How do you explain to a student who is struggling with math that 1/x gets arbitrarily large as x goes to 0?

Check out math4love.com for some creative solutions.

If your approach has not been very successful, no matter how intuitive it seems to you, you have to find another approach. That’s why you’re the tutor and they’re the tutees.

Division is multiplication by the reciprocal

So 1/.1 = 1/(1/10) = 1*10
1/.01 = 1/(1/100) = 1*100

etc.

If they’re having trouble understanding your verbal explanation, ask them to draw a simple line graph. All they need is, say, four points on hand-drawn axes, with x=1, 2, 10, 100 or some such (be sure they don’t make it categorical!). The trend should be obvious.

^^^ As the tutee, I don’t understand what you mean by “categorical.”

@gailcalled No, but @LostInParadise does. He can explain it to his student in a way that will be understood.

I agree with @glacial. Graphing could help. If you show them x=0.5, then 1/x, or 1/0.5, equals 2. If x=100, then 1/x, or 1/100, equals a very small decimal, which is less than 2.

Can’t we bring pizza into it?

If you have one slice out of eight, it’s a normal size slice. But if you cut the pie into four, your one slice is considerably larger. If you cut the pie into two, still bigger. Leave the pie whole, you have a whole pie, which is a lot more pizza than one eighth of a pie.

I agree with @Seek_Kolinahr . I never understood this until my teacher explained it that way.

@Seek_Kolinahr , I can try that. The problem is that we stop at 1 pizza. I want to show that we can generate numbers on to infinity.

The problem with the current pizza method is that it can not demonstrate division by fractions. It works when you divide a pizza into 1, 2, 8 slices, but it doesn’t explain how a pizza divided by ½ is two pizzas.

I think that we can get away with reversing the model.
Instead of asking: As the number of slices decreases, what happens to the size of each slice?
You instead ask: As the size of a slice decreases, what happens to the number of slices?

There are eight slices of pizza. If each person eats two slices, four people can be fed. If each person eats ½ of a slice, sixteen people can be fed.

And if each person eats a slice that is almost infinitely thin, you can feed almost an infinite number of people.

@PhiNotPi Yes, I think that’s what troubles me about the pizza example as well.

@gailcalled Though admittedly, they might not notice that they had been fed!

^^ True, but it is math tutoring and not The Culinary Institute.

No. I’m one of those people who just don’t get math. I can barely do addition and subtraction. I’ve struggled with math my entire life. I have no idea what you are talking about in the Q, and I’m afraid no amount of explanation visual or otherwise is going to help. Sometimes I feel like I have the math equivalence of dyslexia. I just can’t get it. : (

^ Dyscalculia. And I do have it.

Oddly enough, I’ve been pretty good friends with fractions. And if there’s a word problem, I’m golden. Long strands of letters and numbers and computations…. frak it. I’m outsourcing that issue to someone who cares.

“Real-world” examples, like the pizza problem, seem to work just as well if not better when viewed in the absurd (infinite slices for infinite people). Mentally picturing such a strange situation is a fantastic mnemonic device.

An infinite number of mathematicians walk into a bar. The first orders a beer. The second orders half a beer. The third orders a quarter of a beer. Before the next one can order, the bartender says, “You’re all assholes,” and pours two beers.

^^I’ve heard that one with the punchline “you need to learn your limits” – love it.

The pizza analogy that is so useful for teaching fractions doesn’t seem to describe 1/x when x itself is much smaller than one, e.g., how do you divide a pizza into 1/10 pieces, i.e., so the total number of pieces is 1/10? I think the language breaks down.

Say that x = .000001 and we want to consider 1/x. I’d ask something like this: How many millionths do you need to altogether make one? The answer is one million of them. The smaller the size of something, the more of them you need to make one standard unit. The product of x and 1/x is one (x * 1/x = 1) . As one number goes up the other goes down & vice versa, i.e., they are reciprocals, in order to maintain a constant product. If the student understands graphs, reveal the 1/x hyperbola – at least the positive part.

I’d make a formal analogy between the operations of addition and multiplication. This is the way I learned it back in the day…
Additive identity = 0 because a+0=a for all a.
Multiplicative identity = 1 because a*1=a for all a.
Additive inverse of a is -a because a + (-a) = 0 = additive identity.
Multiplicative inverse of a (a<>0) is 1/a because a * (1/a) = 1 = multiplicative identity.
Then you go on to define subtraction and division in terms of the above.

With either operation (addition or multiplication) when one of the pair of inverses goes up, the other goes down & vice versa. In the case of addition their sum is a constant zero. In the case of multiplication their product is a constant one. Hmm, if the student doesn’t comprehend negative numbers either, @LostInParadise, maybe this won’t be so easy!

^ I didn’t understand any of that. I got “blah blah blah, hyperbaric chamber blah.”

I’m not sure why this even requires explanation in such huge words. When the denominator is a smaller number then the numerator, the numerator is thus bigger than the denominator. I accept that. I don’t get the math talk, but I recognize that 1 is bigger than .0005

I think that you are on to something. Just compare numerator and denominator. If you have 10/1 then that is the same as 1 / 0.1 It might help to start with something like 100/1 and then say that we can divide numerator and denominator by the same number and end up with the same value.

@Seek_Kolinahr As I said regarding my “categorical” comment above, there is a difference between the language we are using to talk about teaching the concept and the language that would be used to teach the concept. We can be free to use “huge words” amongst ourselves. What language is appropriate for the student is something that @LostInParadise can decide.

I’m just coming from the perspective of a student that didn’t do well in maths classes, but somehow managed to pull a passing grade.

I’m totally lost since the words How do you explain at the top of this page. It’s like you guys are all speaking Klingon.

You then are the perfect candidate for testing my explanatory skills based on the suggestions made.

We are looking at fractions of the form 1/x. Consider 1/10, ¼, ½, 1/1. As the numbers on the bottom get smaller the value of the fraction gets larger. What happens when we get smaller than 1 on the bottom, using fractions? We expect the same pattern of increasing numbers.

I claim that in fact that we can get the value of 1/x to be as large as you want. For example, let’s say we want it to be 10. Start with 10/1. Now divide both top and bottom by 10 in order to get 1 on top, giving 1/0.1, which will also be equal to 10. We could do the same thing for 100, 1000 or one million. We keep 1 on the top, but for the ratio to get larger, the number on the bottom has to keep getting smaller.

Please let me know if this makes sense to you and, if not, where you got stuck.

100/1 = 1/0.01

Couldn’t the students self-check this with cross-multiplication?

I cannot fathom any of this.