A Staircase InvestigationAuthor(s): Darrell MorganSource: Mathematics in School, Vol. 21, No. 1 (Jan., 1992), pp. 37-38Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216434 .

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Table of convergents for and n p, qn an

b. a x bn

1 2 1 1 1 1 2 3 1 3 2 6 3 14 5 7 5 35 4 17 6 17 12 204 5 82 29 41 29 1189 6 99 35 99 70 6930 7 478 169 239 169 40391 8 577 204 577 408 235416 9 2786 985 1393 985 1372105 10 3363 1189 3363 2378 7997214 11 16238 5741 8119 5741 46611179 12 19601 6930 19601 13860 271669860

A detailed examination of the Table yields a number of conjectures, which can be expressed in the form of a set of equations, listed below as (1) to (6). The proofs of these turn out to provide an interesting exercise in induction whereby the equations have to be done in pairs. Sonia's observations are a consequence of equations (2) and (6), which tell us that q2n,,= a,b,. The fact that all square triangular numbers arise in this way comes from the result that all the solutions of

k2 - 8m2 = 1

are given by k = p2n,, and m = q2n, giving alternate numbers from the third column (1,6,35,204, ...) as those numbers whose squares are also triangular numbers. The Table also shows how quickly the numbers grow.

For all n we have the following list of relationships:

q2n-1 = b2n-1 (1)

2q2n = b2n (2)

P2n - 1 = 2a2n- 1 (3)

p2n = a2n (4)

an b,, + anbn+ 1 -= b2n+1 (4)

b2n = 2anbn (6)

Readers who are interested in continuing fractions might like to provide proofs for (1) to (6) themselves. The recurrence relations which express p, for example in terms of the previous two values of p are needed. In general for a continued fraction of the form [ClIC2,C3, ...] these relations take the form

Pn = cpn-1 + pn- 2

with a similar equation for the q's. The values of the c's come from the continued fractions for /2 and

Readers who would like to see the details of the proofs of these results are welcome to write to the author for an extended version of this article.

Continued fractions are a fascinating area of mathemat- ics, and excellent introductions can be found in Burn2 and Olds4. With a fairly small theoretical background including a grasp of the fundamental recurrence relations one soon encounters many patterns which can be investigated and for which proofs can be constructed. The continued frac- tions for square roots are a particularly rich area. It must surely be the case for example that sets of equations analogous to (1) to (6) govern relationships between con- tinued fractions other than the two particular ones con- sidered in this article, and this could provide an area for students to investigate themselves.

References

1. Burn, R. P. (1991). Square Triangle Numbers, Mathematics in School, 20, 2.

2. Burn, R. P. (1982). A Pathway into Number Theory, CUP. 3. Larsen, M. E. (1987). Pell's Equation: A Tool for the Puzzle-Smith,

Mathematical Gazette, 71, 261-265. 4. Olds, C. D. (1963). Continued Fractions, Mathematical Association of

America.

I nves

by Darrell Morgan

The Staircase This report will consider an investigation using a math- ematical teaching aid I have developed called the staircase. It was designed in order to assist in the teaching of a lesson on Number/Algebra with a mixed ability year seven class.

However, further examination and practice with the stair- case has shown that its use can be extended, both across the National Curriculum Mathematics Attainment Targets, and to differing Levels of Attainment within each target.

The structure of a staircase is fairly simplistic and since it never changes children become accustomed to both its shape and use fairly quickly.

A

8 IC

D E

Each staircase uses the numbers 1-9 once

The answers can be entered in a similar way to doing a crossword. Here are the clues, can you fill in the answers?

Down 9x5=A

142 -2=C 20+30- 15+E=61

50-G= 11

Across 43+ B= 100 35-D=23

9x7=F 50 + 48= H

Fig. 1 A Staircase

Mathematics in School, January 1992 37

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The letters are used to represent clues, as in a crossword, each answer is of length two digits. The use of digits within the staircase in obviously at the discretion of the user, however one use adopted in classroom practise is to use each of the digits 1-9 only once in answering clues.

Once the staircase framework has been introduced the children are given clues and are expected to insert the answers into their correct positions within the staircase.

A specific example to test number work and practise algebraic skills with year seven or eight children might be:

CLUES Down Across

9 x 5 = A 43+B = 100 142+2 = C 35 - D =23

20+30- 15+E =61 9 x 7 = F 50- G= 11 50 + 48 = H

Hence the child may use the first clue to determine that A= 45, this can then be entered into the staircase;

A

4

B IC 5

The child can then progress to the first of the "across" clues where he/she may have problems with 43 + B= 100, however from the first clue he/she has the information that B is fifty something, hence he/she may be able to determine the value of B more easily. As the child answers more of the clues he/she can use the information that each of the digits 1-9 are used only once, to either, assist in answering the following clues, or to aid in checking the answers.

Application Across the Mathematics Curriculum The example illustrated above showed how a staircase might be used in reinforcing algebraic and numerical techniques thus satisfying certain levels within attainment targets 2, 3, 5 & 6. However, since the teacher has the option of selecting appropriate clues any particular area of the mathematics curriculum may be covered. This can be illustrated to a limited degree by the following clues:

CLUES

A. I start at the origin and move along 2 and up 3. Where am I? - AT 7

B. How long is the classroom (to the nearest metre)? - AT 8

C0 C.

C

y= 2x. Find C in degrees. SAT 10

D. Find the mean of 27, 33, 49, 61, 101, 41 - AT 12 etc.

Application to any Level of Attainment The examples using staircases so far have tended to concen- trate on year seven and year eight children, however, their use can be extended to almost every school year and even to university level. The following clues illustrate the use of the staircase at a primary level -

CLUES Down Across

A. 24 B. 100 +15 -9

C. 3x5 D. 30 +27

etc

Whilst the following clues illustrate an extension to sixth-form, university level or into the realms of mathemat- ical puzzles:

CLUES Down Across

A is odd 40 < B < 80 C is prime D is even E is divisible by 13 & E < 55 F>50 2G=3E C+E-2<H<C+E+2

Hence the examples displayed have shown that, with well defined clues, the staircase can be used as a very powerful teaching aid capable of traversing the mathematics curriculum at all levels of attainment.

Extending the Staircase Make up some clues for the following staircase

A 1

B C 9 2

D E 4 8

Try to use each of the following signs +, -, x, =

F G 7 6

H 5 3

Make a staircase for a friend to solve. Use all the digits from 1 to 9 in your answers.

38 Mathematics in School, January 1992

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