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An introduction to graphing for number and algebra
Answers
Exercise 1
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)
Exercise 2
Remember the convention that the numbers go on the
marks
Another convention:
You do not need to write the coordinates next to the point
14)
A convention is ‘the way mathematicians have agreed to do something’. Conventions are
important as they allow everyone to understand what other mathematicians have done
Exercise 3
1)
2)
3)
A is (1, 1)
E is (0, 8)
J is (14, 14)
P is (8, 17)
B is (2, 4)
F is (1, 6)
K is (6, 11)
Q Is (13, 8)
C is (4, 2)
G is (2, 4)
L is (7, 6)
R is ((3, 9)
D is (6, 0)
H is (3, 2)
M is (0, 5)
S is (14, 1)
I is (4, 0)
N is (0, 0)
T is (5, 1)
O is (10, 4)
Exercise 4
Notice how the x axis only has the even numbers
labelled. Is it still obvious where the odd numbers are?
3)
A castle
Exercise 5
3)
A kiwi
Exercise 6
Answers will vary
Exercise 7
Exercise 8
Question to explain
There is a pattern in what each ordered pair starts with, next two ordered pairs start with the next
numbers in the pattern
1)
Whatever the first number is, add two to get the second number
(8, 10)(10, 12)
2)
Whatever the first number is, the second number is the same
(6, 6)(5, 5)
3)
Whatever the first number is, multiply it by two to get the second number (4, 8)(5, 10)
4)
Whatever the first number is, subtract two to get the second number
(8, 6)(7, 5)
5)
Whatever the first number is, halve it to get the second number
(4, 2)(2, 1)
6)
Whatever the first number is, add one to get the second number
(8, 9)(10, 11)
7)
Whatever the first number is, multiply it by three (treble it) to get the second number
(4, 12)(5, 15)
8)
Whatever the first number is, double it and add one to get the second number
(4, 9)(5,
11)
9)
Whatever the first number is, subtract one to get the second number
(5, 4)(6, 5)
10)
Whatever the first number is, double it and subtract one to get the second number (5, 9)(6,
11)
11)
Whatever the first number is, subtract 5 to get the second number
(9, 4)(10, 5)
12)
Whatever the first number is, the second number is the same
(8, 8)(9, 9)
13)
Whatever the first number is, multiply it by itself (square it) to find the second number
(4, 16)(5, 25)
14)
Whatever the first number is, the second number is one
(5, 1)(6, 1)
15)
Whatever the second number is, the first number is always one
(1, 4)(1, 5)
16)
Whatever the first number is, the second number, it and the second number add to 10
(6, 4)(5, 5)
Challenge
1)
4)
7)
10)
13)
15)
16)
n+2
(2)
n
(3)
n2
n–2
(5)
n2
(6)
n+1
n3
(8)
n2+1
(9)
n–1
n2–1
(11) n – 5
(12)
n
nn
(14) 1
This one doesn’t really work like this, so there are better ways of describing such patterns
(and number 14). For example, saying y = 1 for number 14 says that the y value to be
graphed is always 1, likewise x = 1 for number 15 says the x number will always be 1
10 - n
Exercise 9
Exercise 10
Exercise 11
(1)
x
3
4
5
7
x-3
0
1
2
4
(2)
x
0
1
4
5
x2
0
2
8
10
(3)
x
0
4
8
16
x4
0
1
2
4
(4)
x
0
2
5
6
x+3
3
5
8
9
Challenges
1)
2)
You always need one less mark than the number of pieces you want in the interval
Your answers to this challenge will be discussed and marked with the group and the teacher
Exercise 12
4)
The easiest way to do question 1 is to divide the interval up into five equal pieces, then
count along the marks to find the number you want
The easiest way to do number 2 is to halve and halve again,
The easiest way to do number 3 is to halve the interval, then divide the leftover bit into five
equal pieces
Own questions and answers
Swapping work with another student. Make sure you have chosen the ‘best’ question for
use at the next session.
5)
6)
The one that got away
4)



5)
Whose story belongs to which fish picture?
