# Introduction to equations ```Teaching & Learning Plans
Introduction to Equations
Junior Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have and also
to knowledge which may be necessary in order to support them in accessing this new
topic.
Learning Outcomes outline what a student will be able to do, know and understand
having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/or
Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching and
learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing the
topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on teacher input
and gives details of the key student tasks and teacher questions which move the
lesson forward.
ii.
Student Activities – Possible and Expected Responses: Gives details of
possible student reactions and responses and possible misconceptions students
may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
Teaching & Learning Plan:
Introduction to Equations
Aims
• To enable students to gain an understanding of equality
• To investigate the meaning of an equation
• To solve first degree equations in one variable with coefficients
• To investigate what equation can represent a particular problem
Prior Knowledge
Students will have encountered simple equations in primary school. In addition they will
need to understand natural numbers, integers and fractions. They should also be able
to manipulate fractions, have encountered the patterns section of the syllabus, basic
algebra, the distributive law and be able to substitute for example x=3 into 2x+5=11.
Learning Outcomes
As a result of studying this topic, students will be able to:
• gain an understanding of the concept of equality and what is meant by an
equation
• understand the concept of balance (as in a traditional balance or a see-saw)
and how it can be used to solve equations
• gain an understanding of what is meant by solving for an unknown in an
equation
• solve first degree equations in one variable using the concept of balance
Catering for Learner Diversity
In class, the needs of all students whatever their level of ability are equally important. In
daily classroom teaching, teachers can cater for different abilities by providing students
with different activities and assignments graded according to levels of difficulty so that
students can work on exercises that match their progress in learning. For less able
students, activities may only engage them in a relatively straightforward way and more
able students can engage in more open–ended and challenging activities. This will
cultivate and sustain their interest in learning. In this T & L Plan for example teachers
can provide students with the same activities but with variations on the theme e.g.
allow some students to do all the questions in a student activity, while selecting fewer
questions for other students. Teachers can give students various amounts and different
styles of support during the class for example, providing more clues.
In interacting with the whole class, teachers can make adjustments to suit the needs
of students. For example, all students can be asked to solve the equation 3x + 4 = 10,
but the more able students may be asked to put contexts to this equation at an earlier
stage.
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Teaching & Learning Plan: Introduction to Equations
Besides whole-class teaching, teachers can consider different grouping strategies to
cater for the needs of students and encourage peer interaction. Students are also
encouraged in this T & L Plan to verbalise their mathematics openly and share their
work in groups to build self-confidence and mathematical knowledge.
Relationship to Junior Certificate Syllabus
Topic Number
Description of topic
Learning outcomes
Students should be able to
4.5 Equations and
Inequalities
Using a variety of problem
solving strategies to solve
equations and inequalities.
They identify the necessary
information, represent
problems mathematically,
making correct use of
symbols, words, diagrams,
table and graphs.
• consolidate their
understanding of the
concept of equality
• solve first degree
equations in one or two
variables, with coefficients
elements of Z and
solutions also elements
of Z
• solve first degree
equations in one or two
variables with coefficients
elements of Q and
solutions also in Q
Resources Required
A picture that demonstrates a balance, for example the one below
An algebra balance is optional
Whiteboard and markers or blackboard and chalk
Graph paper
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Teaching & Learning Plan: Introduction to Equations
Lesson Interaction
Student Learning Tasks: Student Activities: Possible
Teacher’s Support and Actions
Assessing the Learning
Teacher Input
and Expected Responses
Section A: Introduction to equations and how to solve equations using the concept of balance
»» What do you notice
following?»
6 + 3 = 9»
5 - 3 = 2»
5 + 3 = 1 + 7»
x = 4.
2x = x + x
3x = 2x + x = x + x + x
• The right hand side is equal
to the left hand side.»
»» What does this picture
represent?»
»
»
»
»
»
• A balance or weighing scales
»» Do students recognise
that in order for an
equation to be true
both sides have to be
equal?
»» Demonstrate an algebra balance
if available. Alternatively if no
such balance is available show
the picture of a balance. State
how these balances differ in
appearance from an electronic
balance that students may be
more familiar with.»
»» Do students recognise
when the teacher
speaks of a balance
it is the type in the
picture opposite that
is being referred
to rather than an
electronic balance?
• Both sides are equal.»
• Both sides are balanced.»
• They all have an equals sign
»» What is this apparatus
called (Pointing to an
algebra balance if one
is available)?
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»» Write each of the equations
(opposite) on the board.
Teacher Reflections
»» Relate an equation to a seesaw
if students are happier with the
analogy of a seesaw than that of
a balance.
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Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Support and
Actions
the Law of the Lever in
science and if so what
does it state?
• The weight multiplied by
the length from the fulcrum
is equal on both sides if the
balance is balanced.»
»» Discuss the Law of the
»» Do students understand
Lever and how it works
that for a balance to be
for a balance. (The Law
balanced the weight on
of the Lever states that
the right must equal that
a balance is balanced
on the left provided the
when the distance from
distance from the fulcrum is
the fulcrum multiplied by
the same for both sides?
the weight on that side is
equal for both sides.)
• The balance is balanced if the
weights on either side of the
fulcrum are equal and the
balancing point (fulcrum) is at
the centre.
»» In mathematics we
are going to place the
fulcrum at the centre
of gravity and place
the weights at the
same distance from the
fulcrum on both sides.
Assessing the Learning
»» Draw an empty balance
and show the fulcrum.»
»
»
»
»» Can students verbalise the
Law of the Lever or draw a
diagram to represent it?
»» Do students understand
how a balance of this
nature works?
»» What happens to an
empty balance that is
currently balanced, if we
add a weight to the left
hand side?
• It becomes unbalanced and the
left hand side (the side with
the extra weight) goes down
and the right hand side (the
side without the weight) goes
up.
»» Draw the following
diagram on the board or
demonstrate the action
on an algebra balance.»
»
»
»
to one side, what do we
need to do to the other
side to keep the balance
balanced?
• Add a weight of the same
value to the other side.
