Percentages

Percentages
Teaching & Learning Plans
Percentages
Junior Certificate Syllabus
Leaving Certificate Syllabus
Created by Teachers
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.
Checking Understanding: 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:
Percentages
Aims

To build on the concept of percentage encountered in primary school

To show the equivalence of fractions, decimals and percentages
Prior Knowledge
Students have prior knowledge of integers and operations on integers, fraction concepts and
operations on fractions, decimal concepts and operations on decimals and have met the
concept of percentage in primary school.
Learning Outcomes
As a result of studying this topic, students will be able to

understand that percentages are parts out of 100 i.e. fractions whose denominators are
100

give the equivalent fraction, decimal or percentage given any one of a percentage,
decimal or fraction

work out and become familiar with the percentage and decimal equivalents of “common”
fractions such as ½, quarters, fifths, thirds, eights

calculate percentages in real life contexts

look for the part, the whole and the percentage in problems on percentages and model the
problem using a double number line

find any percentage of a given whole using their knowledge of fraction multiplication

increase/decrease a given whole by a percentage
2009
© Project Maths Development Team 2010
www.projectmaths.ie
1
Teaching & Learning Plan: Percentages
Relationship to Junior Certificate Syllabus
Topic

3.1 Number
Systems
Description of topic
Learning outcomes
Students learn about
Students should be able to

Algorithms used to

calculate percentages
solve problems

use the equivalence of
involving fractional
fractions, decimals and
amounts.
percentages to compare
proportions
Relationship to Leaving Certificate Syllabus
Sub-Topic
Learning outcomes
Students learn
about
Students
working at FL
should be able
to
 3.3 Arithmetic
 calculate
In addition
students
working at OL
should be able
to
In addition
students
working at HL
should be able
to
percentages
 use the
equivalence of
fractions,
decimals and
percentages to
compare
proportions
© Project Maths Development Team 2010
www.projectmaths.ie
2
Teaching & Learning Plan: Percentages
Resources Required
Hundredths disc, 10 X 10 grid, mini whiteboards
Introducing the Topic
Students have met percentages before but mainly from a procedural point of view and
may have some misconceptions........
The following examples could be used to explore misconceptions:

What is the overall percentage change if an item is increased by 50% and then 2 days
later its price is decreased by 50%?

If an item is increased by 50% and two days later is decreased by 20% - what is the
overall percentage increase or decrease

If students answer incorrectly they can reconsider their answers at the end of the class in
light of what they have learned
Real Life Context
The following examples could be used to explore real life contexts.

Percentage increase and decrease on clothes, food, population, numbers applying for the
CAO, popularity of political figures, share prices
3
Teaching & Learning Plan: Percentages
Lesson Interaction
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Checking Understanding
Connection between percentages, decimals and fractions and converting between the different formats.
» How are decimals related to
fractions?
 Decimals are fractions where
the denominator is a multiple
of 10.
 1/10 = 0.1, 1/100 = 0.01 etc.
» When you write a number as
decimal e.g. 2.34 what does
the decimal point indicate?
 The number to the left of the
decimal point is the number
of units and as you go to the
right of the decimal point the
value of the place decreases
by a factor of 10 each time
you move one place to the
right.
» Does anyone know what a
percentage or percent is?
 Per cent means per hundred.
Percentages are fractions
whose denominator is 100
but the denominator is left
out and the symbol % is put
beside the number.
» Remind students of their
work in primary school and
that we will be building on
that work.
» Are many students
volunteering to answer this
question and can they
articulate the concept with
examples? This will
determine how fast the
lesson will proceed.
 It means 1 part out of 100
» Give each student a 10X10
grid. Plastic transparencies
may be used with the grids.
» Do students realise that
each small square on the 10
x 10 grid represents 1%?
» The sign % is derived from
“/” which means “out of” and
the two zeros for 100.
» What does 1% mean?
» Show it on the 10 x 10 grid.
» What does 10% mean?
Outline what it means on
your grid.
parts i.e.
1
100
 It means 10 out of 100 or
1/10
» Ask a student to show it to
you on the grid
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4
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
» What does 12% mean?
Show it on the 10 x 10 grid.
 It means 12 parts out of 100
» Walk around and see if
students are able to show it
on the grid.
» What is the number of parts
and what is the whole/unit?
» Could you write this as a
decimal?
» What one word will I change
when I want to write 12
hundredths as a
percentage?
 It is 12 parts out of 100 parts
»
parts i.e.
12
100
 0.12 = 12 hundredths
 Change hundredths to
percent
» Are students equating the
word hundredths with
percent?
» Are students able to produce
an equivalent fraction?
Can you find an
equivalent fraction to
12
100
which uses smaller
numbers?
» If we were given the fraction
Checking Understanding
12  2
6
3


