# Linear Relationships – first differences

### QUESTION

What does it mean to use “first differences” to determine which of the 4 tables shown at right represent non-linear relationships?

### Finding first differences

To find first differences, look at column 2: subtract the 1st number from the 2nd, the 2nd from the 3rd, etc. If these differences are all the same, then you have a linear relationship. If not, then the relationship is non-linear.

Why? Well, linear relationships have graphs that are straight lines, so their slopes are constant. A linear relationship is like a typical staircase, where all steps are the same height. When you find first differences, it’s like finding the “height” of each of the steps.

We’re making some assumptions here about the numbers in column 1:

• they are in numerical order
• their first differences are all the same: for example, 1,2,3,4,5 or 1,3,5,7,9 would be fine, but 1,2,4,5,7 or 1,3,2,5,4 would not

### Second differences and a context

What about second differences? Let’s look at the table on the right.

• The first differences are in red.
• The second differences are in blue.

Here’s a context for this:

• column 1 could represent time
• column 2 could represent distance travelled by a falling object
• the first differences would then be the speed of the object (notice that it keeps increasing, which makes sense for falling objects)
• the second differences would then be the speed of the speed, or the acceleration of the object, like the acceleration due to gravity, which is constant.

If you studied Calculus:

• the equation for the first differences is the first derivative
• the equation for the second differences is the second derivative

### Finding x- and y-intercepts

Let’s start with a linear equation, like 3x + 2y = 12.

On the x-axis, all points have a y-value of 0. Therefore:

• 3(0) + 2y = 12
• 2y = 12
• y = 6
• the y-intercept is (0, 6)

On the y-axis, all points have a x-value of 0. Therefore:

• 3x + 2(0) = 12
• 3x = 12
• x = 4
• the x-intercept is (4, 0)

### Linear and non-linear relationships across the grades

What are the first differences in the Grade 1 growing pattern shown below?

What linear relationship does this pattern represent?

For more like this, see Growing Patterns:

• linear and other relationships across grades 1, 4 and 9
• an interview with applied mathematician Lindi Wahl, and
• a music video from Grades 1/2 classrooms, with lyrics based on parent comments

### Using computational modelling

Click on the link below to use code to investigate first differences as well as graphs of linear relations.

Investigate first differences:

Investigate linear equations and their graphs: