Mathematics
Stage 4
CENTRE FOR LEARNING
INNOVATION
Gill
Sans Bold
PAS4.1 Algebraic techniques
M7/8 43487
Number: 43487/2007-088-015
Title: Stage 4 Maths – PAS4.1 Algebraic techniques
This publication is copyright New South Wales Department of Education and Training (DET), however it may contain
material from other sources which is not owned by DET. We would like to acknowledge the following people and
organisations whose material has been used:
Extracts from outcomes of the Maths Years 7-10 syllabus
www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_710_syllabus.pd
f
(accessed 04 November 2003).
© Board of Studies NSW, 2003.
pp iii-iv
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Unit contents
Unit overview ....................................................................................... iii
Outcomes ................................................................................. iii
Indicative time........................................................................... iv
Resources..................................................................................v
Icons ......................................................................................... vi
Glossary................................................................................... vii
Part 1 Introducing pronumerals ........................................... 1–47
Part 2 Writing algebraic expressions ................................. 1–52
Unit evaluation ...................................................................................53
Unit overview
i
ii
PAS4.1 Algebraic techniques
Unit overview
In this module you will be given the opportunity to develop your skills in
the use of algebra and the recognition of patterns in mathematics.
Algebra involves the use of letters to represent numbers.
You are probably already well aware of some famous algebraic equations
such as E= mc2.
In this module you will have an opportunity to learn how to develop
algebraic equations. Part of the skill you will be working towards
developing will involve the translation between words and algebraic
symbols and between algebraic symbols and words.
By the end of this module you will have been given the opportunity to
develop skills in how to recognise and use simple equivalent algebraic
expressions. The use of such expressions can save you time when solving
real life problems.
While doing this work constantly ask yourself how you can use the
mathematical learning you are doing to help you now or in the future.
Outcomes
By completing the activities and exercises in this unit, you are working
towards achieving the following outcomes.
You have the opportunity to learn about:
•
using letters to represent numbers and developing the notion that a
letter is used to represent a variable
Unit overview
•
using concrete materials such as cups and counters to model:
•
expressions that involve a variable and a variable plus a constant
eg a, a + 1
•
expressions that involve a variable multiplied by a constant eg 2a, 3a
•
sums and products eg 2a + 1, 2(a + 1)
iii
•
equivalent expressions such as
x + x + y + y + y = 2x + 2y + y = 2(x + y) + y
and to assist with simplifying expressions, such as
(a + 2) + (2a + 3) = (a + 2a) + (2 + 3)
= 3a + 5
•
recognising and using equivalent algebraic expressions
eg y + y + y + y = 4y
w × w = w2
a × b = ab
a ÷b=
•
a
b
translating between words and algebraic symbols and between
algebraic symbols and words.
You have the opportunity to learn to:
•
generate a variety of equivalent expressions that represent a
particular situation or problem (Applying Strategies)
•
describe relationships between the algebraic symbol system and
number properties (Reflecting, Communicating)
•
ink algebra with generalised arithmetic eg for the commutative
property, determine that a + b = b + a (Reflecting)
•
determine equivalence of algebraic expressions by substituting a
given number for the letter (Applying Strategies, Reasoning).
Source:
Extracts from outcomes of the Maths Years 7–10 syllabus
<www.boardofstudies.nsw.edu.au/writing_briefs/mathematics/mathematics_
710_syllabus.pdf > (accessed 04 November 2003).
© Board of Studies NSW, 2002.
Indicative time
This unit has been written to take approximately 8 hours.
Each part should take approximately 4 hours.
Your teacher may suggest a different way to organise your time as you
move through the unit.
iv
PAS4.1 Algebraic techniques
Resources
Resources used in this unit are:
For part 1 you will need:
•
eight cups
•
twelve buttons or counters or peas
•
access to the Internet.
For part 2 you will need:
•
Unit overview
access to the Internet.
v
Icons
Here is an explanation of the icons used in this unit.
Write a response or responses as part of an activity. An
answer is provided so that you can check your progress.
Compare your response for an activity with the one in the
suggested answers section.
Complete an exercise in the exercises section that will be
returned to your teacher.
Think about a question or problem then work through the
answer or solution provided.
Discuss with others.
Access the Internet to complete a task or to look at
suggested websites. If you do not have access to the
Internet, contact your teacher for advice.
Perform a practical task or investigation.
vi
PAS4.1 Algebraic techniques
Glossary
The following words, listed here with their meanings, are found in the
learning material in this unit. They appear in bold the first time they
occur in the learning material. For these words and their meanings
including pronunciation see the online glossary on the LMP website at
http://www.lmpc.edu.au and follow the links to Stage 4 mathematics.
Unit overview
algebraic
equation
A mathematical statement that contains an equal sign
and at least one pronumeral.
algebraic
expression
An expression using numbers, pronumerals and
operations without an = sign.
consecutive
numbers
Numbers that follow in order without jumping for
example 101, 102, 103.
equivalent
Having or being of the same measure or amount.
even
A whole number which when divided by two leaves no
remainder.
odd
A whole number which when divided by two leaves a
remainder of one.
positive
integer
A whole number greater than zero.
pronumeral
A letter used to represent numbers in an algebraic
expression or equation.
vii
Mathematics Stage 4
PAS4.1 Algebraic techniques
Part 1
Introducing pronumerals
Contents – Part 1
Introduction – Part 1 ..........................................................3
Indicators ...................................................................................3
Preliminary quiz – Part 1 ...................................................5
What is algebra? ...............................................................7
The language of algebra .................................................11
Some rules...............................................................................12
Writing algebraic expressions .........................................15
Equivalent expressions ...................................................19
Groups in algebra............................................................23
Suggested answers – Part 1 ...........................................27
Exercises – Part 1 ...........................................................29
Review quiz – Part 1 .......................................................41
Answers to exercises – Part 1.........................................45
Part 1
Introducing pronumerals
1
2
PAS4.1 Algebraic techniques
Introduction – Part 1
This part describes how letters can be used to represent numbers, and
how to write and interpret algebraic expressions. You will explore the
meaning of some simple algebraic sentences through the use of
everyday objects.
Indicators
By the end of Part 1, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
•
using letters to represent numbers;
•
writing algebraic expressions to describe real life and practical
situations; and
•
recognising and use simple equivalent algebraic expressions
•
understanding the meaning of terms such as pronumeral, algebraic
equation, algebraic expression and equivalent expressions.
By the end of Part 1, you will have been given the opportunity to work
mathematically by:
•
Part 1
creating a variety of equivalent expressions that represent a
particular situation or problem.
Introducing pronumerals
3
4
PAS4.1 Algebraic techniques
Preliminary quiz – Part 1
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
For multiple choice questions circle the correct answer.
1
There are four baskets with six eggs in each. Which of the following
calculations will give you the total number of eggs?
a
2
3
6× 4
c
6÷4
d
a
Cost of one book + cost of one pen.
b
Cost of one book + cost of one pen then times by 5.
c
2 × cost of one book + cost of one pen.
d
2 × cost of one book + 3 × cost of one pen.
6− 4
Jill has 864 coins in her collection. She wants to buy some special
folders to store them safely. Each folder can hold 72 coins.
How would she work out how many folders to buy?
864 ÷ 72
b
864 × 72
c
864 + 72
d
864 − 72
c
73 ÷ 4
d
73
73 + 73 + 73 + 73 is the same as
a
Part 1
b
Imran was buying two books and three pens. Which of the
following describes how he would work out the total cost?
a
4
6+4
73 + 4
Introducing pronumerals
b
73× 4
4
5
5
Mary played a game where discs are worth five points each and
triangles are worth two points each.
5
points
each
2
points
each
Mary had three discs and six triangles.
She worked out that she had 27 points.
Describe how Mary would have worked this out.
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
6
PAS4.1 Algebraic techniques
What is algebra?
Algebra is a powerful problem-solving tool, used in a huge variety of
situations. At school you might find it used to calculate speeds of
moving objects in science, to convert between measurements in design
and technology, or to determine formulas needed for spreadsheets in
computing. Engineers, scientists, designers and architects are just a few
of the occupations where algebra is regularly used. In fact most people
use algebraic techniques to solve problems everyday without realising it.