Peter’s story belongs the graph which has two squared between each unit – on both axes
Herewini’s story needs a unit scale on the x axis, and a scale with 2 squares between each
unit on the y axis
Robin’s story needs the x axis to have 2 squares between each unit, and a unit scale on the y
axis
Changing the spacing on just the y axis makes the line look steeper
Changing the spacing of just the x axis makes the line look shallower
Changing the spacing of both axes by the same amount makes the line look the same as the
unit scale
Investigation
You should have produced a poster that shows and explains your findings for this investigation.
Your teacher may give you a marking sheet so you can work in pairs to decide how to mark each of
the posters.
Exercise 13
Note that you may have used different scales when drawing your graphs
Exercise 14
9)
The axis break at the start of an axis chops out part of the graph where nothing was
happening anyway. This allows us to use more space on the bit where the numbers are
placed. If the axis break was in the middle of an axis, there would be numbers on both sides
of the break, and that would change the ‘picture’ the graph shows.
Review
1)
2)
3)
The point where the x and y axes meet is called the origin
An ordered pair can be used as the coordinates of a point
4)
5)
Coordinates are (0, 0)(1, 1)(4, 2)(9, 3)(16, 4)
6) The South Island
7)
(a)
(b)
(c)
(d)
Next two points are (4, 7)(5, 8)
Next two points are (5, 4)(6, 4)
Next two points are (7, 4)(7, 5)
Next two points are (4, 10)(5, 12)
8)
(a)
(b)
(c)
(d)
(e)
First five terms for (x, x + 3) are (1, 4)(2, 5)(3, 6)(4, 7)(5, 8)
First five terms for (x, x  3) are (1, 3)(2, 6)(3, 9)(4, 12)(5, 15)
First five terms for (x, x  2 - 2) are (1, 0)(2, 2)(3, 4)(4, 6)(5,8)
First five terms for (0, x) are (0, 1)(0, 2)(0, 3)(0,4)(0,5)
First five terms for (x, 6) are (1, 6)(2, 6)(3, 6)(4, 6)(5, 6)
There are number of strategies that can be used to find where each number goes on such a
scale.
To locate 2, find halfway between 0 and 6, (3), then divide the interval between 0 and 3 into
3 equal pieces. This strategy can be used for working out where to put the other numbers on
the number line.
11)
(a)
x
10
8
7
5
x-5
5
3
2
0
(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
0
2
3
6
x2+1
1
5
7
13
12)
13)
14)
The coordinates of a point are the numbers used to identify where a point should go on a
graph
15)
On a multiples of ten scale, the missing numbers (1 to 9, 11 to 19, 21 to 29, …) can all be
found between the numbers shown (0, 10, 20, 30, …). This can mean when we plot points,
they may not go on the gridlines. The other effect of using a scale involving multiples of ten
is that the slope of lines and curves change. If this scale is on the x axis, they become
steeper, if the y axis, shallower, if on both axes they stay the same.
16)
Interval – this is the gap between the marks on a number line, or graph
Mark – a small line used to show where a number is
Variable – a letter used to stand for different numbers. For example, in exercise 9, for
(x, x + 2) the x can be any of the numbers 1, 2, 3, 4, 5, 6, 7, …
Horizontal – across the page
You may have other words listed here. For example, Vertical – up the page
17)
Not using a set of multiples for the scales of both axes can change the shape drawn on the
graph. As graphs show information in the form of a picture, by not using multiples the
picture is distorted or even changed.
The same sort of thing can happen if an axis break is put in the middle of an axis, rather than
at the start.
18)
0
4
8
12
To work out where the numbers 1, 2 and 3 go on this number line, the interval between 0
and 4 must be divided into 4 equal pieces. One easy way the do this is to find halfway (2),
then find halfway again
0
4
8
12
0
4
8
12
Thinking time
This question is a really good question to test if you have understood what happens to
the picture drawn when graph scales are changed. Make sure you understand the
answer below if you find that your answer is wrong
In the answers for the review, both graphs look the same, and the line looks to have the same slope
in both. However, this is an illusion – as the graphs both have different scales on each axis. To
compare any graphs they both need to be drawn on the same scales.
For this problem, the scales used for question 12(a) would do, so would the scales for 13(b), or
another set of scales that make it easier to plot both sets of data on the same graph, or even unit
scales if you had a large enough piece of paper!
If you got trapped by this question – redraw both graphs with the same scales and you will see that
the line in 12(a) is much, much