Everyday Example Problems for Solving Linear and Quadratic Equations

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 occur in real life 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

The best example I've found is the question, "What do I need on the final exam to get an [A, B, C, pass] in this class?" at the end of a term -- which really does get asked by one or more students every semester.

I now respond to that with, "Well, you're asking an algebra question, so you should be able to solve that yourself", and I've set up my grading formula specifically to make that tractable for the student. Say the weighted formula is W = 15%Q + 50%T + 35%F, where W is the weighted total, Q the quiz average, T the test average, and F the final exam score. Going into the last week, Q and T are fixed and W is known by the student's goal for overall grade (say, 90% for an A, etc.). Then one can solve for the necessary F score algebraically in a straightforward fashion, but it's really hard to guess otherwise (likely involves decimals).

I get some surprising responses when this comes up and I respond that the asking student should be able to do it on their own. In almost all cases I have to remind them of our grading formula, write it down, and possibly start the substitution step. Some students then rip off the answer in a few seconds. Some can't accomplish it after a day of effort (even at the end of an elementary algebra course). In a college algebra class, I once set up the formula on the board and people were amazed and gasped and took out phones and snapped photos. Other students have complained, "Are you seriously going to be that hard on me?".

So: Be aware that even with this very practical exercise, there's an enormous difference in reaction between doing it as a theoretical exercise in class (mostly the usual "meh" reaction) and doing it in response to an immediate needy student's request (which can be "wow", etc.).

Answer 2

Here is a real-life example that I used in a class with some success. The enrollees were middle school and high school teachers, who were in their first year of teaching (not having arrived with necessarily strong coursework in mathematics, but let us not diverge for the moment).

Based on the interests of the (Boston-based) class, I phrased it in terms of scalping (Red Sox) baseball tickets. Students seemed engaged, and one of them emailed me the next day to say:

For what it's worth: I felt like Will Hunting when I got that Red Sox ticket markup problem yesterday.

(So, some anecdotal evidence that it was received okay!)

I phrase it below in terms of expensive jeans at a vintage clothing store, but I'm sure you can identify the underlying mathematics and modify the context as you see fit.

Real Life A: Your boss at a vintage clothing store tells you a particular brand of jeans have become very popular, and to mark them up by 10%. A few weeks pass, and she asks you to mark them back down by 10%. Is the final price more than, less than, or the same as the original price?

Real Life B: Same as A but the scenario is marking down by 10% first, and back up by 10% later.

Real Life C: Suppose the jean price in A is initially $200; leaving aside whether this is an exorbitant amount to pay for a pair of pants, what will be the final price?

It is a pretty common misconception that the before- and after-prices will be the same, so you could weave this into a tale of your boss (or maybe a nefarious manager, or whomever else) chastising you for messing up the prices. ("I told you to mark it back up by 10%; they should all say $200!" etc.)

Part C is probably the easiest among the three, since it deals with concrete numbers. But I have a habit of wondering about "inverting" problems; in the $200 case of C: Can you use the percentage increase and decrease to determine the final price? With an eye towards inversion, this led me to ask whether you could use the final price to determine the percentage increase and decrease:

Real Life D: A friend of yours is explaining that she was asked to mark a pair of $200 jeans up by some percentage, then back down by the same percentage, and her boss is convinced something went wrong because the final price is now less than the original. The boss is even talking about docking her pay, which seems unfair since your friend has done nothing wrong; she followed directions exactly, and now the jeans are priced 98 cents less than they were at the outset.

In scenario D, this is where you'd tune out and wonder:

Just what was the percentage by which the jeans were marked up and then marked down?

This is the kind of problem that (I think) might fascinate a mathematically inclined individual (especially if there was scrap paper -- or a napkin -- nearby). I think it is a nice place to point out where one might become curious and ask a mathematical question; even if your students do not share in this mathematical curiosity initially, it is a habit of mind that I think is worth fostering. Plus, you can try to work it out and then check correctness by asking your friend for the percentage at the end!

