An introduction to graphing for number and algebra
This unit aims to build your knowledge of graphing. You are expected to work on this on your
own, though at times some exercises will be discussed and marked by the teacher with the rest of
the group.
There are three sections to the unit. Section 1 is straightforward and is suitable for all students.
Section 2 introduces some algebra so is harder, while section 3 looks at using different scales and
spacing on each axis
Copy the notes into your book. Also complete the exercises neatly in your book.
Section 1: Ordered Pairs
Notes
(3 , 2) is an ordered pair
To be written properly, an ordered pair must have both brackets, and the comma
An ordered pair has two numbers in it. The order they are put in is important
Exercise 1
These ordered pairs are missing a few important bits. Write them properly
1)
4, 2
(2)
3, 8
(3)
1, 0
(4)
0, 0
5)
0 5
(6)
6 1
(7)
7 5
(8)
8 8
9)
(4, 6
(10)
(2, 9
(11)
17, 4
(12)
(3, 10
13)
(10 10
(14)
(5 8)
(15)
11 11)
(16)
3, 3)
17)
3, 6
(18)
(9, 17
(19)
8 7
(20)
(3 12)
The following are also supposed to be ordered pairs, but in these letters are being
used to show where the numbers should go. These also have not been written
properly and need to be fixed up.
21)
(a, b
(22) (x y)
(23) (m, n
(24) x, y)
Graphs for number and algebra
Notes
Graphs are drawn on a set of axes
The axes meet at right angles and join at the number zero in the corner
Each axis is numbered 0, 1, 2, 3, . . . from zero in the corner, with the spacing
between each number being equal. Make sure the numbers are put on the lines
rather than in the spaces between the lines
for example
The across axis is called the x axis
The up axis is called the y axis
The place where the two axes meet is called the origin
Different places on the graph are called points.
Putting points on graphs: plotting ordered pairs
Notes
Points are plotted on a set of axes. To plot a point on a set of axes you first need an
ordered pair. (When we use an ordered pair like this it is usually called the
coordinates of the point.)
How to plot an ordered pair
In an ordered pair (a, b)
the first number (a) is how far to go in the x (across) direction
the second number (b) is how far to go in the y (up) direction
for example
To plot the point (2, 3)
(1)
(2)
(3)
(4)
(5)
Draw a set of axes
Start at the origin
Move 2 in the x
direction
Move 3 in the y
direction
Mark the spot with a
cross and the
coordinates
Convention:
Always move in the x direction first
(We read by going across the page so we start plotting points by going the
same way)
Exercise 2
Use the example above to help you to plot these points. Put each question on a new
set of axes.
1)
(3, 4)
(2)
(6, 1)
(3)
(0, 4)
(4) (3, 0)
5)
(3, 3)
(6)
(0, 1)(1, 2)(2, 3)
7)
(4, 7)(1, 1)(2, 3)
(8)
(2, 2)(3, 0)(1, 5)(4, 4)
9)
(6, 2)(0, 0)(4, 0)
(10)
(1, 1)(2, 2)(3, 3)(4, 4)
11)
(0,0)(1, 2)(2, 4)(3, 6)
(12)
(0, 0)(1, 1)(2, 4)(3, 9)
13)
(8, 0)(7, 1)(6, 2)(5, 3)(4, 4)
14)
Explain what a convention is. (If you are not sure, look it up in a dictionary.)
Writing ordered pairs for points
Notes
A point on a graph can be changed into an ordered pair. Each ordered pair will look
like (x, y). This shows the across number (x) goes first and the up number (y) second.
A point on a graph can be labelled with a capital letter. Different points should be
given different letters.
For example
Write down the coordinates of the points on this set of axes.
Point A is (2, 0)
B is (3, 5)
C is (5, 2)
Convention:
Always read in the x direction first, and write this number first
Exercise 3
Use the conventions you have met to write down the coordinates of these points.
Pictures with Ordered Pairs
Sometimes when points are plotted they can be joined up to make a picture or a
pattern.
Exercise 4
1)
Draw a set of axes so it takes up half of a page, and number them.
2)
Plot these points. Join each point to the next with a straight line (Use your ruler to do this.)
