Notes to help Prepare for the Numeracy Test
(for intending school teachers)
About the test
The Numeracy component of the test will consist of 65 questions to be completed in 2 hours. The test
will consist of multiple response and short answer questions. An online calculator will be available to be
used for 80% of the questions, the remaining 20% will require you to answer without using a calculator.
In the numeracy test, there will be questions on three content areas:
• number and algebra
• statistics and probability
• measurement and geometry.
The personal numeracy test has a focus on the active aspect of numeracy, giving most weight to
questions that require candidates to use their numeracy skills: that is, to apply mathematics to solve
appropriate real-world problems.
Three numeracy processes are used to categorise numeracy items:
• identifying mathematical information and meaning in activities and texts
• using and applying mathematical knowledge and problem solving processes
• interpreting, evaluating, communicating and representing mathematics.
Number and Algebra
Multiplying and Dividing by Powers of 10
• There are a lot of man-made conventions in mathematics. Some of them are to enable us to condense
information, one example is the use of exponential form. Look for a pattern then complete the table
below.
10 = 10 = 101
30 = 3 × 10 = 3 × 101
100 = 10 × 10 = 102
3
1000 = 10 × 10 × 10 = 10
4
= 10 × 10 × 10 × 10 = 10
5
100000 =
= 10
1000000 = 10
200 = 2 × 100 = 2 × 102
6000 = 6 × 1000 = 6 × 103
70000 =
× 10000 = 7 × 10
= 9 × 100000 = 9 ×
8000000 =
= 8 × 106
Large numbers are often expressed in exponential form, especially when using a scientific calculator.
Some earlier models of calculators will show the number 50000000 as 5 07 but later models will
show the correct notation of 5 × 107 .
1
• When we divide by ten it is the same as multiplying by 10
th. To find one thousandth of a thing is
the same as dividing by one thousand. Fill in the table below using this information:
2460 ÷ 10 = 246
2460 ÷ 100 = 24.6 = 246 ÷ 10
2460 ÷ 1000 = 2.46 = 24.6 ÷ 10
Proper Fractions
Whole Numbers
1000’s
100’s
10’s
Ones
2
4
6
0
1/10’s
1/100’s
1/1000’s
÷10 or
×1/10
÷100 or
×1/100
÷1000 or
×1/1000
Can you see a quick way to divide by 10’s, 100’s and 1000’s?
• In the same way as we could expand the whole numbers using powers of ten we can also expand the
fractional numbers using negative powers of ten. Exponential notation can also be used for decimals.
So that if we can write 4000 as 4 × 103 we can write 0.002 as 2 × 10−3 . Our short hand notation
allows us to write the following:
= 10−1
10 = 101
1
10
100 = 102
1
100
1000 = 103
1
1000
= 10−2
= 10−3
Complete the following table. Think carefully about what the value of 100 might be.
102
1000
100
10
10−1
10−3
1
100
10−4
1
10000
3 × 4 = 12
3 × 0.4 = 1.2
(Since 3 ×
4
10
3 × 40 = 120
3 × 0.04 = 0.12
(Since 3 ×
4
100
3 × 400 = 1200
3 × 0.004 = 0.012
(Since 3 ×
4
1000
3 × 4000 = 12000
3 × 0.0004 = 0.0012
(Since 3 ×
4
10000
= 12 ×
1
10
= 12 ×
= 12 ÷ 10)
1
100
= 12 ×
= 12 ÷ 100)
1
1000
= 12 ×
= 12 ÷ 1000)
1
10000
= 12 ÷ 10000)
Examples
$3.15 ×10 =
$31.50
$31.50 ×10 =
$315.00
$315.00 ×10 =
$3150.00
2460
÷10 =
246
246
×10 =
24.6
24.6
×10 =
2.46
Suppose I want to calculate 200 × 400. Then:
200 × 400 = 2 × 100 × 4 × 100 = 2 × 4 × 100 × 100 = 8 × 100 × 100 = 800 × 100 = 80000
Order of Operations
Consider the following situation. You have worked from 2 p.m. to 9 p.m. on a major proposal. As you
were required to complete it by the following day you can claim overtime. The rates from 9 a.m. to 5
p.m. are $25 per hour and from 5 p.m. to midnight rise to $37.50 per hour.
