was the car stationary?

2. What does the downward-sloped line indicate?

3. Comment on the speeds for the three sections where the car is moving.

Solution 1. The horizontal sections of the graph are when the car is stationary. From 8.00 am to

8.30 am and 9.30 am to 10.00 am the car was stationary.

2. The downward-sloped line from 10.00 am to 11.30 am indicates the car is travelling backtowards its start point as its distance from start has decreased from 160 km to 40 km.

3. The steepest section and hence the fastest is from 8.30 am to 9.30 am when the cartravels 100 km in one hour (100 kph). The next fastest section is from 10.00 am to11.30 am when the car travels 120 km in 1.5 hours (80 kph). Note that although the slopeis downward, it is still steeper than the first section of the graph. The slowest section,with the least slope, is from 7.00 am to 8.00 am when it travels 60 kilometres in one hour(60 kph). The grid is helpful when judging the slope of sections of the graph. Tracingpaper or a protractor can also be used to help compare slopes.



The travel graph below shows Brady’s bike ride home from school.

On the way home, he stops at Hayden’s house before walking his bike with Hayden to footy training.

After training, he rides home on his own.

Bike ride home

0

2

4

6

8

10

12

14

3.00 4.00 5.00 6.00

Time (pm)

Dis

tanc

e fr

om h

ome

(km

)

. .

. . .

1. How far is it from Brady’s home to school? ____________________________

2. At what time does Brady get to Hayden’s house? ________________________

3. How long is footy training? ________________________

4. During which sections of the graph is Hayden travelling the fastest? ________________

5. Why are all the sloping sections downward-sloping?

_______________________________________________________________________

6. Explain how the five sections of the travel graph match the story at the top of the page.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________




8. Summary A Cartesian plane can be used to graphically represent pairs of values (ordered pairs).

The ordered pair (4, 2) can be represented on a Cartesian plane as point A (plotted inthe diagram above).

The point A in the diagram above has coordinates (4, 2). These are read in the order x-axis, y-axis.

A Cartesian plane has four quadrants.

The horizontal axis is most commonly labelled as the x-axis.

The vertical axis is most commonly labelled as the y-axis.

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-5-4

-3-2-1

12

34

5

A

first quadrantsecond quadrant

fourth quadrantthird quadrant



Representing the ordered pairs in a table enables patterns such as lines to be seen.

x 0 1 2 3 4y 3 5 7 9 11

The graph shows that the values in the table fall on a line.

This indicates that a linear function ( 2 3y x ) could be used to generate values inthe table.

Data that has pairs of measurements or values is known as bivariate data.

Graphs of bivariate data can also be used to make comparisons, follow trends or trackmovement over time. Line graphs and travel graphs are both drawn to enable this.

x-2 -1 1 2 3 4 5

y

-4

-2

2

4

6

8

10

12



Review tasks The following tasks will assist you to consolidate your learning and understanding of the concepts introduced in this resource, and assist you to prepare for assessments.

Task A

Name: _____________________________ Suggested time: 30 minutes

Actual time taken: __________

Instructions Complete this work on your own. You may use a calculator, but show how you got your answer. Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.

1. Write down the coordinates of the points A to E shown below.

A (____, ____), B (____, ____), C (____, ____), D (____, ____), E (____, ____)

2. Plot and label the points F to I on the axes above.

F (9, 0), G (-6, 5), H (0, -8), I (-4, -9)

3. (a) Complete this table of values.

x-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

y

-10

-5

5

10

D

C

B

A

E



x -2 -1 0 1 2 3 4 5

y 5 3 1 -1 -3

(b) Plot the points from the table on the given axes.

(c) Comment on the pattern in the points you have plotted.

____________________________________________________________________

4. Which of these linear functions would fit the graph shown here?

(i) 2 1y x (ii) 2y x (iii): 2 2y x (iv): 3 1y x

x-3 -2 -1 1 2 3 4 5 6

y

-10

-8

-6

-4

-2

2

4

6

x-5 -4 -3 -2 -1 1 2 3 4 5

y

-8

-6

-4

-2

2

4



5. The travel graph below shows the movement of a bus on atrip from Amstad to Dellering.

It departs Amstad at 9.00 am stopping along the way atBudamba before arriving at Dellering.