(1, 1)(1, 9)(2, 14)(2, 17)(4, 16)(2, 16)(2, 14)(3, 9)(3, 5)(4, 5)(4, 6)(5, 6)(5, 5)(6, 5)
(6, 6)(7, 6)(7, 5)(8, 5)(8, 6)(9, 6)(9, 5)(10, 5)(10, 7)(11, 7)(11, 6)(12, 6)(12, 7)(13, 7)
(13, 6)(14, 6)(14, 7)(15, 7)(15, 5)(16, 5)(16, 6)(17, 6)(17, 5)(18, 5)(18, 6)(19, 6)
(19, 5)(20, 5)(20, 6)(21, 6)(21, 5)(22, 5)(22, 9)(23, 14)(23, 17)(25, 16)(23, 16)
(23, 14)(24, 9)(24, 1)(13, 1)(13, 3)(12, 4)(11, 3)(11, 1)(1, 1)
3)
What shape have you drawn?
Exercise 5
1)
Draw a set of axes that go up to 9 on each axis.
2)
Plot these points. Join each point to the next with a straight line (Use your ruler to do this.)
(1, 1)(2, 5)(3, 7)(5, 8)(7, 7)(8, 6)(8, 4)(5, 2)(7, 1)(3, 1)(5, 2)(3, 4)(1, 1)
3)
What shape have you drawn?
Exercise 6
1)
Draw a picture you would like to draw as a graph.
2)
Try to put your picture on a set of axes (so you can work out the coordinates which can be
joined up with lines.)
3)
Write a set of coordinates that would draw your picture.
4)
Give your set of coordinates to a friend. Get them to draw your picture in their book.
Exercise 7
Here are some sets of ordered pairs to plot. In each case, the numbers in the
ordered pairs show the start of a pattern. This pattern links the first number in the
ordered pair to the second.
For example
Plot the set of points (0, 1)(1, 2)(2, 3)(3, 4)(4, 5)(5, 6)…
Notice the arrow by the last point. This
shows that the pattern keeps on going
Also notice that a pattern in the numbers
of the ordered pairs makes a shape on
the graph
Plot these sets of points. Put each question on a new set of axes.
1)
(0, 2)(2, 4)(4, 6)(6, 8) . . .
(2)
(10, 10)(9, 9)(8, 8)(7, 7) . . .
3)
(0, 0)(1, 2)(2, 4)(3, 6) . .
(4)
(12, 10)(11, 9)(10, 8)(9, 7) . . .
5)
(12, 6)(10, 5)(8, 4)(6, 3) . .
(6)
(0, 1)(2, 3)(4, 5)(6, 7) . . .
7)
(0, 0)(1, 3)(2, 6)(3, 9) . . .
(8)
(0, 1)(1, 3)(2, 5)(3, 7) . . .
9)
(1, 0)(2, 1)(3, 2)(4, 3) . . .
(10)
(1, 1)(2, 3)(3, 5)(4, 7) . . .
11)
(5, 0)(6, 1)(7, 2)(8, 3) . . .
(12)
(4, 4)(5, 5)(6, 6)(7, 7) . . .
13)
(0, 0)(1, 1)(2, 4)(3, 9) . . .
(14)
(1, 1)(2, 1)(3, 1)(4, 1) . . .
15)
(1, 0)(1, 1)(1, 2)(1, 3) . . .
(16)
(10, 0)(9, 1)(8, 2)(7, 3) . . .
Did you notice?
Not all of the patterns drew straight lines!
Section 2
Exercise 8
Some of the patterns in exercise 7 involve adding, others, subtracting, and some,
multiplying. Some special patterns are at the end. Look for the pattern in each
question, and once you have worked it out, write it down in words, and try to work out
the next two ordered pairs for the pattern. You may find a few hard. If this
happens, write down what you tried when looking for the pattern and be prepared to
discuss these questions with your teacher.