 100  2 50 25
 We need
3
of 100 parts.
25
3
how would we convert it
25
 We need to know how many
to a percentage?
» What do we want 3/25 of?
» What operation is involved
when you use the word “of
“here?
hundredths it is equivalent to.
 It means multiply
3
300
 100 
25
25

300  5 60



 12%
25  5
5
» Give more examples of this
if necessary or students may
be able to proceed to
Student Activity 1.
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5
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
» Can you suggest another
way of doing this using
equivalent fractions?
 1/25 = 4 out of 100 parts and
3/25 is 12 out of 100 parts
» What is
1
as a percentage?
5
Teacher’s Support and
Actions
Checking Understanding
» Have student been able to
come up with different ways
of working this out?
3
3 4
12
=12%


25 25  4 100
 It is
1
of 100 parts =
5
1
100  20 parts out of 100
5
» Encourage students to use
the grid where they are
having difficulty working this
out.
= 20%
» Read the following as
percentages.
0.75
0.258
0.453
» In other works how many
hundredths in each number?
» Read the following as a
percentage:
0.21
» Have you heard of this
percentage in use?
» Could we write 75
hundredths (75%) any other
way?
 0.75 = 75 hundredths = 75%
 0.258 = 25.8 hundredths =
25.8%
 0.453 = 45.3 hundredths =
45.3%
 0.21 = 21 hundredths = 21%
» Write these numbers on the
board.
» Do students remember that
the second decimal place
represents hundredths and if
we position the decimal
point directly after it, the unit
is hundredths?
» Are students aware that
VAT on goods is 21%
(since 01 January 2010) ?
 It’s the rate of VAT on goods
75  25 3

