Overview: Learning about percentages 1

1. 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

2. 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.

3. 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?

4. 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 ÷ 100 x 86 x 86

5. Section 1: Mathematical Background • Page 4: Why do we have percentages? • Percentages are used in two main ways in everyday life: • 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%

6. 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 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.

7. 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)

8. 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 100, e.g. One quarter is 25 hundredths ( ).

9. 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0% 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: • What percentage is all of something? (Label 100%) • What percentage is nothing of something? (Label 0%) • What percentage is one half of something? (Label 50%) • Find some other percentages that you know the fractions for? 0 1

10. 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).

11. 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 • 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% 100% ÷ 10 10% 1%

12. 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 • 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. ÷ 10 10% ÷ 10 1%

13. 50% 100% 10% 0% 0 10 40 20 30 50 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).

14. 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.

15. 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 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%. Net price Mark up GST 15% of shop price GST Shop price

16. 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: 100% 15% 115% We can also calculate the GST inclusive price by multiplying the 200 by 1.15. 200 x 1.15 = $230 GST = $30 Item costs $200

17. 100% 0% 90% 80% 50% 40% 60% 20% 10% 70% 30% $4.00 40c 20c 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% 5%

18. Section 2: Activity Practice Examples Refer to Section Three, problem examples 4-5, for your students to practise the ideas introduced so far.

19. 10% 20% 50% 80% 100% 0% 0 10 40 20 30 50 60 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?

20. 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?

21. 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.

22. 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.

23. 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?

24. 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?

25. 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?

26. 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?