Warwickshire Middle School Mathematics Project

Warwickshire Middle School Mathematics ProjectAuthor(s): Mrs. C. G. DawesSource: Mathematics in School, Vol. 11, No. 5 (Nov., 1982), pp. 36-39Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214340 .

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WARWI SHIRE MIDDLE SCHOOL MATHEMATICS

PROJECT by Mrs C. G. Dawes, County Mathematics Adviser, Warwickshire

Background to Project "I'd like you to meet a very bright boy in our top class. He's way beyond everyone else in maths. His teacher has given him the book that our local secondary school uses and he's got as far as solving quadratic equations all on his own!" This 11-year- old was obviously fascinated by figures. He hadn't the slightest idea what he was doing, but was thoroughly enjoying manipu- lating the squiggles on the pages of his book. Was the school providing the best course for him? - "We are letting him pro- gress at his own rate". - Was the boy profiting from what he was doing? How would the secondary school cope with him and his superficial knowledge of parts of their 'O'-level course?

Many teachers ask for help in providing challenging mathematics for children who are of above average mathematical ability and national concern about the provision in our schools for these pupils has been expressed in many educational reports. All too often teachers find themselves spending time with those who have problems in learning mathematics and feel that they are neglecting the brighter ones. The easiest remedies are to give these pupils more examples of the same type or to en- courage them to work through texts or schemes more quickly than their peers. The first may result in loss of interest and enthusiasm, "Why should I bother, if all I'm given to do is 10 more sums exactly like the ones I've just done". The second puts too much emphasis on learning more mathematics with the risk of the children's understanding becoming shallow. Instead we should be encouraging them to use the mathematics they already know and to be thinking mathematically. It was with these points in mind that the Warwickshire Middle School Mathematics Project was launched.

Three middle school teachers were seconded to work on the project; one for a period of two terms (March to December 1980), the other two for one term each. So there were two teachers working full-time on the material throughout the time allocated with the co-operation of the third. The project was based at the University of Warwick where all the facilities of the Science Education Department were available to the teachers. Two members of the department's mathematics staff acted as tutors for the project and made themselves available for discus-

sion whenever requested. These two tutors gave superb back- up, found themselves getting more and more involved but never seemed to begrudge the time they gave and were even prepared to say they had learned from the experience.

Our aim for the project was to produce suitable extension material for the top third of the ability range linking with the average mathematics syllabus for the 10-12 year olds in our middle schools; material that would involve the pupils in using the mathematics they already knew and in thinking mathematically. Many differing approaches are used in our schools and they follow many different schemes and texts. Our project had to suit all these and so to begin with the teachers had to look in depth at what was going on in our schools and into the mathematical topics that were thought appropriate to those two years of teaching. Because of the variety within schools some of the material in the pack could be considered by some to be too simple for their brighter pupils, whereas others could think some of it too difficult for all but the top few. The section on Fractions is a good example of this. We found some schools did very little work on fractions until half-way through their final year, whereas others adopted a more spiral approach and built the work on fractions up over the whole of their four years. So we tried to produce material suitable for all stages. The material has been divided under eight main headings: Angles, Circles, Data and Co-ordinates, Directed Number, Fractions, Measures, Number and Symmetry. Out of the 101 worksheets there are far more (19 altogether) on Number than on, say, Directed Number (only 4) reflecting the priority that most teachers give this area of work. The topics had to be limited because of time and although the three teachers had many more ideas for worksheets, indeed they got to the stage where they could see ideas for worksheets in practically everything they did, we had to call a halt in order to get the project published on time. Most of the ideas for the worksheets came from children's interests, e.g. football, darts, snooker, chess, motor racing, flowers, cooking or from familiar objects or happenings in everyday life, e.g. taps, clocks, trains, postage costs, motorways, bridges. Whilst some were planned just to use that delight in figures illustrated by the boy mentioned in the first paragraph, e.g. digit chains, magic squares, equivalence,

36 Mathematics in School, November 1982



familiar fractions, rationals and irrationals, Chinese square roots, others give the children opportunities to meet other ways of doing things, e.g. radar co-ordinates, isometric grids, multiplication by doubling, using powers of numbers. People have asked us why the topics of algebra and sets do not appear. The latter of these is surely not a separate topic and an approach through sets could (and in some cases should!) be used to solve practically any of the worksheets. We just didn't have time to cover everything, but there is a lot of over-lapping between topics and the worksheets could be cross referenced or regrouped, e.g. Fraction Puzzle 1 and 2 and Circle+ Circle = Square could very easily be put in a group headed "Introducing Algebra" whilst Dicing with Dice, Double Top and Soccer Special could be re-titled "Introducing Probability".

