Patterns to Algebra

Yesterday in grade 9 we started looking at patterning. We created several growing patterns that have this as figure 2.

We worked on figuring out the pattern rule. We know that if we increase by the same number of blocks between figures each time, it is a linear relationship. The slope, or rate, for the line is equal to the common first difference. The constant is the value of figure 0. Some equations are not linear, and we notice that those ones grow by more and more and more each time. Those equations can have exponents in them.

Sometimes the pattern looks different, but the rule is the same.

We had a challenge to determine if this pattern is linear or not.

Students built what figure 3 would be, and extended the pattern back to figure 0. We noticed that the pattern grows by 1 each time. Figure 0 is the constant, which is 3. The pattern rule is y=x+3

This pattern is not linear. We know that because when we build figure 2 it will look like a square that is 3×3 and then 2 blocks at the top left. The pattern rule can be written as y=(x+1)^2+x. We know that we are adding different amounts to get from figure 1 to figure 2, than when we go from figure 2 to figure 3.

Our next step today was to start with the equation for a pattern rule, and then build figures 0,1,2,3,4 out of tiles.

Each group had different equations to build.

Groups then worked to dismantle each figure one at a time, and line the blocks up to fit on the 1 inch grid chart paper. We made each figure into a column, and then placed a dot at the top left corner. Figure 1 is then dismantled and all the blocks placed in a column and a dot drawn at the top left.

We can see that the constant, the y intercept, is 2, and the slope in this case is 3, since the “stairs” go up by 3 each time.

After groups had their graph complete, they were given a new pattern rule to do, and they used a different colour marker to make dots, so they ended up with 2 unique lines.

Groups all placed their work on the wall, and we did a consolidation lesson which talked through some new vocabulary, and we learned about slopes, y intercepts, and parallel lines. We looked at points of intersection, and were able to compare steepness of lines using their equations.

I wrote on the graphs as we went through the material. Here are the final chart papers.

We covered a great deal today! It was impressive work on a Friday afternoon.