»» Do students see that equal
»» Explain how in order
for a balance to remain
or removed from each side
balanced the weights on
of a balance, if the balance
the right hand side have
to equal those on the left
is to remain balanced?
hand side.
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Teacher Reflections
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Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible Teacher’s Support and Actions
and Expected Responses
»» Look at the equation »
2x +5=11.
• It becomes unbalanced.
balance what happens to
the equation 2x + 5 = 11
if we remove the 5 from
the left hand side?
»» How can we restore the
balance keeping the 5
removed from the left
hand side?
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Teacher Reflections
»» Do students see that when
subtracted from one side
of an equation it becomes
unbalanced and in order
for it to become balanced
again the same value must
from the other side of the
equation?
»» Distribute Section A: Student
Activity 1.
»» Circulate to see how students
Make sure all students are
aware of the statement at the
top of the worksheet.
»» Watch out for students trying
to give values to the weights of
the shapes.
Note: all the balances
in these questions are
balanced unless told
otherwise.
© Project Maths Development Team 2011
»» Write an equation on the board
for example »
2x + 5 = 11.
• We must also remove the
5 from the right hand
side.
»» Complete question 1
on Section A: Student
Activity 1.
Assessing the Learning
»» Are students able to
successfully complete the
activities?
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Teaching & Learning Plan: Introduction to Equations
»» How do question 2(a) and 2(c) on the
Student Activity differ?
Student Activities: Possible Teacher’s Support and
and Expected Responses
Actions
• Question 2(a) has x and
question 2(c) has 2x.
»» How would we write the problem in
question 2(c) on the activity sheet as an
equation?»
• 2x = 8
»
»
»» What value of x makes this equation
true?»
•
»» How did you get x=4?
Teacher Reflections
x=4
»
• Divided both sides of the
equation by 2.
»» When you have a problem what do you
try to do?»
• Solve it.»
»
»» So if an equation is a problem, what do
we try to do with it?
• Solve it.
»» Sometimes when given an equation like
2x = 8, rather than saying find the value
of x that makes this equation true, the
question will state solve for x.
•
»» Now make up examples of equations.
• Students work in pairs
making up their own
examples.
»» Complete questions 2, 3 and 4 in
Section A: Student Activity 1.
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»» Do students see the
connection between
having a problem
and solving it and
having an equation
and solving the
equation?
x = 9. Divided both sides
»» Now solve 3x = 27 and write in your
exercise book how you did this.
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Assessing the
Learning
by 3
»» Can students make
up examples of
equations?
»» Did the explanations
individual students to
given to question
explain their solutions.
2, 3 and 4 show
i.e. to verbalise their
understanding?
reasoning.
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Teaching & Learning Plan: Introduction to Equations
Student Activities: Possible and Teacher’s Support and
Assessing the Learning
Teacher Input
Expected Responses
Actions
Section B: Backtracking and writing a simple algebraic equation to represent situations and
Teacher Reflections
how to solve these equations
»» We are now going to
play a game. I want you
to think of a number but
do not tell anyone what
it is.»
»» Subtract 3 from the
then divide by 2. Tell
student A what number
he/she first thought of.
»» Can students work out
for themselves what is
happening?
(Student A.)
»» Student A calls out the number
they thought of.
(Student B)»
• Another student calls out their
»» Why are you getting
• We thought of different
numbers in the first place.
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Teaching & Learning Plan: Introduction to Equations
»» Divide into pairs with partners A and B.»
Student Activities: Possible Teacher’s Support
and Expected Responses
and Actions
Assessing the Learning
»» Do an example if
necessary.
»» Can all students
explain how they
were able to return
to the original
number?
»» If necessary do
an example using
division and
subtraction.
»» Do all students
understand that
subtraction by the
same number are
opposite operations?»
»» A is to think of a secret number between 1
and 10.»
Teacher Reflections
»» B is now to tell A to multiply their secret
number by a certain number and add
»» A is now to share their answer with B.»
»» B is now to calculate the number that A
initially thought of and explain to A how
they were able to do this.»
»» A and B are now to swap roles.
»» Now try problems that involve division
»» Do all students
understand that
multiplication and
division by the same
numbers are opposite
operations?
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Teaching & Learning Plan: Introduction to Equations
Input
Student Activities: Possible and Teacher’s Support and
Expected Responses
Actions
»» What you have been doing is an
action called “backtracking”.
Assessing the
Learning
»» Draw on the board:
Teacher Reflections
RULES FOR BACKTRACKING
Original action
Reverse action
+
x
÷
»» Describe backtracking in your
• Start at the last operation and
do the opposite operation to
what was originally done.»
»» Can students
verbalise what
is happening
when they are
backtracking?»
»» Write a definition of
number to get back to the
original number you must
subtract. If you multiply first
then to get back you divide.
»» Do students
understand
backtracking?
»» Complete question 1 on Section
B: Student Activity 2.
»» Distribute Section B:
Student Activity 2.
Note: if we have an equation of
the type 3x = 15, we refer to the
x as the unknown. 3x and 15
are both terms. Terms without
unknowns (15 in this case) are
called constants.
»»
»»
»»
»»
»» If we have an unknown and
multiply it by 3 we get 3x. If
we then add 5 and this is equal
to 11. Write an equation to
express this.
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»» Can students
complete the
table?
Write on the board:
Unknown
Terms
Constant
»» 3x + 5 = 11
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Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
»» What are the terms in
the equation »
2y + 5 = 11?
• 2y, 5 and 11.
»
»
»» What is the unknown in
this equation »
2y + 5 = 11?
•
Teacher’s Support and
Actions
Assessing the Learning
»» Do students understand
the difference between
unknowns, terms and
constants?
y
Teacher Reflections
»» What are the constants in • 5 and 11
this equation»
2y + 5 = 11?
»» How can algebra help
with question 2 in
Student Activity 2?
• Multiply x the unknown by 3
giving 3x, add 2 giving 3x+2,
this equals 11. So we have the
equation 3x + 2 = 11.