Algebra allows us to write mathematical rules and mathematical
sentences quickly and clearly. You represent numbers within these rules
and sentences using letters. These letters are called pronumerals where
pro means in place of.
Consider a problem like this.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Rewrite this number sentence using a pronumeral.
‘Six plus what gives you ten’.
Solution
In earlier stages you used a shape in place of a number.
6+
= 10
To make this a true statement,
must represent four.
However, you can also use a letter in place of the missing
number. You could write 6 + y = 10.
In this case y must represent 4 or y = 4.
This is an algebraic equation. An equation is a mathematical
statement that contains an equal sign.
Part 1
Introducing pronumerals
7
Any letter or symbol can be used but it is easier to read and write this
problem when you use a letter rather than a shape. In this example it is y
that must represent four to make the statement true.
In another problem, y might represent a totally different number.
You can also use algebra to help develop a rule.
In this next example, you are not trying to find a missing number, but
rather you are trying to write a rule you can follow.
Follow through the steps in this example. Do your own working in the
margin if you wish.
My brother, Tom, is five years older than I am.
Write a rule to describe how to find Tom’s age.
Solution
I can always calculate his age by adding five to mine.
This will not change, no matter how old I am.
If I am seven, Tom must be 7 + 5 = 12
If I am 20, Tom must be 20 + 5 = 25
Algebra can be used to describe the process for finding
Tom’s age.
If I am n years old, Tom must be n + 5 years old.
Notice that n stands for a number, in this case my age in years.
The algebraic expression n + 5 is the rule that tells you how to work out
Tom’s age. An algebraic expression is a group of numbers and
pronumerals such as:
n+5
6y + 7
k + t− 3
In this way, you can write rules quickly. The answers will change as
each pronumeral value changes, but the rule always remains the same.
8
PAS4.1 Algebraic techniques
You have seen an example that required you to write an equation and
another that required you to write an expression.
In the equation x + 15 = 20 you can work out what the number x
stands for.
In the expression x + 15 there is no way for you to determine what x
stands for without knowing further information.
Remember, pronumerals stand for numbers not words.
Activity – What is algebra?
Try these.
1
What is the pronumeral used in the expression h + 20?
_______________________________________________________
2
If x represents the number 12, what number would x – 2 represent?
_______________________________________________________
3
Which of these shows the rule for doubling a number?
Circle the correct answer.
(The pronumeral w has been used for the number.)
a
2+w
b
w
2
c
2× w
d
w× w
Check your response by going to the suggested answers section.
You have now been introduced to pronumerals and their use in
mathematics. Now practise using them in the exercise.
Go to the exercises section and complete Exercise 1.1 – What is algebra?
Part 1
Introducing pronumerals
9
10
PAS4.1 Algebraic techniques
The language of algebra
In order to use algebra as a tool to solve problems, you need to become
familiar with reading and writing the language of algebra.
When using algebra, it is vital to understand what the pronumeral
represents. Particular letters are often used that will help you to
remember what the pronumeral stands for, such as using c for number of
cats or t for time in minutes. However, there are some letters you need to
write very carefully when you want to use them.
o
(lower or upper–case oh) it looks like a zero so you shouldn’t
use it at all
l
(lower–case ell) it looks like a one
b
it can look like a six
s
it can look like a five
x
it can look like a multiplication sign
t
it can look like an addition sign
z
it can look like a two
i
it can look like a one
It is common practice to write pronumerals using cursive or running
writing. For example:
l , b , s , x , t , z , i.
This way you are less likely to confuse the pronumerals for numbers or
other signs.
Part 1
Introducing pronumerals
11
Some rules
Some basic rules to follow when writing algebra are:
•
3k means 3× k just as three eights means 3× 8 (the multiplication
sign is the only one that can be left out between a number and a
pronumeral)
•
t × 5 is written 5t and never t5 because t5 can be confused with t 5
•
y by itself means one lot of y or 1× y and you don’t write it as 1y
•
abc means a × b × c and products like this are usually written in
alphabetical order. Notice the multiplication signs can be left out
•
2
2
w means w × w , for example 5 means 5 × 5 . You don’t write ww
•
•
•
y
using the fraction line (this is the same as in
2
1
an ordinary fraction because means 1 divided by 2)
2
y ÷ 2 can be written
2+p
(the fraction bar is a grouping
3
symbol just like the brackets)
(2 + p) ÷ 3 can be written
2 + p ÷ 3 can be written 2 +
p
because here you do not want to
3
group the 2 and the p.
Some examples using these rules and conventions are shown below.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Rewrite these using correct algebraic conventions.
12
a
n × 20
b
5− p ÷7
PAS4.1 Algebraic techniques
Solution
a
n × 20 = 20n
Note the number factor is always written first.
b
5− p÷ 7= 5−
p
7
Note that it is only p that is divided by seven.
5− p
the five is now divided by seven as
7
well as the p, and you don’t want to do that.
If you wrote it as
Now that you have worked through these examples and their solutions try
the activity to practise what you have learned.
Activity – The language of algebra
Try these.
Rewrite these using algebraic conventions.
1
1× t ___________________________________________________
2
k ÷ 3 + 8 _______________________________________________
3
m × m × 9 ______________________________________________
Check your response by going to the suggested answers section.
You have been practising using some algebraic rules on using
pronumerals. Now check that you can use these rules by yourself.
Go to the exercises section and complete Exercise 1.2 – The language of
algebra.
Part 1
Introducing pronumerals
13
14
PAS4.1 Algebraic techniques
Writing algebraic expressions
Writing algebraic expressions using pronumerals may at first appear
daunting but in reality all algebraic sentences are is a type of shorthand
so you don’t have to write a long word sentence.
Follow through the steps in this example. Do your own working in the
margin if you wish.
1
The diagram below represents three bags of lollies and four
loose lollies.
All the bags contain the same number of lollies.
Let b stand for the number of lollies in each bag.
(Remember, b does not stand for the word bag.)
Can you write an algebraic expression to describe the total
number of lollies in the picture including those in the bags?
2
Mary is buying five pencils and three pens.
You can write an expression for the total price Mary will
have to pay:
five × cost of one pencil + three × cost of one pen.
Can you write this as an algebraic expression?
Part 1
Introducing pronumerals
15
Solution
1
2
Here there a three possible answers:
•
b+b +b +4
•
3× b + 4
•
3b + 4 (This last expression is the simplest form.)
To use algebra you need to choose pronumerals to
represent the unknown numbers or the numbers that
will vary.
Let each pencil cost p dollars and each pen cost d dollars.
Notice you could not use the pronumeral p to stand for
both the cost of the pencils and the cost of the pens,
because these numbers are not the same. In any one
problem, each pronumeral can only be used to represent
one type of quantity.
So in algebra,
five × cost of one pencil + three × cost of one pen
becomes
5p + 3d
This is as far as you can go with your answer. You do not
know the cost of either the pencil or the pen, so you cannot
work out a numerical answer.
Algebra is particularly useful when the rules you need to follow,
or the problem you need to solve is too complicated to see a solution
straight away. But to learn to use algebra, you need to start with
simple examples. Try these problems to test your understanding of
this learning.
16
PAS4.1 Algebraic techniques
Activity – Writing algebraic expressions
Try these.
1
Alice runs a shop selling kitchen goods. She has cups in boxes and
some loose cups out of boxes.
There are k cups in each box, and some loose cups. Write an
expression for the total number of cups represented by this picture.
_______________________________________________________
2
There are m marbles in a container. If two more marbles are put in,
write an expression for the total number of marbles now in the
container.
_______________________________________________________
Check your response by going to the suggested answers section.
You should now consolidate your learning in this section by doing some
more practise writing algebraic expressions.
Go to the exercises section and complete Exercise 1.3 – Writing algebraic
expressions.
Part 1
Introducing pronumerals
17
18
PAS4.1 Algebraic techniques
Equivalent expressions
In English there are often several different ways of saying or writing
something. It is the same when using algebra. But always remember
that pronumerals stand for numbers.
Follow through the steps in this example. Do your own working in the
margin if you wish.
1
The diagram below represents the lollies owned by John
and Kim. There are b lollies in each bag.
John
Kim
Write an expression for the total number of lollies in the
picture.