Omitting units, and an explanation around the setup, here is one algebraic approach:

$$200(1+p)(1-p) = 199.02 \iff 200(1-p^2) = 199.02 \iff 200 - 200p^2 = 199.02$$

$$\iff 0.98 = 200p^2 \iff 0.0049 = p^2 \iff 0.07 = p$$

So, the percentage change is 7%.

The initial equation is a true quadratic that can be solved with the quadratic equation or a calculator or whatever else; the solving method above proceeds by drawing from the knowledge around the product $(1+p)(1-p)=(1-p^2)$, and there is a lot more than be unpacked from the given approach: figuring out how to write the initial equation, figuring out how to solve it by hand (how do you hand compute $0.98/200$? well, you could divide by $2$ then divide by $100$), possibly thinking of $0.0049 = 49/1000$ for which finding the sqrt to be $7/100 = 0.07$ might be easier, investigating the other possible "solution" of -7%, and, more generally, this problem has all the components that I think one hopes a student can do by the end of an algebra unit on solving quadratics:

Solve a problem in more than one way (e.g., the method above as well as with the quadratic equation, though a thoughtful version of guess-and-check by recalling C and trying another value like 5% can work, too); represent a real life problem using mathematical notation; and sense-making fluently with decimals, fractions, and percentages in carrying out the algebra.

Answer 3

I once tutored someone and simple linear equations was part of the course.

One example I came up with related to the student's own life was saving up to buy a smartphone.

The problem can be made into different variations, with a basic version such as:

I want to buy a phone which costs $400. With a weekly income of a, how many weeks do I need to save to be able to buy it?

This gives you an equation like y = ax + b where y = 400 (target price), a = your weekly income, x the number of weeks to save and b the starting money.

We can adjust the problem by changing their weekly salary or working more hours, and/or changing the amount they have today in their bank account. You could also change the target variable, for example setting a fixed number of weeks and checking what price I'll be able to pay for a phone by that time.

We can also increase the problem scope by introducing a (real) factor that the price of the phone will decrease over time, giving us a system of two equations, where the price of the phone over time might be for example 400 - 20x, and we'll solve the system for where these two lines will intersect. (When will I be able to buy it, and what will the price be then?)

Since the price won't go to zero in reality, perhaps the price function should rather be a quadratic function with some reasonable minimum. This would also perhaps model the price drop more realistically.

Answer 4

Very difficult algebra (or maybe just longer thinking time) could prove the following is true:

$$\underbrace{1+3+5+7+\dots}_n=n^2$$

Where we simply add the first $n$ odd numbers. Hopefully they will understand what $n^2$ means.

Proof I'd probably follow:

$$\underbrace{1+3+5+7+\dots}_{n-1}=(n-1)^2=n^2-2n+1=\underbrace{1+3+5+7+\dots}_n-(2n-1)$$

And simply see that $2n-1$ would be the last odd number, and when subtracted out, we get what we started with. Then, explain how induction works, so that since we know $1=1^2$, then $1+3=2^2$, since $1+3=2^2$, then $1+3+5=3^2$, etc.

Possibly you might want to give them a hint after 10 minutes, and wink wink point towards the sign on the wall that says "FOIL = First Outside Inside Last" on the wall (a must have for algebra teachers).

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If your students know what $i=\sqrt{-1}$ is, challenge them to find $\sqrt{i}$.

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Challenge your students to uncover a secret message where the letters are all systems of equations of the other letters and solving for just a few letters doesn't necessarily give you the solution to all of the letters.

For example:

$$a=3b+2c$$

$$b=7e+g+h$$

$$c=\dots$$

And the code to break could be

$$5,20,31,13,7,17|4,3,20,4|\dots$$

where $|$ means a space.

Not particularly impossible with substitution, but just time consuming and possibly fun.

Answer 5

How about ray-tracing quadratic equations, e.g., for a ray-sphere reflection?

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Wikipedia has an article detailing the equations here.