STAGE 5 - PAS 5.1.2 COORDINATE GEOMETRY
Lessons are intended for students who have completed Stage 3 by the end of
Year 6 and can be expected to achieve 5.2 outcomes by the end of Year 10.
Syllabus Outcomes
Refer to Stage 5 Syllabus – PAS 5.1.2
Assumed Knowledge






Plotting points on a number plane.
Recognition of the quadrants on a number plane.
Completion of a table of values for a given equation.
Ability to graph lines.
Understanding and the application of Pythagoras’ Theorem.
Concept of average.
References





Maths Zone Level 2 Year 9 (Heinemann).
Maths Quest 5.2 Year 9 (Jacaranda).
Mathscape 5.2 Year 9. (Macmillan).
New Century 5.2 Year 9 (Thomson & Nelson).
Graphic Software www.padowan.dk (available as freeware).
Contents of the Unit


Overview:
- Promotes discussion of key concepts.
Shows the scope of the topic and is used to tick/highlight completed
concepts so the students can monitor their progress.
Assessing prior knowledge:
- Enables the teacher to assess the individual needs of students.
Outcomes
Students will be able to:
 find the midpoint of an interval (using average).
 find the distance between two points (using Pythagoras’ Theorem).
 find the gradient or slope of a line.
 graph and interpret vertical and horizontal lines.
 compare and contrast graphs of linear relationships (table of values).
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 1 of 26
1.
Overview
Intercept
Graphing non linear relationships
Distance
Coordinates
Gradient/slope
Midpoint
Coordinate Geometry
Interval
Pythagoras’ Theorem
Number plane
Graphing vertical & horizontal lines


Graphing linear relationships
The overview is to be copied into exercise books.
Students tick / highlight each concept when understood.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 2 of 26
2.
Pretest
1. State the coordinates of each point:
a) A
b) B
c) C
d) D
e) E
f) F
y
4
B
3
E
2
A
1
-4
-3
-2
-1
F
1
2
3
4
x
-1
-2
D
C
-3
-4
2. In which quadrant would each point lie?
a) (4, -3)
b) (-1, 2)
c) (3, 1)
d) (-4, -4)
3. a) Plot the following points on a number plane (Cartesian plane) and join
them in order, A to B to C to D to A:
A (1, 3)
B (2, 0)
C (-1, -1)
D (-2, 2)
b) Name the shape formed.
4. If x = 3, find the value of y in each of the following:
a) y  x  1
b) y  x  2
c) y  4 x
d) y  3 x  1
5. Complete the table of values using the rule y  x  5 :
x
-3
-2
-1
0
1
2
3
y
6. a) Complete the table of values using the rule y  x  1 :
x
0
1
2
y
b) Plot the graph on a number plane for y  x  1 using the three
coordinates from the table in a).
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 3 of 26
7. Use Pythagoras’ Theorem to find x in XYZ
Y
x
6 cm
X
Z
8 cm
8. a) Find the number that is halfway between –3 and 5.
b) Find the mean (average) of 5 and 21.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 4 of 26
3.
Midpoint of an Interval from a Diagram
Concepts
-
Activity 1
Group Work
-
-
An INTERVAL is a line jointing two points on a
number plane.
MIDPOINT of an interval is exactly half way between
the two end points.
In groups of 4, each group is given 3 questions.
First question A (0,0) B (8,12).
Second question C (2,5) D (6,8).
Third question E (-1,3) F (5,7) in an envelope.
Their task is to plot the points in pairs on the grid
paper provided, and determine a method to find the
midpoint of each line.
Each group reports their results and a common
method is determined by the class. This should lead
to the concept of using average to determine the
midpoint.
Theory
-
Students copy the resulting theory into their exercise
book.
Extension
-
More competent students could develop the formula
for Midpoint.
THEORY
1. Plot the points  2,3 2,6 on a number plane.
22
 0.
2. Add x coordinates and divide by 2
2
3  6  3