The example below is to help show you what to do
(0, 1)(1, 2)(2, 3)(3, 4) . . ←
Look for the pattern in the numbers. Remember
the pattern changes the first number of the
ordered pair into the second number
In this case, the second number is always one more than the first number so we can
write this in words as “whatever number you start with, add one to get the second
number”
The next two in this pattern will start with (4, ) and (5, )
Explain why you think this is
(0, 1)(1, 2)(2, 3)(3, 4)(4, 5)(5, 6)
←
As the second number is always one
more than the first number we can now write the new ordered
pairs
Challenge
Mathematicians are lazy and do not like to write long sentences when they don’t have to. Instead
they use symbols to show what they mean. For example, instead of writing the word three, we
write 3. For the sentences you wrote for exercise 8, mathematicians have invented shorter ways of
writing these.
For example, mathematicians often use the letter n to mean “whatever the number you start with”
This means that the pattern for (0, 1)(1, 2)(2, 3)(3, 4)… could be described by writing n + 1 instead
of the sentence “whatever the number you start with, add one to it.” n + 1 means the same thing.
Try to write your patterns with symbols. Start each pattern by writing the letter n instead of writing
“whatever the number you start with”
Using letters to show a pattern of ordered pairs
Notes
When using letters to show a pattern in a set of ordered pairs, we normally make it
look like an ordered pair itself
for example
(n, n + 1) is how we would write the ordered pair for the pattern
(0, 1)(1, 2)(2, 3)(3, 4)(4, 5)…
In this sort of pattern, the first pair usually starts with 1, the second with 2 and so
on, so the problem “find the first five ordered pairs of the pattern (n, n + 5)” means
you write five ordered pairs that start with (1, )(2, )(3, )(4, )(5, )
The n + 5 part tells you how to change the first number into the second number
This creates the 5 ordered pairs (1, 6)(2, 7)(3, 8)(4, 9)(5, 10)
Because we are using letters to make an ordered pair for a graph, we sometimes use
the letter x as the first number of the ordered pair
Explain why you think this is
This makes (n, n + 5) look like (x, x + 5). They both say the same thing, which is:
whatever the first number in the pair is, add five to it to get the second number’
Another convention we use in mathematics
We tend to write the x’s for mathematics as two back to front c’s (as on the graphs
above). This is so you don’t get them confused with multiplication signs. (Your maths
book may also look like a love letter if you don’t as x × x will look like ×××.)
Another new word
When we use letters in mathematics we call them variables
Another example Find the first five ordered pairs of the sequence (x, x × 2 + 1), and plot
them on a set of axes
Start the ordered pairs with (1, )(2, )(3, )(4, )
To find the second number in each pair, double it and add one, so we get
(1, 3)(2, 5)(3, 7)(4, 9)
Which we can plot on a graph
Exercise 9
Use the following to write the first 5 ordered pairs of each pattern. Then plot the
points you have made for each question on their own set of axes. Use arrows to show
the patterns continue.
1)
(x, x + 2)
(2)
(x, x × 2)
(3)
(x, x - 1)
4)
(x, x + 4)
(5)
(x, x × 6)
(6)
(x, x × 2 - 1)
7)
(x, x × x)
(8)
(x, x + 10)
(9)
(x, x × 4 - 4)
10)
(x, x × 2 + 1)
(11)
(x, x × 3 + 2)
(12)
(x, x × x + 1)
Making Graphs from Tables
Notes
Tables (or charts) like these can be used to make graphs
x
1
2
3
4
x×4-1
3
7
11
15
x
1
2
3
4
y
5
1
6
4
With both tables the second column is the one used for the y numbers (y values) of
each pair
Some people like to make the ordered pairs before they plot them. One way to do
this is to add another column to the table. These ordered pairs can then be plotted
normally.
