Embedded Files
Lesson overview
Lesson overview
In this lesson you will:
practise deciding if two quantities are directly or inversely proportional
explore graphs of direct proportion
explore graphs of inverse proportion.
Activity 1 - Direct or inverse proportion?
Activity 1 - Direct or inverse proportion?
Introducing direct and inverse proportion
Introducing direct and inverse proportion
Two quantities are directly proportional if they change together.
In other words, they both rise together (if one increases so does the other) and they both fall together (if one decreases the other also decreases).
For example, if one quantity is doubled the other is also doubled and if one quantity is halved the other is also halved.
If 𝒚 rises/increases (↑) then 𝒙 also rises/increases (↑).
If 𝒚 falls/decreases (↓) then 𝒙 also falls/decreases (↓).
Two quantities are inversely proportional if an increase in one variable causes a proportional decrease in the other (or if a decrease in one variable causes a proportional increase in the other).
For example, if one quantity is doubled the other is halved.
If 𝒚 rises/increases (↑) then 𝒙 falls/decreases (↓).
If 𝒚 falls/decreases (↓) then 𝒙 rises/increases (↑).
Task
Task
Consider each pair of quantities.
Is it an example of two quantities that are directly proportional or two quantities that are inversely proportional?
Using the definitions above, drag and drop each into the correct definition.
Activity 2 - Graphs of direct proportion
Activity 2 - Graphs of direct proportion
Remember that two quantities are directly proportional if they change together.
What might this look like on a graph?
If the x values changed in the same way that y values changed how would this look on a graph?
The answer – a linear graph. In other words, a graph of a straight line.
Drag the slider to see that when quantity y is in direct proportion to quantity x:
y increases as x increases
y decreases as x decreases.
Task
Task
Let’s explore a graph that compares the amount of water in a pool and the number of minutes that have gone by.
This is an example of direct proportion.
Explore how the graph changes as the speed of the tap increases by moving the blue slider then answer the questions.
Activity 3 - Graphs of inverse proportion
Activity 3 - Graphs of inverse proportion
Remember that two quantities are inversely proportional if as one increases the other decreases and vice versa.
What might this look like on a graph?
If x values were to increase as the y values decrease what might this look like on a graph?
The answer – an exponential graph. In other words, a graph of an exponential curve.
Change the speed by dragging the red dot to see that:
if speed (y) increases, time (x) decreases
if speed (y) decreases, time (x) increases.
Task
Task
Let’s explore the relationship between the height and base of a rectangle where the area is always 40cm squared.
If we started with a base of 10cm, what must the height be?
If we started with a base of 5cm, what must the height be?
Use the slider to determine how this would be graphed as it compares the base and height of any rectangle with an area of 40cm squared.
What do you notice?
What do you wonder?
Activities too easy?
Activities too easy?
In your exercise book or folder, write an arguments to convince me that:
If A is inversely proportional to B then when B is multiplied by 2, A will be halved.
If A is directly proportional to the square of B then when B is multiplied by 3, A must be multiplied by 9.
If two quantities are directly proportional to each other, the graph of them will always pass through the origin.
If two quantities are inversely proportional to each other, the graph of them will not be a straight line.
Handing in your work
Handing in your work
Don't forget to hand in the work you completed today!
Your teacher will have told you to do one of the following:
Upload any digital documents you created and any photos you took of your written work to your Learning Management system (MS Teams, Google Classroom for example).
Email any digital documents you created and any photos you took of your written work to your teacher.
Make sure you keep any hand written work you did in your exercise book or folder as your teacher may need to see these when you are back in class.
Student reflection
Student reflection
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