# 4 mathematical relationship between quantities

### Intro to proportional relationships (video) | Khan Academy mathematical model, and considers its relation to certain aspects of the physical biray (like an oriented line with starting point), for measurements of quantities. It's always possible to find a common multiple for two numbers by multiplying them (4 x Discuss the meanings of the math terms they use and the relationships. Students must learn how numbers are interconnected and how numbers can be used in meaningful ways. Number Relationships is one of the key mathematical principles or “Big Ideas” click to open the list of modules of Numeracy 4 to 6.

What was our corresponding change in y? Well, our corresponding change in y when our change in x was 5, our change in y was equal to 2. And you see that here, when x went from 0 to 5, y went from 0 to 2. So our change in y in this circumstance is equal to 2.

## Intro to proportional relationships

Which if you wrote it as a decimal is equal to 0. So this right over here is your slope. So I'm going to do this with the tool. But first, let's also think about what the equation of this line is going to be. Well, we know that y is equal to some constant times x. And we know that the point 5, 2 is on this line right over here.

### Variables, Functions and Equations

So we could say, well, when x is equal to 5, y is equal to 2. Or, when y equals 2, we have k times 5, or k is equal to-- dividing both sides by 5, you can't see that. We're used to seeing this.

When we have y is equal to something times x, this something right over here is going to be our slope. So the equation of the line is y is equal to 0. So let's fill this in. Let's actually do the exercise now. So we had two points, one was the point 0, 0. So it's y is equal to 2. And you see that right over here. Depending on the number of questions, you multiply it by 5, and you get p. So the dependent variable is the number of points you score.

The dependent variable is the number of questions you answer correctly. No, we've already talked about that. That's the independent variable. The independent variable is the number of points you score. No, that's not right. That's the dependent variable. The independent variable is a number of questions you answer correctly.

And then I wanna graph them to see if we can see anything visually that makes them obviously proportional. And just as a reminder, a proportional relationship is one where the ratio between the two variables, and let's say we took the ratio between Y and X, you could also go the other way around, the ratio between X and Y.

But the ratio between Y and X is always going to be some number, some constant number. Or you could rewrite it another way, If you were to multiply both sides of this equation times X, you could say in a proportional relationship, Y is always going to be equal to some constant times X. So with that out of the way, let's look at these three relationships. So this one over here, let me draw another column here. This is, let me call this the Y over X column. I'm just gonna keep figuring out what this ratio is for each of these pairs. So for this first pair, when X is one, Y is one half, so this ratio is one half over one. Well one half over one is just the same thing as one half. When X is four, Y is two, this ratio is gonna be two over four, which is the same thing as one half.

When X is negative two and Y is negative one, this ratio is negative one over negative two, which is the same thing as one half. So for at least these three points that we've sampled from this relationship, it looks like the ratio between Y and X is always one half. In this case K would be one half, we could write Y over X is always equal to one half.

Or at least for these three points that we've sampled, and we'll say, well, maybe it's always the case, for this relationship between X and Y, or if you wanted to write it another way, you could write that Y is equal to one half X. Now let's graph this thing. Well, when X is one, Y is one half.

When X is four, Y is two.

## Number Relationships

When X is negative two, Y is negative one. I didn't put the marker for negative one, it would be right about there. And so if we say these three points are sampled on the entire relationship, and the entire relationship is Y is equal to one half X, well the line that represents, or the set of all points that would represent the possible X-Y pairs, it would be a line. It would be a line that goes through the origin.

Because look, if X is zero, one half times zero is going to be equal to Y. And so let's think about some of the key characteristics. One, it is a line.

This is a line here.

• Proportional relationships: graphs
• Graphing proportional relationships: unit rate
• Writing proportional equations from tables

It is a linear relationship. And it also goes through the origin.