looking for motivation in teaching algebra

Question

I am going to teach some grade 9 students about solving linear and quadratic equations. I am looking for a question from every day life (of a teenager) or a puzzle which is hard to solve without using algebra. There are of course loads out there in textbooks and the internet, but I haven't yet found one which is really intriguing and which could arouse the interest even of a student who has other things on his/her mind and who has a general dislike for school-mathematics.

For example questions concerning the respective speeds, distances and time-periods of two vehicles with respect to each other are classic examples motivating linear equations, but they don't really seem to be relevant for real life (from the perspective of a teenager) nor are they particularly fascinating (at least for someone who isn't interested in mathematics anyway).

I'd like to begin the subject with such a question and let the students work on it together for maybe half an hour or so (i.e. the question shouldn't be too easy to solve); hopefully this way they will see themselves how useful it can be to introduce variables.

Any ideas?

Answer 1

Here is one type of non-standard question which you might try to sell in a sort of puzzle style.

Can you use the quadratic formula to solve the cubic equation $x^3-26x-5=0$?

The answer obviously should be yes, the trick is to write the equation as $$(-x)5^2-5+(x^3-x)=0.$$ Viewing this a quadratic equation in '$5$', and using the quadratic formula, we get that $5=\frac{1\pm \sqrt{1+4x(x^3-x)}}{-2x}$. Thus $$-10x=1\pm \sqrt{4x^4-4x^2+1}=1\pm \sqrt{(2x^2-1)^2}.$$ Thus $$-10x-1=\pm(2x^2-1).$$ Now this is a quadratic equation and thus can be solved. Now you have two zeroes, and the third can be found by factoring out the other two. In fact you can do this trick for equations of the form $x^3-ax-b=0$ as long as the expression underneath the root is nice.

But then again, there is nothing real-life about this question. It is somewhat surprising to see that the quadratic formula can sometimes be used to find roots of cubics. So in that respect it might be interesting.

Answer 2

Maybe educators shouldn't strive so much to find interesting real-life math problemas at the K-12 level, since there aren't that many. Of course, every student should learn how to solve problems in personal finance, measurements, interpretation of charts & tables, and proportionality in all its guises. But the fact is that interesting and realistic problems in these areas are few and far between. Maybe math classes should also have room for not-so-realistic problems the succumb to some experimentation and basic mathematical reasoning. So, for example, one way to solve the cubic equation above is to plot the graph of the corresponding cubic function, notice that is seems to cross the x-axis at x = -5, check that this is indeed a root by substituting for x, divide the expression by x+5, obtaining a quadratic equation, and solve that by means of the well-known quadratic formula, obtaining the remaining two roots. Of course that requires some theoretical knowledge (at least the polynomial remainder theorem, the quadratic formula and the fact that a cubic polynomial has 3 roots), but the key step, IMHO, is to use experimentation (in the above case, plotting the graph) to get a handle on the problem. Nothing wrong with using the very clever observation (or, if you prefer, a "trick") that the equation can be written as a quadratic equation in 5. But, besides that being a bit too clever for most students, it is also too specific - it works for only a very tiny fraction of algebra problems, as opposed to plotting the graph of an equation, for example.