for example
x
1
2
3
4
(x, y)
(1, 5)
(2, 1)
(3, 6)
(4, 4)
y
5
1
6
4
Exercise 10
Use the tables (charts) below to make graphs. Make a new graph for each chart
(1)
x
1
2
3
5
y
0
6
5
5
(2)
x
2
3
4
5
x+2
4
5
6
7
(3)
x
0
2
4
5
x×3
0
6
12
15
(4)
x
6
7
8
9
x-2
4
5
6
7
(5)
x
0
2
4
6
y
10
8
6
4
(6)
x
0
1
5
7
y
0
1
5
7
(7)
x
0
5
8
10
x+1
1
6
9
11
(8)
x
2
4
5
6
y
7
2
6
7
(9)
x
0
2
4
6
x÷2
0
1
2
3
(10)
x
5
6
9
11
y
0
1
4
6
(11)
x
5
8
9
12
x-4
1
4
5
8
(12)
x
0
3
6
9
y
0
1
2
3
Exercise 11
Here are some unfinished tables. Use the rule for each to fill in the missing numbers,
then draw a separate graph for each
(1)
x
3
4
5
7
x-3
(2)
x
0
1
4
5
x×2
(3)
x
0
4
8
16
x÷4
(4)
x
0
2
5
6
x+3
Section 3: Using multiples as a scale for an axis
Notes
The numbers put on the axis of a graph are called the scale of the axis.
When we use one square for each number on an axis we are using a unit scale.
However other scales can be used on an axis, though this changes the appearance of
what is drawn a little.
x
0
1
2
3
4
for example
x×2
0
2
4
6
8
a y axis scale that uses half a
square for each number still
has the points in a line but it
is not as steep
the unit scale shows the
points in a line
If you can, it is best to use a unit scale when drawing a graph, but sometimes it is
impossible - like when the graph won’t fit on the page if you do!
One way to get a graph to fit on the page is to change the scale on an axis, so instead
of using the numbers 0, 1, 2, 3, 4, . . . use numbers from one of your times tables like
0, 2, 4, 6, 8, . . . or 0, 5, 10, 15, 20, . . . or 0, 25, 50, 75, 100, . . .
for example
With a unit scale the graph of these points would be very long, but with another scale
it will fit nicely on the page
x
0
1
2
3
4
5
x×x
0
1
4
9
16
25
Something to think and talk about
• Why do we always start from zero when we make a scale?
Once you have an answer check with two other people to see if you agree. Be prepared to
discuss this with your teacher.
Some important things to remember:
1)
2)
3)
4)
You don’t have to change the scale on both axes if you don’t want to (the
example above has the one times table on the x axis and the three times table
on the y axis).
It is up to you to decide which times table you want to use on an axis (but you
must use a times table and you can’t miss numbers out).
The three times tables are also called the multiples of three, likewise the five
times tables are the multiples of five and so on.
When using a different scale your points often end up part way between
squares, so you must learn how to plot points in places like this before using
such scales.
for example
On a scale 0, 2, 4, 6, 8, . . .
0
2
4
6
1 is half way between 0 and 2, 3 is half way between 2 and 4, 5 is half way between 4 and 6
and so on, even though you don’t show where they are.
On the scale 0, 3, 6, 9, 12, . . . (multiples of three) the gaps must be spilt up into three equal
pieces if you want to show numbers the missing numbers like 1 or 2 or 4 or 5 on it.
0
3
6
9
On the scale 0, 4, 8, 12, 16, . . . (multiples of four) the gaps have to be spilt up into four
equal pieces to show numbers like 1, 2, and 3.
More words to learn
When we split a gap (interval) into equal pieces we often use marks.
For example
0
10
20
Here each interval has been cut into five equal pieces using marks
Challenges
1)
By looking at the number lines above, work out a rule that tells you how many
marks you would need to put on the scales 0, 6, 12, 18, . . . and 0, 10, 20, 30, . . .
if you wanted to have a mark for each number between the ones that are
labelled.
The table below may help you get started
Number of equal pieces to
split the interval into
2
3
4
5
6
Number of marks needed
1 (in the middle)
2)
Draw a collection of horizontal lines on a blank piece of paper. On the first
line, without using a ruler to help, try to cut the line into exactly two equal
pieces.
• Explain what you did
• How good was your attempt? (Explain how you checked)
On the second line, without using a ruler to help, try to cut the line into exactly
three equal pieces.
• Explain what you did
• How good was your attempt? (Explain how you checked)
On the third line, without using a ruler to help, try to cut the line into exactly
four equal pieces.
• Explain what you did
• How good was your attempt? (Explain how you checked)
Also try cutting lines into five, pieces, six pieces and eight pieces. Each time
explain how you went about doing this, and how good your attempts were.