All the material has been tried out in both the teachers' own schools and in other schools by other teachers. These trials were very helpful because, although the teachers spent a con- siderable amount of time trying to get the wording as clear as possible, there were still misunderstandings when the material was tried out without them present. On one occasion, the teacher was surprised to see steam rising from the classroom sink and quickly realised that the word "cold" must be inserted in the directions "turn on the tap" on the worksheet that introduces children to the idea that angles are measures of turning. Some of the worksheets do appear to be very wordy but, on the whole, we have found them to be usable and under- stood by the majority of our more able pupils.

The Worksheets The worksheets are not presented in any particular order. Some are linked and it would be advisable for pupils to try one before the other, e.g. the pairs of sheets on Die-tossing, Travel by Train and Motor Racing. The teachers' book draws attention to this when it occurs, e.g. Die-tossing and Dicing with Dodeca- hedra. Any equipment needed other than the normal classroom pencils, rulers, etc., is also listed in the teachers' book. There are very few pieces of equipment needed that would not be in the average classroom: a 12 sided die is needed for one worksheet, but another sheet gives instructions for making this; a hinged board would be an advantage for Snookered, but an opening door and wall could easily be used instead. Also in the teachers' book we have tried to give references to similar activities and examples in the most commonly used commer- cial texts. It is hoped that teachers will study the worksheets to see how they will fit into their own programme of work, e.g. the worksheets entitled Aliquot Factors. The idea for this came from what the Greeks called "perfect" numbers. In the worksheet the use of an arrow to represent a relationship is intro- duced and a number chain using arrows as "links" is pro- duced (other worksheets give further work on number chains). The pupil is led to look at the factors of numbers and is asked to look for unusual events or relationships such as those pro- vided by prime numbers or the number 1 184. This worksheet could follow work on factors or prime numbers such as on p. 4 of Beta 5, p. 6 of Beta 6 or p. 3 of Mathematics for Schools Book 8. Alternatively, it could follow work on fractions such as pp. 24-26 in Beta 5 or p. 29 of Mathematics for Schools Book 7.

The worksheet Time on Your Hands could follow after angle work or work on time such as p. 59 of Mathematics for Schools Book 5, p. 38 of Book 6 of the same series or p. 15 of Beta 5, pp. 18 and 19 of Beta 6 or Core Unit 18 of Towards Mathe- matics Set 4. This is a worksheet that makes the pupil think very carefully about joining points on a graph. Many children do not fully understand the difference between discrete and continuous graphs. The first points that the child is asked to plot are collinear but, after further investigation, it becomes clear that the graph is very different from the line joining these points. It may be that the ideas or presentation will need altering before they will be suitable for some pupils in some

ALIQUOT FACTORS

The factors of 15 are 5. Tr startinj with 0.9

ar-, !0. 1, ,

5 and 19. What do jou not ce ? Are there

if we 'gnore 15 itself and add an, other numbers with )hi

up the other -actors,, we havue propert ? 1+ 3+ 5= 9

These factors are called ALIQUOT 6. What happens if o stbart

factors, or divisorG, with a prime number ? 1. Do the same to 9 as was What do jou notice whenjou done to 15. i.e write out its Start With a humber whch is t~Q

factors, ignore 9 itself and add roduct of two primes ?

up the rest. What number do ou

get ? 7.

T, startin9 with 114.

We can write outa cha n It might take jou some irrme, for thi3 using arrows ...... but persevere Are

ther-s ar),: ,, )15 - .... numbers like this ?

where the arrow means 'sum of

factori other than itself.

2. Complete the chain above.

3. Start with other numbers and 8. Ty starLin with 6.

Create their chains. Is the ne-x~t

humber alwajs smaller ? If so, -

The Greek.s called such a

tr starti'ng with 48. number 'a perfect ' number b.c:ause of thi properft d. Can Jou findj

q. a) Will there be annj numbers others ? lhe next three or Tour lrom which there are no

anrows ? can be folnld,

- with

p[Alernce, but

b) Wil there be I at i nrunber,-.s

he !Eevenlrteer-th contz(n:.S to whlich are no arcrws ? 1575 diJit9s

TIME ON YOUR HANDS

io 9

8 -.,

7 5 6

Obtain a clock face.

1. a) What angle does the minute

,and make with tihehour

hand at 1 o'clock .

[Measure

the angle from the minutehard th a clockwfse - direction

towards the hour hkand.] b) ad at 2 o'clodck ?

Complete this table (matrix)

2. a) just b

)okin at the craph,

estimate the arle- made bj the

kanhd at hea past three ......

k) .-.. and at halp past e2ht.

Now check these, usinj jVour clock face and protractor. ( Remember tE0hat, at half past three, the hour hand will be

haif'waj between

3 and 4.) Were your estimates close ?