 100  25 4
» Are students aware that
fractions can look quite
different to their
corresponding percentage
formats?
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6
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Checking Understanding
 Students fill in the activity
and use the discs and grids
where they have any
difficulties.
» Distribute hundredths disc
» Students should keep this
sheet with the 10 x10 grid as
a reference.
» Are students able to convert
between fraction, decimal
and percentage equivalent
forms?
 Some students may have
difficulty with converting 1/3
to a percentage.
» Refer students back to the
activity where they changed
1/3 to a decimal using the
10 x 10 grid.
Student Activity 1
» Fill in the table on Student
Activity 1.
Use the 10 x10 grid and
hundredths disc to help you.
» When you have filled it in
give each other a quiz –
give a fraction/decimal
/percentage and your
partner has to give the
equivalent fraction/decimal
or percentage.
» Which if any did you find
difficult?
Student Activity 2: Percentage Dominoes
» Now using the percentage
dominoes, match each
picture with its percentage
equivalent.
» When finished, each group
is to check with the group
beside them to see if they
have got the same matches.
» Which ones, if any, did you
find difficult? Why?
 Students set up the
dominoes so that one follows
on from the other i.e. the
percentage on one card
follows the percentage
shaded on the previous card.
» Distribute bags with the
percentage dominoes and
circulate to check that
students can match up
correctly.
 The ones which were not
part of a 100 squares picture
– we had to convert to
equivalent fractions with 100
as the denominator.
» Listen to conversations
students are having in
justifying their choices and
ask questions which will
help to clarify difficulties.
» Were students able to
convert percentages to
equivalent fractions when
the fractions were not part of
a “hundredths” square?
Student Activities 3 and 4 on equivalence of fractions, decimals and percentages
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7
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Student Activity 5: Mental calculations using simple percentages
» Work out these problems in
»
 Take off half of 18 /you only
pairs, without the aid of a
pay half i.e. €9
calculator and try to work
them out mentally.
» 1. There is a sale on DVDs
in a local shop – 50% off
selected items. Your
favourite DVD before the
sale cost €18.
How much is being taken
off?
What will it cost you now?
» 2. Another DVD in the shop,  25% is ¼ which is half of
also costing €18 before the
half.
sale, has 25% off. How
 €4.50 off. Cost 18.00 -€4.50
much is being taken off?
= 18-4-0.5=13.50
How much will it now cost?
» 3. The shop next door has a  10% is 1/10
»
clothes sale. A pair of jeans  1/10 of €30 is €3.
which cost €30 has 10% off.  Selling price: €30 -€3 =€27
How much money is being
taken off the price?
What is the selling price?
» 4. Do you think 10% sale is
»
 Students may feel it is
a good saving?
insignificant.
» 5. A house which costs
250,000 last year has
decreased in price by 10%.
How much will you save on
last year’s price by buying
now?
 1/10 of 250,000 = 25,000
saving!
 How significant 10% is
depends on what the whole
is.
Circulate and check that
students are not using
calculators and are able to
figure out the answers
through an understanding of
percentages?
Checking Understanding
» Can students work out these
problems without a
calculator and as far as
possible without pen and
paper to develop a sense of
familiarity with common
percentages?
Ask students to show
answers on mini
whiteboards. Ask students
to explain their answers.
