Guest Post: Ed Barbeau – Exercises that Schmeck

Ed Barbeau took up my recent offer for readers to write something for the blog, and has contributed the following, typically excellent, article. 

 

EXERCISES THAT SCHMECK

Whatever your approach to mathematical pedagogy in the schools, it appears that the pupils have to work through a lot of sludge. At the end of a section in a textbook, they might be faced with a large number of similar low level exercises that are essentially rote and restricted in their application.

In my view, these exercises encourage automatic behaviour rather than having pupils pay attention to what is going on, judge the answer for plausibility and context, and to appreciate mathematics as a network of relationships and analyses in which the topics studied are mutually reinforcing.

While practice is necessary for technical mastery, the amount necessary varies with the student, and we should look for situations where we can slip in something to catch the attention of a kid who otherwise might be a little bored (Multiply 153846 times 3. Who do you know that is a billion seconds old? How many cubic integers can you find which are the sums of three other cubes? Find two different pairs of positive integer for which the sum of each pair is equal to the product of the other pair). Here is a little story to generate discussion:

A man walked into a shop and bought four items costing $1.20, $1.25, $1.50 and $3.16. The clerk pulled out a pocket calculator, pressed some buttons and said, “that’ll cost you $7.11”. However, the sharp-eyed customer noticed that the clerk multiplied the numbers rather than added. Was the total cost correct?

As an add-on, pupils might be asked to convert the dollar amount to vulgar fractions and check the product, and perhaps discuss the merits of the two methods of calculation.

Difference of squares

A remarkably rich topic for an early class in algebra is difference of squares factorisation. There are different ways to see its validity and its application, and to allow us to make connections between different areas. Let me briefly comment on four situations.

1. What positive integers can be written as the difference of two squares?

I like this investigation because it is straightforward for the pupils to construct examples and make a conjecture (all positive integers except those that are twice an odd number) that is easy to express. How many pupils would see that you can pick up all the odd integers by taking the difference to two consecutive squares? One avenue of analysis is to look at the remainders when you divide squares by 4. Another is to relate

to the writing of , where

, .

Think of the mathematics involved in this: the recognition that we are saying something about the divisors, the relationship between the divisors and the numbers and , and the logical realisation for it to work is that has to be the product either of two even numbers or two odd numbers.

The above is not a cut-and-dried recipe for how this might be presented to a class, how much to let the students individually or as a group wrestle with it or what scaffolding is needed to guide the discussion.

 

2. Pupils might be asked to look for patterns when they take the difference of two consecutive integer squares.

This pattern might be expressed in different ways, but the one I want to focus on is illustrated by

.

Of course, this can be checked by working out each side of the equation separately. However, something deeper is going on. We are asked to verify that to get to , we start with and add successively and . Now

and

.

There appears to be an understanding conveyed of the equation that is not granted by the technical verification. A structure is revealed that can be adapted to other instances of differences of consecutive squares; can we believe that the difference of any two consecutive integer squares is equal to the sum of their roots?

While this can be done with elementary pupils not yet exposed to algebra, it can highlight the role of algebra both as a mathematical language for expressing patterns and as an avenue to rigorous proof. The pattern is given by

and the proof given by simple manipulations:

A possible extension of this is to the difference of non-consecutive squares. Briefly, one possibility is illustrated by the calculation of :

There are a number of directions in which this can be developed. For example, there is quite a bit that can be built on the fact that the sum of the first odd integers is equal to . For example, the sum of the first ten odd numbers is equal to 100, and we have

Pupils might be asked to investigate and justify the pattern behind the following equations:

 

3. Pupils might be asked to look at the products of four consecutive positive integers and decide whether the result can ever be a square.

Getting some evidence on the table, we find that

.

We have a situation here where some stored knowledge can be useful. If the pupils are familiar with squares, they will note that each product is less than a square by 1; perhaps higher examples can be checked for this characteristic.

Why might this be so? As in the first example, we can look at the situation numerically or algebraically. Numerically, it is a matter of looking at the sequence , guessing at its continuation and empirically checking it. Algebraically, we examine the product

and want to somehow show that this differs from a perfect square by . Now there is a “trick” here; we have to look at the product in a certain way (an algebraic expression is a carrier of information and manipulating it may make some information more explicit):

.

Why would one do this? Both the pairwise product have 3 as the linear coefficient, and we find that the product is equal to

the product of two factors that differ by 2. Two numbers that differ by 2 can be represented as and . In this case, it is more productive to think of them as and . In this way, we find that

Since two positive squares cannot differ by 1, we see that the fourfold product cannot be square.