We obvious5 need to plot rrn-re points on the graph, so b usi~n3 Jour clock Tace, ry, hke angles made b the two hands

at ever katf hour.

Plot these as ordered pas onr jour graph.

me (rs.) 1 2 3 4 5 7 8 9 10 11 1Z AnsIe ( ") so

Plot the reults s ordered pairs u3sin these axes -

Anje between the hands (*)

- Time (hrs)5

3. Now, just bj a look at the

graph, esbimate the anale at a

qOrater past five. Check this result

with jour clock face. If jour estimate was dose.,

tr a few other imes, then sketch in

the complere graph.

Mathematics in School, November 1982 37



schools. It is certainly not intended that they should be used as an inflexible package, but rather that care should be taken to fit the worksheet to the child and to the mathematical programme in use. The ideas are not exhaustive. An able child may well wish to investigate further an aspect from a particular worksheet or teachers may wish to write their own worksheets de- veloping these aspects or using new ideas. Many of the sheets have a "sting in the tail"; the last section is more difficult than the rest. This section could be omitted by some children but will provide a challenge for the more able ones, e.g. Dicing with Dice begins with straight forward recording of results, develops this into a bar chart, encourages the pupil to interpret results from this chart but finishes with a question involving "unfair" dice which have weights attached to the inside of one of their faces so that they are three times more likely to roll onto those faces than normal. The bar chart showing results for these dice is given and the pupil is asked to decide which face on each die has the weight attached - not quite as easy as it may sound! Other worksheets ask the pupil to generalise from the results obtained, e.g. in Cutting Chords and Arcs and Chords, having drawn a table and filled in results, the pupil is then asked to see if he can see the pattern and predict entries for "n" chords. This could also be added to Regional Chords if the teacher gave hints of intermediate steps to be taken.

Presenting the worksheets in the best possible manner within our limited resources and limited budget was a problem. We were given invaluable help from the Keeper of the Education Service of Warwickshire Museum Service with layout and illustrations. One of the three teachers wrote all the worksheets in her own handwriting and the result, we feel, is attractive to children as well as being clear and concise.

Individual Worksheets and Children's Reactions

GRAPHS FROM DIGIT CHAINS

1. Stud this number chain 6. Look aain a3t the rirst careful( : chaeh :

S )12-)9 )6- 27 ? 5, 12, 9, 36, 27 .........

We shall make ordered pairs To obtain the next number in the Trom these numbers, as folows :- chain, you multipt[ the units dCigit by 4 a&d then add on the tens digit a) pair each number with its

(; therem is one). Following number

So 5 x4 = 12 b) pair the second number of Fhe

end (2 x4)+ 1 - 9 ordered pair with itself 4-x9 = 56 So we get ......

and (6,x 4) + n = 27 ........... ( 5 , 12) ..... fronm(a) (12 , 12 ) f.0-romlb)

2. Continue the chain and ( 12 , 9)

Write down what discover. ( 9, 9)

(9,36) 3.

Repeat the exercise with (36, 36) this chain:- (36, 27)

6 --24 -- -(

27, 27) 6-->24 018 -3 -s? (2n, ?) ...... etc.

4. Does the chain in Ouestion 5 7. Complete the liot: of ordered

behave th the same waj as the pairs, and plot them on squarcd first one ? paper, using axes labelled from

0 to '56 I h multiples o" thee, 5.

Investi0ate other

chai'ns, Join each point to the next,

an. using a sinrle number to start join the last point to the ~rt. each chath i.e. 2, 4-, 7. - What do

,ou discover ?..

Wlhat do jou discover ?

Graphs from Digit Chains This sheet which appears in the Data and Co-ordinates section was written as a linking exercise between the work on digit chains in the Number section and the assignments involving ordered pairs and the plotting of points in the Data section. It uses the child's ability to solve digit chains, examine their con- struction and plot points from ordered pairs made from the chains. The assignment could be developed further by plotting points obtained from other chains or by plotting points found in question 6(a) on the worksheet and introducing the ideas of vectors and "vector journeys".

Graphs from Digit Chains

2. 3 - 12- 9 -36- 27 -30--*3.

The chain comes back at 3 again.

3. 6~ 24 -- 18 -33- 15 -* 2 1 --* 6.

4. Yes it does.

5. 2 8--a 32 - 11 - 5 - 20 -- 2. 4-* 16- 25--22- 10-* 1

--4. 7 2 8 34 19 3 7 3 1 -7.

All the chains come back to the number I started with.

6. ( 3, 12) a (12, 12) b (12, 9) a (9, 9) b ( 9, 36) a (36, 36) b (36, 27) a (27, 27) b (27,30) a (30, 30) b (30, 3) a ( 3, 3) b

7. I get a square spiral (Diagram 1).

36-

33-

30 -

27-

24-

21-

18-

15-

12 -

9 -

6 -

3 "

0 3 6 9 12 15 18 21 24 27 30 33 36

Diagram 1.

