30 www.teachsecondary.com48 SUBSCRIBE AT TEACHSECONDARY.COM

LESSONPLAN

(cancelled down), it is easy to seethat none of them are equal. Acommon way to compare fractionsis to make their denominatorsequal. Another way is to convertthem to decimals. However, insome cases, neither of thesemethods might be needed. Forexample, is bigger than , sowill certainly be bigger than .Encourage students to articulatereasoning like this and only tocalculate when it becomesabsolutely necessary.

No pair of students by themselvesmay be able to find the order of theentire list, but this may be possibleas a whole class if the work ofordering pairs of fractions is dividedup. The order is < < < < < .

Q. Tell me another fraction that isequal to . And another, andanother. Explain how you knowthat they are all equal to .

Students could write their fractionson mini-whiteboards. They willprobably list equivalent fractionssuch as , , , etc.

Then ask them to do the samething with .

Q. How would you add and

Q. What can you say about thesesix fractions?

Students might note that they areall different, that they are all lessthan 1, that they are all positive, thatthey are all expressed in theirsimplest terms, that four are lessthan a half and two are greaterthan a half, that they are not inorder of size, and so on. Encouragestudents to say as many things asthey can think of. Questions likethis are a good way to encouragestudents to be mathematicallyobservant.

Q. Which fraction do you think is the largest? Which is thesmallest? Why?

This next one is hard, and it mightbe easier to begin by asking pairsof students:

Q. Pick two fractions and say whichone is larger.

Since all of the fractions areexpressed in their simplest terms

SUMFRACTIONSWHY

TEACHTHIS?

Adding and subtracting fractions is a procedure which students oftenfind it very difficult to master, despite sometimes having been taught itevery year from upper primary school to the sixth form. They'll do it for afew lessons but then quickly forget it again for another year –particularly in this age of calculators and computers. When revisiting thistopic at Key Stage 4, therefore, it can bring up powerful memories offailure, so it is important to address the area without it feeling like anexact repetition of what they have done before. One way to make thepractice of a technique less tedious than a page of exercises is to embedit in a bigger problem which students are trying to solve. Having theirattention on a grander purpose helps them to internalise the method tothe point where they don’t have to think about it any more, which isalways the goal of mathematical fluency. A problem-based approachalso means that the answers obtained have some significance for thewider problem, so students are more inclined to check their answers andnotice if any of them doesn’t seem to be about the right size.

without a calculator?

Here the equivalent fractions and, with common denominators, are

particularly useful. + = . Theanswer could be written as 1or it could be left as a top-heavy(improper) fraction.

Q. The answer to + is a little bitmore than 1. Is there any way thatyou could have predicted that theanswer was going to be more than1 without working it out exactly?

Since 3 is more than half of 5, and5 is more than half of 8, bothfractions are more than , so theirtotal must be more than 1. This sortof reasoning can be very useful forestimating the size of an answer sothat mistakes can be spotted.Estimation will be important in themain activity (next page).

If students are shaky on theprocess of adding fractions theywill get lots of practice during thelesson. It may be best to ensurethat confident and less confidentstudents are mixed up in the room,so that less confident studentshave some expertise nearby andmore confident students havesomeone to explain to. This shouldbenefit both.

THEY’VE BEEN TAUGHT THE TECHNIQUE A DOZEN TIMES ORMORE... BUT THROUGH EMBEDDING IT WITHIN A RICHER

PROBLEM, ‘ADDING AND SUBTRACTING FRACTIONS’ MIGHTFINALLY MAKE SENSE TO STUDENTS, SAYS COLIN FOSTER...

STARTER ACTIVITY

Developing fluency with aparticular mathematicaltechnique to the pointwhere students no longerneed to think about thedetails can give them apowerful sense of mastery.This enables them to tacklemore demandingmathematical problemsthat rely on that processand contributes to a widerconfidence in the subject.

Lesson plan Maths_Layout 1 16/06/2014 10:15 Page 1

SUBSCRIBE AT TEACHSECONDARY.COM 49

MATHEMATICS | KS4

MAIN ACTIVITY

Q. I want you to add together asmany of the six fractions as youlike to get an answer that is asnear to 1 as possible. You can useeach fraction only once.

Make sure that studentsunderstand that they can choosewhichever of the six fractions theywish and can add just twofractions, or three fractions, ormore. It is probably best for themnot to use calculators, since oneof the intentions is that they gainpractice at carrying out theprocess of adding fractionsthemselves. However, if you wantto focus on making estimates then calculators could be usefulfor checking.

If some students are unsure howto start, you could ask people toshare ideas with the whole classabout how they will begin.Alternatively, you can suggest thatthey sort out an approach withintheir small groups.

Some students will probably beginby keeping either the or the

ADDITIONALRESOURCESFOR STUDENTS WHO WANT SOMETHINGVISUAL TO HELP THEIR THINKING, AFRACTION NUMBER LINE IS AVAILABLE ATwww.mathsisfun.com/numbers/fraction-number-line.html

STRETCH THEMFURTHERSTUDENTS WHO WANT ADDITIONALCHALLENGES COULD MAKE UP A SIMILARPROBLEM USING THEIR OWN SET OF SIXFRACTIONS. WHAT CHOICES OF FRACTIONSMAKE THIS A GOOD PROBLEM AND WHATCHOICES DON’T? THEY SHOULD TRY TOMAKE THE PROBLEM NEITHER TOO EASYNOR TOO DIFFICULT. ANOTHER PROBLEMSTUDENTS COULD CREATE WOULD BE ASET OF 10 FRACTIONS IN WHICH A SUBSETADD UP TO 1 EXACTLY. WHAT WOULD BE AGOOD CHOICE OF FRACTIONS TO USE FORTHIS PROBLEM SO THAT THE ANSWERISN’T OBVIOUS?

INFORMATIONCORNER

Colin Foster is a SeniorResearch Fellow inmathematics education inthe School of Education atthe University ofNottingham. He has writtenmany books and articles formathematics teachers (see www.foster77.co.uk).

ABOUT OUR EXPERT

SUMMARYand adding on a smaller fraction.Others might start with 1 as thegoal and subtract fractions from itto see what is left. Others mightjust choose two or three fractionsat random and add themtogether to see what will happen.Some might try to find a lowestcommon denominator for all sixfractions (600) and express themall with this same denominator.Estimation comes into play injudging the size of theanticipated answer so thatsensible combinations offractions are chosen.

Students will probably end upquite close to 1 fairly soon. If theyobtain an answer like (from +

+ ), they may think that theyare as close as possible, as theiranswer is “only 1 away”, butbecause the “one” is “onetwelfth” they are not really thatclose ( is more than 8%), so theyshould aim to get even closer!

If students find the best possiblesolution, another problem theycould tackle is to find the set offractions which add up to as nearto as possible. (The answer tothis is + + = .)

You could conclude the lesson with aplenary in which the students talkabout what they have found out andlearned. Is it possible to make 1exactly? How do you know? Howclose to 1 did different groups get?What strategies did they follow?

Although the task may appear to bemainly about addition of fractions, inorder to decide whose answer isclosest to 1 some subtraction may benecessary. If the answer is greaterthan 1, then subtracting 1 is quiteeasy, especially if the answer iswritten as a mixed number. If theanswer is less than 1, it is probably abit harder to calculate the difference.

It is not possible to make 1 exactly.The five nearest sums, in order, are:

+ + = 1+ + + =

+ + = 1+ + = 1+ + + =

Lesson plan Maths_Layout 1 16/06/2014 10:16 Page 2