Modelling quadratic patterns

Using the website visualpatterns.org we explored a few patterns. We looked at them in a similar way to patterns we explored in grade 9, linking the visual pattern to an equation, a table and a graph.

We remembered how to look for groups of “n” and squares of “n” to help us build an equation. We also remembered that the data in the tables correspond to (x,y) coordinates for the graph. We looked at the first differences, and second differences. For linear growth the first differences are constant. For quadratic (parabolic) growth the second differences are constant. We looked at how the 2nd difference is related to our equation (our “a” value is always half of the constant second difference). We looked at how figure 0 is related to our constant term in our equation, and the y intercept of the graph.

We had a look at a cubic function too, just to notice that the 3rd differences are constant, and that the graph looks different from a parabola.

There are sometimes some tricks to use when visualizing and making equations. For this pattern it helps to add another set on top to create a rectangle that is (x+1)(x+2) and then since we only need half of that we can multiply by 1/2 or divide the whole thing by 2.

For this pattern some people looked at groups of “n” to model the yellow squares. We see 8n^2+4n+1 this way.

Others looked at creating expressions for the yellow squares.

And some looked at an expression for the entire square, with white rectangles removed.

And in the end they all simplify to the same thing! Algebra is neat that way.

We will continue to explore these trends over the next few weeks.