»» Now complete question 2 »» Students should try this
around the class and have a
»» What does it mean to
solve an equation?
• To solve an equation means to
find the value of the unknown
and if the unknown is replaced
by this value the equation is
true.
»» Monitor students'
difficulties.
»» If students appear to be
to talk through their work
so that they can identify
the area of weakness.
»» Do students see that an
equation presents them
with a problem that
requires a solution?»
»» Do students see that
solving the equation
involves finding the
unknown?
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Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Support and
Actions
Assessing the Learning
»» Present one or both of
these methods.»
»» There are two possible
ways to write out the
solution to an equation:»
Teacher Reflections
»» Talk students through
each step of one or both
methods.»
Method 1
»» What is the first step
when solving»
2x + 3 = 11?
• Subtract 3 from each side.»
»» How do we write this?»
• 2x+ 3 - 3 = 11 - 3
»» What is the next step?
• 2x = 8
»» What is our solution?»
»
»
• Divide each side by two.»
»» How do we know if this
value is correct?
• Replace the x in the original
equation with 4 and check if
the equation is true.»
•
x=4
»» Write the following on
the board:
Method 1
»» Let students suggest each »» Do students understand the
step in the solution.
concept of balance as we
solve this equation?
»» Write on the board:
2x + 3 = 11
2x + 3- 3 = 11 - 3
2x = 8
x=4
2(4) + 3 = 11. True.
• 2(4) + 3 = 11. True.
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Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
»» When you get good at
solving the equations
you can abandon the
stabilisers.
»» Once again how do we
check if this value is
correct?
Assessing the Learning
Method 2 Stabilisers
Method»
Method 2 Stabilisers Method
»» Draw lines at the side of
the equation as on the
board. These are referred
to as stabilisers. This is
a similar idea to bike
stabilisers.»
»» When you got good at
riding a bicycle, what
did you do with the
stabilisers?»
Teacher’s Support and
Actions
Teacher Reflections
»» Let students suggest each
action.
- 3»
• Abandon them.
÷2
2x + 3 = 11»
2x = 8»
- 3»
÷2
x=4
»» Replace the x in the original
equation with 4 and check if it
is true.
»» 2(4) + 3 = 11. True.»
»» Emphasise replacing the
unknowns with their
»» Do students understand the
steps involved in each line
of the solution irrespective
of the approach that is
»» Use an algebra balance if
available at this point
»» If this was not true, what
would it tell you?
»» You made an error solving the
equation.»
»» This value does not satisfy the
equation.
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Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible
and Expected Responses
»» Solve the equation»
»
2x + 3 = 7 using either or
both methods.
Teacher’s Support and
Actions
Assessing the Learning
»» Write 2x + 3 = 7 on the
board.»
»» Does students’ work show
their understanding of
solving equations»
»» Check students’ work.»
»» If a laptop and data
projector are available in
the classroom show some
Appendix A page 40.
Note: The first link is very
useful if students are
experiencing difficulty
grasping the basic concept.
»» Complete questions 3-7
Student Activity 2.
»» Parts a and c give the same
»» How are parts a and c of
these questions related?
»» Circulate around the room,
checking if students can
assistance when needed.»
Teacher Reflections
»» Did students get the correct
»» Was students work laid out
properly?»
»» Did students check their
»» Are students using the
correct layout for part c of
the questions?»
»» Can students do
backtracking?»
»» Students have to be
encouraged to replace the
unknown in their equations »» Can students write the
problems as equations?»
in order to check their
solutions.»
»» Can students relate
backtracking to equations?
»» If students are having
difficulty allow them to
talk through their work so
that misconceptions can be
identified.
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Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Support and
Actions
Reflection:
»» Write down what you
today.»
• What it means to solve an
equation»
»» Circulate and take
note of any difficulties
students have noted and
them.
• To find the value of the
unknown that makes the
»» Write down anything you
equation true.»
found difficult today.»
• Do the same to both sides
»» Complete Student
of the equation, to keep it
Activity 2.
balanced.
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Assessing the Learning
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Teacher Reflections
14
Teaching & Learning Plan: Introduction to Equations
Student Activities: Possible
Teacher’s Support and
Assessing the
and Expected Responses
Actions
Learning
Section C: Dealing with variations in layout of equations of the form ax + b = c
Teacher Reflections
Dealing with equations of the form ax + b = cx + d and variations of this layout
»» Ask a student to write a
solution on the board.»
»» How do we solve the equation »
4p + 3 = 11
»» What is the solution of 4y + 3 = 11
•
y=2
»» What is the solution of 4p + 3 = 11
•
p=2
»» Does the unknown always have to be
x?
• No, it can be any letter of the
alphabet.
- 3»
÷4
»» Do students
realise any letter
of the alphabet
can be used for
- 3»
the unknown?
4p + 3 = 11
4p = 8
÷4
p=2
4(2) + 3 = 11 True»
Note: A student may use
Method 1 if this is the
preferred method.
»» What do you notice about these
equations?»
• They are all the same.
»» Do students
appreciate that
these equations
are all the same?
• It would mean that the
solution has to be a natural
number.»
»» Do students
recall what a
natural number
and an integer
is?
3x = 6
6 = 3x
3x = 2 + 4
»» If the question had been written in
the form: solve 2x + 3 = 11, x ∈ N,
what would it mean?»
»» If the question stated solve »
• It would mean that the
2x + 8 = 4, x ∈ Z, what would it mean?
solution has to be an integer.
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Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Support and
Actions
Assessing the Learning
C: Student Activity 3.
»» Students should compare
why the answers are not all
agreeing.
»» Distribute Section C:
Student Activity 3.
»» Are students checking
»» Circulate to check if
students are solving the
equations correctly and
that the layout of their
work is correct.
»» If students are having
difficulty allow them to
talk through their work
so that they can identify
their misunderstandings
and misconceptions.
• The unknown is now on
the right hand side of the
equation.»
• Write the following on
the board:
the equation 9 = 2x + 5 in
comparison to the ones we
have dealt with earlier?»
