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I was tutoring a student today and we were doing basic factoring of quadratics and expanding terms like $(x+2)(x+5)$. Now he ended up being able to do this by the end of our 2 and a half hour session, but it was clear that he didn't understand why he should be learning this and didn't see much of a point to it. Now I could try and tell him that factoring is useful later on for simplifying things, partial fractions for integration, etc, but I don't think this is useful as these students don't know about integration and its uses. Is there any application which a high-schooler could understand that would help motivate learning this material (or other applications which just require high school algebra)?

The only thing I could think of was shooting a projectile off a cliff horizontally and trying to figure out how far it goes.

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I think at this level, and for a tutor (not a teacher), the main carrot (really more of a stick) is getting good grades, getting into a good college, etc. Most people really will never use this later, and for most others it's mostly a stepping stone to more advanced topics.

As for how to present it to someone, try the handshake method (every term in one factor "shakes hands" with every term in the other factor) for how to carry it out, and try the rectangle area method for a justification at their level. By "rectangle area method" I mean, when expanding $(x+2)(x+5),$ draw a rectangle with one side of length $x+2$ (the side consists of abutting intervals on the same line, one labeled as length $x$ and the other labeled as length $2),$ and an adjacent side of length $x+5$ (the side consists of abutting intervals on the same line, one labeled as length $x$ and the other labeled as length $5).$

For basic trinomial factoring, look up something called the AC Method. I like this method for weaker students because it makes things a little more systematic for weaker students, and it reinforces the idea of "factoring by grouping", which students are often weak with.

However, if you really want to give applications, maybe psudo-applications like the following will be enough:

Example 1: $(997)(1003)$ equals $(1000-3)(1000 + 3) = (1000)^{2} - 3^2$

Example 2: A $17$ percent increase followed by a $17$ percent decrease is a net decrease since $\left(1 + \frac{17}{100}\right)\left(1 - \frac{17}{100}\right)$ equals $1 - \left(\frac{17}{100}\right)^{2} < 1.$

Example 3: To find the prime factorization of $12^{11} + 12^{12} + 12^{13},$ factor as $12^{11}\left(1 + 12 + 12^{2}\right),$ which equals $4^{11}3^{11}(157) = 2^{22} \cdot 3^{11}\cdot 157.$

Example 4: Determine the exact value of $a^2 - b^2$ if $$a = 7.0241132301442003123012230341430201$$ $$b = 6.0241132301442003123012230341430201$$ Calculators are allowed for #4.

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    $\begingroup$ Ha. Calculators should especially not be allowed on 4. With all the digits post decimal under 5, the answer should be spat out left to right with no pause. $\endgroup$ – JoeTaxpayer Jun 20 '14 at 21:59
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    $\begingroup$ @JoeTaxpayer: Perhaps the point of allowing a calculator is to have the students briefly struggle with trying to fit all those $35$ digits for each number into the calculator. When they realize that a calculator cannot do some purely numeric problems they will appreciate other approaches more... especially when they see that the other approach is so easy, as you note. $\endgroup$ – Rory Daulton Jun 10 '15 at 14:26
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    $\begingroup$ @Rory Daulton: This was intended as a joke, somewhat like saying "pun not intended" to call attention to an intended pun. $\endgroup$ – Dave L Renfro Jun 10 '15 at 15:00
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I was always honest:

You won't use this. If you don't like math by now, it's probably not going to be useful to you in your career or life, or ever again once you graduate. If you're like most people, you're going to completely forget this.

That's not why we're doing it. Ever watch an athlete workout? Maybe do a bench press? Regardless of the sport he or she plays, how many times do they have to lie on their back during a game and heave a heavy weight over their chest? They're not doing bench presses because that's a specific skill they need: they're doing it to get stronger so they'll be better at what they do need to do. You're learning this advanced math to make your brain stronger and to be a smarter, more analytical person.

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    $\begingroup$ I used this sports analogy (even the specific weight lifting part) all the time when teaching! It was something I "discovered" on my own, probably as long ago as when I was an undergraduate in the 1970s, but I've later heard it from others. I imagine many teachers/tutors independently come up with this analogy. In my case, being at one time a fairly serious tennis player and a somewhat serious runner (sub 3 hour marathon), the analogy seemed pretty obvious. What never did seem very obvious to me, however, is why it's so much more accepted in athletics than in academics. $\endgroup$ – Dave L Renfro May 6 '14 at 21:10
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    $\begingroup$ Yeah, it's a fairly straight-forward analogy. I came up with it on my own but I have no doubt that many others did the same before me.... as to why athletics and academics are treated so differently: that's a much longer response than 600 characters allows ;) $\endgroup$ – thumbtackthief May 6 '14 at 21:12
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How many of your students are interested in space exploration?

The quadratic equations became a lot more interesting to me when I learned paths about the sun are ellipses, parabolas or hyperbolas with the sun as a focus. Paths of moons or satellites about a planet are also conic sections with the planet center at a focus.

For me, the visual presentation of quadratics as conic sections also made them more tangible and less remote and abstract.

And if the student becomes interested in quadratics, he might also become interested in factoring them.

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Redwing, In "Conceptual physics", Twelfth Edition, by Paul Hewitt, he makes the following comment on page 181, "Why study this material since you"ll forget most of it? My answer is that whether or not you use this knowledge, in the act of learning how to connect concepts and solve problems you're establishing connections in your brain that didn't exist before. It's the wiring in your brain that makes you an educated person. That wiring will be useful in areas you can't dream of right now".

In the March 17,2014 New York Times article "The Walls Close In", VW invested 1 Billion in a car assembly plant in 2011 in Chatanooga,Tn. The plant offered 2,000 job positions paying an average of 19.50/hour. 80,000 people applied for these jobs in an area where the average pay scale was/is $9.00 an hour. Many low wage workers wanted to work there but could not pass the basic math tests. In a phone interview I had with UT Economics Professor Matt Murray, I told him I contacted the VW plant to determine what the basic level of math was and I never got a reply. He told me that the 78,000 people that didn't get the jobs also failed their drug tests and that VW wouldn't release that math info because there is fierce competition between every company in the state of Tennessee for skilled and educated workers. Please share this information with your students. This is the way that the job market seems to be going. Allen Seay

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