2
Alice runs a shop selling kitchen goods. She has cups in
boxes and some loose cups out of boxes.
Each box hold k cups.
Alice sells two boxes and one loose cup.
Write an expression that describes the following diagram.
Part 1
Introducing pronumerals
19
Solution
1
You can write the total number of lollies in several ways.
•
b + b + b + 2 + b + 5 (John’s plus Kim’s)
•
3b + 2 + b + 5 (John’s plus Kim’s with the bags
collected together)
•by collecting all the bags together and all the loose lollies to
The expressions above are equivalent because they
describe the same thing.
2
Five boxes and seven loose cups take away two boxes and
take away one loose cup can be written as 5k + 7 – 2k – 1.
This leaves 3k + 6.
Having worked through the example you should now test yourself on
whether you can identify and write equivalent algebraic expressions.
Activity – Equivalent expressions
Try this.
Write at least two algebraic expressions for the total number of lollies in
the following picture. There are b lollies in each bag.
_______________________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
20
PAS4.1 Algebraic techniques
You have been practicing using and determining equivalent expressions.
Now check that you can solve these kinds of problems.
Go to the exercises section and complete Exercise 1.4 – Equivalent
expressions.
Part 1
Introducing pronumerals
21
22
PAS4.1 Algebraic techniques
Groups in algebra
When working with numbers, groups can be created using brackets.
For example: 2 × (5 + 3) .
The brackets group the five and the three together meaning you must do
that bracketed operation before you multiply by two.
You can also use brackets in algebra to represent groups.
Just as in using number this indicates an order in which
operations occurs.
Part 1
Introducing pronumerals
23
The example below illustrates that point.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Angela has shared out some lollies equally between Harry and
herself.
Harry
Angela
Write the total number of lollies in several ways.
Solution
There are many ways to answers this problem:
•
b +b+b +4 +b +b+b +4
•
3b + 4 + 3b + 4
•
6b + 8
A final way of answering this problem is to use grouping
symbols. Since there are two bundles with 3b + 4 in each you
can write an expression using grouping symbols.
2(3b + 4) [This means 2 × (3b + 4) or two lots of (3b + 4) .]
Having worked through this example try to use the learning in the
following activity.
For this activity you will need to get:
•
Eight cups
•
Twelve buttons or counters or peas
Imagine each cup contains the same number of small objects.
Assume there are k small objects in each cup.
24
PAS4.1 Algebraic techniques
So in front of you there should now be eight cups each containing k
pretend small objects, as well as twelve loose object that are not in cups.
Please use paper to cover the answer column on the right. Slide the
paper down one step at a time. That way you will have a chance to do
your own thinking before you see the answers.
Step 1
Step 2
Write an algebraic expression
for the total number of small
objects you have in front of you,
including the imagined ones in
the cups.
Solution: (don’t look until you have had a go)
Using your cups and small
objects, share them equally into
two groups.
Solution: (don’t look until you have had a go)
Use grouping symbols to write
the total number of objects on
the table.
8 cups of k + 12 loose ones = 8k + 12
You should have 4 cups and 6 objects in each
group.
Therefore you have two lots of 4k + 6.
This can be written as
2 X (4k + 6) or 2 (4k + 6)
Step 3
Collect all the cups and objects
together.
Solution: (don’t look until you have had a go)
4 groups with 2k + 3 in each = 4(2k + 3)
Now share them out into 4 equal
groups.
Write an expression using
grouping symbols to describe
the arrangement in front of you.
Step 4
Collect all the cups and objects
together again.
Now share them out into 3 equal
groups, with any left-overs put to
the side but still on the table.
Solution: (don’t look until you have had a go)
3 groups with 2k + 4 in each but also with 2 cups
left over = 3(2k + 4) + 2k
Notice that the 2 cups left over are not included
in the grouping symbols.
Write an expression using
grouping symbols to describe
the new arrangement.
Part 1
Introducing pronumerals
25
It is important to realise that no matter how you rearrange the cups and
objects, you always have the same total. You didn’t lose any or add
any more. Similarly in algebraic expressions if you rearrange the
expression to an equivalent expression it still represents the
same amount.
Step 1 gave a total of
8k + 12
Step 2 gave a total of
2(4k + 6)
Step 3 gave a total of
4(2k + 3)
Step 4 gave a total of
3(2k + 4) + 2k
Therefore, you have four equivalent expressions, since they all represent
the same thing: the total.
8k + 12 = 2(4k + 6) = 4(2k + 3) = 3(2k + 4 ) + 2k .
Access related sites to investigate this type of activity further using an
interactive computer animation by visiting the LMP webpage below.
Select Stage 4 and follow the links to resources for this unit PAS4.1
Algebra techniques, Part 1.
<http://www.lmpc.edu.au/mathematics>
It is now time for you to make sure you understood this work by doing
the exercise.
Go to the exercises section and complete Exercise 1.5 – Groups in
algebraic.
Congratulations you have completed the learning for this part. It is now
time for you to complete the review quiz so you can show your teacher
how much you have learned.
26
PAS4.1 Algebraic techniques
Suggested answers – Part 1
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1
b
6× 4
2
d
2 × cost of one book + 3 × cost of one pen
3
a
864 ÷ 72
4
b
73× 4
5
5 + 5 + 5 + 2 + 2 + 2 + 2 + 2 + 2 or 3 × 5 + 6 × 2
work out 3× 5 and work out 6 × 2 , then add the two answers together.
Activity – What is algebra?
1
h
2
10
3
c
2× w
Activity – The language of algebra
1
t (you do not need to write 1t because t by itself means one t)
2
k
+ 8 (it is only the k that is divided by 3)
3
3
2
9m (the number factor must be written first, and m × m is better
2
written as m )
Part 1
Introducing pronumerals
27
Activity – Writing algebraic expressions
1
2
There are three possible solutions:
•
3k + 6
•
k+ k+ k+6
•
3× k + 6 (3 lots of k plus 6 loose)
m + 2 (m marbles plus two more)
Activity – Equivalent expressions
There are several expressions that can be used to describe the total
number of lollies shown in the picture. It is important to realise that you
must have included four bags and a total of six singles.
The most likely expressions are:
28
•
b+b+b+b+6
•
4b + 6
•
b + 2 + 2b + 3 + b + 1
PAS4.1 Algebraic techniques
Exercises – Part 1
Exercises 1.1 to 1.5
Name
___________________________
Teacher
___________________________
Exercise 1.1 – What is algebra
1
2
3
What pronumeral is used in these expressions?
a
m + 12 _____________________________________________
b
40 – k ______________________________________________
Write true (T) or false (F) for each of these.
a
x + 7 is an algebraic equation ___________________________
b
y – 5 is an algebraic expression _________________________
c
k –2 = 5 is an algebraic equation ________________________
d
k – 2 is an algebraic equation ___________________________
Find the number the pronumeral is standing for.
a
y + 10 = 18
y = ________________________________________________
b
5–k=3
k = ________________________________________________
4
Part 1
If b = 7, what would be the value of b + 1? ____________________
Introducing pronumerals
29
5
Circle the expression below that shows the rule for finding the
square of any number? (The pronumeral n is used here to stand for
the number.)
a
30
n×2
b
n× n
c
n+n
d
n+2
PAS4.1 Algebraic techniques
Exercise 1.2 – The language of algebra
Rewrite these using correct algebraic rules and conventions.
1
x ÷5
_______________________________________________________
2
7×k
_______________________________________________________
3
m×b
_______________________________________________________
4
r × 10
_______________________________________________________
5
(v + 2) ÷ 4
_______________________________________________________
6
b× 4× a
_______________________________________________________
7
8+p÷3
_______________________________________________________
8
m×m
_______________________________________________________
9
5× t× t
_______________________________________________________
Part 1
Introducing pronumerals
31
Exercise 1.3 – Writing algebraic expressions
1
Write an algebraic expression for the number of lollies in the
diagram below. Use b to represent the number of lollies in each bag.
_______________________________________________________
_______________________________________________________
2
John has a bag of lollies with b lollies in it. He opens the bag and
eats one lolly. How many lollies are left?
_______________________________________________________
3
32
Draw a simple diagram to represent 2b + 5 where b represents the
number of lollies in a bag.