Mathematically, this can be expressed as 3 × $25 + 4 × $37.50. How much do you think you earned?
Which attempt below is most reasonable?
Attempt 1
Attempt 2
3 × 25 + 4 × 37.50
= 75 + 4 × 37.50
= 79 × 37.50
= 2962.50
3 × 25 + 4 × 37.50
= (3 × 25) + (4 × 37.50)
= 75 + 150
= 225
Mathematical expressions can be read by anyone regardless of their spoken language. In order to avoid
confusion certain conventions (or accepted methods) must be followed. One of these is the order in which
we carry out arithmetic.
Brackets:
Evaluate the expression inside the brackets first eg. (3 + 5) = 8
Indices:
Evaluate the expressions raised to a power eg. (3 + 5)2 = 82 = 64
Division:
Divide or multiply in order from left to right.
Multiplication:
eg. 6 × 5 ÷ 3 = 30 ÷ 3 = 10
Addition:
Add or subtract in order from left to right.
Subtraction:
eg. 4 + 5 − 2 + 4 = 9 − 2 + 4 = 7 + 4 = 11
This can easily be remembered by the acronym BIDMAS.
Modern calculators have been programmed to observe the order of operations and also come equipped
with a bracket function. You just enter the brackets in the appropriate spots as you enter the calculation.
Estimating
When we go shopping we usually have some idea of how much to put in the trolley so that we do not
embarrass ourselves at the checkout by not having enough money. What we are in fact doing, as we drive
the trolley up and down the aisles, is estimating. As very few people take a calculator into the store,
mental arithmetic plays a crucial role in estimating the cost.
• For each of the monetary amounts below round to the nearest dollar. That is, if the digit after the
decimal point is ≥ 5 then increase the dollar amount by 1 and replace all the digits after the decimal
point with zeros. If the digit immediately after the decimal point is < 5, leave the dollar amount
the same and replace all the digits after the decimal point with zeros. For example;
$2.05 is close to $2.00, $6.85 is close to $7.00 and $1.35 is close to $1.00.
So,
$2.05 + $6.85 + $1.35 ≈ $2.00 + $7.00 + $1.00 = $10.00
Other Examples
Usually when we make an educated guess or estimate, we round the numbers involved off to the nearest
leading digit. It is just a matter of multiplying the non-zero digits together and multiplying the powers
of ten together.
Suppose we need to estimate 2.39 × 12.98.
Then we think of 2.39 as approximately 3.
12.98 is approximately 13. So
2.39 × 12.98
is approximately
2 × 13 = 26
Thus we are expecting 2.39 × 12.98 to be near 26. In fact we expect it to be a bit bigger than 26. Why?
Well 2.39 is quite a bit bigger than 2, even though it is still less than 3.
Strategies for Mental Arithmetic
The ideas of multiplying and dividing by powers of 10 and using estimation are both strategies that we
can employ to do mental arithmetic.
Examples
Our number system increases by powers of 10 as we move to the left and decreases by powers of 10 as
we move to the right. Given the number 123 456.78, the 1 tells us how many 100,000’s there are, the 4
indicates the number of 100’s and the 7 indicates the number of tenths. When we multiply two numbers
together such as 200 × 3000 we can use our knowledge of the place value system to represent this as
(2 × 100) × (3 × 1000). Since the order is not important in multiplication (i.e. 2 × 3 × 4 = 4 × 3 × 2 ) we
can rewrite this as:
200 × 3000
=
(2 × 100) × (3 × 1000)
=
(2 × 3) × (100 × 1000)
=
6 × 100000
=
600000
Similarly we can calculate:
4000 × 0.05
1
)
100
1
= (4 × 5) × 1000 ×
100
= (20 × 10)
=
(4 × 1000) × (5 ×
=
200
Suppose we want to multiply 345 by 89, then
Step 1: Round 345 to the nearest hundred:
Step 2: Round 89 to the nearest ten:
Step 3: Multiply the leading digits together:
Step 4: Multiply the powers of ten together:
Step 5: Complete the multipliaction:
Finally we have:
300
90
3 × 9 = 27
100 × 10 = 1000
27 × 1000 = 27000
345 × 89 ≈ 300 × 90 = 27000
Percentages
Percentages are fractions with a denominator of 100.