Bus Trip

0

20

40

60

80

100

120

140

160

180

200

9.00 10.00 11.00 12.00

Time (am)

Dis

tanc

e fr

om s

tart

(km

)

(a) At what time does it arrive at Budamba? _________________________

(b) How long does it stop at Budamba? _____________________________

(c) At what time does it arrive at Dellering? _________________________

(d) How far away is Dellering from Amstad? ________________________

(e) In the first hour it travels at the same speed for the whole hour. What speed is that?

_________________________

(f) Does it travel faster from Budamba to Dellering than it did in the first hour?

_________________________

(g) How long does the section from Budamba to Dellering take?

_________________________

(h) By looking at the graph from 10.30 to 11.30 the speed for this section can becalculated. What is this speed?

_________________________



Task B

Name: _____________________________ Suggested time: 30 minutes

Actual time taken: __________

Instructions Complete this work on your own. You may use a calculator, but show how you got your answer. Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.

Going walking

Kitty and her mum go for a walk every evening. They leave the house together at 5.00 pm, walk along a trail for 4 km and then return home.

The travel graph here shows the movements of both Kitty and her mum during this walk.

Going walking

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5.00 6.00 7.00

Time (pm)

Dis

tanc

e fr

om h

ome

(km

)

Mum Kitty



1. At what time does Kitty get home?

_______________________

2. At what time does Kitty’s mum get home?

_______________________

3. How long does Kitty take to complete the walk?

_______________________

4. How long does Kitty’s mum take to complete the walk?

_______________________

5. What is Kitty doing from 5.30 pm to 5.45 pm?

______________________________________________

6. When they leave home, who is walking faster?

_______________________

7. When do Kitty and her mum walk together?

_______________________

8. Does Kitty’s mum walk faster on the way out or the way back?

_______________________

9. Write a story explaining Kitty’s walk.

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________



Self-evaluation task Please complete the following.

How well did you manage your own learning using this resource?

Always Usually Rarely Not sure

Each section took approximately 45 minutes to complete.

I needed extra help.

I marked and corrected my work at the end of each section.

I made the journal entries and summaries when asked.

I have kept to my work schedule.

How much mathematics have you learnt using this resource?

Always Usually Rarely Not sure

Understanding I understand and can interpret the meaning of graphs of bivariate data such as line graphs and travel graphs.

Fluency I can plot points and identify coordinates on a Cartesian plane.

Problem Solving I can investigate patterns in data by representing the data graphically.

Reasoning I can analyse graphs of bivariate data and reason conclusions from that analysis.



Write a list of topics for which you need additional assistance. Discuss these with your teacher.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________



Solutions 1. Reviewing the Cartesian plane

Solutions to Warm-up 1

1. 1 and 7 are the factors of 7.2. 173. (-6)

4.2

35. 12.56. 5.37. 18.3

8.2

39. 10510. 90°

Solutions to Review 1

The diagram here shows the solutions to questions one to four.

5. A (4, 6), B(-3, 5), C(-9, -6), D(7, -1), E(-5, 0)

x-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

y

-11-10

-9-8-7-6-5-4-3-2-1

123456789

10

D

C

BA

E

1st quadrant

2nd quadrant

4th quadrant

3rd quadrant



2. Plotting points

Solutions to Warm-up 2

1. 52. 83. (-3°)

4.1 2

4 5

5. 256. 77. 3.218. 0.79. 4.510. (4, 4)


x-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

y

-5-4

-3-2-1

12

34

5A

B

C

D

E



Solution to Focus problem 2

Solutions to Skills development 2

A(3, 10), B(-10, 11), C(-5, 0), D(-12, -3), E(-8, -11), F(2, -5), G(0, -10), H(10, 0), I(10, 4), J(8, -9), K(0, 5), L(-3.5, 7.5), M(-6.5, -5.5), N(5.5, -3.5)



3. Tables of valuesSolutions to Warm-up 3

1. 252. 423. (-3)4.