PAS4.1 Algebraic techniques
The square below has an area of a square centimetres.
4
The semi-circle has an area of b square centimetres.
a
b
Example
This shape has an area of
a + b square centimeters
a
b
Write an algebraic expression for the area of each of these
new shapes.
a
___________________________________________________
___________________________________________________
b
___________________________________________________
___________________________________________________
c
___________________________________________________
___________________________________________________
Part 1
Introducing pronumerals
33
5
a
Fred had twenty cards but lost five of them.
How many does he have now?
___________________________________________________
b
Fred had k cards but lost five of them.
How many does he have now?
___________________________________________________
c
Fred had k cards but lost m of them. How many does he have now?
___________________________________________________
6
a
Wendy saved $5 each week for 10 weeks. Write a number
sentence to show how much money she saves altogether.
___________________________________________________
b
Wendy saved $5 each week for y weeks. Write an algebraic
expression to show how much money she saves altogether.
___________________________________________________
c
Wendy saved x dollars each week for y weeks.
Write an algebraic expression to show how much money she
saves altogether.
___________________________________________________
7
a
I have twenty-four marbles to share between three people.
Write a number sentence to show how many marbles each
person will receive.
___________________________________________________
b
I have twenty-four marbles to share between p people.
Write an algebraic expression to show how many marbles each
person will receive.
___________________________________________________
c
I have m marbles to share between p people. Write an algebraic
expression to show how many marbles each person will receive.
___________________________________________________
34
PAS4.1 Algebraic techniques
8
a
How many cents in seven dollars?
___________________________________________________
b
How many cents in p dollars?
___________________________________________________
9
There are five apples on a scale. The algebraic expression below
gives total weight of the five apples.
5a grams
Circle the correct ending for this statement.
The pronumeral a in this expression stands for:
Part 1
a
the word apple.
b
the number of apples.
c
the weight of each apple.
d
the total weight of all the apples.
Introducing pronumerals
35
Exercise 1.4 – Equivalent expressions
1
Write at least two algebraic expressions for the total number of
lollies in following diagram. Remember there are b lollies in
each bag.
________________________________
________________________________
________________________________
2
Each jug holds j mL of juice.
Each glass holds g mL of juice.
Write at least two expressions for the total amount of liquid shown in
the diagram.
___________________________________________________
___________________________________________________
36
PAS4.1 Algebraic techniques
3
The diagram below shows some containers of counters and some
loose counters.
Each container holds w counters.
a
Write at least two expressions for the total number of counters
in the picture.
___________________________________________________
b
Write an expression to show taking away one container and
three loose counters.
___________________________________________________
c
Write an expression for what is left.
___________________________________________________
Part 1
Introducing pronumerals
37
Exercise 1.5 – Groups in algebra
1
Three people have the same number of lollies each.
Adam
Peter
Sally
There are b lollies in each bag. Write an expression using grouping
symbols to represent the total number of lollies in the diagram.
_______________________________________________________
2
To help you with this set of questions, you might like to gather
six circles (like buttons or washers) and 18 triangles (pieces of paper
will do).
Each circle is worth c points and each triangle is worth t points.
a
Write an expression for the total number of points shown in the
diagram.
___________________________________________________
b
Share the circles and triangles evenly into two groups.
Write an expression using grouping symbols for the total
number of points.
___________________________________________________
38
PAS4.1 Algebraic techniques
c
Gather all the shapes together, then share them evenly into
three groups. Write an expression using grouping symbols for
the total number of points.
___________________________________________________
d
Gather all the shapes together, then share them evenly into six
groups. Write an expression using grouping symbols for the
total number of points.
___________________________________________________
You have now completed the exercises and tasks for this part.
Complete the review quiz section and return it to your teacher.
Part 1
Introducing pronumerals
39
40
PAS4.1 Algebraic techniques
Review quiz – Part 1
Name
___________________________
Teacher
___________________________
When answering multiple choice questions, circle the correct answer.
1
The pronumeral in 2f + 7 is
a
2
4
b
2f
c
f
d
f+7
b
3+t
c
3÷t
d
t× t× t
3t is the same as
a
3
2
3× t
Write the number the pronumeral is standing for.
a
y − 10 = 6
b
2k = 10
If m = 4 what would be the value of m + 5?
_______________________________________________________
5
Kim rewrote h × 7 as h7. Does this follow normal algebraic
conventions? Give your reasons.
_______________________________________________________
_______________________________________________________
6
Which of the following expressions is the same as v ÷ 5 + 2
a
Part 1
v +5
2
Introducing pronumerals
b
v +2
5
c
v+
5
2
d
v
+2
5
41
7
Write an algebraic expression for the total number of lollies in the
diagram below. Use b to represent the number of lollies in each bag.
8
Jo bought seven videos each costing the same amount.
The total cost was 7v dollars.
In this expression v stands for:
9
a
the number of videos
b
the cost of one video
c
the total cost
d
the word video.
I have $50 to share between k people. Write an algebraic expression
to show how much money each person will receive.
_______________________________________________________
10 One of these expressions is NOT equivalent to all the others.
Which one is it?
42
a
x +x+x +x+7
b
3x + x + 7
c
xxxx + 7
d
4x + 7
PAS4.1 Algebraic techniques
11 Each bus has seats for p passengers and each car has seats
for c passengers.
seat for p
passengers
a
seat for c
passengers
Write an expression for the number of passengers that can
travel in:
i
six buses ________________________________________
ii
five cars _________________________________________
iii two buses and six cars _____________________________
b
Why could you not use p to represent the number of passengers
in each bus and also the number of passengers in each car?
___________________________________________________
___________________________________________________
___________________________________________________
12 Each biscuit tin contains f funny-faced biscuits. There are also
some loose biscuits not in the tin.
Write two algebraic expressions to describe the total number of
biscuits in the diagram.
_______________________________________________________
_______________________________________________________
Part 1
Introducing pronumerals
43
13 In each basket there are p peanuts and w walnuts.
a
Write an expression for the number of nuts in one basket.
___________________________________________________
b
Write an expression using grouping symbols to show the total
number of nuts in the diagram.
___________________________________________________
14 Which of the following is equivalent to 2(h + 3) ?
[Remember that this means 2 lots of (h + 3) ].
a
2h + 3
b
2h + 6
c
(h + 3)( h + 3)
_______________________________________________________
44
PAS4.1 Algebraic techniques
Answers to exercises – Part 1
This section provides answers to questions found in the exercises section.
Your answers should be similar to these. If your answers are very
different or if you do not understand an answer, contact your teacher.
Exercise 1.1 – What is algebra
1
a
m
2
a
F (it is an expression)
b
T
c
T
d
F (it is an expression)
3
a
y=8
4
8
5
B
b
b
k
k=2
Exercise 1.2 – The language of algebra
1
x
5
2
7k
3
bm (writing the pronumerals in alphabetical order is best but not
essential so mb would also be correct).
4
10r
5
v +2
4
6
Part 1
4ab
4ba
Introducing pronumerals
45
7
8+
8
m
9
5t 2
p
3
2
Exercise 1.3 – Writing algebraic expressions
1
4b + 2
b +b+b +b+2
2
b–1
3
4
a
a+a+b
2a + b
b
a–b
c
a+a+a+a–b–b
4a – 2b
2a + 2(a – b)
5
a
20 – 5 = 15
6
a
5 × 10 = 50 dollars
b
5 × y = 5y dollars
c
x × y = xy dollars
a
24 ÷ 3 =
b
24 ÷ p or
24
p
c
m ÷ p or
m
p
8
a
700 cents
9
c
the weight of each apple
7
46
b
k–5
b
100p cents
c
k–m
24
=8
3
PAS4.1 Algebraic techniques
Exercise 1.4 – Equivalent expressions
1
2
There are two answers.
•
b+b+1
•
2b + 1
Here are three possible answers.
•
j + g + 2j + g + 2 j + g
•
5 j + 3g
•
j + j+ j + j+ j +g+g+g
(There are other possible answers).
3
a
There are two answers.
•
•
w+w+ w+9
3w + 9
b
3w + 9 − w − 3
c
2w + 6
Exercise 1.5 – Groups in algebra
1
2
There are two answers.
