**Linear Relationships – ***first differences*

*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:**