Bring along your sheet and answers to the next teaching session, and be
prepared to discuss, with the group and the class, what you tried and how
accurate your answers were. There are also some additional resources to help
you learn how to accurately divide an interval into equal pieces. You may want
to talk about them with your teacher.
Exercise 12: Finding numbers between the marks
For this exercise, use a piece of blank paper, and draw a new axis for each question
1)
Draw an axis with the scale 0, 5, 10, 15, 20, . . . Do not use a scale of one square for each
whole number. On the axis show where the numbers 2, 8 and 16 can be found
2)
Draw an axis with the scale 0, 8, 16, 24, 32, . . . Do not use a scale of one square for each
whole number. On the axis show where the numbers 2, 10 and 20 can be found
3)
Draw an axis with the scale 0, 10, 20, 30, 40, . . . Do not use a scale of one square for each
whole number. On the axis show where the numbers 4, 7, 13, 29 and 43 can be found
4)
For each question, explain how you worked out where to put the numbers
5)
Make up three questions of your own. Each question should have a different scale, and a set
of numbers to show between the marks. Write answers for each question, and what you
think is the best strategy for locating the numbers.
6)
Get and partner, and swap questions - you try his three and he tries yours
(a)
Compare your answers and strategies. Are the numbers in the same places? If not,
why not, and who is right?
(b)
Did you use the same strategies on each question? If not, decide which is the best
strategy for the question, and explain why it is the best.
(c)
Between the two of you, choose the best question of the six, and be prepared to share
it with the group at the next session. Why is it the best question?
(d)
Have a copy of this best question to hand out to everyone next session. Also have a
model answer, so you can mark their work.
The one that got away
Herewini, Peter and Robin all went out fishing with Robin’s dad. When they came
home they all had a story about “the one that got away”.
The points below can be used to draw a picture of the fish that got away.
(3, 3)(3, 4)(5, 5)(6, 5)(10, 4)(12, 6)(11, 4)(12, 1)(10, 3)(6, 2)(5, 2)(3, 3)
Task
For this task you will need two copies of the material master. Ask your teacher for
these.
1)
Plot the points above on the axes with unit scales. Join each point to the next
with a ruler.
2)
Now draw a picture of the fish on the other 3 sets of axes provided.
3)
All these drawings show the same fish, plotted with the same points. Explain
the effect that stretching the spacing between numbers has on a graph you
draw.
4)
Each boy told a story about “the one that got away”.
Peter, who nearly caught this fish said “it was a really huge fish, it was very
long and very fat – and heavy, that’s why it got away”.
Herewini said no, “it was a short fat fish. That was why it got away”
Robin said they were both wrong – “it was a long and skinny fish”
5)
Match the fish stories to the fish drawings.
You need the second copy of the material master for this problem
(a)
Plot the points (0, 0)(1, 1)(2, 2)(3, 3)(4, 4)(5, 5) on each of the four sets
of axes. For each graph, join up the points with a line
(b)
Explain the impact that changing the spacing on a scale has on a line.
Investigation
1)
You have just learned that changing the spacing between numbers changes
what a graph looks like. For this investigation, you are expected to investigate
what changing the scale on an axis does to the graph. For example, start with a
shape drawn on a graph (it could be the same fish you used in ‘the one that got
away’). From there investigate what would happen to the graph if, for example,
the scale on the x axis was changed to 0, 2, 4, 6, 8, … from 0, 1, 2, 3, … Try
changing the scale on the x axis first, then on the y axis, then on both at the
same time.
2)
3)
When the scale on a graph is changed we can say that the graph is
transformed. Examples of mathematical transformations are reflections,
rotations, stretches, enlargements and translations. Describe the changes to
each graph using the language of transformations.
Now copy the set of axes drawn below and plot the points for the fish used in
‘the one that got a way’.
For the second part of this investigation you are to explore what can happen to
curves when scales are changed like this. The points (1, 1)(4, 2)(9, 3)(16, 4)(25, 5)…
are part of the pattern that shows the square roots of numbers. Plot them on a
graph with unit scales and see if you can work out the curve they are all on. Then plot
them again on a set of axes like the ones below.
4)
Explain what has happened to the fish and the curve when the scale were not
properly drawn, and why a set of multiples must be used when creating a scale
for an axis
Present your findings from the investigation as a poster that can be put on the wall.