•
3(b + 2)
•
3(2 + b)
a
There are two answers.
b
c
d
Part 1
•
6c + 18t
•
18t + 6c
There are two answers.
•
2(3c + 9t)
•
2(9t + 3c )
There are two answers.
•
3(2c + 6t)
•
3(6t + 2c )
There are two answers.
•
6(c + 3t)
•
6(3t + c )
Introducing pronumerals
47
Mathematics Stage 4
PAS4.1 Algebraic techniques
Part 2
Writing algebraic expressions
Contents – Part 2
Introduction – Part 2 ..........................................................3
Indicators ...................................................................................3
Preliminary quiz – Part 2 ...................................................5
Translating into words .......................................................9
Translating into algebra...................................................15
The language of mathematics .........................................19
Algebra in action..............................................................23
Exploring number rules ...................................................27
Suggested answers – Part 2 ...........................................31
Exercises – Part 2 ...........................................................35
Review quiz – Part 2 .......................................................45
Answers to exercises – Part 2.........................................49
Part 2
Writing algebraic expressions
1
2
PAS4.1 Algebraic Techniques
Introduction – Part 2
This part continues to develop the concepts involved in the reading and
writing of algebraic expressions. You will explore the relationship
between words, numbers and algebraic symbols.
Indicators
By the end of Part 2, you will have been given the opportunity to work
towards aspects of knowledge and skills including:
•
translating between words and algebraic symbols
•
translating between algebraic symbols and words
•
creating algebraic expressions that can be used to solve problems.
By the end of Part 2, you will have been given the opportunity to work
mathematically by:
•
describing relationships between the algebraic expressions and
number properties
•
linking algebra with rules in number
•
determining equivalence of algebraic expressions by substituting a
given number for the letter
•
Part 2
interpreting the meaning of algebraic statements.
Writing algebraic expressions
3
4
PAS4.1 Algebraic Techniques
Preliminary quiz – Part 2
Before you start this part, use this preliminary quiz to revise some skills
you will need.
Activity – Preliminary quiz
Try these.
1
Draw a line to match each word on the left with a symbol on the
right. Symbols may match with more than one word.
sum
difference
product
share
take away
total
minus
subtract
times
divide
add
Part 2
Writing algebraic expressions
5
2
Calculate the answers to:
a
the sum of 8 and 12 __________________________________
___________________________________________________
b
the product of 5 and 6 _________________________________
___________________________________________________
c
the number that is 5 less than 20 _________________________
___________________________________________________
d
the difference between 35 and 40 ________________________
___________________________________________________
e
half of 28 ___________________________________________
___________________________________________________
3
4
6
Write T (true) or F (false) for each of these number sentences.
a
6 + 7 = 7 + 6 ________________________________________
b
6 × 7 = 7 × 6 ________________________________________
c
6 – 7 = 7 – 6 _________________________________________
d
6 ÷ 7 = 7 ÷ 6 ________________________________________
e
5 × 2 + 3 = 3 + 5 × 2 __________________________________
f
5 × (2 + 3) = 5 × 2 + 3 _________________________________
g
10 ÷ 2 + 3 = 10 ÷ (2 + 3) ______________________________
h
3 + 3 + 3 + 3 + 3 = 5 × 3 _______________________________
Answer T (true) or F (false) for each part. These pairs of algebraic
statements are equivalent.
a
m + m + m and 3m __________________________________
b
4y and 4 + y
c
k
and k ÷ 2 _______________________________________
2
d
2w and w
2
______________________________________
________________________________________
PAS4.1 Algebraic Techniques
5
Each bag below contains b lollies, with several loose lollies also
shown. Circle all the algebraic expressions below that describe the
total number of lollies in the diagram.
a
b +b+ 3
b
2b + 3
c
b×b +3
d
2(b + 3)
Check your response by going to the suggested answers section.
Part 2
Writing algebraic expressions
7
8
PAS4.1 Algebraic Techniques
Translating into words
Using words to describe rules in the world around you can sometimes
lead to lengthy statements. Look at this example:
The velocity (or speed) of an object that has a constant acceleration
can be calculated by multiplying its acceleration by the time it has
been travelling, and adding this to its starting velocity.
This statement, like many others that describe real situations, can be
written more concisely using algebra. Using pronumerals, the statement
above reduces to:
v = u + at
In this section you will learn to read algebraic statements and translate
them into English. This skill will move you a step closer to using algebra
as a problem-solving tool.
Part 2
Writing algebraic expressions
9
Statements in English can often be interpreted in different ways.
Take this statement for example:
‘Double eight plus five.’
Does it mean double
eight then add five:
2 × 8 + 5 = 21?
or
Does it mean double the
answer to eight plus five:
2 × (8 + 5) = 26?
Mathematical statements avoid this confusion by being precise.
By using grouping symbols and other tools, you can explain exactly what
you mean.
Algebra is just one tool you can use to communicate exact meanings.
In these examples, you will translate algebraic statements into simple
English instructions.
Follow through the steps in this example. Do your own working in the
margin if you wish.
1
Give, in words, the meaning of 3p + 1.
2
Give, in words, the meaning of 3( p + 1) .
Solution
1
You must first examine the steps shown in the expression,
being very careful of the order.
Start with a number ⇒ p
Multiply it by 3 ⇒ 3p
Then add 1 ⇒ 3p + 1
So describing this statement in words you get: start with a
number, multiply by three and then add one.
You could reword this in many ways. You might prefer to
write: triple a number then add one.
10
PAS4.1 Algebraic Techniques
2
Examine the steps in order.
Start with a number ⇒ p
Add 1 to it ⇒ p + 1
Then multiply by 3 ⇒ 3( p + 1)
So describing this statement in words you get:
Start with a number, add one and then multiply by three.
Pause here to look back on the two examples. The brackets change the
order of the steps and therefore change the meanings of the expressions.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Give, in words, the meaning of 15 − 2a .
Solution
Examine the steps in order.
Start with a number ⇒ a
Multiply it by 2 ⇒ 2a
Then subtract this from 15 ⇒ 15 − 2a
So describing this statement in words you get: start with a
number, multiply by two then subtract this from 15.
Another way of saying this is: double a number and take this
answer away from 15.
You do not have to show all the working as seen in the above examples.
Your answer might only include the statement written in English.
Part 2
Writing algebraic expressions
11
Follow through the steps in this example. Do your own working in the
margin if you wish.
Give, in words, the meaning of
2x − 5
.
3
Solution
There are many ways of describing the meaning of this
statement. Here are two possible answers.
Start with a number, multiply it by two, take away five and then
divide by three.
Or double a number, take away five and divide the answer by
three.
Now that you have worked through these examples try some yourself in
the activity below.
Activity – Translating into words
Try these.
Give, in words, the meaning of:
1
y + 6 __________________________________________________
_______________________________________________________
2
6k − 4 _________________________________________________
_______________________________________________________
3
m
+ 8 _________________________________________________
5
_______________________________________________________
4
(Harder)
4(3 − 5t )
______________________________________
10
_______________________________________________________
12
PAS4.1 Algebraic Techniques
5
2
(Harder) w − w ________________________________________
_______________________________________________________
Check your response by going to the suggested answers section.
Translating algebraic expressions into words can look a little complex
but you can see that the benefits in terms of the shorthand and the
precision make the skill worthwhile. Try the examples in the exercises to
see if you have mastered this skill.
Go to the exercises section and complete Exercise 2.1 – Translating into
words.
Part 2
Writing algebraic expressions
13
14
PAS4.1 Algebraic Techniques
Translating into algebra
To use algebra as a tool for solving problems, you often need to create
algebraic statements from a problem expressed in words.
In these examples, you will explore writing algebraic expressions to
describe written processes. .
Follow through the steps in this example. Do your own working in the
margin if you wish.
1
Start with a number, triple it and take away 5.
2
Take a number from 7, then multiply by 8.
Solution
1
You may choose any pronumeral.
Here, let y represent the number.
Start with
a number
Triple it
Take away 5
y
3y
3y − 5
So the answer is 3y – 5.
2
Let the number be n (any pronumeral will do).
Start with
a number
Take it away
from 7
n
7 − n
Multiply by 8
8 × (7 − n)
So the answer is 8 × (7 − n) which can be written 8(7 − n) .
