Direct and inverse proportions | Class 8 (India) | Math | Khan Academy
Two values are said to be in direct proportion when increase in one results in an increase in the other. Similarly, they are said to be in indirect proportion when. The greek letter "alpha" (α) for both. If x is directly proportional to y, you could say x α y. For inversely proportional, you would say something like. Understand the concept of proportionality in real-world and mathematical from other relationships, including inversely proportional relationships (xy = k or y = k.
We are going to keep going until we have 10 books' height. 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? 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. Now using your pattern in the table, can we write a rule that would find the height of any stack of books? So our book was only 3.
We need to translate that to an equation with only numbers, variables, and mathematical symbols. Who thinks they have an equation that would work here? We have a proportional relationship because the height of each book is our constant rate of change, and it increases by the same amount each time we add one book to the stack.
So we can say that the height of the stack is 'proportional' to the number of books in the stack. How would you do that?
I think the stack would have 15 books.
How does it change the rule? Is it still proportional? Think about that tonight and we will start there tomorrow. It is important that teachers help students realize the importance of scalar thinking in proportionality and that it will appear in many places throughout the year.
When looking at tables and graphs of proportional relationships, remind students to keep in mind the labels that match with the numbers variables.
Direct and inverse proportions
Doing this will help them remember which way to divide to find the constant of proportionality slope. In a table of gallons of gasoline used and miles traveled it would be logical to divide miles by gallons because they are familiar with the phrase, miles-per-gallon. Students think that just because a relationship between variables increases or decreases by the same value, it is proportional.
They need to know that that is not true. The graph of the relationship must pass through the origin as well as change by a constant amount. Thus, using an example like miles per gallon is a good way to illustrate this concept, because when gallons is 0 the independent variable is zero then the number of miles is also zero 0. Using a graphical representation of equivalent fractions to develop student thinking about proportionality and a constant rate of change slope makes a good transition from rational numbers to proportionality.
Using slopes as fractions provides opportunities for students to practice working with fractions at the same time that they are working with the concept of slope.
As a result, students are given the opportunity to learn the difficult concept of fractions creatively. Give students real life examples of direct proportional relationships and inverse proportional relationships and brainstorm similarities and differences.
Proportional Relationships | Minnesota STEM Teacher Center
Some examples of direct proportional relationships might include: Some examples of inverse proportional relationships might include: Allow students to become familiar with both general forms for a proportional relationship: Remind students that the y-intercept needs to be zero in a proportional relationship's graph. Using the following examples: The temperature is dropping proportionally to the time passing.
The amount of water and juice are proportional to each other in the punch, and both are proportional to the amount of punch. Lanius, Cynthia and Williams, Susan E. Write a linear equation on the overhead. Give each row of the class different x or y values to use in solving the equation. Let one row choose their own values. When all have finished, have each row plot their points on a wall coordinate grid.
In other words, when quantity A changes by a certain factor, quantity B will change by the same factor.
The relationship between the money you spend and the amount of chocolate you get is called a proportional relationship, and that the amount of chocolate you get is proportional to or sometimes directly proportional to the amount of money you spend.
Inverse Proportionality Some relationships don't increase or decrease at the same time, but are still related proportionally. Think about a long journey to see some family. The speed you travel at is related to the time the journey takes, but not in the same way as above. As you increase your speed, the journey time decreases instead. In fact travelling twice as fast will cut the journey time in half.
We call this type of relationship inverse proportionality. If one quantity increases by a certain factor, the other quantity decreases by that same factor. We use the same symbol as for proportionality, but represent one quantity by its inverse, so: Proportion Equations All proportional relationships can be rewritten in the form of an equation with a constant of proportionality.
So our proportional relationship can be rewritten from: If we look at our previous two examples: We will use this fact in several parts of the course. Other Proportional Relationships Proportional relationships can get more complicated than simply directly or inversely proportional.