A proportional relationship graphs as a line through the origin, and its unit rate is the slope.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Why they are the same number
£3 an hour means y rises 3 for every 1 across. That is exactly rise over run.
Comparing two
Draw both on one grid. The steeper line is the faster rate, whatever units are used.
Across different forms
One relationship may be given as a table and another as a graph. Turn both into a rate and they can be compared.
A unit rate is a slope
A car travelling 60 miles per hour has a distance-time graph with slope 60. The rate and the steepness are the same number viewed two ways. Every unit rate you met in Grade 6 and 7 was a slope waiting to be graphed.
The units of a slope
A slope is a change in y per unit change in x, so its units are the y-unit divided by the x-unit: miles per hour, pounds per kilogram, degrees per minute. Reading a slope's units tells you what it measures.
Comparing rates by steepness
On the same axes, the steeper line has the greater rate. That makes comparison visual rather than arithmetic, which is why graphs are used to compare plans, speeds and prices in practice.
Negative rates
A tank draining at 5 litres per minute has slope −5. Negative rates describe decrease, and the sign is part of the number, not an afterthought. Dropping it turns a description of emptying into one of filling.
Step 2: Try It Yourself
Tap and try it out.
The slope is 3: for every 1 across, the line goes 3 up.
Step 3: Watch an Example
One step at a time.
Watch Leo Compare Two Jobs
One job pays £24 for 3 hours. Another graphs as a line through (2, 18).
- Step 1
The first job’s rate is 24 ÷ 3 = £8 an hour.
Step 4: Your Turn
Practice makes it stick.
The Table
Problem 1 of 2
5 hours earns £40. What is the rate, in pounds per hour?
The Graph
Problem 2 of 2
A proportional line passes through (4, 36). What is its slope?
Rate and Steepness
1 of 4
Through (3, 15). Slope?
2 of 4
6 kg costs £30. Rate in pounds per kg?
3 of 4
Of slopes 3 and 7, which line is steeper? Give the number.
4 of 4
A proportional line through (5, 20). What is y when x is 1?
Step 5: Quick Check
Show what you know.
Question 1 of 1
A proportional line through (7, 28). What is its slope?
What You Learned
- For a proportional relationship, the unit rate and the slope are the same number.
- A steeper line means a faster rate.
- Converting both to rates lets a table and a graph be compared.