»» How can we solve the
equation 9 = 2x + 5.
• Students verbalise how to solve
this equation.
»
»
- 5»
÷2
9=2+5
9 = 2x + 5
4 = 2x
»
2=x
Teacher Reflections
»» Do student see this as
being the same as earlier
equations except that the
unknown is now on the
»
right hand side of the
»
equation?
- 5»
÷2
»» Check 9 = 2(2) + 5 True
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Teaching & Learning Plan: Introduction to Equations
Student Learning
Student Activities: Possible and
Expected Responses
Teacher’s Support and Actions
Assessing the Learning
»» What is different
equation?»
•
»» Give students time to solve this
equation and offer assistance
where needed.»
»» Are students gathering
the unknowns to one side
of the equation and the
constants to the other side,
while keeping the equation
balanced?»
2x + 5 = x + 9
»» How do you think
you would solve
this equation?»
x appears on both sides and
there are constants on both
sides.»
»» Give students time to explore
• Bring the terms with x (the
possibilities and to discuss what
unknown) to one side, but
is happening.»
keep the balance and then
bring the constants to the
other side keeping the balance.» »» Encourage students to explain
their reasoning.»
2x + 5 = x + 9
2x + 5 – 5 = x + 9 -5
2x = x + 4
2x – x = x – x + 4
x=4
Note: The stabiliser
method can also be
used if preferred.
• Check 2(4) + 5 = 4 + 9 True
• Students work on questions
chosen from Section C: Student
Activity 3.
Teacher Reflections
»» Are students still using
stabilisers?
»» Write the following on the
board:
2x + 5 = x + 9
2x + 5 – 5 = x + 9 -5
2x = x + 4
2x – x = x – x + 4
x=4
Check 2(4) + 5 = 4 + 9 True
»» Select questions to do from
Section C: Student Activity 3.
»» Circulate and check the
students’ layout of their
»» Are students using a clear
layout for these questions
and doing the calculations
successfully?»
»» Are they differentiating
between the unknowns
and the constants?»
»» Pay particular attention to
students’ work in questions 2, 3,
4, 6, 7, 8, 10, 11 and 12, if these »» Are students using
questions were chosen.
mathematical language in
their discussions?
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Teaching & Learning Plan: Introduction to Equations
Student Learning
Student Activities: Possible and Expected
»» If we get an
equation like
2x + 3 = 5x + 6,
words how you
would solve this
equation.
Teacher’s Support and Actions
• Gather the terms with x (unknown) to one
side and the constants to the other side,
keeping the equation balanced.
Assessing the
Learning
»» Are students’
written
explanations
showing their
understanding
of how to solve
equations?
Teacher Reflections
Section D: Forming an equation given a problem and relating a problem to a given equation
»» Think of a story
represented by
the equation »
4x = 8.
• Mary has 4 times the number of pets she
had last year and she now has 8.»
»» Look for a selection of stories
that this equation could
represent.
• This week John saved four times the
amount of money he saved last week. This
week he saved €8.
»» Can students
develop
appropriate
stories?
• Michael is 4 times as old as Karen. Michael
is 8.
»» Think of a story
represented by
the equation »
4x + 5 = 53.
»» Students compose and compare equations.»
• Think of a number, multiply it by 4, add 5
• A farmer has 4 times the number of sheep
total number of sheep he now has is 53.
© Project Maths Development Team 2011
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18
Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible Teacher’s Support and Actions
and Expected Responses
»» Answer questions 1 12 Section D: Student
Activity 4.
»» Distribute Section D: Student
Activity 4
»» Circulate and check students’
work. Engage students in
Assessing the Learning
»» Are students capable of
forming equations to
represent the problems
posed in Section D: Student
Activity 4?
Teacher Reflections
»» Ask individual students to do
questions on the board. They
should explain why they are
doing each step.
»» Could the following story
be represented by this
equation x + 2 = 25?
“2 new students enter a
class and the class now
has 26 students”.»
• No.»
»» Why?
»» Write the equation and
students’ suggestions on the
board.
• The equation should be
x + 2 = 26 or the problem
should state the class
now has 25 students.
»» How does this differ from • The first situation is x + 2
saying the number of
and the second is 2x.
students double?
»» In pairs develop
problems that could
be represented by the
equations given in
questions 13-16 of the
Student Activity 4. Write
© Project Maths Development Team 2011
»» Can the students relate the
equation to the problem
and can they see that
there is often a different
equation for each problem?
• Students explore Section
D: Student Activity 4.
• Students should compare
and have a discussion.
www.projectmaths.ie
»» Can students verbalise the
difference between 2x and
x + 2?
»» Check the examples that
students are devising for
the questions which can be
represented by the equation.»
»» Can the students develop
problems that could
be represented by the
equations?
»» Allow students to share their
problems.
KEY: » next step
19
Teaching & Learning Plan: Introduction to Equations
Student Activities: Possible and Teacher’s Support and
Assessing the Learning
Teacher Input
Expected Responses
Actions
Section E: To show that equations can also be solved graphically
»» In pairs discuss question
1 of Section E: Student
Activity 5.
»» Distribute Section E:
Student Activity 5.
»» Can 2x = 2x + 1 be solved? • No it is not an equation. The
left hand side does not equal
the right hand side.»
»» Why do you give this
•
y=0
»» Circulate and see what
misconceptions.
»
»
»» Where did the line cut the
x axis?
• At the point (-3, 0).
»» Solve the equation »
x + 3 = 0.
•
»» Do you see any
relationship between
where the line cuts the x
axis and the solution got
by algebra?
• Yes the solution was x = -3 and
the point where the line cuts
the x axis had an x value of -3.
© Project Maths Development Team 2011
»» Are the students’
explanations showing
that they understand why
the equation cannot be
solved?
• The equation is not balanced.
»» In pairs answer question 2
on Student Activity 5.
»» What is the value of x
when the line cuts the x
axis?»
»» In pairs allow students to
discuss question 1 of this
activity.
