I wanted to introduce my year 8 class to Geometric Sequences. I had originally planned to show them a task similar to the task I had used for Quadratic Sequences:
but then last minute i changed my mind and thought i would given them a starter based on Sierpinskis Triangle. My decision to switch was partly down to the fact i wanted them to have an opportunity to view the effect of geometric progression on an image. The task was simple; find how many triangles of each type there is in the image but it doesn’t include rotated versions of the triangle.
Students quickly spotted things about the diagram like, there was a copy of 3’s of each triangle and the connection between each number was x3. They spotted things like 3 squared was 9 and 3 cubed was 27 so the numbers were all powers of 3.
I asked students what the next term in the sequence would be and they worked out 243. One Student commented on the fact that the smallest triangles in the image would have 81 x 3 triangles because the image shown would be repeated 3 times. It was nice when a student suddenly said ‘i’ve got it! its 3 to the power of n, divide by 3’ (I hadn’t asked for a rule or anything)
As everyone in the class had time to check this general rule worked (nth term) a student noted that it would be 3 to the power but its the second term starts at 3 not the first. We then moved to discussing 3 to the power n-1
Next I gave Students the task below. Students either started to double the values or tried short cuts like; multiplying 8p by 5 as 8p was day 4 or other similar ideas. After a while a student raised their hand and realised that there was a pattern. That all the numbers in the second column ended with a 2, the third column ended with a 4 and 4th with a 8 but the 1st column had a 1 then 6 after. Her idea was soon passed around the classroom like wildfire and students discussed why it had to work and more generally what effect doubling has on numbers. Student discussion turned to the idea that the rule could be 2 to the power of n. suddenly the connection to the triangle starter was made and that the rule was in fact 2 to the power n but it needed to be halved after.