3. Add y coordinates and divide by 2
2
2
  3
 the midpoint is  0,
.
2 

(-2,3)
(2,-6)
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 5 of 26
4. Distance Between Two Points
Concepts
-
Investigation
(Teacher directed)
B
A
-
C
-
Distance between two points is the length of the
interval joining the two points.
It can be calculated using Pythagoras’ Theorem.
Equipment needed:
 grid paper
Plot the points A (2,5) & B (6,8) in your book. Join the
points.
Form a right angle triangle by drawing a vertical side
from the higher point and a horizontal side from the
lower point.
Find the distance of AC.
Find the distance of BC and add to diagram.
Distance of AB, can then be determined by using
Pythagoras’ Theorem.
REMEMBER:
hypotenuse2 = (side 1)2 + (side 2)2
Applying Pythagoras’ Theorem
AB 2  3 2  4 2
AB 2  9  16
AB 2  25
AB 2  25
 AB  5units
hypotenuse
side 1
side 2
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 6 of 26
Activity 2
Sequencing
Exercise
-
Equipment needed:
 5 A4 sheets (coloured card) each containing 1
question.
 Prepare Activity 2 worksheet by cutting &
laminating each card.
 Blutak to adhere cards to A4 sheets.
-
Two students are given a question and a mixed
solution. Their task is to order them correctly and
stick onto the corresponding question sheet (sheets
are then displayed around
the room).
Alternately, each student is given 1 card that needs
to be positioned correctly under the corresponding
question (which are placed at stations around the
room). If there are insufficient students for the
number of cards, the initial diagram in each solution
set could be positioned by the teacher.
-
Appraisal
1. Students were handed a card as they entered the room.
2. The 5 questions were written on 5-A4 sheets that were posted around the
room. The students were told they each had 1 step in the solution of the 5
questions – the aim was to get their card on the right A4 sheet and in the
correct order. They could change anyone else’s card to change the order,
but all students had to remain with their card to argue/defend their
position.
3. I did question 1 as a cooperative class exercise (any student who
contributed a step was given a replacement from the spares so they could
still be involved). The students were then set the task of solving the other
4 questions. Students enjoyed the experience – they found “it made them
think”.
4. I found the lesson noisy, but interesting and students still tended to refer to
me for adjudication instead of solving the problem together – this will
come with practice. Some students (2) still didn’t contribute, but other
students worked around this.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 7 of 26
ACTIVITY 2
1. Find the distance
between the points
A(3,1) and B(6,5)
y
2. Find the distance
between the points
A(-1,1) and B(3,-1)
3. Find the length of the
interval joining
A(0,10) and B(5,-2)
y
B(6,5)
A(0,10)
y
B(6,5)
5
A(-1,1)
8
4
6
1
3
2
1
4
x
A(3,1)
-1
x
-1
1 2 3 4 5 6
2
1 2 3
x
1 2 34 5
B(3,-1)
B(5,-2)
-2
y
B(6,5)
A(-1,1)
5
4
3
2
1
A(0,10)
y
y
6
4
A(3,1)
8
12
-1
x
2
3
1 2 3
x
4
4
2
x
B(3,-1)
-2
1 2 3 4
5
5
B(5,-2)
AB 2  3 2  4 2
AB 2  2 2  4 2
AB 2  5 2  12 2
AB 2  9  16
AB 2  4  16
AB 2  25  144
AB 2  25
AB 2  20
AB 2  169
AB  25
AB  20
AB  169
AB  5
Distance between A and AB is 13 units long
B is 20 units
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 8 of 26
4. Find the distance between the
points A(1,-1) and B(-1,1)
5. Find the distance between the
points A(4,1) and B(-2,-7)
y
x
B(-1,1)
y
-1 x
-2 -1
-1
-2
-3
-4
-5
-6
