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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?

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  • $\begingroup$ Try to link the matter to probability, combinatorics and games. I dont know something more funny than this. The problem is that real valued equations models physical problems in general, but the "general life things" (aka human mind) are modeled in natural numbers generally. $\endgroup$
    – Masacroso
    Aug 17, 2016 at 11:00
  • $\begingroup$ An idea: some geometric identities on the plane, areas and perimeter of polygons by example. $\endgroup$
    – Masacroso
    Aug 17, 2016 at 11:09
  • $\begingroup$ Linear functions are quite good at getting quick estimates. If you walk 1 mile in 15 minutes, then you could walk 4 in an hour. Randall Munroe's What If series contain many great and entertaining examples of estimating things, I'm sure you will get some inspiration there. $\endgroup$
    – dtldarek
    Aug 17, 2016 at 11:25
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    $\begingroup$ You might find some great suggestions at matheducators.stackexchange.com $\endgroup$
    – amWhy
    Aug 17, 2016 at 22:30
  • $\begingroup$ You may look for some inspiration in the old book for school pupils "Algebra can be fun" by Yakov Perelman (original publication 1933 in Russian). There are copies available on the net if you google it. $\endgroup$
    – A.Γ.
    Aug 29, 2019 at 6:38

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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.

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  • $\begingroup$ nice! that is indeed interesting. I'll keep it for a bit later though, since it might be confusing when one is not used to working with variables... $\endgroup$
    – Larry
    Aug 17, 2016 at 14:25
  • $\begingroup$ Do you know a reference (a name, perhaps) for this method in general? I'd surprisingly never seen it before. $\endgroup$
    – YiFan Tey
    Oct 5, 2019 at 15:05
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    $\begingroup$ @YiFan: Unfortunately no, I can't even remember writing this answer or where I got this method from. I'm as surprised as you are this moment! $\endgroup$ Oct 6, 2019 at 12:20
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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.

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When it comes to teaching algebra especially in solution guide to the quadratic equation I would definitely suggest the story of Indian Mathematician Sridhara Acharya, who invented the magical formula for calculating quadratic roots and separated algebra from arithmetic [google him]. As suggested by Mathematician 42 proceed like that and then try to show some graphs in Desmos [app is also available] and try to visualize them that what exactly zero means and also show that the result comes by using formula used to find roots coincide with the graphs. Hope they will enjoy that.

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