Lessons taught; Lessons learnt

One of the teachers in my school regularly provides a ‘problem of the day’ to his students. This one, from last week, caught my eye, as containing some ‘hidden’ maths that would make a good lower-secondary investigation:

The average of a set of 64 numbers is 64.
The average of the first 36 numbers is 36.
What is the average of the last 28 numbers?

Solution

The total of all 64 numbers is .
The total of the first 36 numbers is .
The total of the last 28 is therefore ,
which makes their average .

Comments

In its ‘raw’ form this is just a basic test that someone understands how to calculate means. It’s nice that the final number is an integer, though… and is it a coincidence that ?

Let’s try the same problem with different numbers:

The average of a set of 51 numbers is 51.
The average of the first 17 numbers is 17.
What is the average of the last 34 numbers?

The average of the remainder is .

Not only do we get a whole number, but the answer is .

Can we conjecture that the answer will always be the sum of the two set sizes? Well, it’s happened twice, so it must be true (!). Looking at the general problem:

The average of a set of  numbers is .
The average of the first  numbers is .
What is the average of the last  numbers?

The average of the remainder is .

Here we might get stuck, unless we remember (or discover, or are taught) at this point the difference of two squares identity: .

This makes our average , and our conjecture is proven.

Extensions

Does a similarly nice thing happen if we specify extra conditions? For example, we could say that there are 5 numbers with an average of 5, and another 9 numbers with an average of 9.

Can we say anything about the median in these situations?

1% of a day plus 1% of an hour is exactly 15 minutes.

Check with Google if you don’t believe me!

It’s been a busy term, and it’s not over yet, but I promised that I would post something this week. So, a question for you to explore:

The second term of a geometric sequence is 2. What’s the sum to infinity?

If your first thought is ‘it could be anything’, then you’re not alone, but you’re also not correct!

If you get an answer algebraically, can you give a geometric interpretation?

The Seed

I recently found myself reflecting, while solving a mathematical problem that occurred to me on a car journey, on the differences between mathematics as I experience it, and the mathematics that my students experience.

This post introduces the problem, and gives some reflections on the problem solving process, as well as suggestions for ways to simplify or extend the problem.

The Problem

What’s the question?

Fix two points. Consider all the quadratics which go through those two points. What is the locus of the stationary points of these quadratics?

If the two fixed points are (2, 0) and (0, 2), for example, here are four quadratics passing through those points, with their stationary points highlighted in red:

To solve the problem we need to find what happens to the stationary points of all the quadratics which pass through the two fixed points.

The Motivation

Why is this interesting?

First, I think this is an interesting problem, with a result that isn’t immediately obvious.

Second, I have an emotional investment in it — the problem arose naturally in the course of an investigation I was doing, and I spent time thinking about and proving the result.

Third, I believe this is an example of a problem where dynamic geometry can be used to help make conjectures and build understanding.

However, the problem isn’t some new area of maths. It isn’t a significant result with many corollaries or consequences. It doesn’t shed light on other mathematical objects. It’s a mathematical chew toy.

The Answer

What’s the answer?

Really, then, the answer isn’t important. The point is the journey, and considering how to continue the journey once we’ve reached our initial destination, the answer.

And that’s why I’m not going to tell you what the answer is.

It’s actually harder for me to withhold than it would be to write up the page of working I have in front of me. The culture of the mathematics classroom is one where answers are all important. Throughout school and university, and not just in mathematics, the way we assess students is based on the following assumption:

“Good students are those who can answer questions on lots of unrelated topics in a set time limit.”

As someone who passed through this system with flying colours, I have an unhealthy need to demonstrate my facility in tests.

This buying into, and even enjoyment of tests as an end in themselves is a hallmark of the more ‘nerdy’/scientific subjects. And I can’t ignore it, because it’s been built into my personality, into my job, and into society.

So, no answer.

What I am going to do is to describe some of the ways which I investigated the problem, and some further avenues that suggested themselves to me.

The Journey

How do I solve it?

The first thing I did was to build a tool to help me explore the problem. In this case, it was the Geogebra worksheet which appears at the end of an earlier post on this problem.

I then picked two fixed points, and generated the following:

There are a number of features to notice in this example: the locus goes through both the fixed points; it seems to have a vertical asymptote roughly half-way between the fixed points; it seems to have another, oblique asymptote.

How many of these features are generic, and how many are artifacts of the particular example? I notice that the fixed points I’ve chosen are fairly symmetrical: (0,2) and (2,0). Does it alter the features if I destroy this symmetry?

Here’s a less symmetrical example, and all the features I noticed are preserved. So we have here a particular example which is serving as an exemplar for the general case.

We can be led astray, though, into thinking that the general case always happens. Are there any special cases where these features would break down? What happens, for example, when the chosen fixed points have the same x coordinate? There are no quadratics at all in this case. What about fixed points with the same y coordinate?