Ask for opinions and record.
Review after the next
problem.
» Will some students say that
it depends on what you are
getting 10% of?
» Do all students now realise
a judgement on the
significance of 10% is not
possible unless you know
what you are getting 10%
of?
KEY: » next step
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8
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
» 6. Your local supermarket
has a special price reduction
this week. When you go to
the cash register you are
offered a choice”10% off or
€10 off.”
Should your decision be the
same in the following
situations:
(i) Your shopping amounts
to €70
(ii) Your shopping amounts
to €120
(iii) your shopping amounts
to €200
(iv) Your shopping amounts
to €11
» What should your strategy
be?
» Have you any advice for the
manager?
» 7. OPEC decides to raise
the cost of a barrel of crude
oil by 40%. The current cost
is $45. What will the new
cost be?
 Under €100 the €10 is the
one to choose.
 Customers who bought their
shopping in lots of €10 would
get it free!
Teacher’s Support and
Actions
Checking Understanding
» Given whole and %, find the
part.
» Allow students to come up
with their own scenarios.
» Do some students initially
say that it makes no
difference?
» Do students see the
difference between fixed
discounts and percentage
discounts?
 The manager needs to
perhaps ensure that you can
only avail of this offer once
 10% of 45 is €4.50. 40% is
4x 4.5 = 4x4 +
 4 x 0.4=16 +2 =$18
 New cost is $45+ $18 = $(45
+20-2)=$63 or 40+10+5+3
etc
» Given the whole and the %
find the part.
» Encourage students to use
10% first.
» Encourage students to add
by deconstructing numbers.
No calculators!
» Are students able to use
10% to find other
percentages without having
to convert them to fractions
or decimals? Can they
decompose a percentage so
that it can be calculated
mentally/ without a
calculator?
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9
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Checking Understanding
» 8. Kay visits her local
hardware shop and sees
that there is 50% off a
kitchen whose list price is
€5000. Two days later the
same kitchen has a sign on
it “Further reduction of 20%
on last week’s price”. She
visits the shop again at the
weekend and sees that
there is a new sign on the
kitchen “Weekend special –
further reduction of 10%”.
She decides to buy - what
will she pay?
» Is she getting 80% off?
 50% of €5000 = €2500
Kitchen now costs €2500.
20% = 1/5 (or twice 10%)
 20% of 2500 = 500
 Kitchen now costs 2000
 10% of 2000 = 200
 Kitchen now costs 1800.
» Ask students for answers on
the mini whiteboards.
» Take answers from
individual students and write
the answers on the board.
Ask for class approval and
discuss points arising.
» If a student suggests taking
80% of the original price off
– this must be brought to a
whole class discussion to
see why it is incorrect.
Again as with fractions place
emphasis on the “whole”.
» Are students recognising
what the “whole” is in these
situations?
» 9. Kay finally bought a
kitchen which originally cost
€5000 for €1800. How much
did she save?
» What fraction of the original
price was this?
» What % of the original price
was this?
» If you were the shopkeeper
would you tell your
customers 50%, then 20%,
then 10% or just say 64%
off? Explain.
 She saved €3200
» Ask students who originally
thought she was getting
80% off to explain this.
» Do students who had
misconceptions now
understand why their
original answer was not
correct?
 80% of 5000 = 8x10%= 8x
500= 4000
 If 80% off, kitchen cost is
1000 or if 80% off you only
pay 20% which is 1/5 which
is 1000.
 This is
3200 32 16
of


5000 50 25
the original price.

