Download Tutor version of the unit - National Centre of Literacy and Numeracy

Overview: Learning about percentages 1
Key words:
Percentage, discount, mark up, tax, GST, increase, decrease, difference, wastage
Increase, decrease
Purpose: This unit is designed to help tutors who teach courses that require
calculations with percentages, e.g. GST, discounts, wastage
Tutor Outcomes:By the end of the unit tutors should be able to:
1. Recognise contexts and problems that involve percentages
2. Develop lessons in their teaching context that help learners to solve
problems with percentages
Section 1: Mathematical Background
Page 1: What does % mean?
The symbol % is a combination of the two zeros
from 100 and the sign / which means “out of”. So
% means “out of one hundred”.
This can be quite misleading for learners because
in most contexts the percentage operates on a
quantity that is not 100, e.g. Find 35% of $86
means you are actually working with $86 not $100.
Another way to look at it is through the word
“percent”. Per means “for every” and cent is the
prefix for 100, like a century is 100 years or 100
runs. So percent means for every hundred.
Section 1: Mathematical Background
Page 2: Percentage as a rate
One way to think about a percentage is as a special rate.
At 35% off you pay 65% or $65 in every $100.
At the same rate how much do you pay for something that normally costs $86?
Normal Price
Discount Price
100
65
86
?
Section 1: Mathematical Background
Page 3: Percentage as a rate
All of the things you can do with other rates, like kilometres per hour, you can do
with percentages. Both numbers in the rate can be multiplied or divided by the
same number.
÷ 100
x 86
Normal Price ($)
Discount Price ($)
100
65
1
0.65
86
55.90
÷ 100
x 86
Section 1: Mathematical Background
Page 4: Why do we have percentages?
Percentages are used in two main ways in
everyday life:
1. As operators
In many real life situations you find a percentage
of an amount. For example, if you buy something
at 30% discount you pay 70% of the usual price.
70% operates on the usual price, e.g. 70% of $60
is $42
2. As proportions
Percentages are often used to compare two or
more proportions. For example, to compare two
shooters in a netball game you might convert the
statistics into percentages.
Selma gets 32 out of 40 shots so her shooting
percentage is 80%
Niki gets 33 out of 44 shots so her percentage is
75%
Section 1: Mathematical Background
Page 5: Are percentages always less or equal to
100%?
Most situations involve percentages less than
100.
In a sale a percentage is taken off the full price
so you pay less than the full price, less than
100%.
When you toss a coin at the start of a sporting
match your chances of winning the toss are onehalf or 50%.
Comparison situations can involve percentages
greater than 100%.
For example the price of a house was $200,000
in 2000 and $280,000 in 2010.
Compared to the $200,000 the house is now
worth 140% of what it was in 2000.
Section 2: Activity
Page 1:
What is a percentage?
Write 35% on the board.
What does this mean?
Discuss this in small groups of 3-4 learners.
Record the ideas from each group as they report back.
Discuss things like:
•% means “out of one hundred” (/ means “divide by”, 00
comes from 100)
•“Per” means for every, “Cent” means one hundred, e.g.
Century is 100 years or 100 runs
•35% is less than one half but bigger than one quarter
because 50% is one half and 25% is one quarter
•35% is about one third because one third is 33.3%
•35% of something, what is the something? (Whole needs
to be given, e.g. 120 kg)
Section 2: Activity
Page 2: When do we use percentages (examples)?
Provide each group of learners with a copy of copymaster 1.
This provides possible real life situations in which percentages may be involved.
Ask the learners:
•How might percentages occur in each of these situations?
•Can you think of other situations in which percentages are used?
Share the ideas from each group.
Important points are:
•Percentages are used in situations where the whole varies, e.g. Goalkickers take
different numbers of shots, people borrow different amounts of money.
•Percentages can be more than 100% in comparison situations, e.g. Lambing
percentages are usually between 150-200% where the number of lambs is
compared to the number of ewes
•Percentages must be no more than 100% in “out of” situations, e.g. Jenny goals 35
out of 60 shots in netball.
•Percentages are special types of fractions with denominators (bottom numbers) of
25 ).
100, e.g. One quarter is 25 hundredths ( 100
Section 2: Activity
Page 3: Common Percentages
Provide the learners with one strip of 100 beads (Copymaster 2).
Ask, “How many beads are one the string in total?”
“What has this got to do with percentages?”
Percentages are out of 100 and this is a model of fractions out of 100.
Pose the following problems and tell the learners to label their strip as they go:
1.What percentage is all of something? (Label 100%)
2.What percentage is nothing of something? (Label 0%)
3.What percentage is one half of something? (Label 50%)
4.Find some other percentages that you know the fractions for?
0%
0
10%
1
10
20%
30%
1
4
40%
50%
1
2
60%
70%
80%
3
4
90%
100%
1
Section 2: Activity
Page 4: Percentage to Fraction Snap
Play a game of snap with cards made from
copymaster 3.
This game is designed to practise simple
percentage to fraction knowledge.