In the diagram below, what fraction of the total number of small squares have black blobs on them?
i.e. what percentage of the squares have black blobs in them?
yyyy
y
yyyy y
y
y
y y y y
y
y
y
y
y
y
Often there will not be 100 squares or 100 people out of which to express a fraction or a percentage.
When this is the case you will need to find an equivalent fraction out of 100 by multiplying by 100% or
by multiplying by 1.
In the next figures, what percentage of the squares have circular shapes drawn in them?
~
~ ~
~
~ ~
~ ~
~
~ ~ ~
~ ~
Percentage circles =
~
Percentage circles =
Examples
• Say for example that we know that 74% of students enrolled in a biology major at Macquarie
University are female. Then to calculate the percentage of males we note that altogether the males
and females make up 100% so
The percentage of biology majors that are male
= 100% − 74% = 26%
• Suppose 31% of students in Year 9 at a certain high school achieved at Band 8 or above, and 33%
achieved at Band 7. What percentage achieved at Band 6 level or below? In order to calculate this
we do the following
31% + 33% = 64%
So we want to know what percentage remains:
100% − 64% = 36%
So 36% of students performed at Band 6 or below.
Another kind of percentage question that we can ask is, ”What percentage of the whole is the part we
are interested in?”. We have the relationship
percent
part
=
100
whole
or alternatively we can write
percentage =
part
× 100
whole
Example
At a certain primary school there are 425 enrolments. 209 girls are enrolled at the school. What percentage
of the enrolments are girls?
209
× 100 = 49.18 (rounded to two decimal places)
425
So approximately 49% of the enrolments are girls. can we now say what percentage of the enrolments are
boys? Yes we can.
100% − 49% = 51%
So approximately 51% of the enrolments are boys.
The final kind of percentage example that we will look at is finding a percentage of a quantity. We
rely on the fact that a percentage is a fraction with a denominator of 100 to do these calculations.
Examples
• Firstly a straightforward example. Imagine that you see a dress that you really like. It costs $200
but the shop has a sale on and everything is 60% off. What is the discount on the dress, i.e. how
much money can you save if you purchase the dress while the sale is on?
What we need to find out here is what is 60% of $200. When calculating this sum we replace the
word ’of’ with the symbol times. (There are examples like this in fraction work as well. If we wanted
to find one quarter of 8, the sum we would do would be 14 × 8 = 2.)
Now for the calculation:
60% of $200 = 60% × $200 =
60
× $200 = $120
100
• The numeracy test consists of 65 questions. We have been told that for 80% of the questions students
will have access to a calculator. How many of the 65 questions will you be able to use a calculator
for?
In other words what is 80% of 65?
80% of 65 = 80% × 65 =
80
× 65 = 52
100
So you will access to a calculator for 52 of the questions. The remaining 13 questions will require
you to use mental arithmetic. (I will do some examples like this in a moment.)
• This one is a bit tougher. Jane is doing a course that has three assessment tasks including a project
which is worth 70% of the total mark. Jane gets 80% for her project. How many marks does this
contribute to her final grade in the unit?
In other words we want to find 70% of 80%.
80% of 70% = 80% × 70% =
80
× 70% = 56%
100
Percentage calculations without a calculator
In the sample test one of the non calculator questions is a percentage calculation. I would like to remind
you of some strategies you can use to do percentage calculations without a calculator. Becasue finding a
percentage of something involves multiplying by a fraction out of 100 we can use ideas a bout dividing
and multiplying by 100 to simplify calcultions. Let me demonstrate this using some examples.
Examples
• Suppose you have a classroom budget of $300 to spend on classroom resources. You have been told
that you need to keep 30% of it for use in term 4. The rest you can spend on things you need at the
beginning of the year. How much do you need to set aside for Term 4?