Something different is happening here. A moment’s reflection reveals that this particular example is equivalent to saying that a quadratic with roots at 0 and 1 always has a stationary point at 1/2. This isn’t interesting to me, because I’m well aware of the properties of quadratics — but even just investigating this restricted problem could be interesting to someone who hadn’t thought about quadratics in this way before.

Does it make a difference if the points are not on the x-axis?

No. So another feature of the locus should be: a translation of the fixed points leads to a translation of the locus (are you convinced? if not, what would make you convinced?).

So, just through playing with some fixed points, we’ve developed an intuition about the form and properties our answer should take and hold. How do we actually find this answer? It will almost inevitably involve algebra: we need to assign values to the two fixed points, and try to figure out the equation of the locus.

We could just call one point (p,q), the other point (r,s), and dive in from there… but once we’ve noticed the translational property of the locus, why not fix one of the points to be something simple, like (0,0)? To do this requires me to have the experience to know that fixing one of the points will make my life easier (I now have two variables instead of four), and that (0,0) is likely to be a good choice for the fixed point (do I have any non-intuitive reason for this?).

[Actually, the first time I looked at this problem, I didn't fix one of the points to be (0,0) -- I had one of the points on the x-axis, and the other on the y-axis.]

If I wasn’t comfortable with abstracting straight away, I could have been even more specific, and just looked at the problem for one particular set of fixed points.

Diving into algebra, then — what is it I actually want to do? I want to find a set of equations which represent all the quadratics that go through (0,0) and (p,q). This will be a one parameter family of equations. I then want, for each choice of parameter, to figure out where the stationary point of that quadratic is. This will give me a parametric equation for the locus. I could stop with that, but I’ll probably try to shake things around until the parameter disappears, and I have an equation for the locus in the standard ‘y=f(x)’ style.

That’s what I did, and you do get a ‘nice’ answer.

Nothing here should be beyond a decent A-level student, although it’s not a typical A-level question, so I’d be interested to see how many of my students would be able to do this.

The Destination

Once we’ve got an answer, what do we do with it?

This is a question which is outside the comfort zone of most students, across all age ranges, school types, and abilities. The answer is the goal — once we have something that matches what the back of the book says, it’s time to move on to the next question.

Some students are so focused on ‘the answer’ that they don’t even check whether the answer they get makes sense. In the context of this question, checking our answer involves verifying that it holds the properties that we found in our initial investigation: what are the asymptotes? does it really go through the fixed points? does it account for the special behaviour when the points are horizontal?

We should also look for any check we’ve actually answered our initial question. For example, I should make sure that I can give the formula for any two points, rather than requiring that one point is the origin.

The View

Where now?

This is the question that is almost never asked at a meaningful level in school mathematics. It’s true that discussing extensions and further work was supposed to be a key point of GCSE coursework, but that soon degenerated into teachers giving students fixed lists of phrases to use: “I’d try this with different shapes/higher numbers/different data, etc.”

Here are some things which occurred to me, after I answered the initial question:

• Perhaps I could introduce this to students by asking them to generate the equation of a quadratic which passes through two given points, with a stationary point at a given value of x.
• Even just the question: “How do we generate the quadratic which goes through three specific points?” is an interesting one for them to investigate, although quite a standard one.
• What if only one point was fixed? Would the locus be the whole plane?
• What about cubics? We’d have to fix three points to get the locus to be a line. Would it look similar to the quadratic case, or does new behaviour emerge?
• What about if I restrict the fixed points in some way?

These are a combination of thoughts on how to ’scaffold’ this question so that it would be accessible to others, and thoughts on how to alter the question to create further interesting questions.

As promised, here is an animated Geogebra visualisation. Click ‘Start’, and six moving lines will appear.

• What behaviours do the lines exhibit?
• What relationships are there between the different lines?
• How are these lines defined?

Geogebra applet (enable Java to see it).

(Source: movinglines.ggb and movinglines.html.)

Click on ‘Show Values’ and three values will appear — all six of these lines are generated from these three values. If you click on ‘Stop’, and move the sliders, then the lines will be automatically updated to reflect those new values.

• Can you make any conjectures now about how the lines are defined?
• What values would we need to make one (or more) pair of the lines
  • perpendicular?
  • parallel?
  • vertical?
  • horizontal?
• Can we generate any line we wish by setting the values appropriately?

I managed to captivate both a lower school class and an A-level class with this visualisation for quite a while last year, and the A-level class made good progress in making and justifying conjectures about what was going on.

Technical Note

One of the few downsides of Geogebra compared to other dynamic geometry software, is that it contains no in-built facility for animation. Thus, to create any animations using Geogebra, you have to create the animations manually in Javascript. It’s not particularly difficult to do, particularly if you know some Javascript, as you can see if you view the source of the .html file above. Putting the ability to animate inside Geogebra itself is, though, my one big wish for future versions.