Teacher Reflections
x + 3 - 3= 0 - 3.
x=-3
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»» Do students see that the
algebraic solution to the
equation will be the x
value of the point where
the line cuts the x axis?
KEY: » next step
20
Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
»» Answer the rest of the
questions on Student
Activity 5.
Teacher’s Support and
Actions
Assessing the Learning
»» Check to see if the
to these questions
demonstrate that they
understand how to solve
equations graphically.
»» Are students able to do
questions 5 and 6 without
referring to what they were
asked to do in the previous
questions?
Teacher Reflections
Section F: To solve equation involving brackets
»» Write 2(3) + 7 = 13 on the »» Do students know what is
board.
meant by trial and error?
»» So we can now solve
equations by algebra and
by graph.»
»» How could you solve
2x + 7 = 13 by trial
and improvement
(Inspection)?»
»» How do you prove that
the solution you got is
correct?
»» That was a simple one.
What about 2x + 5 = -1.
• Try x = 1, if it does not work try
x = 2 and if that does not work
try x = 3 etc.
»
• Substitute x = 3 as follows 2(3)
+ 7 = 13.
• This is more difficult and not as
easy to predict the solution.
•
x=-3
»» So while trial and
improvement (Inspection)
is a possible method of
solving an equation, it is
often very difficult to use
unless the answer is 1, 2,
3 etc.
© Project Maths Development Team 2011
»» Write on the board:
2x + 5 = -1
2(-3) + 5 = -1
-6 + 5 = -1 True
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»» Do the students realise that
it is not sufficient to guess,
but the proposed solution
must be checked using
substitution?
KEY: » next step
21
Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Support and Actions
»» How is the value of »
2(3 + 4) found?
Note: Knowledge of the
distributive law is important
here.
»» Add the 3 and 4 first and then
»» Write on the board
2(7) = 14»
»» We can also have
brackets in an equation
for example: »
2(x + 4) = 18.
• Multiply each term inside the
bracket by 2 and get 2x + 8 = 18
»» How do you think we
would solve »
2(x + 4) = 18?
»» First multiply each number in the
brackets by 4 and then add the
Assessing the
Learning
Teacher Reflections
2(3) + 2(4)
=6+8
= 14
»» Allow students time to adopt an
investigative approach here. Delay
giving the procedure.
• A student may write on the
board:»
– 8»
2x + 8= 18
÷2
2x = 10
– 8»
÷2
x=5
»» How would we solve»
3(x – 2) = 2( x – 4)?
• Multiply each term inside the
bracket by 3 and get 3x – 6.
Multiply each term inside the
other bracket by 2 to get »
2x – 8. Then do what you would
normally do.»
»» Allow students time to adopt an
investigative approach here. Delay
giving the procedure.
»» Do students
understand to
remove the
brackets from
both sides?
• A student may write on the
board:
3x – 6 = 2x - 8
3x – 6 + 6 = 2x – 8 + 6
3x = 2x – 2
3x – 2x = 2x – 2x – 2
1x = – 2
3(–2 –2) = 2(–2 –4) True
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22
Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
»» Can you put in words
what you do if brackets
are present in an
equation?
»» Remove all the brackets by
multiplying out before we start
to solve the equation.
in Section F: Student
Activity 6.
»» Students should compare
have a discussion.
Teacher’s Support and
Actions
Assessing the Learning
Teacher Reflections
»» Distribute Section F:
Student Activity 6.
Circulate and check
students’ work.
»» Are students removing
the brackets before they
commence solving the
equations?»
»» Are students clearly
showing all steps involved
in solving an equation?»
»» Are students continuing to
Reflection:
»» Write down what you
an equation if there are
brackets present.»
»» Remove the brackets and then
solve the equation.
»» Circulate and note any
difficulties or questions
students have.
»» If students are noting
difficulties that they have
allow them to talk through
them so that can identify
for themselves their
misconceptions.
»» Write down any
questions you may have.»
»» Write down anything you
found difficult today.
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23
Teaching & Learning Plan: Introduction to Equations
Student Learning Tasks: Student Activities: Possible
Teacher’s Support and Actions
Teacher Input
and Expected Responses
Section G: To solve equations involving fractions
Assessing the
Learning
»
»» Write the answer on the board.»
»
»
»
»
»
»
»» Do students
remember how
fractions?
»» Allow students time to adopt an explorative
approach here. Delay giving the procedure.»
»» Are students
extending their
knowledge
fraction?»
• Get a common
denominator, which is the
Least Common Multiple of
2 and 3 and is equal to 6.»
Note: Allow students to
articulate and explain how to
»» Equations can also
involve fractions for
example:»
»
© Project Maths Development Team 2011
»» Write the equation and its solution on the
board as it evolves:»
Solve the equation:
»
»
»
»
»
»
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KEY: » next step
Teacher Reflections
Note: Students
are more likely
to learn with
understanding
if they have
tried to extend
their existing
knowledge rather
than be prescribed
a “rule” to follow
from the start.
24
Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
Teacher’s Support and Actions
»» Let’s compare our
and see if we agree or
not.
• Students offer their solutions
and explain how they arrived
at them.
»» Write varied solutions on the
board and allow students to
talk through their work so
that they can identify areas of
misconceptions.
question 1 in Section G:
Student Activity 7.
»» Distribute Section G: Student
Activity 7.
»» Circulate and check students’
Assessing the
Learning
Teacher Reflections
»» Are students
getting the
correct common
denominator and
getting the correct
solutions?
»» Ask individual students to do
questions on the board when the
class have done some of the work.
Students should explain what
they are doing in each step.
»» Circulate and check students’
work ensuring that all students
»» In pairs do questions 2-12
from Section G: Student
Activity 7. The equations
formed from these
questions will mostly be
in fraction format.
© Project Maths Development Team 2011
»» Are students
forming the
correct equations
and solving them?
»» Ask individual students to do
questions on the board when the
class have done some of the work.
Students should explain what
they are doing in each step.