In other words we want to calculate 30% of $300. How would we do this without a calculator?
1
10
= 10
, to find 10% I divide by 10. (you can
I would firstly work out what 10% was. Since 10% = 100
get the same result by multiplying by 10 and then dividing by 100 if that makes more sense.) recall
that to divide by ten you shift the columns of numbers in the number to the left by one column,
so the hundreds column becomes the tens column, the tens column becomes the ones column, etc
(alternatively you can think of this as shifting the decimal place to the left by one column). thus
300 ÷ 10 = 30.0 = 30
Since 10% of $300 is $30,
30% × $300 = 3 × 10% × $300 = 3 × $30 = $90
• We can use a similar method to find 20% of 150. 10% of 150 is 15. So 20% is 2 × 15 = 30.
• In the Mt Druitt local government area 4% of the population is from an indigenous australian
background. If the Mt Druitt local government area has a population of 300,000, how many of the
population in Mt Druitt are from an indigenous australian background?
So we want 4% of 300,000. In this case we first work out 1% and then multiply that answer by 4 to
get 4%. So
1
1% × 300, 000 =
× 300, 000 = 300, 000 ÷ 100 = 3000.00 = 3000
100
So 1% of 300,000 is 3000. Therefore 4% will be
4 × 3000 = 4 × 3 × 1000 = 12 × 1000 = 12, 000
So 4% of 300,000 is 12,000.
Fractions
To be added
Statistics and Probability
Reading and Interpreting Graphical Information
Firstly, why do we present information in graphical ways? Because once you know how to read them they
convey a lot of information very quickly. Its the old saying, a picture paints a thousand words. Graphs
are widely used to convey information, in newspapers, by government agencies, in reports and of course
in schools. There are many different kinds of graphs but the two that I want to look at are bar charts and
line graphs. These are probably the most common ones, along with pie charts, which while easy to draw
using excel can be quite difficult to interpret. (Abar chart has been used
Four steps in reading information from a graph
1. Identify what the graph represents. Most graphs have a clearly labelled horizontal and vertical axis,
which gives you information about what the graph represents. Most graphs will also have a title
which will give you information as well.
2. Check the scale for each of the graph axes. For example if you are looking at a graph about the
change in average house prices over time, each increment in the vertical axis may represent thousands
of dollars or even hundreds of thousands of dollars. You won’t know which apply until you check
the graph.
3. Locate the part of the graph that you want information on. For example you might want to know
what the average house price in Sydney was in 1996. You need to locate the part of the graph that
refers to 1996, and possibly even, depending on the graph, the part that refers to Sydney. Time is
often found on the horizontal axis and quantity measurements on the vertical axis.
4. Read up from the horizontal axis until you find the line or the top of the bar or a dot and across
to the vertical axis where you can read the labelled amount, or, across from the labelled amount on
the vertical axis until and down to the horizontal. This is difficult to explain without a picture so I
will run you through the process using and example.
Examples
Pupil/Teacher Ratio in Primary Schools, 1997
16
14
12
Percentage
10
8
6
4
2
0
less than
16
16
17
18
19
20
21
22
23
24
25
26
27
over 27
Pupil/Teacher ratio
• This type of graph is called a bar chart
• The title of this graph tells us that it is about Pupil/Teacher ratios in Primary Schools in January
1997.
• The vertical scale tells us the percentage of classes that had a particular Pupil/Teacher ratio and
notice that it is going up in 2s.
• The horizontal axis shows the different ratios, starting from less than 16 and going u to over 27.
• For example if we wanted to find out what percentage of classrooms has a ratio of 22 pupils to one
teacher, we locate the 22 on the horizontal axis and then move across from the top of this bar to
find that about 10% of class rooms had this ratio.
Changes to Mode of Transport 50 45 Percentage of Commuters 40 35 30 1960 25 1980 20 2000 15 10 5 0 train car bus tram Modes of Transport • This type of graph is called a clustered car chart and it can be useful to compare information across
different categories.