If you take all the ordered pairs made in question 6 part (a) and plot the points and join them up you get Diagram 2.

38 Mathematics in School, November 1982



36-

33-

30-

27-

24-

21-

18-

15-

12-

9-

6-

3-

0 3 6 9 12 15 18 21 24 27 30 33 36 Diagram 2.

Familiar Fractions

Quite oftcn, when we are pfrcsentcd it a frction hic loo FAMILIAR FRACTiuONSE

unfaniliar 8rndJ we cannot- easily

irr3ine its size, it is Useful to

approximate it to the nearest Write these statemrnt6, puttinrq I~ farniliar

-ractihon that we know. the most accurate

famrniliarract-on These faymilar fmractions usual 9. From 9.30 a-m. Io Come from the set :- 10.13 a.m. is

approximate.i 1 1 5 1 2 1 2 3 - of an hour. 2

- T T

'

3 '

5 ' '

4 5 1 3 7 9-

106 0. A b3 ' of swceis vie;

L-jI rna,

include oiLhers vhich are 794, weighs aboul-fdvl .;v.r

well known to ,ou. For _mpI0

ii. A one ltre boEle Let's approximate tothe containin 290 ml of

'0ater !'s

nearest f~amiliar fraction. rtought_ Suli. 3 is Very close in value to 1 1I 12

and is e-pal to 1 12. An an le measurinq 27 "

12 4 v '2 t is near - of a right-anga & is nearest to 1 Check by

drawin&, or bk lmookj at a . A pack OF cards Yvith

number line. EWelve cards missing is about complete.

Now tr to give the nearest ifamiliar

fraction to these--

14. A child with 67p ccc :l

,.) 9 ) 8 mone is a6boub of e

w(', if 19 towards haviq 1 o

4) - 21 15. An athlete w-ho has run

5) 1_5

6) b 1'9

km of a 5'km

rece h!s 22 17 completed about

_--of the

7) 7 _1- course. 2Z 33

Children often learn to manipulate fractions in a very formal way. This worksheet helps them to equate or approximate lesser known fractions with more well-known or "familiar" ones. The APU Survey of Mathematics in Primary Schools found that children often had difficulty in ordering fractions, i.e. in recog- nising their comparative values. The worksheet begins by sug-

gesting ways of approximating fractions. In the accompanying child's work we see that he found that many of the examples could have several likely answers. This led him to convert them and the original example to decimal fractions using a calculator in order to see which was the nearest approximation. Questions 9 to 15 present difficulties to some children. They can cope with the periods of time rounding these off to quarters, halves and three-quarters of an hour but those on other aspects such as weight, length and capacity some find less easy.

Familiar Fractions

Fraction

1. 9 14 2. 1 3. 21

4. 7 5.

13

6. 617 7. 7

8 12 . 33

Nearest

9 _3 15-5

20 - 5

20 -4 7 1

14--2 11 _1 22 - 2 6 1

18 -3 7 -1

21--3

113 1 33 --3

9. 3 of an hour. 10. 4 of a kilogram.

11. 3 full. 12. 13

13. - 14.

155.

I thought some other answers might be better.

Fraction

1. 9

14 2.

4. 175

5. 1

17__ 7. 7

23

8 12 . 33

My answer

3 5 2 5 1 4 1 2 1

2

1 3 1

3

1 3

Other answers

14-7 7 1 or = or_ = 21 -3 Or18 9 r18 -2 7 1

21 -3

15-5 14 - 12 6

21 - 3 O22 1 1

6 3 16--8

24 - 3 or 22 -11 12 3 32 --8

I did not know if some of the other answers were nearer or not, so I used a calculator to divide them and change them into decimals, and I had to change the first fractions into decimals too, so I could compare them.

Fraction

1. 0= 0.642

2. 1-89=0.421

3. = 0.238 4. 15= 0.466 5. 3= 0.590 6. 0= 0.352 7. 7=0.304 8. 12= 0.363

My answer

T =0.600

-0.400 5 ~

=0.250 0=0.500 0=0.500 0=0.333

= 0.333

0=0.333

Other answers

-=0.571 A=0.444 1-0.333 =0.4 3-

= -0.500

=0.333 S=0.600 2= 0.666 0.545

S=0.375 = 0.333 -A= 0.363

3=0.375

I have underlined the best answers. Most of the answers were right.

Copies of the WMMP available at 05 including postage from The Education Offices, 22 Northgate Street, Warwick.

Mathematics in School, November 1982 39



Documents

Warwickshire Middle School Mathematics Project