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LESSONPLAN MATHEMATICS | KS3

ADDITIONAL RESOURCESA NICE PERCENTAGE INCREASE PROBLEM ISAVAILABLE AT HTTP://NRICH.MATHS.ORG/515AND ANOTHER, FOR THOSE WHO LIKE CODEPUZZLES, IS AVAILABLE ATHTTP://NRICH.MATHS.ORG/4799.

STRETCH THEM FURTHERSTUDENTS COULD EXPLORE HOW LONGIT WOULD TAKE TO “DOUBLE YOURMONEY” IN A BANK ACCOUNT WITH AFIXED RATE OF INTEREST. FOR EXAMPLE,IF THE INTEREST RATE IS 10% PER YEAR,HOW MANY YEARS WILL IT TAKE BEFOREYOU DOUBLE YOUR MONEY? (ASPREADSHEET WOULD BE USEFUL.) ITWILL TAKE SOMEWHERE BETWEEN 7 AND8 YEARS, BECAUSE 1.17 < 2 < 1.18.

HOW DOES THIS ANSWER DEPEND ONTHE INTEREST RATE? IN FINANCE, THEREIS A ROUGH RULE, CALLED THE “RULE OF70” (OR A NUMBER CLOSE TO 70), WHICHGIVES AN APPROXIMATE ANSWER TO THISPROBLEM. YOU JUST CALCULATE 70DIVIDED BY THE INTEREST RATE TOOBTAIN THE APPROXIMATE NUMBER OFYEARS NEEDED TO DOUBLE YOURMONEY. IN THIS CASE 70/10 = 7, WHICH ISABOUT RIGHT. WHEN THERE IS INFLATION,YOU CAN USE THIS RULE TO WORK OUTHOW LONG IT WILL TAKE FOR THEMONEY’S BUYING POWER TO HALVE BYCALCULATING 70 DIVIDED BY THEINFLATION RATE.

INFORMATIONCORNER

Colin Foster is an AssistantProfessor in mathematicseducation in the School ofEducation at the Universityof Nottingham. He haswritten many books andarticles for mathematicsteachers (see www.foster77.co.uk).

ABOUT OUR EXPERTWHAT’S THE DEAL?

However, there are only threeways to fit in all 12 numberswithout any repetitions. (Ofcourse, the two “increase” linesand the two “decrease” linescould be swapped in eachsolution.) One is:

£20 increased by 75% = £35£50 increased by 40% = £70£80 decreased by 25% = £60£100 decreased by 10% = £90

IS 50% OFF A BETTER DEAL THAN 50% EXTRA FREE? USING NUMBERPUZZLES CAN HELP STUDENTS GET TO GRIPS WITH THE ISSUES

SURROUNDING PERCENTAGE CHANGE, SAYS COLIN FOSTER...

STARTER ACTIVITY

You could conclude the lessonwith a plenary in which thestudents talk about what theyhave learned. Has anythingsurprised them? Did they find anyquick ways of finding percentageincreases or decreases? Did they become better at estimatingthe answers without doing the calculations?

Students may have noticedthat, for instance, although £50increased by 40% is £70, it is notthe case that you can simplyreverse this and say that £70decreased by 40% is £50. This isone of the features that makes thepuzzle hard! In fact, £70decreased by 40% is £42, which isless than £50, because the 40%decrease is 40% of £70, which is a

larger amount than 40% of £50.(In fact, because £70 is £20 morethan £50, the answer is 40% of£20 = £8 smaller.) It can bechallenging for students to try to explain these differences but a good opportunity for them to try to make sense of howpercentage change works and develop precise mathematical communication.

As soon as you enter ashopping precinct or high streetanywhere, you will see signsadvertising “50% off” or “50%extra free”. Unless studentsunderstand what thesepercentage changes mean, theywill be vulnerable to being misledand exploited. Is “50% extra free”better or worse than “50% off”?Perhaps it depends on how muchof the item you want and howmuch money you have availableat the time. In this lesson,students explore what happensto a certain amount of moneywhen it undergoes a particularpercentage increase or decrease.They will discover that a 50%increase followed by a 40%increase is not a 90% increase,and a 50% increase followed by a50% decrease does not get youback to where you started! In themain part of the lesson, studentswill work on a puzzle that willinvolve them in a lot of intelligenttrial and error, generating plentyof practice at percentageincreases and decreases but alsohelping them to understandmore deeply how these work. Bythe end of the lesson they shouldbe able to handle percentagecalculations confidently, have asense of the misconceptions toavoid, and have developed astronger understanding aboutpercentage change.