32 64
=64%

50 100
 At a glance it might look like
80% if you say it the first
way, and appear more
attractive to the customer.
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10
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Checking Understanding
» 10. Last year the local IT
accepted 120 students for
its science course. Due to
increased interest in the
course, the number of
places this year has been
increased by 15%. How
many students can enrol for
the course this year?
» 11. Shares in a
pharmaceutical company
were worth $100 each. The
share price fell by 95%
when one of the company’s
drugs caused patients to
become ill.
» What were the shares now
worth?
» Next day the share price
rose by 100%.
» What was the share price
after these two days of
trading?
» Thomas has $1000 worth of
shares in the company.
What was the value of his
stock after these two days of
trading?
» Does anyone know of a
similar situation?




» Encourage students to
break this down into 10%
and 5%.
» Are students able to work
this out without the aid of
calculators?
» Ask students to give current
examples of falls in share
prices or to look them up on
the internet or in the
business section of
newspapers. Encourage
discussion on this.
» Do students see that the
inverse of a decrease by a
percentage is not an
increase by the same
percentage and that they
must always check to see
what the “whole” is?
10% of 120 = 12
5% of 120 =6
15% of 120 =18
120+18 = 138
 Shares now worth only 5% of
100 =$5
 100% of $5 (x 100/100)= $5
 On 100% increase shares
now worth $10 each.
 Every original $100 worth of
shares is now worth $10.
 $1000 = 10 shares of $100
each which have become 10
shares worth $10 each =
$100
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11
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Checking Understanding
 Estimate 1: 62% is more
than 50%. 50% is 62,000 so
the estimate is more than
62,000
 Estimate 2: Choose “nice”
near percentages and car
sales figures.
 62% is close to 66 2/3 %
which is 2/3.
 124,000 is reasonably close
to 120,000 which it is easy to
get 2/3 of.
 1/3 of 120,000 = 40000.
 2/3 of 120,000 = 80000
or
 10% of 120,000 =12,000
and 60% is 72,000
 1% is 1240
 2% is 2480
 10% is 12400
 60% is 6 x12400 = 72,000+
2400=74,400
 62% = 74400+2480=76,880
» Allow students to use a
calculator to do the exact
value only after they have
made an estimate without
the calculator.
Estimating percentages
» 12. According to SIMI, the
Society of the Irish Motor
Industry 124,000 cars were
sold in the first six months of
2008. There was a drop of
62% on this figure for the
first six months of 2009.
» Estimate the drop in car
sales for the first six months
of 2009.
» Calculate the exact value of
the drop using the figures
given.
» Are students able to make
estimates?
 Exact Value using calculator:
124,000 x 62/100 or
124,000x0.62=76,880
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12
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Checking Understanding
» The school office has
ordered a new filing cabinet
for €152.99. Vat at 21 %
must be added to this price.
The secretary needs a quick
estimate of the final price.
What will she do?
» Now calculate the exact
value.
 €152.99 is approximately
€150 and
21 % is close to 20% which
is 1/5.
 Vat is 1/5 of €150 which is
€30. Final cost is €150 +€30
= €180
 Exact value of VAT =
»
» Do students find that in this
case the decimal equivalent
of the percentage is easier to
work with than the fraction
equivalent?
» Do students understand why
multiplication by 1.21 will
give them the final price?
152.99 
21
100
=152.99x.0.21=32.13
Total cost
 152.99 x 1.21 = €185.12
or 152.99+ 152.99x 0.21
= €185.12
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Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Checking Understanding
 Student A: get 20% of 400
and add it on.
 Student B: You cannot do
this as it was 20% of a
different number which was
taken originally and 20% of
400 will not be the same
amount.
 Divide the original price into
5 parts (5 lots of 20%) and
take 1/5 off.
» Where students have
difficulty give them a hint to
represent the original price
with a rectangle as they did
when working with fractions.
» Are students approaching
this problem by first asking
what the “whole” is?
Common misconception
»
A shop has a 20% sale on
its TVs for one week. At the
end of the week the sales
assistant wishes to change
the prices back to their
original values. The original
price is missing from one of
the TVs marked €400 for the
sale. How will the assistant
work out the original price?
» Is it ok to add 20% onto
€400?
» Are students tending to draw
a diagram when they don’t
immediately know how to
solve a problem?
» Does the pictorial
representation convince
students that the part which
was 1/5 of the original price
is now ¼ of the sale price?
-20%
» How many equal parts is it
made up of?
 This represents the €400.
 4 equal parts. These parts
are ¼ of 400 and not 1/5 –
the whole has changed. If
we had one of those added
on we would be back to the
original price.
¼ of 400 = 100
 Hence the original price is
€500
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14
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
» What percentage is €400 of
the original price?
 It is 80% of the original price.
Hence 10% of the original
price is 400/8=50 and 100%
is 50 x 10 = €500
» Using rectangles to help you
as above what if 25% was
taken off the original price,
what fraction of the new
price would this represent?
-25%
25% of the original price is the
same as 1/3 of the new price.
» What % is 1/3?
»
» What if 1/6 was taken off the
original price – what fraction
is this of the new price?
» What are these values as
percentages?
» Generalise this and test your
generalisation for unit
fractions?
Teacher’s Support and
Actions
» Are students able to use