-7
1
x
-1
1
-1
A(1,-1)
A(4,1)
x
1 2 3 4
B(-2,-7)
y
x
y
x
B(-1,1)
-2 -1
-1
-2
-3
-4
-5
-6
-7
1
x
2
-1
1
-1
A(1,-1)
2
A(4,1)
-1
x
1 2 3 4
8
B(-2,-7)
6
AB 2  2 2  2 2
AB 2  6 2  8 2
AB 2  4  4
AB 2  36  64
AB 2  8
AB 2  100
AB  8
AB  100
AB is
8 units
Henry Kendall High
AB = 10
PAS 5.1.2 Coordinate geometry unit
Page 9 of 26
5. Gradient
Concepts
-
Activity 3
Group Work
-
Gradient means slope.
The steeper the line, the greater the gradient.
The gradient is the same at any point along a straight
line.
Negative gradients slope to the left, positive
gradients slope to the right.
rise
Formula gradient =
run
Horizontal lines have zero gradient.
Vertical lines have undefined gradient.
change in y
m
change in x
Each group is given a gradient, eg. m = 2
The task of the group is to determine four pairs of
points that would satisfy this gradient and draw each
one, eg. (0,0) & (1,2) is one pair.
Textbook Reference (suggestion only):
Maths Zone 9 Level 2 Exercise 7.2 pg 300.
Maths Zone 9 Level 3 Exercise 8.3 pg 385.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 10 of 26
6. Extension
Activity 4
Given :(a)
The midpoint of the interval AB is M(0, 0) and one endpoint is B(3, 4),
find:
(i)
the coordinates of the end point A
(ii)
the distance AB
(iii)
the gradient of the interval AB
(b)
The midpoint of the interval AB is M(5, 0) and one endpoint is B(10,12),
find:
(i)
the coordinates of the end point A
(ii)
the distance AB
(iii)
the gradient of the interval AB
(c)
The midpoint of the interval AB is M(0, 9) and one endpoint is B(-2, 1),
find:
(i)
the coordinates of the end point A
(ii)
the distance AB
(iii)
the gradient of the interval AB
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 11 of 26
7. Graphing Vertical and Horizontal Lines Using Tables of Values
Whole Class
-
-
Discussion
-
On your paper graph the following points (1, -1),
(1, 1) (1, 0), (1, 2). What do you notice about these
points?
All x values the same; forms a vertical line  x  1.
On your paper graph the following (-1, 2) (0, 2) (2, 2).
What do you notice about these points?
All y values the same; forms a horizontal line y  2
Why does the x axis have an equation of y = 0?
Why does the y axis have an equation of x = 0?
What point is the intercept for x = 3?
Examples:
Board Work
x=1
y = -1
Activity 5
-
Students complete the sheet by labelling each
straight line with its correct equation.
Textbook Reference (suggestion only):
Mathscape 9: Exercise 10:3 pp 374-376.
Extension: Parallel lines –
Maths Zone 9 Level 3: p 387 Q’s 6, 7.
Perpendicular lines – Maths Zone 9 Level 3 Investigation p 391.
New Century 9 Stage 5.2 / 5.3 p 330.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 12 of 26
ACTIVITY 5
Complete by labelling each line with the correct equation:
A
B
y
C
y
3
8
8
0
0
D
x
0
x
E
y
F
y
x
y
3-
-2
0
G
y
0
x
9 x
0
x
-3
y
0
Henry Kendall High
2
x
PAS 5.1.2 Coordinate geometry unit
Page 13 of 26
8. Graphing Linear Relationships
Line y  2 x  5
x
y
0
5
1
7
2
9
Table of values
Plot the points and draw the line
Questions (oral):
1. Does the point (3, 12) lie on this line?
2. What is the x intercept?
3. What is the y intercept?
Activity 6– Silent Card Shuffle
- The class is divided into 7 groups of 4.
Each group is given a shuffled set of cards
(printed on laminated card ) to work on.
- Rules and cards are on pages15 - 17.
Textbook Reference (suggestion only):
Mathscape 9 p 387.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 14 of 26
ACTIVITY 6
Silent Card Shuffle: Rules for Players
1.Silently sort cards into 4 groups:

Equations

Tables of Values

Graphs

Descriptions
2. Silently re-arrange cards to match like information.
3. Discuss your reasoning - You are allowed to change
the work of someone else.
4. All students (except one) walk to next group (in a
clockwise direction) and compare results. The student
who remains behind explains what reasoning the group
used to arrive at their answers.
5. Return to your group to check your group’s results
(teacher directed).
6. Work in pairs, see-sawing (i.e. taking it in turns to tell
each other something you have noticed):
Compare and contrast the features of each graph. How
does the equation of the graph affect the y-intercept?
How does the equation of the graph affect the slope?
The aim of this section is to work towards the
gradient /intercept form of the straight line  y = mx + b
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 15 of 26
Cards
x
y
y=x–4
y=x+4
y= x
-1
0
-1
1
0
x
y
1
-1
0
3
4
y
x
1
y
5
-1
0
-5
y
y=x–3
y=x+3
-4
1
x
-1
0
1
x
-1
-3
y
2
3
4
y
-4
y
y
0
-3
1
-2
y
x
x
x
x
x
The gradient of
the line is 1.
The y-intercept of
the line is 0.
The gradient of
the line is 1.
The y-intercept of
the line is 4.
The gradient of
the line is 1.
The y-intercept of
the line is -4.
The gradient of
the line is 1.
The y-intercept of
the line is 3.
The gradient of
the line is 1.
The y-intercept of
the line is –3.
y = 2x
y = 2x + 4
y = 2x - 4
y = 2x + 3
y = 2x - 3
x
-1
0
1
x
-1
0
1
x
-1
0
1
x
-1
0
1
x
-1
0
1
y
-2
0
2
y
2
4
6
y
-6
-4
-2
y
1
3
5
y
-5
-3
-1
y
y
y
y
y
x
x
x
x
x
The gradient of
the line is 2.
The y-intercept of
the line is 0.
The gradient of
the line is 2.
The y-intercept of
the line is 4.
The gradient of
the line is 2.
The y-intercept of
the line is -4.
The gradient of
the line is 2.
The y-intercept of
the line is 3.
The gradient of
the line is 2.
The y-intercept of
the line is –3.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 16 of 26
y=4–x
y = -x
y = -x - 4
x
-1
0
1
x
-1
0
1
y
1
0
-1
y
5
4
3
y
y=3-x
x
-1
0
y
-3
-4 -5
y
1
y = -x - 3
x
-1
0
1
x
-1
0
y
4
3
2
y
-2
-3 -4
y
y
y
x
x
x
The gradient of
the line is -1.
The y-intercept of
the line is -4.
The gradient of
the line is -1.
The y-intercept of
the line is 3.
The gradient of
the line is –1.
The y-intercept of
the line is –3.
y = -2x - 4
y = 3 – 2x
y = -2x - 3
x
x
The gradient of
the line is -1.
The y-intercept of
the line is 0.
The gradient of
the line is -1.
The y-intercept of
the line is 4.
y = -2x
y = 4 – 2x
x
-1
0
1
x
-1
0
1
x
-1
0
y
2
0
-2
y
6
4
2
y
-2
-4 -6
y
y
1
1
y
x
-1
0
1
x
-1
0
y
5
3
1
y
-1
-3 -5
y
1
y
x
x
x
x
x
The gradient of
the line is -2.
The y-intercept of
the line is 0.
The gradient of
the line is -2.
The y-intercept of
the line is 4.
The gradient of
the line is -2.
The y-intercept of
the line is -4.
The gradient of
the line is -2.
The y-intercept of
the line is 3.
The gradient of
the line is –2.
The y-intercept of
the line is –3.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 17 of 26
9. Investigating Gradient
Before you begin:
1. Go to network programs.
2. Click on maths graphs.
3. Maximise the screen.
4. Go to edit options. Tick the box: calculate complex, then ok.
5. Go to axis box and tick the boxes in both the y axis and x axis to show
units.
6. You can now get started. Each time you wish to enter a new function,
press the function box and change the colour each time.
(To change the colour double click on the box).
Using the graphs package on the computer, complete the following questions:
1. a) On the same number plane draw the straight lines with equations:
y  2x
y  2x  1
y  2x  3
y  2x  3
b) What do you notice about these lines?
________________________________________________________
________________________________________________________
________________________________________________________
2. a) On the same number plane draw the straight lines with equations:
y  3 x  1
3x  y  2  0
6 x  2y  3  0
b) Rewrite the second two equations into gradient – intercept form:
__________________________________________________________
__________________________________________________________
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 18 of 26
c)
What do you notice about these three lines?
__________________________________________________________
__________________________________________________________
__________________________________________________________
3. a) Complete the following sentence:
Straight lines are ____________________ if their ___________ are equal.
4. a) On the same number plane draw the straight lines with equations:
y  2x  1
y 
1
x2
2
b) What do you notice about these lines?