5. 106. 87. 9.48. 75%

9.7

6

10.1

3


1. (a)

x 1 2 3 4 5 6 7 8

y 1 4 7 10 13 16 19 22

(b)

x 1 2 3 4 5 6 7 8

y 19 17 15 13 11 9 7 5

(c)

x 1 2 3 4 5 6 7 8

y 14 11 8 5 2 -1 -4 -7

(d)

x 1 2 3 4 5 6 7 8

y -3 1 5 9 13 17 21 25

(e)

x 1 2 3 4 5 6 7 8

y 1 4 9 16 25 36 49 64

0 1210




1. (a)

x 1 2 3 4 5 6 7 8

y 3 3.5 4 4.5 5 5.5 6 6.5

(b)

x -3 -2 -1 0 1 2 3 4

y -1 1 3 5 7 9 11 13

x-1 1 2 3 4 5 6 7 8 9

y

-1

1

2

3

4

5

6

7

x-3 -2 -1 1 2 3 4

y

-2

2

4

6

8

10

12

14

16



(c)

x -3 -2 -1 0 1 2 3 4

y 11 8 5 2 -3 -6 -9 -12

x-3 -2 -1 1 2 3 4

y

-12-10-8-6-4-2

2468

1012



4. LinesSolutions to Warm-up 4

1. 22. 93. (-4°)

4.1

105. 26. 0.72637. 7.4

8.9

109. 6210. 61°


1. Complete these tables of values for each of the functions.

(a) 5y x

x -3 -2 -1 0 1 2 3 4

y 2 3 4 5 6 7 8 9

(b) 2 2y x

x 0 1 2 3 4 5 6 7

y 2 4 6 8 10 12 14 16

(c) 3 3y x

x 1 2 3 4 5 6 7 8

y 0 3 6 9 12 15 18 21

(d) 4y x

x -3 -2 -1 0 1 2 3 4

y -7 -6 -5 -4 -3 -2 -1 0




1.

x 1 2 3 4 5 6 7 8

y 4 6 8 10 12 14 16 18

2, 3, 4 and 6.

5. The eight ordered pairs all fall along a line.

7. Reading from the graph, 11 tables would require 24 chairs.

8. (a) 42 (b) 202

9. Linear is used as the points all fall on a line.

x-2 -1 1 2 3 4 5 6 7 8 9 10 11 12

y

-3-2-1

123456789

10111213141516171819202122232425




1. (a)

x -3 -2 -1 0 1 2 3 4

y -1 0 1 2 3 4 5 6

(b) and (c)

(d) The value of y is 2.5 when x = 0.5.

x-3 -2 -1 1 2 3 4

y

-2

-1

1

2

3

4

5

6



2. (a) (-2, 3) and (2, 5)

(b)

(c)

x -4 -3 -2 -1 0 1 2 3 4

y 2 2.5 3 3.5 4 4.5 5 5.5 6

(d) III: 0.5 4y x

x-4 -3 -2 -1 1 2 3 4

y

-2

-1

1

2

3

4

5

6



5. Bivariate dataSolutions to Warm-up 5

1. 2 and 32. 1863. (-5)

4.5

85. 76. 13 mm7. 18. 0.3759. 1.910. (4, 3)


(a)

(b) G (8, 3), H (-6, 7), I (-8, -7), J (0, -5), K (8, -7)

x-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

y

-11-10

-9-8-7-6-5-4-3-2-1

123456789

10

D

B

A

C

E F

G

H

I

J

K




1. 22. 33. Dysprosium and Terbium4. Neodymium and Dysprosium5. Gallium6. Cerium and Lithium have the same supply risk.7. Gallium is of more importance to clean energy than Cerium.8. Terbium


1. (a) Earlyn(b) Earlyn(c) Caeleah and Jennifer(d) They are the same age.(e) They read fewer books.

2. (a) Dakota(b) Alex and Cadell get $14 per hour.(c) Alex and Bailey(d) Dakota(e) 12 × $8 = $96(f) Shown on graph here.

Hours worked (h)8 9 10 11 12 13 14 15 16 17 18 19 20

Pay rate ($/h)

2

4

6

8

10

12

14Alex

Bailey

Cadell

DakotaEmile

Fraser



6. GraphsSolutions to Warm-up 6

1. 20.32. 83. 9°

4.5

75. 56. 79007. 48. 90%

9.9

10

10.1

6

Solutions to Review 6.