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KEY: » next step
25
Teaching & Learning Plan: Introduction to Equations
Student Activities: Possible
Teacher’s Support and
Assessing the Learning
Teacher Input
and Expected Responses
Actions
Section H: Note the activities to date are for students taking ordinary level in the Junior
Teacher Reflections
Certificate where the variables and solutions are elements of Z. For students taking higher
level in the Junior Certificate the variables and solutions can be elements of Q. Hence students
taking higher level will need to cover the following activities. Students taking ordinary level
can progress to the Reflection section of this T&L Plan.
»» Give examples of numbers
that are elements of Z?
• -3, -2, -1, 0, 1, 2, 3, 4 etc.»
»
»
»» What is another name
for the numbers that are
elements of Z?
• Integers
»» Give examples of numbers
that are elements of Q?
• ½, ¼, ¾ , etc»
»
»
»» What is the name for
numbers that are elements
of Q?
• Fractions»
»
»
»» Are negative and positive
whole number elements of
Q?
• Yes
»» Solve the equations that are
on the board.
• Students solve the
equations.
© Project Maths Development Team 2011
»» Are students recognising
the differences between
an integer and a rational
number?»
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»» Can students verbalise
the differences between
natural numbers, integers
and rationals?
»» Write the following
equations on the board:
2x + 5 = 8
3x – 7 = 17
3x/5 = 13
2.5x = 45
1.5x + 3 = 22
KEY: » next step
26
Teaching & Learning Plan: Introduction to Equations
Teacher Input
Student Activities: Possible and
Expected Responses
»» Complete the questions
on Section H: Student
Activity 8.
Teacher’s Support and
Actions
Assessing the Learning
»» Distribute Section H:
Student Activity 8.
»» Are students using a clear
layout for these questions
and doing the calculations
successfully?
»» If students are having
difficulties allow them
to talk through them
so that can identify
their misconceptions for
themselves.
Reflection:
»» Work in groups and
summarise what you
solving an equation and
solutions.
• Both sides of a balanced
equation are equal.»
• When solving an equation
you must perform the same
operation to both sides of an
equation.»
• To solve an equation means to
find a value for the unknown
that makes it true.»
»» Circulate the class,
necessary and listen to
students’ conclusions.
Teacher Reflections
»» Do students know how to
solve equations and what is
meant by this action?»
»» Do students understand the
terms:
• Equation
• Solve
• Solution?
• The solution is the value that
makes an equation true.
»» Make a list of key words
you have learned and
write an explanation for
each word.
© Project Maths Development Team 2011
• Students write the key words
into their copybooks and an
explination of each one.
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»» Can students write
explanations for these
words or verbalise this to
the class?
KEY: » next step
27
Teaching & Learning Plan: Introduction to Equations
Section A: Student Activity 1
1. Describe the balances labelled a, b, c and d below in two ways:
(i) using words and
(ii) using mathematical symbols.
(i) Words: the weight of three spheres is balanced by the weight of one cylinder
(ii) Symbols: 3s =c
(Assume all balances in these questions are balanced unless told otherwise)
a. b. d. c. 2. What can you tell about the value of x or y in the following
a. b. c. d. e. f. 3. If we know this balance is not balanced, what number can x not
be?
4. If x = 8, what will we do to achieve balance?
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28
Teaching & Learning Plan: Introduction to Equations
Section B: Student Activity 2
1. Complete the table of rules for backtracking.
RULES FOR BACKTRACKING
Original action
Reverse action
+
x
÷
2. John thinks of a number, multiplies it by 3 and adds 2 to his
a. Using backtracking, what number did he think of?
b. Write an equation to represent this problem.
c. Solve the equation.
3. Sarah thinks of a number, multiplies it by 4 and adds 5 to her
a. Using backtracking, what number did she think of?
b. Write an equation to represent this problem.
c. Solve the equation.
4. Dillon thinks of a number, multiplies it by 3 and subtracts 5 from
his answer. The result is 7.
a. Using backtracking, what number did he think of?
b. Write an equation to represent this problem.
c. Solve the equation.
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29
Teaching & Learning Plan: Introduction to Equations
Section B: Student Activity 2 (cont.)
5. Saoirse thinks of a number and divides it by 2 and adds 5 to her
a. Using backtracking, what number did she think of?
b. Write an equation to represent this.
c. Solve the equation.
6. Susan thinks of a number and divides it by 3 and subtracts 5 from
her answer. The result is 14.
a. Using backtracking, what number did she think of?
b. Write an equation to represent this.
c. Solve the equation.
7. Solve the following equations and check solutions (Answers):
a. 2x = 4
b. 3x + 1 = 13
c. 5x - 4 = 21
d. 4x - 4 = 44
e. 11x - 5 = 39
f. 3x - 4 = 11
© Project Maths Development Team 2011
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30
Teaching & Learning Plan: Introduction to Equations
Section C: Student Activity 3
1. Solve the following equations and check solution which will be a
natural number in each case:
a. 2x = 8
d. 2s + 1 = 9
c. 40z = 360
f. 5r - 8 = 17
b. 40y = 160
e. 2t - 1 = 7
g. 2x - 9 = -1
h. 2y - 15 = 31
i. 1 - 2c = -5
j. 8d -168 = -16
2. Solve the following equation 4s + 7 = 19, x ∈ N.
3. Does the equation 6x + 12 = 8, x ∈ N have a solution? Explain.
4. Does the equation 6x + 12 = 8, x ∈ Z have a solution? Explain.
5. Is x = -1 a solution to the equation 2x + 10= 8? Explain your answer.
6. Is x = 4 a solution to the equation 2x + 5 = 10? Explain your answer.
7. Is x = 2 a solution to the equation –x + 3 = 1? Explain your answer.
8. Examine this student’s work. What do you notice?
3x + 6 = 21
3x + 6 – 6 = 21
3x = 21
x=7
© Project Maths Development Team 2011
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31
Teaching & Learning Plan: Introduction to Equations
Section C: Student Activity 3 (cont.)
