Posts Tagged ‘Resources’

Taught: A Pascal’s Triangle wall-display

“What are those numbers on the wall, sir?”

“An interesting question. What do you think they are?”

This isn’t a single taught lesson, but something which impacted many of my lessons for at least a week. It’s the story of one of the first pieces of decoration I put up in my new classroom. A wall-display… to be precise, this one:

Pascal\'s Triangle Wall Display

(as with most images on this site, click for full size)

Background

Pascal’s Triangle has an enormous number of properties, and can be the starting point for a large number of investigations. I think it should be mandatory for all maths classrooms to feature it in a prominent place!

The Story of the Wall-Display

The above image isn’t how the display started, though. When I first put the (individually cut out and laminated) numbers on my wall, there were no questions, no header, and several rows missing. All the questions which subsequently appeared were ones which had been asked by students, mainly during the ‘dead time’ at the start of lessons, while waiting for a critical mass to arrive.

Before the questions arrived on the board, I gave the numbers their heading, but only after every group I teach had had the chance to see the numbers ‘naked’. These groups vary both in age and ability, but the initial questions asked by the students were all very much the same (and notice that I say ‘asked by the students’ rather than ‘asked of the students’).

Most of the sets figured out the generating relationship very quickly, but the colours took a little extra time, particularly as the focus of the lessons being taught that week was never on the display itself. Several of the students I discussed it with quickly saw that the white numbers were those divisible by three, but were unsure what the other colours meant. For several, it was their first exposure to the ideas which will lead into modular arithmetic, which is not actually in any UK GCSE or A-level syllabus.

Although it took a significant amount of time to cut out and laminate each number individually, I feel it was worth it for the increase in flexibility you get: you can easily remove several numbers, or a row, and ask students to put them back; you can remove a diagonal column, and lay it out somewhere else for inspection, so that students can see (for example) the trianglular numbers without getting distracted by the rest of the triangle; you can slowly build up the triangle over several days, so that the display isn’t seen as something static and fixed from the very start.

The display remained on my wall for the rest of the year for several reasons — one being that the numbers are actually useful in several different contexts. In statistics, for example, they give the number of ways of choosing k elements from an n-element set, and in pure mathematics they form the coefficients needed in the Binomial theorem. It also looks pretty!

The Quest for Ownership

By the time this display arrived, I had had the following poster on my walls for several weeks:

Pascal\'s Triangle Poster

As you can see, it contains essentially the same information as my wall display, although in a slightly less colourful format. It actually goes much further, revealing a large number of patterns in the Triangle which my students never guessed at. Despite this, not a single student demonstrated any interest in Pascal’s Triangle until my display appeared… and I certainly didn’t get the impression that anyone in any of my sets had read anything on this poster.

To me, this emphasises the importance of getting students engaged in the material by feeling that they are making a contribution to something dynamic and developing. The static, over detailed poster with all the information already there ellicited no interest, while the simple, changing, colourful display (which didn’t intially have a title imposted on it) was very successful at engaging a wide variety of pupils.

Future Development

Next year, I would love to use this wall display as the basis of a mathematical investigation with one of my groups. It might lead to combinatorics, or to fractals (via the Serpinski gasket), or to the Stirling numbers, or to elementary number theory… I need to spend some time over the summer thinking about how best to structure the activity.

I also feel there is a need to put this in some historical context, both European (who was Pascal, and what else did he do?) and international: Indian commentaries on poems, Chinese diagrams with rod numerals, etc. This historical context is something that I will be spending much more time on with all the areas I teach next year (expect a post on this in the future!).

Sources

Here are the files I used to create the wall display:

Both of these were originally created using the excellent free vector drawing program Inkscape, which I plan to write about in the future. Inkscape’s native format is .SVG, and I have uploaded in this format for anyone wishing to edit the files.

Through the magic of embedded media, you can also browse them:

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Lessons Taught, Resources, The Classroom Comments 7 Jul 13th, 2008