both strategies for solving
this problem always keeping
in mind what the “whole” is?
» Summarise pattern of
getting back to the original
price on the board as given
by the students using
rectangles:
Original price
Operation on
new price to
get back to
the original
price
Down by 1/5
Up by 1/4
Down by 1/4
Up by 1/3
Down by 1/6
Up by 1/5
Up by 1/5
Down by 1/6
Up by ¼
Down by 1/5
Up by 1/6
Down by 1/7
33 1 %
3
-1/6
 1/6 ( 16 2 % ) of the original
3
price = 1/5 (20%) of the new
price.
 If I decrease by 1/n then to
get back to the original figure
I must increase by 1/(n-1)
Checking Understanding
» Can students see that the
inverse of a decrease by a
percentage is not the same
as an increase by the same
percentage and vice versa?
» Can students see a pattern
very easily using fractions
but not so easily with
percentage figures and
hence realise that it is useful
to be able to switch between
the different
representations?
 If a price x goes down by 1/5 »
this results in 4/5 x. To get
back to x multiply by 5/
which is increasing by ¼.
Student Activity 6 : Increasing and decreasing by a given percentage
» Can you justify this with
fraction multiplication?
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15
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
Teacher’s Support and
Actions
Checking Understanding
» Extension: Ask HL
» Generalise this for an
» Were HL students able to
 If I increase by 1/n then I
increase of 1/n – what is the
students to show that
prove the pattern which they
must decrease by 1/(n+1)
corresponding decrease in
had seen using algebra?
1
1
» (1  )(1 
) 1
the new price to return to the
n
n 1
original price?
» Check student work on this
» Show results on mini white
boards
» If a person puts €500 into
» Check student work on their » Do students realise that you
 100% (500)+ 10%(500) =
the bank for one year at
mini white boards
can only subtract the
110%(500)
10% what percentage of the  110% of 500 = 110/100 (500)
percentages because the
€500 does he have at the
whole was the same for
= 1.10(500) = €550
end of the year? Write this
both?
as a decimal and use
decimals and percentages
to work out the amount.
» If he invests €500 and then
100%(500) - 10%(500) =
loses 10% after one year
90%(500) = 90/100(500) =
what percentage of the €500 0.90(500) = €450
does he now have? Write
this as a decimal and use
decimals and percentages
to work out the amount.
»
»
» If a number is increased by
 (1+r)% of the original value
r% what is its new value?
 (1-r)% of the original value
» If a number is decreased by
r% what is its new value?
Possible strategy for solving percentage problems: Double number line with percentages on one side and quantities on
the other side. It helps students to organise information and see what is missing – part, whole or %.
See Appendix 2 for double number lines.
» Q. 20% of the 60 students in
first year walk to school.
How many first years walk to
school?
 Students draw a line and
mark 0% at one end and
100% at the other end.
» Draw a number line on the
board and follow through the
procedure with the students
KEY: » next step
• student answer/response
16
Teaching & Learning Plan: Percentages
Student Learning Tasks:
Teacher Input
Student Activities: Possible
and Expected Responses
» Draw a line any length to
represent the total number
of first years. What
percentages will you mark at
each end?
» Do you know what the total
number of students in first
year is? What % of students
in first year is this?
» What other information are
you given? What is the
question asking for? Put a
question mark on the line to
indicate this.
 Students look for the whole
(60) from the question and
mark it above 100%.
 Students place a question
mark over 20% on the
percentages side of the line.
 Given 20% and this is
marked in approx 1/5 of the
way along the line on the
percentage side.
 20% is 1/5 of 100% so we
need 1/5 of 60 =12 or
 60 is 6/10 of 100 so we need
6/10 of 20 =12 or
 10% is 6 children so 20% is
12 or
 1% is 60/100 =0.6, and 20%
is 20 x 0.6 = 12 or
» 12 of the 60 students in first
year walk to school. What
percentage of first year
children walk to school?
 This time students mark in 12
along the students line and
note the relationship
between 12 and 60 i.e. 12 is
1/5 of 60 so therefore they
need 1/5 of 100% = 20%
» 20% of the students in first
year walk to school. 12 first
year students walk to
school. How many students
are in first year?
Do Student Activity 7
 Again students see the
» Summarise in groups what
you know about
percentages.
relationship between the
percentages is the same as
that between the quantities.
100 = 5x20. Hence we need
5 x12 =60
 Students list what they
learned and report back.
Teacher’s Support and
Actions
Checking Understanding
» Are students checking to
see if they have been given
the “whole”?
» Do they see that they have
been given the whole and
the % and need the “part”?
» Are students noticing the
ratio between the
percentages and using the
same ratio between the
corresponding quantities?
» Are students noticing that
the direction of the arrows is
important?
» Are students able to use
proportional reasoning to
figure out the answer?
Circulate and check that
students are using the double
number line correctly.
» Are students identifying that
they are now given the
whole and the part and are
being asked for the %?
» Is the double number line
» Distribute Student Activity 7
model helping students to
solve problems with missing
“wholes”?
Walk around, asking questions
where necessary and listen to
students’ conclusions.
KEY: » next step
• student answer/response
17
Teaching & Learning Plan: Percentages
Student Activity 1
Student Activity 1: Write the following fractions as decimals and as percentages. You may use the hundredths disc or the
10 x 10 grid to confirm your answers.
Fraction
Fraction in hundredths
1
2
1 1 50 50


