 New Zealand
Level 6 - NCEA Level 1

Graphs and Direct Proportions

Lesson

Constant or variable?

Rates can either be constant or variable.

A constant rate means that the relationship between two things remains the same.  For example, if I earn $850 a week and this amount doesn't change, this is an example of a constant rate. A variable rate is when one variable is affected by another. For example, if I earn$20 per hour, the amount I earn will will vary depending on how many hours I work.

What is direct proportionality?

A very common misconception is that two variables are directly proportional if one increases as the other increases. This is not the case. We can only say that two variables are directly proportional if, and only if, the ratio between the variables stays constant. In other words, both variables increase or decrease at a constant rate.

Example of Direct Proportionality

If I pay $\$6$$6 for 1212 eggs and \10$$10 for $20$20 eggs, are these rates directly proportional?

Think: $\$6$$6 // 1212 eggs == 50c50c // egg \10$$10 $/$/ $20$20 eggs $=$= $50c$50c $/$/ egg

Since both variables show a rate of $50$50 c per egg, the prices are directly proportional.

Example of Variables that are not Directly Proportional

If one car travels at $54$54 km/hour and another travels at $62$62 km/hour, the speed between these cars are NOT directly proportional because the rates are different.

Graphs that show direct proportionality

If we graph direct proportionality, we will see a linear graph (in other words, a straight line graph) that passes through the origin $\left(0,0\right)$(0,0).

The diagram below show a linear graph, where $A$A is directly proportional to $B$B. The next picture explains why $A$A and $B$B are directly proportional. Firstly, we can see that this is a straight line that passes through the origin. Now let's look at the ratios between the $A$A and $B$B coordinates. Look at the point $\left(1,1\right)$(1,1), marked by the black lines. If we write these points as a ratio of $A$A to $B$B, it would be written as $1:1$1:1. Now look at the point $\left(2,2\right)$(2,2), marked by the blue lines. If we write these point as a ratio, we would write it as $2:2$2:2, which simplifies down to $1:1$1:1, which is the same as the first point. Example

If the amount of petrol a car uses is directly proportional to the distance it travels, and it uses $12$12L/$100$100km, how much petrol does the car use when it travels $350$350km?

Think: $350$350km is $3.5$3.5 times bigger than $100$100, so we also need to multiply $12$12 by $3.5$3.5 to keep the variables in direct proportion.

Do: $3.5\times12=42$3.5×12=42

The car uses $42$42L of petrol over $350$350km.

Worked Examples

QUESTION 1

Ivan paints $10$10 plates every $6$6 hours.

1. Complete this proportion table:

 plates painted hours worked $0$0 $10$10 $20$20 $\editable{}$ $40$40 $\editable{}$ $6$6 $12$12 $18$18 $\editable{}$
2. Using the data from the table above, plot a graph.

QUESTION 2

Frank is making cups of fruit smoothie. The amount of bananas and strawberries he uses is shown in the proportion table.

 Strawberries Bananas $4$4 $8$8 $12$12 $16$16 $20$20 $3.5$3.5 $7$7 $10.5$10.5 $14$14 $17.5$17.5
1. Graph this proportional relationship.

2. For every $1$1 additional strawberry, how many additional bananas need to be added?

3. Let $x$x represent the number of strawberries and $y$y represent the number of bananas used to make the smoothie. State the equation relating $x$x and $y$y.

4. Which of the following describes the proportional relationship?

There may be more than one correct answer.

For every $3.5$3.5 bananas Frank uses, he adds $4$4 strawberries.

A

The unit rate of bananas in respect to strawberries is $\frac{8}{7}$87.

B

The unit rate of bananas in respect to strawberries is $\frac{7}{8}$78.

C

For every $4$4 bananas, Frank uses $3.5$3.5 strawberries.

D

For every $3.5$3.5 bananas Frank uses, he adds $4$4 strawberries.

A

The unit rate of bananas in respect to strawberries is $\frac{8}{7}$87.

B

The unit rate of bananas in respect to strawberries is $\frac{7}{8}$78.

C

For every $4$4 bananas, Frank uses $3.5$3.5 strawberries.

D

Outcomes

NA6-1

Apply direct and inverse relationships with linear proportions

91026

Apply numeric reasoning in solving problems