__________________________________________________________
__________________________________________________________
__________________________________________________________
5. a) On the same number plane draw the straight lines with equations:
y  3x  1
x  3y  6  0
b) Rewrite the second equation in gradient – intercept form and find the
gradient:
__________________________________________________________
c) What do you notice about these lines?
__________________________________________________________
6. Complete the following:
Straight lines are ___________________ if the product of their gradient is
__________________.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 19 of 26
10. Investigating Transformations
1. a) On the same number plane draw the graphs:
y  3x
y  3x  1
b) Which transformation, rotation, reflection or translation would be used
to superimpose the graph of y  3x over that of y  3x  1 ?
__________________________________________________________
__________________________________________________________
__________________________________________________________
c) Describe the transformation that would superimpose the graph of
y  3x over the graph of y  3x  5 .
__________________________________________________________
__________________________________________________________
__________________________________________________________
2. a) On the same number plane draw the graphs:
y  2x
y  2 x
b) Describe the transformation that would superimpose the graph of
y  2 x on the graph of y  2 x
__________________________________________________________
__________________________________________________________
__________________________________________________________
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 20 of 26
3 a)
On the same number plane draw the graphs:
1
y  x
3
y  3x
b) Describe the transformation that would superimpose the graph of
1
y  3x over the graph of y   x
3
__________________________________________________________
__________________________________________________________
__________________________________________________________
4.
Describe the transformation that would superimpose the graph of
y  4 x over the graph of:
(i)
y 
1
x
4
y  4 x
(ii)
(iii)
y  4x  7
__________________________________________________________
__________________________________________________________
__________________________________________________________
5.
Describe the transformations needed to map:
y  3x
y  5  3x
a) y  3x
to
to
__________________________________________________________
__________________________________________________________
__________________________________________________________
b) y  4 x
to
y
1
x
4
to
y
1
x2
4
__________________________________________________________
__________________________________________________________
__________________________________________________________
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 21 of 26
c) y  1 x
to
to
y 1 x
y 5 1
2
2
2
__________________________________________________________
__________________________________________________________
__________________________________________________________
d) y  3x  5
to
y  3x
to
y  3x
__________________________________________________________
__________________________________________________________
__________________________________________________________
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 22 of 26
11. Graphing Non Linear Relationships
Create a table of values and graph these equations:
y  x2
y  x2  2
y  2x
What statement can we make about these equations?
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 23 of 26
12. Coordinate Geometry Bingo
This is a meta-language exercise.
In this the students have to recognise the words and cross them off as they
occur during a general review lesson.
Alternatively the sheet could be used throughout the whole topic with awards
for “five in a row” etc.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 24 of 26
Coordinate Geometry Bingo
origin
axes
rise
run
right
positive
Number
plane
length
intersect
slope
Pythagoras
x-axis
midpoint
negative
y intercept
linear
ordered pair
distance
y-axis
Cartesian
gradient
left
points
equation
right
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 25 of 26
13. How Much Do You Know?
Papers for each group headed by:







linear relationships
non-linear relationships
number plane
Pythagoras’ Theorem
gradient
midpoint
interval
Students write down as many ideas as they can.
On a signal papers are swapped.
Students can only add new ideas and points – “See-Saw”.
Each group makes up 4 questions.
Henry Kendall High
PAS 5.1.2 Coordinate geometry unit
Page 26 of 26