When translating English into algebra, you must know both the meaning at
the word and the order in which you must do it. So subtract ten from the
sum of x and 3 is ( x + 3) − 10 not ( x + 3) .
Part 2
Writing algebraic expressions
15
Activity – Translating into algebra
Try these.
Write algebraic expressions for these processes.
1
Start with a number, halve it then add five.
Let the number be ________________________________________
(use this space for your working if needed)
The answer is ___________________________________________
2
Double a number, subtract four then divide by ten.
Let the number be ________________________________________
The answer is ___________________________________________
16
PAS4.1 Algebraic Techniques
3
(Harder) Square a number, subtract this from ten then divide by the
number you started with.
Let the number be ________________________________________
The answer is ___________________________________________
Check your response by going to the suggested answers section.
The replacement of a sentence with a simple equation adds precision and
means a complex problem is reduced to its main elements. This is a
mathematical skill that provides great benefits if mastered. Try these
exercises to see if you are on your way to mastering this important skill.
Go to the exercises section and complete Exercise 2.2 – Translating into
algebra.
Part 2
Writing algebraic expressions
17
18
PAS4.1 Algebraic Techniques
The language of mathematics
There are terms used in mathematics that can be represented using
symbols such as sum (+) or square root ( ).
Understanding these words and symbols is essential for communicating
in mathematics.
The following examples will examine translating mathematical
statements from words to symbols.
Part 2
Writing algebraic expressions
19
Follow through the steps in this example. Do your own working in the
margin if you wish.
Write these statements using algebraic and mathematical
symbols.
1
The sum of x and y
2
Consecutive numbers are whole numbers that follow one
another. For example 34, 35, 36 and 37 are four
consecutive numbers starting at 34.
a
Write three consecutive numbers starting with d.
b
Write four consecutive even numbers starting with m
(m is even).
Solution
1
Sum means to add, so the answer is x + y or y + x.
2
a
Each consecutive number is one more than the
previous, so you need to add one each time.
add 1
p
add 1
p+1
p+2
The answer is therefore p, p + 1, p + 2
b
To move from one even number to the next, you must
add two.
add 2
m
add 2
m+2
add 2
m+4
m+6
So the answer is m, m + 2, m + 4, m + 6.
Now that you have had a chance to look at the language of mathematics as
applied to algebra see if you have learned some of the skills required to
use it.
20
PAS4.1 Algebraic Techniques
Activity – The language of mathematics
Try these.
Write an expression for the following.
1
The product of 4 and y.
_______________________________________________________
2
The product of m and m.
_______________________________________________________
3
The difference between k and j if k is the larger number.
_______________________________________________________
4
(Harder) Three consecutive numbers ending with t. (You need to
find expressions for the two number that come before t.)
_______, _______, t
Check your response by going to the suggested answers section.
The exercises following will reinforce your skills in using the language
of algebra.
Go to the exercises section and complete Exercise 2.3 – The language of
mathematics.
Part 2
Writing algebraic expressions
21
22
PAS4.1 Algebraic Techniques
Algebra in action
In this section you will write algebraic expressions and equations that
relate to real-life problems. Each problem will require you to think of the
method needed to solve it.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Louise earns w dollars a week delivering papers. She also gets
p dollars pocket-money each week from her parents.
a
How much does she earn altogether in one week?
b
Write an expression using grouping symbols to show how
much she will earn altogether in 10 weeks.
c
Write an equivalent expression without grouping symbols
to show her total earnings in 10 weeks.
Hint: if you are unsure how to start, you can replace the
pronumerals with numbers. This will let you explore the steps
you need to follow to solve the problem.
For example, you could decide that Louise earns $20 per week
delivering papers and gets $5 pocket money. Using these
numbers you would work out that the answer to part a would
be $25, because you have to add the two values.
Using the pronumerals, this becomes w + p (adding the two
values). You can use this technique with the rest of the
question.
Part 2
Writing algebraic expressions
23
Solution
a
w + p dollars (add the two amounts)
b
10(w + p) dollars (ten lots of one week’s money)
c
10w + 10 p dollars (ten lots of paper-delivery money and
ten lots of pocket-money)
Note that the answers to part b and part c are equivalent
expressions because they both represent the same thing.)
Now that you have worked through these examples do the following.
Follow through the steps in this example. Do your own working in the
margin if you wish.
A rectangle has length L cm and width W cm.
What is the perimeter of the rectangle?
Solution
Drawing pictures often helps solve problems.
L cm
W cm
W cm
L cm
The perimeter is the total distance around the shape.
So you can write this as.
•
Perimeter = L + W + L + W cm
•
Perimeter = 2L + 2W cm
•
Perimeter = 2(L+W) cm
Using algebra to solve real life problems is a skill once learned that can
serve you well.
24
PAS4.1 Algebraic Techniques
Activity – Algebra in action
Try these.
1
Write the rule for finding the area of the rectangle shown in the
example above.
Area = _________________________________________________
2
Ajit had $100 in his money box. He then put in $5 each week.
a
How much money would he have in his money box after
3 weeks?
___________________________________________________
b
Write an algebraic expression for the money he would have in
his money box at any time. (You can use any pronumeral you
like to stand for the number of weeks he has been depositing.)
Let ____ represent ____________________________________
Check your response by going to the suggested answers section.
To get more practise using algebraic expressions to describe other
situations see the Internet based activities below.
Access related sites on animation that demonstrates different ways of
solving a problem and writing expressions by visiting the LMP webpage
below. Select Stage 4 and follow the links to resources for this unit
PAS4.1 Algebraic techniques Part 2.
<http://www.lmpc.edu.au/mathematics>
The learning for this section is now complete and you need to reinforce
the skills learned. Do the exercises to assist with that.
Go to the exercises section and complete Exercise 2.4 – Algebra in action.
Part 2
Writing algebraic expressions
25
26
PAS4.1 Algebraic Techniques
Exploring number rules
You can use algebra to describe rules that work for all operations using
numbers. In this section, you will determine the meaning of some
algebraic statements, and explore some patterns in numbers.
Follow through the steps in this example. Do your own working in the
margin if you wish.
What does the following expression mean, and is it true?
a + b = b + a.
Solution
To explore this you can choose any numbers for a and b.
If a = 5 and b = 4 you get
5 + 4 = 4 + 5.
This is true.
If a = 20 and b = 6 you get
20 + 6 = 6 + 20
This is true.
By replacing the pronumerals with numbers, you can often get an insight
into the meaning of the statement, and decide whether it is true.
Looking at the number sentences above, you can see that a + b = b + a
really means: you can add any two numbers in either order and still get
the same answer.
It appears that it doesn’t matter what numbers you choose; the statement
will always be true. But be careful! Just because it works for these
numbers does not mean it will work for all numbers. For example, will it
still work if the numbers are fractions, or negative numbers, or zero?
Part 2
Writing algebraic expressions
27
Use a calculator to explore whether this statement is true for all types of
numbers.
Follow through the steps in this example. Do your own working in the
margin if you wish.
If n is an odd positive integer (meaning an odd counting
number like 1, 3, 57 or 1009 ), determine whether these will be
odd or even.
a
2n
b
n +1
c
3n
Solution
a
To examine each question you can try various odd
numbers for n like in Alex has.
If n = 5
then 2n = 10
That’s even.
If n = 101
then 2n = 202
That’s even.
But these two examples don’t prove 2n will always be even.
If you think about it further, two lots of any odd number will
always be even because the answer can be divided by two with
no remainder.
Therefore, 2n is even when n is any odd number.
b
28
Let n = 3
n+1=3+1=4
That’s even.
Let n = 35
then n + 1 = 36
That’s even.
PAS4.1 Algebraic Techniques
Again you have not proven that it will work for all numbers,
but the answers determined so far are even.
If you think about it further, n + 1 means ‘one more than’ or
‘the next number after n’. The number after an odd number is
always even.
Therefore, n + 1 is even when n is any odd number.
c
Try n = 7
3n = 3 × 7 = 21
That’s odd.
Try n = 21
3n = 3 × 23 = 69
That’s odd.
The answer will always be odd because you have an odd
number multiplied by an odd number.
Therefore, 3n is odd when n is an odd number.
In this section you have seen how it is valid to check an algebraic
expression by substituting numbers for the pronumerals as a sort of test.