Include examples, and use the following sentence somewhere on your poster
All graphs show information as a picture. This picture is changed by …
Drawing graphs with non-unit scales
Notes
Now you have had a little practice with finding numbers on an axis it is time to start
plotting points on a graph with axes like this. The example below should help show
you what to do.
example
Plot the points (0, 3)(2, 11)(3, 15)(6, 21)(8, 27)(11, 36) . . .
First decide which scale to use on each axis
the x axis has the numbers 0, 2, 3, 6, 8, and 11, so a unit scale or multiples of 2 (0, 2,
4, 6, 8, 10, 12) will fit them nicely
the y axis has the numbers 3, 11, 15, 21, 27 and 36 so multiples of 4 (0, 4, 8, 12, 16,
20, . . . ) or multiples of 5 (0, 5, 10, 15, 20, 25, .. . ) will stop the axis getting too
long - the choice is up to you.
Next draw the axes and work out where to plot the points
the point (3, 15) means you
go half way between 2 and 4
on the x axis and three
quarters of the way
between 12 and 16 on the y
axis
for the point (0, 3) you may
want to put marks on the y
axis between 0 and 4 so you
put the 3 in the right place
Exercise 13
Plot the following points by using multiples as scales on the axes. Remember you can
use different scales on each axis and that it is up to you to choose a scale that makes
the graph a sensible size and the points easy to plot.
(1)
x
0
10
20
30
y
0
20
20
60
(2)
x
5
10
15
20
5)
(0, 0)(4, 8)(8, 12)(12, 24)
6)
(10, 8)(10, 6)(10, 4)(10,2)(10, 0) . . .
7)
(6, 40)(12, 20)(18, 35)(36, 40)
8)
(100, 8)(80, 16)(60, 32)(40, 64) . . .
x×3
15
30
45
60
(3)
x
5
10
15
20
y
20
20
20
20
(4)
x
7
14
21
28
x×2
14
28
42
56
(9)
x
0
2
4
6
8
10
y
7
5
21
34
8
6
(10)
x
1
2
3
5
6
y
0
1
13
43
17
13)
(1, 6)(2, 1)(3, 15)(4, 54)(5, 7)
14)
(3, 24)(8, 36)(11, 43)(17, 82)(23, 61)
15)
(2, 22)(5, 25)(8, 38)(11, 27)(14, 18)
(11)
x
1
2
4
6
7
8
y
8
19
5
6
21
28
(12)
x
4
5
10
11
16
y
3
18
7
31
37
A convention to check:
Did you notice the … on some of the problems in exercise 13? …means that the pattern carries on.
To show this on a graph you put an arrow next to the last point (as we did for some patterns earlier).
If you don’t put an arrow there it means that only the points shown are supposed to be plotted.
Breaking an axis
Notes
There is one more way of changing a graph to plot difficult numbers: that is to put a
break in the start of an axis.
for example
x
0
1
2
3
4
y
27
30
33
36
39
plot the points in the table below
Things to note: more conventions
1)
2)
3)
4)
The y values are all multiples of 3, so I used these as the scale on the y axis.
However, as there are no values between 0 and 27 I am allowed to break the
axis by putting a small earthquake at the start of the axis. This shows there
was not enough room to draw in this piece of the axis.
A earthquake is officially called an axis break
You are only allowed to put in an axis break at the start of an axis (so if the
points you are trying to plot look like (1, 1)(2, 5)(3, 24)(9, 106)(11, 267) you are
not allowed to have an axis break. Instead you would have to use a long y axis
with, say, multiples of 10, 12, 15 or 20.
You can put axis breaks at the start of both axes if you want to, but we try to
avoid this as the graph looks messy!
Something to talk about with a friend
Why can we not use an axis break in the middle of an axis – only at the start?
