I used to teach high school. I would often see students who struggled to see:

- How to generalize patterns and extrapolate
- The meaning and usefulness of a function
- That the cartesian plane was just two number lines stuck together, useful for comparing two quantities changing at the same time
- The difference between a linear relationship and an exponential relationship

Thanks to teacher and activity creating genius Fawn Nguyen, we have a way to address this problem. I now run a visual pattern activity at the start of my lessons, every Wednesday. Just a few weeks into the semester, I am already seeing the above gaps filled!

We have started mainly by mainly using linear patterns with constant differences. Now that this link is pretty strong I have started to introduce increasing differences and they are quickly getting the idea of a curved relationship on the graph. I have created this sheet to help organize the students thinking. I give them 4 minutes to think on the problem by themselves and 2 minutes to discuss their ideas. Then as a class we ask and discuss:

- Can you describe in words, how this pattern is growing?
- What do you notice about the numbers in the table?
- What would be a really slow way of calculating the number of blocks/objects in step 43?
- What would be a quicker way of calculating the number of blocks/objects in step 43?
- Using this rule, what would step 1000 look like?
- If I saw a step with [ ] blocks, which step would I be looking at?
- If I were to graph steps against blocks/objects, what would the shape be? Why?
- Ext: What would step 0, step -1 look like?

I am quickly finding that, by accident, students are solving equations and building up a sense of the need for processes such as factoring, finding the inverse and finding the slope of a line. I have found I am able to coherently validate the need for calculus, 5 years before they take it. I believe this will really help my students when I run lessons such as Dan Meyer’s toothpick activity, later in the year. Sure this is just similar to the explicit/recursive rule section of particular algebra textbooks, spread over a year, but I think a regular discussion on this idea is crucial to making connections and getting the deep understanding needed for algebra and beyond.

I am excited to hear if it has made much of a difference, next year and into the future. I suspect it really will.

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A British Math(s) Teacher now living in Indianapolis, USA, aiming to show my students how questions are often more important than answers. Presenter of the MathEd Out Podcast and contributor for the Guardian Teacher Network.

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