9. Solve the following equations and check your solutions which will
be an integer in each case:
a. 3x - 7 = 2x
b. 4t + 6 = 2t
c. 1 + 2c = 7
d. 42 = 7 – 5c
e. -42 = 5m – 7
f. –p = 72 + 2p
g. 6 -3k= 0
h. -9y = -y - 48
i. j. 10.Is x = 9 a solution to the equation 5 - 2x = -13?
11.Is r = 2 a solution to the equation –6r + 3 = r?
12.Examine this student’s work. What do you notice?
5 – x = 21
5 – 5 – x = 21 - 5
-x=16
13.Is t = 4 a solution to the equation 5t - 2 = 3t - 3? Explain your
14.Is x = -2 a solution to the equation –6x + 3 = -x + 13? Explain your
15.Examine this student’s work. Spot the errors, if any, in each case.
Student A
Student B
Student C
4x + 4 = 6x - 6
4x + 4 = 6x - 6
4x + 4 = 6x - 6
4x = 6x - 2
4x = 6x - 10
4x = 6x - 10
- 2x = -2
2x = - 10
- 2x = - 10
x=1
x=-5
x=5
4x + 4 - 4 = 6x - 6 + 4
4x - 6x = 6x - 6x - 2
2x = 2
4x + 4 - 4 = 6x - 6 - 4
4x - 6x = 6x - 6x - 10
2x = - 10
4x + 4 - 4 = 6x - 6 - 4
4x - 6x = 6x - 6x - 10
2x = 10
4(5) + 4 = 6(5) - 6 True
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32
Teaching & Learning Plan: Introduction to Equations
Section D: Student Activity 4
1. Brendan thinks of a number, adds 3 and the answer is 15. Represent this statement
2. Joanne thinks of a number then subtracts 5 and the answer is 10. Represent this
3. A farmer has a number of cows and he plans to double that number next year,
when he will have 24. Represent this statement as an equation. Solve the equation
4. A new student enters a class and the class now has 25 students. Represent this
5. The temperature increases by 18 degrees and the temperature is now 15. Represent
this statement as an equation. Solve the equation and check your answer.
6. A farmer doubles the amount of cows he has and then buys a further three cows.
He now has 29. Represent this as an equation. How many did he originally have?
7. Emma and her twin brother will have a total age of 42 in 5 year’s time. Represent
this as an equation. How old are they at the moment?
8. A table’s length is 6 metres longer than its width and the perimeter of the table
is 24 metres. Allow x to represent the width of the table write an equation to
represent this information and solve the equation to find the width of the table.
x
9. Mark had some cookies He gave half of them to his friend John. He then divided his
remaining cookies evenly between his other three friends each of whom got four
10. Chris has €400 in his bank account and he deposits €5 per week thereafter into his
account. His brother Ben has €582 in his account and withdraws €8 per week from
his account. If this pattern continues, how many weeks will it be before they have
the same amounts in their bank accounts?
11. The sum of three consecutive natural numbers is 51. What are the numbers?
12. A ribbon is 30cm long and it is cut into three pieces such that each piece is 2cm
longer than the next. Represent this as an equation? Solve the equation to discover
how long each piece of ribbon is.
13. Write a story that each of the following equations could represent:
i. 2x = 10
ii. 2x + 5 = 11
iii. 3x – 5 = 13
iv. 3x – 5 = 2x + 13
© Project Maths Development Team 2011
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33
Teaching & Learning Plan: Introduction to Equations
Section E: Student Activity 5
1. Can you solve the equation 2x=2x+1? Why or why not?
2.
a. Make a list of 4 points on this line.
b. What is added to each x to give the
y value?
c. So is it true to say the line has equation y = x + 3?
d. Solve the equation x + 3 = 0 by algebra.
e. Can we read from the graph the point where y = 0 (or x + 3 = 0)?
f. Do you get the same answer when you
graph the line y = x + 3 and find where it
cuts the x axis as you get when you solve
the equation x + 3 = 0 by algebra?
3. Complete the following table and draw the resulting line on graph
paper.
x
y = 2x + 2
-2
-1
0
1
2
3
a. Where does the line y = 2x + 2 cut the x axis?
b. What is the x value of the point where this line cuts the x axis?
c. Solve the equation 2x + 2 = 0 using algebra.
d. Do you get the same answer for the x value of the point where the line y = 2x + 2
cuts the x axis and from solving the equation 2x + 2 = 0 using algebra?
© Project Maths Development Team 2011
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34
Teaching & Learning Plan: Introduction to Equations
Section E: Student Activity 5 (cont.)
4. Complete the following table and draw the resulting line on graph
paper.
x
y = 2x - 1
-2
-1
0
1
2
3
a. Where does the line y = 2x -1 cut the x axis?
b. What is the x value of the point where this line cuts the x axis?
c. Solve the equation 2x – 1 = 0 using algebra.
d. Do you get the same answer for the x value of the point where the line y =2x -1
cuts the x axis and from solving the equation 2x -1 = 0 using algebra?
5. Given the table below find the solution to the equation 2x-3=0.
x
-3
-2
-1
0
1
2
3
2x - 3
-6
-5
-4
-3
-2
-1
0
6. Solve the equation 2x - 6 = 0 graphically.
7. Solve the equation x + 5 = 0 graphically.
8. Describe in your own words how to solve an equation graphically.
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35
Teaching & Learning Plan: Introduction to Equations
Section F: Student Activity 6
1. Solve the following equations and check your solutions:
a. 3(y – 2) = 3
b. 4(x - 2) = 8
e. 4(x - 1) + 3(x - 2) = 4
f. 4(p + 7) + 5 = 5p + 36
c. 2(4 - x) = 6x
d. 5(t - 2) + 6(t - 3) = 5
g. 5(q - 4) + 12 = 3(q - 3)
i. 2(s - 1) + 3(s - 3) + s = 1
k. 2(d + 3) + 3(d + 4) = 38
h. 2(x + 3) - 3(x + 2) = - 2
j. 3(x + 1) - (x + 5) = 0
l. (x + 1) + 5(x + 1) = 0
2. Is y=5 a solution to the equation 2(y - 4) + 5 = 3(y + 2)? Explain your answer.
3. Is y=2 a solution to the equation (y - 4) + 6 = 3(y + 2) - 7? Explain your answer.
4. a. These students each made one error, explain the error in each case.
