Proportionality (mathematics) - Wikipedia
You will also be able to apply the inverse variation relationships to solve Take a look at the plot when y is inversely proportional to x with a. This formula can be rewritten/rearranged to y = c x and x = c y. Examples are y Table. The table below corresponds to an inversely proportional relationship. In mathematics, two variables are proportional if there is always a constant ratio between them. The statement "y is inversely proportional to x" is written mathematically as "y = c/x." This is equivalent . The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of.
And so this is a proportional relationship and its graph is represented by a line that goes through the origin. Now let's look at this one over here, this one in blue. So let's think about whether it is proportional.
And we could do the same test, by calculating the ratio between Y and X. So it's going to be, let's see, for this first one it's going to be three over one, which is just three. Then it's gonna be five over two. Five over two, well five over two is not the same thing as three. So already we know that this is not proportional. We don't even have to look at this third point right over here, where if we took the ratio between Y and X, it's negative one over negative one, which would just be one.
Let's see, let's graph this just for fun, to see what it looks like. When X is one, Y is three. When X is two, Y is five.
X is two, Y is five. And when X is negative one, Y is negative one. When X is negative one, Y is negative one.
- 7.2.1 Proportional Relationships
- Inverse Proportion and The Hyperbola Graph
- Proportional relationships: graphs
And I forgot to put the hash mark right there, it was right around there. And so if we said, okay, let's just give the benefit of the doubt that maybe these are three points from a line, because it looks like I can actually connect them with a line. Then the line would look something like this. The line would look something like this.
So notice, this is linear. This is a line right over here. But it does not go through the origin. So if you're just looking at a relationship visually, linear is good, but it needs to go through the origin as well for it to be proportional relationship. And you see that right here. This is a linear relationship, or at least these three pairs could be sampled from a linear relationship, but the graph does not go through the origin. And we see here, when we look at the ratio, that it was indeed not proportional.
So this is not proportional. Now let's look at this one over here. Let's look at what we have here. So I'll look at the ratios. So for this first pair, one over one, then we have four over two, well we immediately see that we are not proportional. And then nine over three, it would be three. So clearly this is not a constant number here. We don't always have the same value here, and so this is also not proportional. But let's graph it just for fun. When X is one, Y is one. When X is two, Y is four.
This actually looks like the graph of Y is equal to X squared. When X is three, Y is nine. At least these three points are consistent with it. So one, three, four, five, six, seven, eight, nine. So it's gonna look something And so, if this really is, if these points are sampled from Y equals X squared, then when X is zero, Y would be zero. Multiply, simplify and scale up fractions. Change in the independent variable leads to change in dependent variable.
Know the parts of tables and graphs.
Minimal use of graphing calculators. Correlations NCTM Standards Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
Analyze proportional relationships and use them to solve real-world and mathematical problems. Recognize and represent proportional relationships between quantities. Decide whether two quantities are in a proportional relationship, e.
Identify the constant of proportionality unit rate in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Student Misconceptions Student Misconceptions and Common Errors Students attempt using an additive, not a multiplicative relationship, to create a scaled relationship students add values to side lengths to get similar figures instead of multiplying by the scale factor. When the scalar rate is non-integer for example, 3. Students mistakenly think slope is only found on graphs, but slope is a way of describing a constant rate of change that can be graphical, tabular, or symbolic.
Sometimes students are plotting points incorrectly, switching around the x and y coordinates, or not including the negative sign if there is a negative coordinate. Thinking products are always greater than factors ex. Thinking that all relationships that increase or decrease by a constant value are proportional.
If students are unclear about what they are dividing or what they are trying to find out, they often can get a constant that is not appropriate; they could be satisfied just getting an answer and moving on, not testing the reasonableness of their answer.
Today you can see the 10 stacks of different books I have at the front of the room. Each group gets one set of 10 of the same kind of book. Do not use magazines, or if you do, use that as an added topic after the activity is over. If I told you how many of your books are in a stack, can you tell me the height?
Or, if I told you the height of your stack of books, can you tell me how many books are in the stack? Your group's task is to find out how tall different stacks of books are. You will need to make a two-column table for number of books and height of the stack. Count from 0 to ten for the number of books.
The independent variable will be what? What do we already know? The number of books is the independent variable because the height of the stack depends on the number of books. So height is my dependent variable.
Make sure your table starts at 0 books. Your task is to use the same type of book and figure out what different stacks' heights are. Do not take any shortcuts here. I want to know the height of each of the 10 stacks of books. The teacher is walking around the room questioning and assisting groups of working students.
Tell me about that. Then we took 2 books and measured how tall those books were. We are going to keep going until we have 10 books' height.
Inverse Proportion Graph | Zona Land Education
Students to one another in their group: Maybe we're doing it wrong? How can this be? If one book is 3. How about we re-measure, and this time let's have the same person measure, ok? Teacher walks around the room. After some time, she hears the following: Why do you think that would work? Well, each book is the same height as the others, so we can just multiply. Wow, that is easy!
So by actually doing the activity and thinking through it, you have solved the problem. Like 1 book was 2. Our table doesn't change by exactly the same value, but our graph is almost a perfectly straight line going up. Where does your graph start?
Who else graphed this? What can we conclude then about our graphs?
Proportional relationships: graphs (video) | Khan Academy
Now let's talk about the table. One group said the values did not change by exactly the same value. Let me ask this, were you putting the same type of book on every time? If you were putting the same type of book on the stack every time and the size of the book doesn't change, then what would you say should happen to the height as you increase the number of books on the stack by 1?
Someone could've read the tape measure wrong, or the book could have been bent or something to make it seem a little thicker than the others. We all know that if we do an experiment, sometimes conditions aren't perfect. But here again, if I would have just told you that each book is exactly the same, would have we seen this type of error?
No, we wouldn't have. This makes for good conversation, and we learn from measuring and counting to see the discrepancies.
Back to the problem.