LESSONPLAN

LEARNING MATHEMATICS IS ABOUT MAKING SENSE OF MATHEMATICAL SITUATIONS. THESEPUZZLES PROVIDE AN INTERESTING CONTEXT IN WHICH STUDENTS CAN MAKE PREDICTIONS AND

CONJECTURES AND DEVELOP THEIR POWERS OF MATHEMATICAL REASONING, SAYS COLIN FOSTER...

MATHEMATICS | KS3

TODAY YOU WILL...> INVESTIGATE THERELATIONSHIP BETWEENNUMBERS IN A DIAGRAM ANDJUSTIFY YOUR REASONING

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Often in mathematics lessonsstudents are presented with lotsof short closed questions totackle, one after the other, wherethe reasoning length in eachquestion is just one step. Bycontrast, in this lesson studentsare invited to explore one biggerproblem for the whole lesson.Arithmagons (also known asaddagons) are polygons with acircle number placed at eachvertex and a square number oneach side so that each squarenumber is the sum of the twoadjacent circle numbers. Findingthe missing square numbers in atriangular arithmagon when thecircle numbers are given justinvolves addition, but finding thesquare numbers when the circlenumbers are given is much morechallenging. Although theproblem is equivalent to solvingthree simultaneous linearequations in three unknowns, itis not necessary for students touse algebra to make sense of thestructure. Some carefulreasoning enables them to solvethe general problem and see whyit will always work. However, analgebraic approach offersconsiderable benefits whenworking with arithmagons withmore than three vertices, whichsome students might exploretowards the end of the lesson.

ARITHMAGONS

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LESSONPLAN

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QThese diagrams are calledarithmagons. What have

you learned about them fromwhat you have done so far?When are they easy and whenare they hard?Students will notice that if you aregiven the circle numbers it is quiteeasy to work out the squarenumbers, but starting with thesquare numbers and working outthe circle numbers is much harder!Arithmagons are also easier to doif all of the numbers are smallpositive integers.

Q If I just give you the threesquare numbers, can you

always find the circle numbers?What other questions can youask about arithmagons?Students might wonder whetherthere is always just one possibleset of three circle numbers for anygiven three square numbers.Might there sometimes be morethan one? Might there sometimesbe no possible circle numbers thatwill work?

There is lots here for students toinvestigate and they will needplenty of time to explore thepossibilities. Encourage studentsto begin by trying specificnumbers and recording carefullywhat they have done. Encouragethem to notice what happenswhen they change the numbersand to try to explain their findings.When testing out theirconjectures, one student couldinvent three circle numbers, workout the square numbers(carefully!), and then give just thesquare numbers to the personnext to them to see if they canrecreate the original circlenumbers.

MAIN ACTIVITIES

Have the diagram below on the boardat the start of the lesson.

STARTER ACTIVITY

Q Watch what I do. You might haveseen this before – that’s OK – wecan do new things with it.

Give students a moment to take inwhat you have done and then leavethe completed diagram on the boardfor them to look back to later.

Q. Now here are four for you to try.Can you find the missing numbers?Each time the square number is thesum of the circle numbers on eitherside of it.

3

8 7

3

8 7

11 10

15

5

6 2

1

9 5 19 10

15

10 11

6

Fill in the box numbers as below,saying:Q. 3 plus 8 is 11, 8 plus 7 is 15, 3 plus7 is 10.

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MATHEMATICS | KS3

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You could conclude the lessonwith a plenary in which thestudents talk about what theyhave found out. They maynotice patterns to do with oddand even numbers andinteresting results, such as thatthe sum of the square numbersis always twice the sum of thecircle numbers. This makessense, because each circlenumber is counted twice bycontributing to two differentsquare numbers. Students couldtry to explain this carefully,perhaps with a diagram showingwhy it will always work.

This is a useful result, becauseone way to solve an arithmagonin which you are given just thethree square numbers is to startby adding these together andhalving the answer, to give youthe total of the three circlenumbers. You can write this inthe middle of the arithmagon.For example, for the arithmagonshown below, the total of thesquare numbers is 19 + 10 + 15 =44, so the circle total is 22(shown in green at the centre).

SUMMARY

Students could try out thearithmagon puzzle on theirparents/carers and see ifthey can find the circlenumbers given the threesquare numbers! Can thestudent then explain tothem how to find the circlenumbers and why it works?

HOME LEARNING

If we now focus on the bottomsquare number, for instance, thistells us that the bottom twocircles add up to 15. Since weknow that all three circles add upto 22, we know that the top circlemust be 22 – 15 = 7.

We could use the same processto find the values for the othertwo circles, but we don’t need to,since once we have one circle itis easy to work out the values ofthe other circles by subtractingfrom the squares adjacent tothem. This gives the circle valuesof 7, 12 and 3.

The same process could berepresented more formally by

19 10

15

22

S1 S2

S3C3 C2

C1Give students a few minutes to work

on these. The first one should be

straightforward, the second involves

subtraction and the third and fourth

are more difficult. The answers, in

order, are {11, 7, 8}, {8, 12, 4}, {7, 12, 3}

and {7.5, 2.5, 3.5}. The last one may

be particularly challenging, given the

halves in the answers. If no one can

do it at the moment, you could leave

it up on the board as a challenge for

later, rather than spoiling it by giving

the answer.

labelling the six numbers as inthe diagram below and writingand solving the three equations:c1 + c2 = s2c1 + c3 = s1

c2 + c3 = s3

(If you use this notation, makesure that students understandthat the subscript numbers arejust labels for which c-number ors-number we are referring to, andnot indices.)

Colin Foster is a SeniorResearch Fellow inmathematics education inthe School of Education atthe University ofNottingham. He haswritten many books andarticles for mathematicsteachers (seewww.foster77.co.uk).

ABOUT OUR EXPERT

ADDITIONAL RESOURCESA GOOD WEBSITE FOR PLAYING AROUNDWITH THE NUMBERS IN AN ARITHMAGON ISHTTP://NRICH.MATHS.ORG/2670. THE PAGEHTTP://NRICH.MATHS.ORG/7447 EXTENDS THISTO ‘MULTIPLICATION ARITHMAGONS’, ANDHTTP://FLASHMATHS.CO.UK/VIEWFLASH.PHP?ID=24 OFFERS BOTH IN ONE.

STRETCH THEM FURTHERWHAT HAPPENS IF YOU MAKE A SQUAREARITHMAGON, WITH FOURCIRCLE NUMBERSAND FOURSQUARE NUMBERS? THINGSWORK OUT VERY DIFFERENTLY THIS TIME!WHAT HAPPENS WITH ARITHMAGONS WITHFIVE OR MORE VERTICES?

INFORMATIONCORNER

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