You should now consider the validity of this through further exploration
with the help of others.
Discuss these questions with others, exploring all types of numbers such
as fractions, negative numbers, whole numbers and zero. You may use a
calculator if you wish.
1
Will 2n always be even? That is, when I double a number is the
answer always an even number?
2
If you only let n be a positive integer (meaning a positive whole
number) will 2n now always be even? Why?
3
Can you write another expression that will always be even if you use
any counting number?
In this way, algebra can be used to describe facts about numbers without
the need for words. All you need is the confidence and training to read
the language of algebra.
Part 2
Writing algebraic expressions
29
Activity – Exploring number rules
Explore these.
1
Is this true?
k + k + k = 3k (remember, 3k means 3× k )
2
Is this true?
2d + 5 = 2(d + 5)
Check your response by going to the suggested answers section.
You have been exploring number rules through algebra.
Now check that you can use these rules.
Go to the exercises section and complete Exercise 2.5 – Exploring number
rules.
Congratulations you have completed the learning for this part. It is now
time for you to complete the review quiz so you can show your teacher
how much you have learned.
30
PAS4.1 Algebraic Techniques
Suggested answers – Part 2
Check your responses to the preliminary quiz and activities against these
suggested answers. Your answers should be similar. If your answers are
very different or if you do not understand an answer, contact your teacher.
Activity – Preliminary quiz
1
To mark these, check that the line you have drawn from each word
connects to the symbol shown here.
2
Part 2
sum
+
difference
–
product
×
share
÷
take away
–
total
+
minus
–
subtract
–
times
×
divide
÷
add
+
a
20 (‘sum’ means add)
b
30 (‘product’ means multiply)
c
15 (‘5 less than’ means take away 5)
d
5 (‘difference’ means take the smaller number from the larger)
e
14 (‘half’ means divide by 2)
Writing algebraic expressions
31
3
4
5
a
T
b
T
c
F
d
F
e
T
f
F
g
F
h
T
a
T
b
F ( 4y means 4 × y )
c
T
d
2
F ( 2w means 2 × w but w means w × w )
A and B
Activity – Translating into words
There are many different ways of describing these. Two typical answers
have been given to each question.
1
Start with a number then add six or add six to a number.
2
Start with a number, multiply it by six then subtract four.
Or multiply a number by six then take away four.
3
Start with a number, divide it by five then add eight. Or divide a
number by five then add on eight.
4
Start with a number, multiply by five, subtract this from three,
multiply by four and then divide by ten. Or multiply a number by
five, take this away from three, multiply by four then divide by ten.
5
Start with a number, square it then take away the original number.
Or multiply a number by itself then subtract the number you started
with.
Activity – Translating into algebra
In these practice questions, you may have chosen to use any pronumeral.
When marking these, substitute your pronumeral into the answer.
1
32
Let the number be k. To halve a number, you divide it by 2.
k
The answer is + 5 .
2
PAS4.1 Algebraic Techniques
Note that the 5 is not included in the fraction. This can also be
written k ÷ 2 + 5 although describing it as a fraction is preferable.
2
Let the number be d;
2d − 4
10
Note that the fraction line groups the top. This could also be written
as (2d − 4 ) ÷ 10 but the fraction line is preferable.
3
Let the number be x.
The answer is
x 2 − 10
x
Activity – The language of mathematics
1
Product means multiply, so the product of 4 and y is 4 × y better
written as 4y .
2
m × m better written as m
3
Difference means subtract the smaller from the larger.
On the number line, it means the distance between the two numbers.
So the answer is k − j .
4
To end at t the other numbers are one less that t and two less than t.
2
So the answer is t – 2, t – 1 and t. (For example, three consecutive
numbers ending in 9 are 7, 8 and 9.)
Activity – Algebra in action
1
Area = L × W square centimeters or Area = LW cm2 .
(Area is measured in squares and describes the number of squares
that would be needed to cover the rectangle. To find the area of a
rectangle you multiply the length by the width.)
2
a
Ajit has $100 plus three lots of $5.
Total = 100 + 3 × 5 = 115 , so Ajit has $115 after three weeks.
Part 2
Writing algebraic expressions
33
b
Let w represent the number of weeks Ajit has been saving.
(You may have chosen a different pronumeral.)
Total amount in the bank after w weeks
= 100 + w × 5 dollars
= 100 + 5w dollars
(The number factor 5 is written before the pronumeral.)
This can also be written 5w + 100 dollars.
Activity – Exploring number rules
1
(You may have chosen different numbers to explore the statement.)
Let k = 5, 5 + 5 + 5 = 3 × 5 (True).
Let k = 10, 10 + 10 + 10 = 3 × 10 (True).
(Translating this into English, the statement means `adding the same
number three times is the same as multiplying the number by three`.)
This will always work, so k + k + k = 3k is true.
2
Let d = 8
2d + 5 = 2 × 8 + 5 = 21
2(d + 5) = 2(8 + 5) = 2 × (8 + 5) = 2 × 13 = 26
They do not give the same answers, so the statement is not true.
(Exploring this further, this shows that the order in which you work
out a problem matters. On the left-hand side, you double the number
first then add five. On the right-hand side you add five first then
double.)
34
PAS4.1 Algebraic Techniques
Exercises – Part 2
Exercises 2.1 to 2.5
Name
___________________________
Teacher
___________________________
Exercise 2.1 – Translating into words
Give, in words, the meaning of:
1
m − 5 __________________________________________________
_______________________________________________________
2
5 − m __________________________________________________
_______________________________________________________
3
7t ____________________________________________________
_______________________________________________________
4
10
____________________________________________________
c
_______________________________________________________
5
c
____________________________________________________
10
_______________________________________________________
6
20x − 50 _______________________________________________
_______________________________________________________
7
k+7
__________________________________________________
2
_______________________________________________________
Part 2
Writing algebraic expressions
35
8
40( f + 3) ______________________________________________
_______________________________________________________
9
(Harder) 5(r + 4 ) + r _____________________________________
_______________________________________________________
10 (Harder)
x + 5 ________________________________________
_______________________________________________________
36
PAS4.1 Algebraic Techniques
Exercise 2.2 – Translating into algebra
Write algebraic expressions for these processes. You may choose to use
any pronumeral.
Part 2
1
Add five to a number
2
Five less than a number
3
Start with a number, multiply by four and add seven
4
Start with a number, add seven then multiply by three
5
Take seven from a number
6
Half of a number
7
Any number times itself
8
The square root of a number
9
Start with a number, multiply by eight and divide by five
Writing algebraic expressions
37
10 (Harder) Add seven to a number, then take the square root of the
answer
11 (Harder) Start with a number, take away ten then divide by the
number you started with
38
PAS4.1 Algebraic Techniques
Exercise 2.3 – The language of mathematics
Write an algebraic expression for these processes.
1
The sum of a and b. ______________________________________
2
Six lots of t. ____________________________________________
3
Subtract eight from the product of three and q.
_______________________________________________________
4
Take five from the sum of x and y.
_______________________________________________________
5
The sum of a, b, c, and d.
_______________________________________________________
6
The difference between eight and k if k is the smaller number.
_______________________________________________________
7
Three consecutive numbers beginning with r.
_______________________________________________________
8
Four consecutive odd numbers starting with d.
_______________________________________________________
9
(Harder) Three consecutive multiples of ten starting with t.
(Hint: consecutive multiples of three are 3, 6, 9, 12, etc)
_______________________________________________________
10 (Harder) Three consecutive multiples of eight ending with p.
_______________________________________________________
Part 2
Writing algebraic expressions
39
Exercise 2.4 – Algebra in action
1
A box of six coffee mugs costs b dollars. Write an expression for the
cost of one mug.
Cost of one mug = _______________________________________
2
Write an algebraic statement for the perimeter of this triangle.
y cm
w cm
x cm
Perimeter = _____________________________________________
3
Joshua has f fish in a tank. Only g of them are goldfish.
What can you say about the numbers that f and g might stand for?
_______________________________________________________
_______________________________________________________
_______________________________________________________
4
A farmer has a square field with a side length of f metres.
a
What is the area of the field?
___________________________________________________
b
What would be the length of a fence around the field?
