Confusion about Linear Equations
May. 10th, 2005 04:54 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
livingdebOf course my readers would never be confused about linear equations, but apparently many prospective teachers are.
Surprisingly, they are not confused about how to write an equation for a real life situation that can be expressed as a linear equation. For example, if a gym charges a one-time $200 membership fee plus $30 per month, most prospective math teachers can come up with a usable equation describing total cost. Examples:
Cost = $200 + $30 x (# of months) or
y = 30x + 200 or
f(m) = 200 + 30m.
But they run into trouble graphing the equation. For future reference, a graph of a linear equation is a line. A straight line. With no end points. Yes, I know that you do not have infinitely large paper: just use arrowheads.
Apparently the y-axis is very confusing. It's true that if you don't join the gym, you don't have to pay any money. But if you are graphing this equation, the graph at the y-axis does not curve down to zero. (Straight line, remember?) It does not disappear before reaching the y-axis, say at 1 month. (Infinitely long, remember?) It hits the y-axis at the value of the membership fee.
You can designate the parts of the line that are useless for your problem by drawing them in dotted lines or explaining them in a footnote. Mention that people don't join gyms for negative months. Mention that if you quit the first day or even the first month, the gym may refund all your money or that the contract may require that pay for a full year, so your actual costs won't exactly match the line. These things do not change what a graph of a linear equation looks like.
Another thing about a line is that it contains infinite points. Although you may choose to calculate a few points, the line consists of all the points in the equation, not just the random few you have chosen to calculate. And not a bar graph! So connect your points with a line, as straight as you can draw it, with arrowheads.
One of the cool things about a straight line is that you only need two points to be able to draw one. Five points might give you more confidence that you have done it correctly, but you really only need to calculate two. I recommend the one where x = zero and, if different, the one where y = zero. Or use any values that look easy to calculate. Math doesn't have to be horrible. Really.
Now let's say there's another gym with a lower membership fee, but higher monthly rate, and you want to join a gym but expect to graduate in two years, but it might take you longer, and you expect to move away, but you might end up staying or you might find out you are the type who never goes to gyms even after paying a lot of money, so you want to graph both equations to see what the story is. And you realize that a good thing to know is the break-even point: the point in time at which both gyms would cost the same. The best way to find the break-even point is not to do calculations for random months until you find the month where the answers match for both. Yes, it works. Especially if the break-even point is month 3 or month 6. But what if it's month 17? Don't waste your time doing 17 calculations for each gym. (Or if you're good at narrowing in on the number, perhaps a mere 8 tries. For each gym.) And what if it's month 3.4245? Are you randomly going to hit upon that number? Just find out when the two equations are equal to each other (set them equal and solve for your variable) and be done with it. Yeesh. One calculation--that's really all you need. (And remember, this can be one of your two points for graphing purposes.)
Then, when you draw the lines, they will intersect at the break-even point. If your graph does not show the intersection, it is not an ideal graph. It is okay to have hundreds on one axis and ones on the other, because they are each measuring different things. So, keep messing with the scales until you can fit all the interesting parts on the graph. In this case, you want to show from zero months to at least a bit beyond two years, and you'd want to show the break-even point unless it's far in the future.
Most prospective teachers seem to know to put dollars on the y-axis and months on the x-axis. It works the other-way around, too, but following the convention of having the dependent variable on the y-axis and the independent one on the x-axis shows a certain degree of experience in the field. It's just like the way spelling phonetically instead of properly will allow your readers to figure out what you are trying to say, but it also alerts them that there may be a lack of experience with reading and writing.
Whatever you do, don't fall into the trap of putting the sign-up fee on one axis and the monthly rate on the other. Notice how it's hard to plot the lines then. After calculating a lot of points for an equation, be assured that these points should be findable on your graph.
Surprisingly, they are not confused about how to write an equation for a real life situation that can be expressed as a linear equation. For example, if a gym charges a one-time $200 membership fee plus $30 per month, most prospective math teachers can come up with a usable equation describing total cost. Examples:
Cost = $200 + $30 x (# of months) or
y = 30x + 200 or
f(m) = 200 + 30m.
But they run into trouble graphing the equation. For future reference, a graph of a linear equation is a line. A straight line. With no end points. Yes, I know that you do not have infinitely large paper: just use arrowheads.
Apparently the y-axis is very confusing. It's true that if you don't join the gym, you don't have to pay any money. But if you are graphing this equation, the graph at the y-axis does not curve down to zero. (Straight line, remember?) It does not disappear before reaching the y-axis, say at 1 month. (Infinitely long, remember?) It hits the y-axis at the value of the membership fee.
You can designate the parts of the line that are useless for your problem by drawing them in dotted lines or explaining them in a footnote. Mention that people don't join gyms for negative months. Mention that if you quit the first day or even the first month, the gym may refund all your money or that the contract may require that pay for a full year, so your actual costs won't exactly match the line. These things do not change what a graph of a linear equation looks like.
Another thing about a line is that it contains infinite points. Although you may choose to calculate a few points, the line consists of all the points in the equation, not just the random few you have chosen to calculate. And not a bar graph! So connect your points with a line, as straight as you can draw it, with arrowheads.
One of the cool things about a straight line is that you only need two points to be able to draw one. Five points might give you more confidence that you have done it correctly, but you really only need to calculate two. I recommend the one where x = zero and, if different, the one where y = zero. Or use any values that look easy to calculate. Math doesn't have to be horrible. Really.
Now let's say there's another gym with a lower membership fee, but higher monthly rate, and you want to join a gym but expect to graduate in two years, but it might take you longer, and you expect to move away, but you might end up staying or you might find out you are the type who never goes to gyms even after paying a lot of money, so you want to graph both equations to see what the story is. And you realize that a good thing to know is the break-even point: the point in time at which both gyms would cost the same. The best way to find the break-even point is not to do calculations for random months until you find the month where the answers match for both. Yes, it works. Especially if the break-even point is month 3 or month 6. But what if it's month 17? Don't waste your time doing 17 calculations for each gym. (Or if you're good at narrowing in on the number, perhaps a mere 8 tries. For each gym.) And what if it's month 3.4245? Are you randomly going to hit upon that number? Just find out when the two equations are equal to each other (set them equal and solve for your variable) and be done with it. Yeesh. One calculation--that's really all you need. (And remember, this can be one of your two points for graphing purposes.)
Then, when you draw the lines, they will intersect at the break-even point. If your graph does not show the intersection, it is not an ideal graph. It is okay to have hundreds on one axis and ones on the other, because they are each measuring different things. So, keep messing with the scales until you can fit all the interesting parts on the graph. In this case, you want to show from zero months to at least a bit beyond two years, and you'd want to show the break-even point unless it's far in the future.
Most prospective teachers seem to know to put dollars on the y-axis and months on the x-axis. It works the other-way around, too, but following the convention of having the dependent variable on the y-axis and the independent one on the x-axis shows a certain degree of experience in the field. It's just like the way spelling phonetically instead of properly will allow your readers to figure out what you are trying to say, but it also alerts them that there may be a lack of experience with reading and writing.
Whatever you do, don't fall into the trap of putting the sign-up fee on one axis and the monthly rate on the other. Notice how it's hard to plot the lines then. After calculating a lot of points for an equation, be assured that these points should be findable on your graph.
no subject
on 2005-08-01 10:55 am (UTC)