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 KEY: » next step • student answer/response 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. KEY: » next step • student answer/response 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? KEY: » next step • student answer/response 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 KEY: » next step • student answer/response 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 • student answer/response 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? KEY: » next step • student answer/response 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. KEY: » next step • student answer/response 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 KEY: » next step • student answer/response 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 KEY: » next step • student answer/response 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 KEY: » next step • student answer/response 13 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 KEY: » next step • student answer/response 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? KEY: » next step • student answer/response 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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