Overview: Learning about percentages 1Key words: Percentage, discount, mark up, tax, GST, increase, decrease, difference, wastageIncrease, decrease Purpose: This unit is designed to help tutors who teach courses that require calculations with percentages, e.g. GST, discounts, wastageTutor 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 BackgroundPage 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 BackgroundPage 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 Price1006586?

Section 1: Mathematical BackgroundPage 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 100x 86x 86

Normal Price ($)Discount Price ($)1006510.658655.90

Section 1: Mathematical BackgroundPage 4: Why do we have percentages?Percentages are used in two main ways in everyday life:

As operatorsIn 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 proportionsPercentages 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 BackgroundPage 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 one-half 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: ActivityPage 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 runs35% is less than one half but bigger than one quarter because 50% is one half and 25% is one quarter35% 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: ActivityPage 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 ewesPercentages 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 100, e.g. One quarter is 25 hundredths ( ).

Section 2: Activity

Section 2: ActivityPage 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: ActivityPage 5Finding 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

Use this method to find 10% of:Find 10% of:8075136589Ask learners to find 5% of 24Record students methods.Look for methods such as finding 10% then halving to find 5%100%1%

hundredstensonestenthshundredths4646046

Section 2: ActivityPage 5Finding 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

Use this method to find 10% of:Find 1% of:8075136589Ask learners to find 3% of 24Record students methods.Look for methods such as finding 10% then dividing by ten.

hundredstensonestenthshundredths4646046

Section 2: ActivityPage 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).

Section 2: ActivityPage 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: ActivityPage 8: Adding on GSTAsk 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 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%.

Adding on GSTGST is 15%To add on GST we can mentally workout 10% plus 5%.Look at the following example:

100% 15%115%We can also calculate the GST inclusive price by multiplying the 200 by 1.15. 200 x 1.15 = $230Section 2: ActivityItem costs $200 GST = $30

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?

Section 2: Activity10%5%

Section 2: ActivityPractice Examples

Refer to Section Three, problem examples 4-5, for your students to practise the ideas introduced so far.

Section 3: ExamplesPage 1: Shopping Spree

Mareea wants to buy a top that usually costs $60The shop has a 20% off sale.How much will Mareea save?How much will she pay for the top?

Section 3: ExamplesPage 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: ExamplesPage 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: ExamplesPage 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: ExamplesPage 5: Honest Phils Car Dealership

The shop price of a car you want is $13,500Honest Phil forgot to tell you about the GST.

How much GST needs to be added?

Section 4: AssessmentPage 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: AssessmentPage 2: Weed SprayingThe 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: AssessmentPage 3: Brakes

Ralph has fixed your car brakes. The bill is $280 but GST has to be added. What will the total bill be?

Be aware that out of describes a part-whole (part of a whole) relationship. Most percentage problems involve either finding the part, e.g. 25% of $36 means the part which is one quarter of the whole ($36). Rarely do we find the whole given the part, e.g. 25% of an amount is $9. What is the amount?*Rate thinking is very easy in that in comes from the most basic type of multiplication problem. It is natural to look for a unit rate, e.g. What is 1% of $86? Or What is 65% of $1? In the next slide we find 65% of $1.

*Here the unit rate $1 (price): $0.65 is scaled up by 86. This is like multiplying 86 x 0.65 = $55.90. Alternatively you can work out 1% of $86 = $0.86 then calculate 65 x 0.86 = $55.90.Either way it works.*When we use a percentage as an operator it is easy to think of it like a whole number even when it isnt. Its actually a fraction, e.g. 25% means 25/100.Treating it like a whole number lets you use rate thinking to make finding the answer easier. For example, 35% of something can be found using 10% + 20% + 5% of the something.Thinking about percentages as proportions, part-whole relationships, is usually harder to do.*Percentages equal of less that 100% usually, but not always, involve a part-whole relationship that can be described as out of.Percentages greater than 100% always involve a comparison of one whole with another whole, e.g. Comparing the value of something with its value 10 years ago.*Be aware that there are many legitimate ways to think about percentages that can prove obstacles later on. For example out of 100 can restrict the person in terms of comparisons where the percentages are greater than 100. Ask the learners to provide a situation that matches their idea about percentages and ask connecting questions, e.g. What is the same about the Kathmandu discount situation and the 95% fat free label on the cereal box? *You may find that drawing strip diagrams of the situations in copymaster 1 may help students to see the structural similarity between different problems.Refer to the webinar of percentages to see how to do this.*Benchmark fractions such as 25%, 50%, 10% are very useful for calculation and estimation of percentage problems. These common benchmarks need to be remembered so they are accessible.*You may find other ways to help learners remember the benchmark percentages. Bingo is another game that may work.*Learners often have rules for multiplying by ten, e.g. Add a zero. Trusting a rule that works is useful but bear in mind that such rules can also cause confusion unless supported by ideas about why it works. For example, mistakes with adding zero are common, e.g. 4.3 x 10 = 4.30. Connect multiplying be ten and dividing by ten as inverse operations that undo one another. Practising with anticipation using a calculator helps learners considerably.Enter is 3.7. If you divide by ten what do you get? (think ahead then check) What multiplication operation returns 0.37 to 3.7? *In general:Dividing 100% by 10 gives 10% so dividing an amount by 10 gives 10% of the amount, e.g. $56 10 = $5.60 so 10% of $56 is $5.60Dividing 100% by 100 gives 1% so dividing an amount by 100 gives 1% of the amount, e.g. $56 100 = $0.56 so 1% of $56 is $0.56*It is normal to call the shop price the GST exclusive price to show that GST is not included.Naturally the total price is called the GST inclusive price to indicate GST has been added.*10% of $4.00 is $0.40 so 5% of $4.00 is $0.20.So 15% of $4.00 is $0.60.The GST inclusive price is $4.00 + $0.60 = $4.60*20% is one fifth so Mareea pays one fifth of $60 which is $12.80% is one fifth off so Mareea pays $60 - $12 = $48.Pose another question such as:Mareea wants to buy a something that usually costs $60The shop has a 35% off sale.How much will Mareea save?How much will she pay for the something?

*60% of 550 is 0.6 x 550 = 330 kilograms.Another way to do it is:10 % of 550 = 55kg6 x 55 = 330*18% of $1760 can be solved as 0.18 x 1760 = $316.80It could be solved as 10% of $1760 = $1765% of $1760 = $881% of $1760 = $17.60 so 3% of $1760 = $52.80176 + 88 + 52.80 = $316.80*15% of $120 is 0.15 x 120 = $18This could be worked out as 10% of $120 = $12 so 5% of $120 = $6, $12 + $6 = $18*15% of $13,500 is 0.15 x $13,500 = $2,025*Shoe sale solutionThere are two main ways to work this out: Method One 25% is one-quarter. 1/4 of $160 is $40. So you pay $160 - $40 = $120 Method Two 10% of $160 is $16 5% of $160 is $8 20% of $160 is $32 $32 + $8 = $40 (25% of $160) So you pay $160 - $40 = $120 *Weed Spray SolutionThere are two main ways to solve this problem. Method One 10% of 5 litres is 0.5 litres So 80% is 8 x 0.5 = 4 litres Method Two 20% is one-fifth 1/5 of 5 litres is 1 litre 5 litres - 1 litre = 4 litres

*Car Bill SolutionYou can find the total bill this way: 10% of $280 is $28 5% of $280 is $14 $28 + $14 = $42 So the bill will be $280 + $42 = $322 On a calculator you could go: $280 + 15% = or $280 x 1.15 = *