• The graph is telling us something about changes to modes of transport.
• The horizontal axis is marked with different modes of transport.
• The vertical axis is again in percentages (but this time percentages of commuters).
• The different coloured bars are telling us about different years and we can see this from the key on
the right hand side of the graph. You might not be able to see the colours if you print this out but on
the screen you can see that the blue represents information from 1960, pink represents information
from 1980, and the green from 2000.
• So if we wanted to know what percentage of commuters used trains in 2000 we would need to locate
trains on the horizontal axis and then the 2000 column which is the right most one and then we
read up the graph and then from the top of the bar go across to the vertical axis and get 21% of
commuters used trains in 2000.
• We could also ask about combined information, so for example, what percentage of commuters used
buses and trams in 1980? looking for buses and trams and locating the middle column, we see that
22% used buses and 28% used trams so in total 22% + 28% = 50% used buses and trams.
• We could also ask which mode of transport was most popular in 1960. Here we locate the tallest
bar for 1960 and then moving down to the horizontal axis, we see that the most popular mode of
transport in 1960 was trams.
• We could also ask, in which year did the largest percentage of commuters use cars, this time we
locate the columns referring to cars and then we choose the largest column, which is the right most
one. This corresponds to the year 2000. So the year that cars were used by most commuters was
2000.
CPU Load by Days Percentage 100% CPU Load week 15 80% 60% 40% 20% 0% Monday Tuesday Wednesday Thursday Friday Saturday Sunday Days CPU Load by Days Percentage 100% CPU Load week 16 80% 60% 40% 20% 0% Monday Tuesday Wednesday Thursday Friday Saturday Sunday Days • These graphs are called line graphs.
• We have two here that we are comparing. Notice that the graph titles are the same, CPU load
by days, so the graphs are telling us about the percentage load on the CPU over different days.
In the top right hand corner of each graph is some extra information. the first graph is giving us
information about week 15 and the second about week 16.
• The horizontal axis is days of the week.
• The vertical axis is again about percentages, this time going up by 20% for each horizontal line on
the graph.
• These two graphs allow us to talk about differences between week 15 and week 16.
• We can see for example that the CPU load was lower in week 16 on the weekend days. We see this
by looking at Saturday and Sunday in each graph and noting that on Saturday in week 15 the CPU
load was about 40% and then for Sunday in week 15, the load was about 20%. In week 16 the load
on both Saturday and Sunday was less than 20%.
• Suppose we wanted to know what percentage of the CPU was NOT being used on Monday in week
16. We go to Monday on the horizontal axis on the graph for week 16. We go up from Monday until
we reach the line and then across to the vertical axis to see that on Monday in week 16, the CPU
load was about 35%. So what percentage was NOT being used? 100% − 35% = 65% was not being
used.
• We could ask on what day across week 15 and week 16 was the load the highest. Looking at both
graphs we see that the highest point on the line for week 15 and week 16 is Friday. then we compare
the percentages for those two days. On Friday of week 15 we go up the graph til we hit the line then
look across to the vertical axis to see that the load was about 90%. On friday of week 16 we do the
same thing and see that the load was a bit more than 60%. So the load was the highest on Friday
of week 15.
Summarising Information
to be added
Measurement and Geometry
to be added
Additional Resources and Websites
Reading Graphs
Online practice tests and summary lessons
• Reading graphs practice test
• Online lesson on reading graphs
• Practice questions on interpreting graphs
• Quiz on interpreting graphs
• Graph quiz and under fact sheets some explanations
Powers of ten and percentages
• A lesson on the basics of percentage calculations plus there are some exercises for practice
• In general the Khan academy has many videos and practice sheets. Below are some links to relevant
topics at the Khan academy
Some more on percentages
Some Notes from the Numeracy Centre
• Quick overview of basic maths
• Maths Assertiveness Course notes
• Notes on percentages
• Notes on fractions and decimals
Other online resources
• And finally as an extra resource here is the link to the Numeracy Centre’s bridging course textbooks.
The first book of the text for our Bridging course - this contains more detailed explanations and lots
of worked examples.