Student A
Student B
Student C
2x + 3 – 7 = 3x – 9 + 4
2x + 6 – 7 = 3x – 9 + 12
2x + 6 – 7 = 3x – 9 + 4
2x– 1 +1 =3x + 3 + 1
2x – 1 + 1 = 3x – 5 - 1
2(x + 3 ) -7 = 3(x - 3) + 4
2x – 4 = 3x - 5
2x – 4 + 4 = 3x - 5 + 4
2x – 1 = 3x + 3
2(x + 3) -7 = 3(x - 3) + 4
2x – 1 = 3x - 5
2x = 3x - 1
2x = 3x + 4
2x = 3x + 6
-1x = -1
-1x = 4
-1x = 6
2x - 3x = 3x - 3x - 1
x=1
2(x + 3) -7 = 3(x - 3) + 4
2x - 3x = 3x - 3x + 4
x = -4
2x - 3x = 3x - 3x + 6
x=6
b. Solve the equation correctly showing all the steps clearly.
5. Mary is 5 years older than Jack. Twice Mary’s age plus 3 times Jack’s age is 125.
Write an equation to represent this information and solve the equation to find
Mary’s age.
6. The current price of an apple is x cents. The price of an apple increases by 4 cents
and Alan goes to the shop and buys 4 apples plus a magazine costing €2. His total
bill came to €4.44.
7. Half of a number added to a quarter of the same number is 61. Write an equation to
represent this information. Solve the equation to find the number?
8. Erica went shopping. She spent a quarter of her money on books, half of her money
on shoes and €5 on food. She had €12 left. Write an equation to represent this
situation. Solve the eqution to find how much money she had at the beginning of
the day?
a. Write an equation in terms of x to represent her total bill in cents?
b. Solve the equation. What does the answer tell you?
© Project Maths Development Team 2011
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36
Teaching & Learning Plan: Introduction to Equations
Section G: Student Activity 7
1. Solve the following equations and check your solutions:
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
p.
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37
Teaching & Learning Plan: Introduction to Equations
Section G: Student Activity 7 (cont.)
2 Martha has a certain number of sweets in a bag and she gives half to Mary and
Mary gets 20. How can this be represented as an equation? Solve the equation and
3 A father is x years of age and is twice the age of his daughter, who is now 23. Find
an equation in terms of x to represent this situation and solve the equation.
4 There are three generations in a family: daughter, mother and grandmother. The
daughter is half the age of the mother and the grandmother is twice the age of the
mother. The sum of their ages is 140. Write an equation to represent this situation
and solve the equation to find the ages of each member of the family.
5 A carpenter wished to measure the length and width of a rectangular room, but
forgot his measuring tape. He gets a piece of wood and discovers the length of the
room is twice as long as the piece of wood and the width of the room is half that
of the wood. The owner says that the only information he can remember about
the room is that its perimeter is 50 metres. Write an equation to represent this
information, letting x equal the length of the piece of wood. Solve the equation
6 Jonathan is half Jean’s age and Paul is 3 years older than Jean. Given that the sum
of their ages is 43, write an equation to represent this situation and solve the
equation. What age is each person?
7 A student took part in a triathlon which involved swimming, running and cycling.
He spent ½ the time swimming that he spent running and 3 times the time cycling
as he spent running. His total time was 45 minutes. Write an equation to represent
this situation. Solve the equation and state how long he spends at each sport.
8 Kirsty has just bought a new outfit consisting of a skirt, a shirt and shoes. She will
not tell her mother the cost of the shoes, but her mother knows she spent all her
pocket money of €220 on the outfit. Through a series of questions her mother
discovers that she spent 4 times the amount she spent on the skirt on the shoes and
she spent half the amount she spent on the skirt on the shirt. Write an equation to
represent this information and find the cost of the shoes using this equation.
9 There are x chocolate buttons in a bag. Dan ate 6 chocolate buttons. Eamon then
ate a quarter of the remaining chocolate buttons in the bag. There were now 90
chocolate buttons left in the bag. Write an equation to represent this information
and solve the equation to find the number of chocolate buttons originally in the
bag?
10 Simplify 11 Simplify © Project Maths Development Team 2011
and hence solve for x.
and hence solve for www.projectmaths.ie
for x.
38
Teaching & Learning Plan: Introduction to Equations
Section H: Student Activity 8
Higher level only
Solution may be elements of Q.
1 Solve the equation 2x = 9.
2 Solve the equation 2x – 5 = -3x – 7.
3 Solve 4 Solve the equation 2(x - 3) - 3(x - 2) = 15.
5 Solve the equation 5(x - 5) - 3(x - 2) + 4 = 0.
6 Solve the equation 7 An electric supplier has a fixed charge of €48 for every two months and also
charges 9 cent per unit of electricity used.
(a). Write an equation to represent this information. (b). The Gallagher family got a bill for €77.97 for the last 2 months. Use your
equation to find how many units of electricity they used during this period .
8 3 is taken from a number and the result divided by 4. This is then added to half of
the original number giving an answer of 47. Find the original number?
9 Julie went shopping. She spent one sixth of her money on books, an eight of
her money on shoes and €5 on food. She had €13.50 left. Write an equation to
represent this information. Solve the equation to find how much money she had at
the beginning of the day?
10 The difference between a half of a number and a third of the same number is 34.5.
What is the number?
11 The difference between one third of a number and 2 sevenths of the same number is Find the number.
© Project Maths Development Team 2011
www.projectmaths.ie
39
Teaching & Learning Plan: Introduction to Equations
Appendix A
Internet sites that will aid the teaching of this topic:
http://nlvm.usu.edu/en/nav/frames_asid_201_g_4_t_2.htm
l?open=instructions&from=category_g_4_t_2.html