These results are the exact values. Students may be approximate with the temperatures. 1. A (8, 21) B (10, 21.5) C (3, 24.2) D (13, 16.9) 2. (1, 19.6) (6, 17.4) (11, 20.8) (9, 19.2) 3. D (13, 16.9)4. C (3, 24.2)


These results are the exact values. Students may be approximate with the water levels. 1. Date and Water level2. (9 March, 1830 mm), (14 March, 1800 mm)3. 1813 mm4. 15 March and 18 March5. 1852 − 1793 = 59 mm6. The trend is a decrease in the water level over time.7. 5 to 6 mm per day is a reasonable estimate of the evaporation rate.8. In both cases the water level rose which was probably due to rainfall.


These results are the exact values. Students may be approximate with the number friends. 1. (a) The overall trend shows an increasing number of friends.

(b) The horizontal axis is measuring time and the vertical axis, the number of friends.(c) (Apr, 191)(d) (Jan, 126), (Aug, 194), (Sep, 205), (May, 172)(e) It is bivariate as there are two variables, time and number of friends.(f) April and September(g) June



7. Travel graphsSolutions to Warm-up 7

1. 2.722. 1923. 4°

4.1

65. 106. 97007. 148. $609. Subtract two each time to get the next number.10. Circle


1.

2.

3.

4.


Questions 1 to 5 required the student to label the graph. 6. 7.12 am7. 7.21 am8. 4 km9. 20 km10. 25 minutes11. 2 minutes12. These sections are horizontal as there is no movement.13. The train is travelling faster.


5.205.155.10 5.25 5.30

5.11

5.12

5.13

5.14

5.16

5.17

5.18

5.19

5.21 5.23

5.245.26

5.27

5.28

5.295.22

3.002.001.00 4.00 5.00

1.30 2.30 3.30 4.30

9.008.007.00 10.00 11.0010.45

10.30

10.15

9.45

9.30

9.158.45

8.30

8.157.45

7.30

7.15

6.005.00 7.00 8.00

5.10

5.20

5.30

5.40

5.50

6.10

6.20

6.30

6.40

6.50

7.10

7.20

7.30

7.40

7.50



1. 12 km2. 3.30 pm3. 1 hour4. From 3 to 3.30 and from 5.30 to 6.5. The downward slope is due to all travel being towards home.6. From 3 to 3.30 Brady is riding from school to Hayden’s house.

From 3.30 to 4 he is at Hayden’s house.From 4 to 4.30 he is walking with Hayden to footy training.From 4.30 to 5.30 he is at footy training.From 5.30 to 6 he is riding his bike home.

Solutions to Review tasks Solutions to Task A

1. A ( 3, 7), B (-2, 6), C (-4, 0), D (-5, -6), E (6, -2)2.

x-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

y

-10

-5

5

10

D

C

BA

E

F

G

IH



3. (a)

x -2 -1 0 1 2 3 4 5

y 5 3 1 -1 -3 -5 -7 -9

(b)

(c) The plotted points from the table all fall along a line.

4. (ii) 2y x 5. (a) 10.00 am

(b) 30 minutes(c) 11.45 am(d) 185 km (within 5 km is acceptable)(e) 60 kph(f) Yes(g) 1 h 15 min(h) 100 kph

Solutions to Task B

1. 6.45 pm2. 7.00 pm3. 1 h 45 min4. 2 h5. She is stationary.6. Kitty7. From 5.45 pm to 6.00 pm8. Same speed for both sections9. (This will vary – an example is given)

Kitty leaves home with her mum at 5.00 pm for a walk. She walks faster than her mumwalks for 30 minutes, then stops and waits while her mum catches up. She then walkswith her mum for 15 minutes until they turn around to walk home. Kitty then walksfaster than her mum does on the way home, arriving home at 6.45 pm.

x-3 -2 -1 1 2 3 4 5 6

y

-10

-8

-6

-4

-2

2

4

6

MATHSAC022 CARTESIAN COORDINATES

ISBN: 9780730744429

australian curriculum mathematics year 7

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