Points that may arise:
•Nine tenths is one tenth less than the
whole. This is because the whole is ten
tenths. So nine tenths is 90% (100% - 10%)
•Four fifths is one fifth less than the whole.
This is because the whole is five fifths. So
four fifths is 80% (100% - 20%)
•33.3% is another name for one third. This
is because 100 ÷ 3 = 33.3 (recurring).
Section 2: Activity
Page 5
Finding a percentage using place value knowledge.
To find 10% is the same as dividing by 10.
When we divide be 10 the number gets 10 times smaller. The digits
move one place to the right, e.g. 46 ÷ 10 = 4.6
hundreds
÷ 10
tens
ones
4
6
tenths
hundredths
100%
4
6
0
4
10%
6
Use this method to find 10% of:
Find 10% of:
•80
•75
•136
•589
Ask learners to find 5% of 24
Record students methods.
Look for methods such as finding 10% then halving to find 5%
1%
Section 2: Activity
Page 5
Finding a percentage using place value knowledge.
To find 1% is the same as dividing 10% by 10.
When we divide be 10 the number gets 10 times smaller. The digits
move one place to the right, e.g. 46 ÷ 10 = 4.6
hundreds
÷ 10
÷ 10
tens
ones
4
6
tenths
4
6
0
4
Use this method to find 10% of:
Find 1% of:
•80
•75
•136
•589
Ask learners to find 3% of 24
Record students methods.
Look for methods such as finding 10% then dividing by ten.
hundredths
10%
6
1%
Section 2: Activity
Page 6: Finding percentages of something
Present this problem to your learners or pose a problem with the
same numbers but a different story.
Kegs hold 50 litres of beer.
There is 10% allowance for wastage. What a shame!
How much beer is wasted out of each keg?
Note: Wastage is loss of beer through pouring overflow, clearing
the hose lines when kegs are changed and the beer left behind in
the keg.
Ask the learners to solve the problem and share their strategies.
For example, “I know that 10% is one tenth and one tenth of 50 is
5 litres” or “10% is ten out of 100 so it must be 5 out of 50 litres.”
Present the problem using the strip diagram (Copymaster 4).
0%
0
50%
10%
10
20
100%
30
40
50
Section 2: Activity
Page 7: Practice Examples
Refer to Section Three, problem examples 1 - 3, for your students to practise the
ideas introduced so far.
You will need to run off copies of Copymaster 4 for your students to use.
Section 2: Activity
Page 8: Adding on GST
Ask your learners what they understand by GST (Goods
and Service Tax).
The total price you pay for any item includes net price,
mark up and GST.
Net price
Mark up
GST
Net price is how much the shop pays for the item and
the mark up is the profit the shop makes. These two
parts add up to the shop price.
GST is charged on top of the shop price at a rate of
15%.
15% of shop price
Shop price
GST
Section 2: Activity
Adding on GST
GST is 15%
To add on GST we can mentally workout 10% plus 5%.
Look at the following example:
Item costs $200
GST = $30
100%
15%
115%
We can also calculate the GST inclusive price by
multiplying the 200 by 1.15. 200 x 1.15 = $230
Section 2: Activity
Pose the following problems:
Before GST is added the bottle of
milk costs $4.00.
How much do you pay for the milk
after GST is added on?
10%
0%
10
%
20
%
30
%
40
%
50
%
$4.00
60
%
70
%
80
%
90
%
100%
40c
20c
5%
Section 2: Activity
Practice Examples
Refer to Section Three, problem examples 4-5,
for your students to practise the ideas
introduced so far.
Section 3: Examples
Page 1: Shopping Spree
Mareea wants to buy a top that usually costs $60
The shop has a 20% off sale.
How much will Mareea save?
How much will she pay for the top?
0%
0
10
%
20
%
10
50
%
20
30
80
%
40
100
%
50
60
Section 3: Examples
Page 2: Horsing Around
A horse eats about 60% of its own body
weight each month.
This horse weighs 550 kilograms.
How much does it need to eat this month?
Section 3: Examples
Page 3: Credit Crunch
Warren has $1760 owing on his credit
card.
He pays 18% interest per month on
what he owes.
How much will Warren pay in interest
this month if he does not pay
anything off his card.
Section 3: Examples
Page 4: Credit Crunch
The shop price of a pair of jeans is
$120.
Add the GST and find out how much
you pay for these jeans.
Section 3: Examples
Page 5: Honest Phil’s Car Dealership
The shop price of a car you want is
$13,500
Honest Phil forgot to tell you about
the GST.
How much GST needs to be added?
Section 4: Assessment
Page 1: Shoes
At Shoes 4 Less there is a 25% off sale.
This pair of shoes normally costs $160.
How much will the shoes cost on sale?
Section 4: Assessment
Page 2: Weed Spraying
The instructions say that the
spray should be 80% water and
20% concentrate.
Your sprayer takes 5 litres of
liquid.
How much water should you put
in before topping it up with
concentrate?
Section 4: Assessment
Page 3: Brakes
Ralph has fixed your car brakes.
The bill is $280 but GST has to be
added.
What will the total bill be?