Ämne / Topic

Proportional functions.

Inlärningsmål / Learning Goals

Be able to identify when a function is proportional.

Google Slides & Videos

3. Aktiviteter / Key Activities

In this lesson the teacher will guide learners on how to apply their knowledge of linear functions and finding if they are in proportion.

To do this, the teacher will go through two examples with the learners, before giving them an example to go through with a partner.

Proportional Relationships

A proportional relationship is one in which two quantities vary directly with each other. We say the variable y varies directly as x if:

y=kx

for some constant k , called the constant of proportionality .

(Some textbooks describe a proportional relationship by saying that " y varies proportionally with x " or that " y is directly proportional to x.")

This means that as xincreases,y increases and as x decreases, y decreases-and that the ratio between them always stays the same.

The graph of the proportional relationship equation is a straight line through the origin.

Graphing Proportional Relationships - Example 1

Let us use the relationship between U.S. Dollars and U.K. Pounds to illustrate this. The exchange rate used in this example is 0.69 U.S. Dollars per 1 U.K. Pound. (Note that this, and all currency exchange rates, change all the time).

The table of values and their graph show above a straight line that passes through the origin. This indicates that the relationship between the two currencies is in direct proportion. Think about what this means in real terms – if you have ten times more dollars than another person, when you both exchange your money, you will still have ten times more money. Notice also that the graph passes through the origin; this makes sense as if you have no dollars you will get no pounds!

We can express these relationships algebraically as well as graphically.

Graphing Proportional Relationships - Example 2

Another common example of directly proportional relationships is that between time and distance when travelling at a constant speed. The graph below shows the relationship between distance and time for a vehicle travelling at a constant speed of 30 miles per hour. Note that this means the unit rate is 30 miles per hour.

Notice below a similar graph. In this example the vehicle is travelling at a constant speed of 50 miles per hour. The slope of the graph is steeper. The steepness of the slope for directly proportional relationships increases as the value of the constant m (y = mx) increases. In our two speed examples, the change in steepness of slope represents the change in speed or the change in the unit rate.

Graphing Other Linear Relationships

Directly proportional relationships always pass through the origin (0,0). There are other linear relationships that do not pass through the origin. The example below is for a taxi fare that has a standing charge (a fee payable no matter how far is travelled) and a cost per mile.

Notice where the line intercepts the y-axis in the above example. This point is known as the y-intercept and there is more on this here.

4. Kolla vad eleverna kan / Check for understanding / Exit ticket

Learners will complete questions as pairs to check if they understand the concept.

Question 1 :

The graph shows the relationship between the weight of an object on the Moon and its weight on Earth. Write an equation for this relationship.

5. Stängning / Closure / Summary of lesson

Review the definition of proportionality.

Resurser & Materiel / Resources & Materials

http://webbapp.liber.se/matematikboken-z/#

http://webbapp.liber.se/matematikboken-z/#/4-samband-och-forandring

Textbook practice

Ett pg. 198. no. 4114 abc, 4115 ab, 4116

Två pg. 199 no. 4120, 4121 abc

Tre pg. 200 no. 4123 abcd, 4124 abc, 4125, 4126 abc

Fyra pg. 201 no. 4128, 4129, 4130, 4131 ab