2 2  50 100
1
4
3
4
1
10
Decimal
50
= 0.50
100
(or 1  2  0.5 )
Percentage
50% ( 1  100 parts =50parts)
2
You may have learned
previously, that to change
a fraction to a
percentage, you multiply
the fraction by 100/1
which is what we are
doing in the third column.
1
100
1
5
7
10
2
5
3
5
4
5
8
10
9
10
1
8
3
8
You are free to
choose!
50% =
50
1
 0.5 
100
2
Sometimes one
format is more
suitable than
another.
2
8
6
8
7
8
8
8
1
3
2
3
1
50
1
2
1
2
18
Teaching & Learning Plan: Percentages
Student Activity 2 Percentage Dominoes
http://www.teachingideas.co.uk/maths/files/percentagedominoes.pdf
19
Teaching & Learning Plan: Percentages
Student Activity 2 Percentage Dominoes
20
Teaching & Learning Plan: Percentages
Student Activity 3 Equivalence of fractions, decimals and percentages
Fractions, decimals and percentages activity
Fill in the gaps in the following table to give 3 equivalent forms of each number i.e. fraction, decimal and
percentage. Simplify the fractions to their lowest terms.
Fraction
Decimal
Percentage
0.54
54%
65%
13
20
0.5625
19
200
0.095
5.431%
5431
100000
2
17
20
2.85
0.01034
1
3.047
613
10000
1.034%
162.5%
5
8
7
56.25%
304.7%
7.0613
Percentage games
http://nrich.maths.org/public/viewer.php?obj_id=6028
21
Teaching & Learning Plan: Percentages
Student Activity 4
Dominoes for fractions, decimals and percentages including recurring decimals
0.4
3
5
0.16
37.5%
2
7
83.3%
0.7
0.4
0.25
60%
5
6
0.3
4
9
25%
29
100
7
10
0.285714
0.875
29%
3
8
22
Teaching & Learning Plan: Percentages
Student Activity 5
Q1 – Q12 Mental questions on percentages – no calculator – work out calculations in your head.
1. There is a sale on DVDs in a local shop – 50% off selected items. Your favourite DVD before the sale cost
€18.
a) How much is being taken off?
b) What will it cost you now?
2. Another DVD in the shop, also costing €18 before the sale has 25% off.
a) How much is being taken off?
b) How much will it now cost?
3. The shop next door has a clothes sale. A pair of jeans which cost €30 has 10% off.
a) How much money is being taken off the price?
b) How much will you pay for them in the sale?
4. Do you think 10% sale is a good saving?
5. A house which costs 250,000 last year has decreased in price by 10%. How much will you save on last year’s
price by buying now?
6. Your local supermarket has a special price reduction this week.
When you go to the cash register you are offered a choice of ”10% off or €10 off.”
Should your decision be the same in the following situations?
Your shopping amounts to €70
Your shopping amounts to €120
Your shopping amounts to €200
Your shopping amounts to€11
Explain
e) Would you hire the supermarket manager?
a)
b)
c)
d)
7. OPEC decides to raise the cost of a barrel of crude oil by 40%.
The current cost is $45.
What will the new cost be?
8. Kay visits her local hardware shop and sees that there is 50% off a kitchen whose list price is €5000.
Two days later the same kitchen has a sign on it “Further reduction of 20% on last week’s price”. She visits
the shop again at the weekend and sees that there is a new sign on the kitchen “Weekend special – further
reduction of 10%”. She decides to buy - what will she pay?
Kay bought the kitchen which originally cost €5000 for €1800. How much did she save?
a) What fraction of the original price was this?
b) What % of the original price was this?
c) If you were the shopkeeper would you tell your customers 50%, then 20%, then 10% or just say “64%
off”? Explain.
23
Teaching & Learning Plan: Percentages
Student Activity 5
10. Last year the local IT accepted 120 students for its civil engineering course.
Due to increased interest in the course, the number of places this year has been increased by 15%.
How many students can enrol for the course this year?
11. Shares in a pharmaceutical company were worth $100 each.
The share price fell by 95% when one of their drugs caused patients to become ill.
a) What were the shares now worth?
Next day the share price rose by 100%.
b) What was the share price after these two days of trading?
c) Thomas has $1000 worth of shares in the company. What was the value of his stock after these two
days of trading?
12. According to SIMI, the Society of the Irish Motor Industry 124,000 cars were sold in the first six months of
2008.
There was a drop of 62% on this figure for the first six months of 2009.
a) Estimate the drop in car sales for the first six months of 2009.
b) Calculate the exact value of the drop using the figures given.
13. The school office has ordered a new filing cabinet for €152.99.
Vat at 21% must be added to this price.
The secretary asks you for a quick estimate.
a) Explain how you will do this and give the estimate you came up with.
b) Now calculate the exact price of the filing cabinet including VAT.
24
Teaching & Learning Plan: Percentages
Student Activity 6
More dominoes on Percentages including questions on increasing and decreasing by a certain %
€9.90 is 75% Decrease
125% of €13
of ...?
€12 by 20%
€16.25
€13.60
1
2
of €36
3
€13.20
€3
€9.60
40% of 80%
of €25
€24
Increase €9
by 5%
€9.80
€10.80
