first differences

Linear Relationships – first differences

QUESTION

This image has an empty alt attribute; its file name is Screen-Shot-2019-10-16-at-10.29.45-AM.pngWhat does it mean to use “first differences” to determine which of the 4 tables shown at right represent non-linear relationships?

 

ANSWER

This image has an empty alt attribute; its file name is Screen-Shot-2019-10-16-at-10.29.45-AM-1.pngTo 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

BIGGER PICTURE

First differences (and second and third differences) help determine whether there is a pattern in a set of data, as well as the nature of the pattern.

For example, let’s say you invest $1,000. Below are 3 growth options, tracked for 5 years. Which would you prefer?

Which option would you prefer?

Let’s look at how your investment grows in each option:

  1. In the first option, your investment grows each year by $0. (first difference is always 0)
  2. In the second option, your investment grows each year by $1000. (first difference is always 1,000)
  3. In the third option, your investment grows by $1,000 in the 1st year, $1,100 in the 2nd, $1,210 in the 3rd, and so on. The annual growth increases as the years pass. (first difference keeps changing/growing)

Options 1 and 2 have constant growth ($0 and $1,000 annually, respectively).  Option 3 does not.

Data that has constant growth represent linear relations.

Linear relations, as the name suggests, have graphs whose points are in a straight line.

If you plotted the 3 options as a scatter plot (and extend for 20 years), it would look like this:

Surprised by this graph?

Why does the 3rd option so much better than the 2nd option, when at first it seemed so similar to the 2nd option?

  • Option 1 gives you an additional $1,000 every year, which is constant growth.
  • Option 2 gives you an additional 10% every year, which is exponential growth.

Here are the equations for the 3 options.

(x represents the year)

 

SECOND DIFFERENCES

This image has an empty alt attribute; its file name is Screen-Shot-2019-10-16-at-11.16.49-AM.png

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:

  • This image has an empty alt attribute; its file name is dreamstime_xs_26786345.jpgcolumn 1 could represent time
  • column 2 could represent distance travelled by a falling object
  • the first differences relate to the speed of the object (notice the speed keeps increasing, which makes sense for falling objects)
  • the second differences would relate to 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 first differences relate to the first derivative (y’)
  • the second differences relate to the second derivative (y”)
Linear and non-linear relationships across the grades

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

This image has an empty alt attribute; its file name is train-base-3x3.jpg

What linear relationship does the above pattern represent?

See http://learnx.ca/growing-patterns/ for more like this, such as:

  • 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

COMPUTATIONAL MODELLING

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

https://colab.research.google.com/drive/12qzXANGGr-pYx6T6ImvsJuOzAFgvUXig#scrollTo=fQw9CSdbqg-k&forceEdit=true&sandboxMode=true

Investigate first differences:
This image has an empty alt attribute; its file name is Screen-Shot-2019-10-16-at-4.41.59-PM-1.png    This image has an empty alt attribute; its file name is Screen-Shot-2019-10-16-at-4.41.48-PM.png
Investigate linear equations and their graphs:
This image has an empty alt attribute; its file name is Screen-Shot-2019-10-16-at-4.45.54-PM.pngThis image has an empty alt attribute; its file name is Screen-Shot-2019-10-16-at-4.41.25-PM.png