Most of the time, when I did this with a group, that was the end of the matter. But on one occasion, a pupil asked why it was true that two positive squares could not differ by 1. So I put it to the class, who had a lot of trouble nailing it down. Most were convinced, of course, since the difference between consecutive squares widened as the numbers grew larger. But expressing it neatly was another thing. An application of difference of squares came to the rescue:

is an expression of 1 as a product of two integers.

 

4. Suppose we have a round robin sports tournament in which each pair of the participating teams competes exactly once and each match results in a win (W) or a loss (L).

If there are teams, there are matches and each team plays competitors. Make up a table. Leave the diagonal entries blank. Otherwise, in the th row and th column, enter the symbol when team vanquishes team and when team vanquishes team .

For each team, we record the number of wins and the number of losses. Two facts are apparent: is constant; and the sum of all the s is equal to the sum of all the s.

For example, suppose that we have six competitors, labelled {\bf 1, 2, 3, 4, 5, 6}. Then there will be 15 matches; we can draw up a table recording the results. This table has six rows and six columns indicating the plays and we give the outcomes of the matches.

A possible outcome of a tournament is given by the following table:

The perhaps surprising outcome is that the sum of the squares of the numbers of wins over all the players is equal to the sum of the squares of the numbers of losses. This is not a particularly easy matter for most groups to settle, but an elegant resolution is afforded using the difference of squares factorisation:

 

Concluding remarks

My role of a mathematician is not to dictate classroom practice, but to provide some raw material, ideas to be considered and then used appropriately, as desired. The probability that the material might be effective lies strictly between 0 and 1; how much it can be augmented depends very much on the mathematical experience of the teacher, her reading of the pupils in her class, the strategic choices in how it might be presented and the extent to which the discussion can be directed or left to evolve among the students. The goal is to make pupils into engaged participants in the mathematical enterprise.

In my experience, most children like to know and do stuff, and are proud of what they know and can do. If you can get them past their conception of themselves as regurgitators of routines, then things can get quite interesting. Sometimes, you are surprised at what they do not know, when their instruction takes for granted things that need to be opened up. On other occasions, they can come up with interesting insights that surprise the teacher.

I had an instance of the latter in one of the class discussions of the problem of finding all arrangements of four distinct points in the plane for which the six distances between pairs of them  took exactly two distinct values. This is a nice problem, because one example (usually, four points at the vertices of a square) is found early on, and various of the four examples with three points at the vertices of an equilateral triangle are winkled out. However, there is a sixth example (vertices of an equilateral 72-108 degree trapezoid) that most groups miss. However, on one occasion, the first example proffered was this elusive case. How did the student find it? He thought of a regular pentagon and removed one of its vertices; I did not think of this.

The key to this sort of stuff working with a class is very much dependent how the teacher owns the situation. A good teacher is a witness to her subject: what happens in the classroom is refracted through her own interests, perspectives, knowledge and experience. Her formation should lay a foundation that goes beyond compiling course credits.

There are some things I have found useful with teachers taking one of my courses. When you assign problems, before marking them yourself, hand the scripts for each to one of the students who will have the responsibility to read through them and prepare an oral report for the class on the solutions. The only markings they can put on the scripts are to indicate mathematical errors. This is designed to give them practice in understanding and presenting other points of view; they might realise that some solutions are more elegant and revelatory than others.

Teachers may be asked, not only to solve a problem, but to consider how it might be used with students and come up with a solution at the appropriate level. Practising teachers might be asked to take a standard textbook exercise, where the author clearly expects the exercise to be done in a certain way, and find an alternative approach. For example, a Year 9 problem might posit a man in a theatre line with 5/6 of the line in front of him and 1/7 behind and ask for the number of people in the line. (Let be the number of people and solve ). Looking at it in number theoretic terms, we see that the answer has to be divisible by both 6 and 7, and, with a little thought, has to be their smallest common multiple. (There is a bit of subtlety here. If there is an answer, it has to be 42. But this needs to be checked. There are other pairs of fractions for which the problem does not have an answer.)

Many years ago, I was in a discussion group that included a Russian mathematician, who mentioned some of the challenging problems that were given to middle school students. When he was asked how the students were able to solve such problems, his response was that you keep giving them and “they get the idea”. I think the moral of this comment is that, if we are to escape from the current doldrums in mathematical education, we need a consistent environment in which children encounter exercises and problems that respect their potential and stimulate a desire to explore and understand. Such exercises can arise from natural situations and do not have to be difficult.