Exercise 14
Plot each of the following questions on a different set of axes. Use an axis break if
this helps you draw the graph easily on the page
(1)
x
0
2
4
6
y
1000
1006
1004
1008
(2)
x
0
5
10
15
x × 2 + 50
50
60
70
80
(3)
x
15
16
17
18
y
44
55
44
30
(4)
x
50
60
70
80
y
3
5
7
8
5)
(6, 48)(4, 62)(10, 50)(8, 56)
6)
(18, 85)(24, 65)(30, 70)(36, 80)
7)
(2, 24)(5, 40)(13, 66)(18, 45)
8)
(30, 32)(41, 53)(43, 56)(50, 63)
9)
What do you think that using an axis break does to the picture shown by the graph? Give an
example as part of your answer.
Review
This exercise takes you through a lot of the things you have learned in this unit.
1)
The point where the x and y axes meet is called the ________
2)
An ordered pair can be used as the __________ of a point
3)
Plot these points on a set of axes
(3, 0)(1, 7)(5, 4)(2, 2)(0, 6)
4)
Write down the coordinates of these points
5)
Draw a set of axes that each go up to 13. Plot the following points. Join each point to the
next by using a ruler.
(1, 2)(1, 4)(3, 6)(6, 8)(7, 10)(8, 11)(8, 12)(9, 13)(9, 12)(10, 11)(11, 12)(12, 11)(12, 10)
(10, 7)(10, 6)(9, 6)(8, 5)(7, 3)(5, 1)(3, 1)(1,2)
6)
What shape have you drawn?
7)
Here are some sets of ordered pairs to plot. Use a new set of axes for each question
(a)
(0, 3)(1, 4)(2, 5)(3, 6) . . .
(b)
(1, 4)(2, 4)(3, 4)(4, 4) . . .
(c)
(7, 0)(7, 1)(7, 2)(7, 3) . . .
(d)
(0, 2)(1, 4)(2, 6)(3, 8) . . .
Work out the next two ordered pairs for each sequence.
8)
Use the general terms below to write the first five points of each sequence. Once you have
done this plot the points for each question on its own set of axes.
(a)
(x, x + 3)
(b)
(x, x × 3)
(c)
(x, x × 2 - 2)
(d)
(0, x)
(e)
(x, 6)
9)
Copy the axis below, then show where the numbers 2, 7, 10, 15 and 19 should be plotted.
0
6
12
18
10)
Draw an axis with the scale 0, 12, 24, 36, . . . and on it show where the numbers 2, 8, 11, 16
and 20 are
Explain how you worked out where each of the numbers should go
11)
Use these charts to make graphs. (If there are missing numbers work out what they should
be). Make a new graph for each chart.
(a)
x-5
5
3
2
0
x
10
8
7
5
(b)
x
0
1
3
4
x×4
0
4
12
16
(c)
x
1
2
3
4
x×4-1
3
7
11
15
(d)
x×2+1
1
5
7
13
x
0
2
3
6
12)
Make a new graph for each set of points
(a)
(0, 30)(1, 45)(2, 60)(3, 75)(4, 90) . . .
(b)
(1, 1)(2, 20)(3, 51)(4, 82)(7, 16)
(c)
(4, 17)(7, 14)(16, 5)(21, 13)(22, 15)
13)
Use these tables to make graphs. (Don’t forget to use an axis break if it will make the job
easier.) Make a new graph for each question.
(a)
x
0
1
2
3
4
5
y
0
4
15
65
10
1
(b)
x
0
3
6
9
12
y
20
21
22
23
24
(c)
x
120
130
140
150
160
y
7
15
6
9
13
(d)
x
1
2
3
4
5
y
1
8
27
64
125
14)
Explain what the coordinates of a point are
15)
Explain what happens when we use multiples of ten as the scale on the y axis of a graph,
instead of a unit scale
16)
Write out meanings for these words
(a)
interval
(b)
mark
(c)
variable
(d)
horizontal
List any other new words you met in this unit, and explain what each means
17)
Why must we use a set of multiples as the scale of an axis, and why can’t we put an axis
break in the middle of an axis?
18)
Summer was away from school and missed a lesson about how to divide an interval up into
four equal pieces. Explain to her how to do this, showing the examples you would use.
Thinking time
For an assignment, Marcus has to compare two graphs, in particular the slopes of the lines. The
ones he is given are from the unit review, 12 (a) and 13(b). Have a look at these graphs in the unit
answers, then write what you can to compare these graphs.
The one that got away
Material master