WHY TEACHTHIS?

MAIN ACTIVITY

Write these four statements on theboard:

£___increased by ___ % = £ ____£___increased by ___ % = £ ____£___decreased by ___ % = £ ____£___decreased by ___ % = £ ____

Q. Can you suggest some possiblenumbers to go in the gaps here?

Students should be able to choosevalues for the two gaps on the left-hand side of each equation andthen calculate the answer on theright-hand side. If this is too easy,you could challenge students tomake all of their numbers wholenumbers, or to make all of theirnumbers multiples of 10, or to makeall of their numbers different, or todo it without a calculator.

Q. Now I have a puzzle for you. Iwant you to find where these 12numbers can go in the 12 spaces.You have to use each numberonce.

10, 20, 25, 35, 40, 50, 60, 70, 75, 80,90, 100

A task sheet for students to writeon is available atteachsecondary.com/downloads/maths-resources

There are many possible ways offitting the numbers into the spaces ifrepetition is allowed. Someexamples are:

DISCUSSION

followed by a 50% increase is a110% increase, whereas doubling isjust a 100% increase.

Students might think thatpercentages greater than 100 areimpossible, but this is not the case.A percentage increase of more than100% simply means that the finalamount is more than twice thestarting amount.

Q. What happens if you switcharound the order, so you have a50% increase followed by a 40%increase? What happens with otherpairs of percentage increases?

Students may be surprised that theorder makes no difference. It is still a110% increase. Increasing by 40% isequivalent to multiplying by 1.4, andincreasing by 50% is equivalent tomultiplying by 1.5, so the order

£20 increased by 25% = £25 £40 decreased by 50% = £20

£25 increased by 40% = £35 £60 decreased by 25% = £45

£40 increased by 50% = £60 £60 decreased by 50% = £30

£40 increased by 75% = £70 £70 decreased by 50% = £35

£50 increased by 20% = £60 £80 decreased by 50% = £40

£50 increased by 80% = £90 £80 decreased by 75% = £20

£60 increased by 25% = £75 £100 decreased by 75% = £25

COLIN WOULD LIKE TO THANK TOM SPILLANEFOR ESTABLISHING THAT THERE ARE EXACTLYTHREE SOLUTIONS TO THE PUZZLE.

Another is obtained by justswapping the 10 and the 90 in the final line. (You could ask students why the second and thirdnumbers in the other lines can’t besimilarly swapped.)

And the third is:

£40 increased by 100% = £80£75 increased by 20% = £90£25 decreased by 60% = £10£70 decreased by 50% = £35

Students will need some time toexperiment in order to obtain these,and lots of useful practice andthinking will take place during theprocess. If anyone succeeds infinding all three solutions, you couldask them to make up their own setof 12 numbers that will work and seeif someone else can solve it. Canthey make it have just one solution?

Giving 110%Q. Look at these two statements.Which is the bigger increase and why?

“A 40% increase followed by a50% increase.”

“Double your money.”

Most students will probably thinkthat “Double your money” isbetter. They are likely to assumethat a 40% increase followed by a50% increase is a 90% increase,which is less than doubling whatyou start with. But this is wrong. Ifthey think this, don’t tell themthey are wrong – insteadencourage them to try it out withsome numbers.

Q. What happens if you startwith £100?A 40% increase makes £140, butthen a 50% increase on top ofthat means adding £70, not £50,because the 50% refers to 50% ofthe £140, not 50% of the original£100. So the final amount is £210.In general, a 40% increase

doesn’t matter, since 1.4 x 1.5 = 1.5x 1.4. Both are equal to 2.1, which isa 110% increase. In general, an a%increase followed by a b%increase is equivalent to an (a + b+ ab/100)% increase.

Q. What happens withdecreases? What if you have a40% decrease followed by a 50%decrease? What is that the sameas? What happens with otherpairs of percentage decreases?

A 40% decrease followed by a50% decrease is equivalent to a70% decrease. In general, an a%decrease followed by a b%decrease is equivalent to an (a + b– ab/100)% decrease. Students willprobably not approach thisalgebraically but instead willexplore what happens with somedifferent starting numbers.

Solving puzzles can be agood way to get lots ofpractice with amathematical techniquewhile something a bit morethought-provoking is goingon. Percentages crop up allover the place in daily life,and many peoplemisinterpret andmisunderstand them. This lesson helps to clarifystudents’ thinking aboutpercentage change.