# Simple middle school math problem...

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I was helping my little cousin with his math hw, and one of the questions was predicting points on a graph. They gave you a table:

X | Y

.5 | 2

1 | 1

2 | .5

4 |.25

5 | ?

10| ?

Now, obviously this is just y=1/x just from looking at it. It's easy to see what happens to Y as you increase X.

But I didn't know how to explain it to him algebraically if he gets a similar problem on a test, and he is not able to see trends in his head like more experienced people can. I also don't know if they are going to give him unknown values that are unrealistic to every number out to.

This is embarrassing, but how do I explain this to him?

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That's a nice challenge. It took me a bit to think about how to explain it, but then I thought about fractions. Middle schoolers should know fractions and in general it's usually better to translate decimal to fractions anyways seeing as it's easier to see trends.

So, that would mean that:

$x = 0.5 = \frac{5}{10} = \frac{1}{2}$ WHEN $y=2$

$x = 1$ WHEN $y=1$

$x = 2$ WHEN $y=0.5=\frac{5}{10}=\frac{1}{2}$

$x = 4$ WHEN $y=0.25=\frac{25}{100}=\frac{1}{4}$

This will show a trend. The only "different element" here is the first one, when x=0.5, but ask your cousin to put it in the same 'trend' as the rest, like this:

$y=\frac{1}{x}=\frac{1}{0.5}=\frac{1}{\frac{1}{2}}=\frac{2}{1}=2$

And he can see that it works out. From here, finding the rest is easy.

Did that help?

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Graph it?

Plot it roughly if need be, the relationship should become apparent.

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Hm the graph of 1/x contains an asymptote and doesn't quite look intuitive... I am not sure that it would help in this case unless the student knows to recognize such graphs....

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That's a nice challenge. It took me a bit to think about how to explain it, but then I thought about fractions. Middle schoolers should know fractions and in general it's usually better to translate decimal to fractions anyways seeing as it's easier to see trends.

So, that would mean that:

$x = 0.5 = \frac{5}{10} = \frac{1}{2}$ WHEN $y=2$

$x = 1$ WHEN $y=1$

$x = 2$ WHEN $y=0.5=\frac{5}{10}=\frac{1}{2}$

$x = 4$ WHEN $y=0.25=\frac{25}{100}=\frac{1}{4}$

This will show a trend. The only "different element" here is the first one, when x=0.5, but ask your cousin to put it in the same 'trend' as the rest, like this:

$y=\frac{1}{x}=\frac{1}{0.5}=\frac{1}{\frac{1}{2}}=\frac{2}{1}=2$

And he can see that it works out. From here, finding the rest is easy.

Did that help?

The problem is, its difficult to go from a trend like that to y=1/x Sure, once you point it out to them they see why it works, but its hard for them to recognize it on their own.

I think I may be overestimating the diffcutlty of potential problems. Most of the time you will not have to extend the table out to arbitrarily large numbers, and you can look at the trend to get it without finding an equation. I guess I was just wondering if I was forgetting some algebraic method of determining the equation of a line from a table of values.

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I know what you mean, I was tutoring middle-school kids for a while when I was in high-school. It's a matter of practice. The more your practice, the easier it is to see the 'tougher' problems and you gain experience on what to look for. That, by the way, is true for everyone in math and sciences (and probably in many other types of subjects). I see it today as a junior in College. Practice makes perfect - the more you deal with new types of problems, the more experience you gain to solve them in the future.

What I would recommend, though, is to switch to fractions regardless; with fractions you can see patterns much better than with decimals, specially in the level of middle school.

And practice. Practice practice practice.

~moo

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