€8
1
1
1
of of of 27
3
3
3
Increase €8
by 70%
35% of €28
€9.45
30% of €28
3
of 60% of €30
5
Decrease 50%
of €7.20 by
1
6
€8.40
25
Teaching & Learning Plan: Percentages
Student Activity 6
2. In an ideal world as you get older your pocket money increases by 10% each year. Your pocket money
varies from year to year so we call the amount of your pocket money a variable.
a) If P is your pocket money this year, what will be the increase in your pocket money next year?
b) Write it as a decimal multiplication.
c) Write it as a fraction multiplication.
What will be the total of your pocket money next year P1 compared to your pocket money this year P?
a) Write it as a decimal multiplication
b) Write it as a fraction multiplication
If you are having any difficulty with this exercise substitute a specific amount of money for P. Work out the
amount of pocket money for next year given this year’s pocket money and then substitute the letter P for this
year’s pocket money.
Q3. Select from the following statements and insert the appropriate statement above each arrow between the
values in the given cycle: It is not necessary to use all the statements.
.
(i)
(ii)
(iii)
(iv)
(v)
Up by 25%
Down by 60%
Up by 20%
Up by 50%
Down by 250%
(vi)
Up by 33 1 %
(vii)
Up by 30%
€100
€120
3
€250
€150
€200
Q4. For each of the stages given work out the percentage increase or decrease involved if we reversed all the
arrows.
26
Teaching & Learning Plan: Percentages
Student Activity 7
1. There are students from 160 different families in third year. 75 % of those families were represented by one
parent each at a Transition Year information evening.
How many parents attended the meeting?
Use the double number line if necessary.
Number of
Families
0
0%
% of families
160
50%
100%
2. Jamie scored 80% of the penalties he took last year. He took a total of 25 penalties during last year.
How many penalties did he fail to score?
Number of
Penalties
0%
% of penalties
25
80%
100%
1
3. 10 students, or 33 % of Mr. Daly’s Maths class, were absent last Friday morning due to a heavy snowfall.
3
How many students are in this Maths class?
1
Draw a number line and mark 0%, 100% and 33 % underneath the number line.
3
1
Above the number line mark the number corresponding to 33 % .
3
Hence find the total number in the class.
4. Elaine bought a new laptop with a 20% discount on it. She paid €640 for the laptop. How much did the laptop cost
originally? Use the double number line below.
Money
100%
0%
Percentages
27
Teaching & Learning Plan: Percentages
Student Activity 7
5. John has travelled 120km of the 180 km to the airport from his home. What percentage of the journey has
he covered?
6. A jar contains 16 cubes. 12 ½ % are white, 37 ½ % are red, and 50% are orange. How many cubes of
each colour are in the jar?
7. Croke Park has a capacity for 82,300 people.
At a recent match it was reported that the stadium was 73% full.
a)
b)
c)
d)
e)
f)
Estimate how many people attended and explain the estimate.
What is 1% of the capacity of Croke Park? (Did you think 1% was small?)
Using this value what is 3% of the capacity of Croke Park?
What is 10% of the capacity of Croke Park?
Using this value what is 70% of the capacity of Croke Park?
Using these answers what is 73% of the capacity of Croke Park?
Use double number lines to help you work out the following problems
8. 25 out 40 students turned up for a practice for the school concert.
What % of students came to the practice?
9. 15 students turned up for athletics training.
The coach reported to the manager that he had 75% attendance.
How many students should have turned up to give 100% attendance?
10. In trials for the local team 42 players attended.
The coach said later that he got 84% of the attendance he expected.
How many players was the coach expecting?
28
Teaching & Learning Plan: Percentages
Appendix 1 - 10 X 10 Grid
29
Teaching & Learning Plan: Percentages
Appendix 2 – Hundredths disc
Hundredths Disc – Cut out two of these discs. Cut along one of the radii otf each and enmeshthe two discs together.
Estimate a fraction of the disc on one side and read the corresponding % on the other side.
30
Teaching & Learning Plan: Percentages
Appendix 2 – Hundredths disc
Hundredths discs
One quarter showing on one side
On the reverse, 25 hundredths (colours reversed)
31
Teaching & Learning Plan: Percentages
Appendix 2 – Hundredths disc
Eighths disc which could be meshed with the hundredths disc to find equivalent percentages for eighths.
32
Teaching & Learning Plan: Percentages
Appendix 3 – Double number line
Q. 20% of the 60 students in first year walk to school. How many first years walk to school?
Students
0
?
20%
0%
5
60
5
100%
Percentages
Q. 12 of the 60 students in first year walk to school. What percentage of first year children walk to school?
Students
0
12
?
0%
5
60
5
100%
Percentages
Q. 20% of the students in first year walk to school. 12 first years walk to school. How many students are in
first year?
0
Students
12
20%
0%
5
60
5
100%
Percentages
33
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