___________________________________________________
c
A two-metre gate will be put in one side of the fence, so the
farmer will need less fencing. Write a new expression for the
amount of fence needed.
___________________________________________________
40
PAS4.1 Algebraic Techniques
d
(Harder) The fence costs c dollars for each metre.
The gate costs $250 complete. Write an expression for the total
cost of fencing and gate.
___________________________________________________
5
(Harder) John is paid the same amount each hour for working in a
shop. Write down an algebraic expression for the total amount that
he earns. (You must choose your own pronumerals and describe
what they stand for.)
(If you don’t know how to start, decide what numbers are missing
from the question. You can even rewrite your own similar question
with numbers in it to decide how to solve this problem.)
Let ______ stand for ______________________________________
Let ______ stand for ______________________________________
Total earnings = _________________________________________
Part 2
Writing algebraic expressions
41
Exercise 2.5 – Exploring number rules
1
Explore these to discover whether they are true or false.
a
y– 4=4–y
b
10 k
=
k 10
c
f + f + f + f + 5 = 4 f +5
d
42
2(h + 3) = 2h + 6
PAS4.1 Algebraic Techniques
2
e
y
1
×y=
4
4
f
(Harder) 5x 2 = (5x)2
Write an expression for the number that is
a
ten more than x
___________________________________________________
b
ten less than x
___________________________________________________
c
x less than ten
___________________________________________________
d
half of x
___________________________________________________
e
ten less than half of x
___________________________________________________
Part 2
Writing algebraic expressions
43
3
If p is an even number, write an expression for the next even
number.
_______________________________________________________
4
(Harder) For 5y to always be even, what type of number does y have
to be?
_______________________________________________________
You have now completed the exercises and tasks for this part.
Complete the review quiz section and return it to your teacher.
44
PAS4.1 Algebraic Techniques
Review quiz – Part 2
Name
___________________________
Teacher
___________________________
When answering multiple choice questions, circle the correct answer.
1
Which of these means ‘the product of p and 3’?
a
2
b
3p
c
p× p× p
d
p3
Which of these means ‘add three to x then divide by two’?
a
3
p+ 3
x +3
2
b
x
+3
2
c
x+
3
2
d
x +3÷2
Give, in words, the meaning of:
a
k + 5 _______________________________________________
b
m
_________________________________________________
2
___________________________________________________
c
8(t − 12) ____________________________________________
___________________________________________________
d
x2
+ 7 _____________________________________________
3
___________________________________________________
Part 2
Writing algebraic expressions
45
4
Write an algebraic expression for.
a
Seven less than a number. _____________________________
b
Start with a number, double it then add ten. ________________
c
Add twenty to a number then divide by five. _______________
d
Start with a number, take the square root then divide by eight.
___________________________________________________
5
A shop-keeper buys b boxes of books.
Each box has fifteen books in it.
a
Write an algebraic expression describing the total number of
books the shop-keeper bought.
___________________________________________________
b
She sold twelve books. Write an expression for the number of
books remaining.
___________________________________________________
6
Write an expression for:
a
the number of metres in x kilometers
___________________________________________________
b
the number of days in w weeks.
___________________________________________________
7
There are 25 students in a class. There are g girls and b boys.
What can you say about the number that g and b stand for?
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
46
PAS4.1 Algebraic Techniques
8
Explore this and determine if it is always true. You may use a
calculator to explore. Write down your working.
x + 10 =
x + 10
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
9
Consecutive numbers are whole numbers that come after one
another. For example, seven, eight and nine are three consecutive
numbers.
a
Write three consecutive numbers beginning with 45.
___________________________________________________
b
Write three consecutive numbers beginning with x.
___________________________________________________
Part 2
Writing algebraic expressions
47
48
PAS4.1 Algebraic Techniques
Answers to exercises – Part 2
This section provides answers to questions found in the exercises section.
Your answers should be similar to these. If your answers are very
different or if you do not understand an answer, contact your teacher.
Exercise 2.1 – Translating into words
There are many ways to write English translations for these processes.
Two typical answers are given here for each question. Your answers
may be slightly different to these. Check that the meaning is the same.
1
Take five from a number or start with a number and take away five.
2
Take a number from five or start with a number and take it away
from five.
3
Multiply a number by seven or start with a number and multiply it by
seven.
4
Divide a number into ten or start with a number and divide ten by it.
5
Divide a number by ten or start with a number and divide it by ten.
6
Multiply a number by twenty, then subtract fifty or start with a
number, multiply it by twenty and then take away 50.
7
Add seven to a number then divide by two or start with a number,
add seven then divide the answer by two.
8
Add three to a number, then multiply by forty or start with a number,
add three then times by forty.
9
Add four to a number, multiply by five then add on the original
number or start with a number, add four, multiply by five then add
on the number you started with.
10 Take the square root of a number, then add five or start with a
number, take the square root of it then add five.
Part 2
Writing algebraic expressions
49
Exercise 2.2 – Translating into algebra
You may have chosen to use different pronumerals than those shown
here. Simply substitute your pronumeral and mark.
1
x +5
2
x−5
3
4x + 7
4
3( x + 7)
5
x−7
6
x ÷ 2 or
7
x × x or x
8x
or 8x ÷ 5
5
x +7
10
11
2
x
8
9
x
2
x − 10
or ( x − 10) ÷ x
x
Exercise 2.3 - The language of mathematics
50
1
a+b
2
6t
3
3q – 8
4
x+y–5
5
x–7
6
a+b+c+d
7
8–k
8
r, r + 1, r + 2
9
d, d + 2, d + 4
PAS4.1 Algebraic Techniques
10 t, t + 10, t + 20 (each number is ten more than the previous)
11 p – 16, p – 8, p
Exercise 2.4 – Algebra in action
b
dollars (It is important to include the
6
dollars otherwise someone might mistake the answer for cents.)
1
Cost of one mug = b ÷ 6 or
2
Perimeter = w + x + y centimetres (It is important to include the
units.)
3
f must be a whole number, and it is likely to be smaller than
50 unless Joshua has a huge tank. If g is a whole number, then it
must be smaller than the number that f is standing for.
1
1
or , or it
4
2
could be a percentage less than 100%, like 25% or 40%.
However, g could be a fraction smaller than one like
You may have also realised that f − g = number of other types of
fish in the tank.
4
a
Area = f × f square metres or Area = f 2 m 2
b
Length of fence = f + f + f + f metres or
Length of fence = 4 f metres
c
Length of fence = 4 f − 2 metres
(The total perimeter less two metres for the gate.)
d
The total cost is calculated by multiplying the number of metres
in the fence by the cost per metre and then adding the cost of the
gate.
Total cost = (4 f − 2) × c + 250 dollars
This can also be written as
Total cost = c (4 f − 2) + 250 dollars
Part 2
Writing algebraic expressions
51
5
The numbers you need to work with are the number of hours John
worked and how much he was paid each hour.
Let h stand for the number of hours worked.
Let p stand for the pay received each hour in dollars.
(You may have chosen to use different pronumerals.)
Total earnings = hp dollars or ph dollars
Exercise 2.5 – Exploring rules
1
2
a
false
b
false
c
true
d
false
e
true
f
false
a
x + 10
b
x − 10
c
10 − x
d
There are three possible answers for this
•
•
•
e
52
x ÷2
x
2
1
×x
2
x
− 10
2
3
p+2
4
y must be an even number
PAS4.1 Algebraic Techniques
We need your input! Can you please complete this short evaluation to
provide us with information about this module. This information will
help us to improve the design of these materials for future publications.
1
Did you find the information in the module clear and easy to
understand?
_______________________________________________________
_______________________________________________________
_______________________________________________________
2
What sort of learning activity did you enjoy the most? Why?
_______________________________________________________
_______________________________________________________
3
Name any sections you feel need better explanation (if any).
_______________________________________________________
_______________________________________________________
4
Were you able to complete each part in around 4 hours? If not
which parts took you a longer or shorter time?
_______________________________________________________
_______________________________________________________
5
Do you have access to the appropriate resources? This could include
a computer, graphics calculator, the Internet, equipment and people
to provide information and assist with the learning.
_______________________________________________________
PAS4.1 Algebraic techniques
53
Centre for Learning Innovation
NSW Department of Education and Training