### Math Corner

I’ve been starting to help my neice with math homework, and there are a million questions popping up. I thought this would be good place to pose my questions–which will not just be about solving specific problems, but also exploring math concepts and terms more deeply. (The posts may be sort of sprawling and stream of consciousness, so please bear with me.)

Here’s a question to consider:

What happens when a line has an irrational slope?

Is it possible to accurately graph it?

What type of “real-life” context would it represent?

All I can think of is y=πx

If you’re talking about a real irrational slope, then you could graph it but you wouldn’t be able to use the standard integers as increments on the axes; think of the graph for a sine wave and you’ll see what I mean: the y-axis is marked between -1 and 1, but the increments along the x-axis are fractions of pi over six.

I’m still thinking about where it might happen in real life, but my guess is that it has something to do with squaring the circle, one of the traditional “solutions” to calculating pi. Not really sure, though. That’s obviously a question for a much more intelligent math teacher, such as my own high-school calculus instructor. 🙂

What is an irrational slope?! Dang, I was going to ask what a slope is. 🙂 I actually did have questions about that. From what I understand the slope is the rate of change in a line on a graph. The equation for the slope is y2-y1 over x2-x1. How do we explain the reason for this equation? For example, (x2, y2) is the point that where “more change” has occured than (x1, y1), right? Is that correct and is this an appropriate way of describing it?

So the difference between y2 and y1 would represent the amount of change on the y-axis, while the difference between x2 and x1 would represent the amount of change on the x-axis. y2-y1 over x2-x1 is a ratio of the differences (changes).

I have a hard time grasping the concept of a ratio. Can someone define or explain the concept?

It’s been a long time since I’ve thought about slopes and graphing and xs and ys. Since slope is the rate of change in a line, wouldn’t a line with a slope of pi result in a straight line, with a slope of pi/1? Would an irrational slope be represented by a vertical line with a slope of 1/0?

Two questions.

My niece is having trouble remebering NOT to add a number to a number multiplied by a variable–e.g. 4+5x=9x. Of course, I told her she can’t do that, but she did it several times after my saying that, so obviously she’s not undrestanding something.

When she adds numbers to both sides of an equation, she writes the number that is going to be added underneath the equation like so,

4x+y=7

-4x -4x

———–

y=-4x+7

So I thought this might be confusing, so I tried to have her write the solution this way:

4x+7-4x=7=4x

But I don’t think that was working.

I showed her what would happen if you add the number in front of the variable to a number by itself by assigning a value for the variable, but I’m not sure she understood the concept.

What’s a better way of explaining the reason you can’t add a number to a number that is being multiplied by a variable (and what’s the correct terminology to use–that I’m not using–i.e. integer?)

You’re talking about “like terms.” You can only add (and subtract, since addition and subtraction are the same thing) terms that have EXACTLY the same variables with EXACTLY the same exponents.

Don’t get too stressed if this is a tough concept for her to get. I’ve seen it a million times and it just takes practice. I have an enormous zip-loc bag full of colored index cards on which are written monomial terms (such as x, 2x^2, and 3xy). Sometimes I take that out and have students “collect” like terms, which they have no problem with. They sort through the whole stack and line up cards that have exactly the same variables and exactly the same exponents. Then I make them make equations using them, showing what they are if combined (added and subtracted). You’re free to borrow mine if you want. It’s a pretty good activity if the student can remember the point to the whole thing. I would like to think it helps tactile learners, but I don’t honestly know.

Thanks, Mitchell. Next time you’re over the house, why don’t you bring them (along with those black and white cards). Speaking of which, I would still like to understand the concept of “addition and substraction being the same thing.” What might help is to understand when understanding this concept would be the most useful.

Addition and subtraction is the same thing because in subtraction, what you’re doing is adding a negative. x-y can be expressed as x + -y.

Right but to say that “addition and subtraction is the same thing” seems a bit odd. So addition is subtraction and subtraction is addition, then what exactly is the “addition/subtraction” process? (We should have another term for whatever it is.)

And as I mentioned to Mitchell, there’s probably some specific context(s) where thinking of addition/subtraction in this way is useful. What is that context(s)?

I’ve always found it easier to think of subtraction as addition when subtracting a negative number (not that I do a lot of subtracting of negative numbers anymore. The expression 7 – (-5), can be written as 7 + 5.

My son is working on fractions now, and in talking to him the one thing that I feel like I should discuss with him is the notion of the “whole” represented by the number 1. Fractions involve taking something whole and dividing into parts and then showing a relationship between those parts relative to the whole. That’s not the greatest way to explain the concept, but I think that’s basically correct.

The aspect that I find confusing (or potentially confusing and strange) is the way “1” represents two things: a) a specific quantity and b) the concept of “wholeness.” (I don’t even know if “wholeness” is the right or best word here, but that’s what I’m going with now.) “Wholeness” could actually be a single entity or it could be many. One cookie could be the “wholeness” we are dividing or a bag of twenty cookies could also be a “wholeness” we are dividing.

Let me back up. Just to be clear, I’m thinking of the way the numerator and denominator being equal equals one–e.g., 4/4 = 1. I think explaining this to a child can be confusing. How can four parts being equal to “1?” I think the problem is that “1,” in this case, isn’t a specific quantity but represents that concept of wholeness or 100%. The fact that “1” represents 100% seems like it could trip someone up as well–or at least a fairly significant abstract leap.

I don’t know, maybe kids don’t need to understand what I’m talking about, but my initial impulse is to try and explain this aspect of fractions. I wouldn’t be surprised if this is a really bad way of going about it, though.

My son is working on decimal points. I’m not sure he understands the concept, and I don’t think I understand it well enough to explain it. Any ideas?

Does he understand place value?

I think he does, but I don’t know how well…Indeed, maybe his father doesn’t understand that. Place value is the value of number based on its position to other numbers–e.g., 32. The place value of “3” is thirty. The place value of the “2” is two. Yes?

Yes, but it’s more useful to think of 32 as 3 tens (3 in the tens column) and 2 ones (2 in the ones column).

Each column to the left is ten times the column to the right. Decimals work the same way. 32.1 is three tens, two ones, and 1 tenth. That first place right of the decimal is tenths. And that “ten times the column on the right” rule still holds true. 1 is ten times .1 . .1 is ten times .01.

43.21 is four tens, three ones, 2 tenths, and 1 hundredth. Is he clear on that? Is his dad?

His dad is clear, but I’m not sure my son will be. That’s a good explanation, but what if he has trouble going from the ones column (This always seems like an awkward name) to the tenths. Would a good way be to start talking about dividing up the ones place by tens?

That’s what I would do. For best results I’d use a sheet of graph paper (with the tiny squares). Then show him a few sample decimal numbers. Then you could create simple addition problems where, when the tenths column gets full, you have to carry the one to the next column and begin again in the tenths. Something like that.

The ones column may seem awkwardly named but it’s maybe the most important concept in all of math. Just about everything I can think of except geometry (and the math that’s derived from it) is based on the concept of one.

The only thing that might rival it is its opposite: the concept of zero.