Patterning and Algebra

I never understood the connection between patterning and algebra until this past lesson; we seek out patterns, and then algebra is how we communicate patterns to each other. We start in elementary school by describing and representing patterns, then in middle school we model and predict patterns (through algebra), and then in high school we solve and analyze those patterns. 

Since I'm observing a grade 7 class right now, I'm going to focus on strategies at the grade 7 level. In grade 7 students move beyond extending a pattern to the next three items, and start being able to calculate patterns at any point, and students begin to see patterns as relationships. For example, if a car is moving at 50km/h consistently for 3.5 hours, how far will they have travelled? If we chart it on a 'T' chart we can see the patterns: 

MacCuish (2015). 

Based on the above diagram, students can easily see that as time increases by 30 minutes (0.5 hours) then the distance travelled increases by 25km. 

The next step is asking students how long it would take to travel 400km. The T chart is likely the most time consuming strategy to answer this questions, so students would likely look for a quicker alternative. Two strategies are making a graph or making an equation. 

The graph is simple for this question since both variables (time and distance) change at a constant rate, so students could create a graph like the following, either using paper and pencil, or a web calculator: 

MacCuish (2015). 

The above chart is a representation of what it might look like if students graphed the information provided in the T Chart. The red dotted lines show the point of intersection between 3.5 hours and 175 km. Based on the above chart, students could extend the line and create a chart like the one below. I used a graphing software for these graphs so they're slightly different than what a student would likely produce (for example they would likely mark each increment along the horizontal and vertical axises and would also be encouraged to start the graph at 0,0).

MacCuish (2015). 

This chart shows a student extending the straight line at the same angle as it was at previously, then finding where on the chart 400 km intersects the blue line (the horizontal dotted red line). From there, students can bring down a second red dotted line and see that 400 km is 8 hours.

These visual representations of patterns leads into teaching students about algebra. There are plenty of ways of developing an equation to answer this question such as thinking about the question theoretically, using the units to help you figure out the question, or creating an equation with the variables you already have (which is what I did below).

MacCuish (2015). 

This method of using algebra to explain the relationship between distance and time can be applied to any distance or time and students can get the answer. For example, they can use this equation to know how long it would take to travel 1000km at this speed (in theory, of course), or how long it would take to travel 400 km at a speed of 60 km/h.

This type of scaffolding to make sure students grasp the foundation before moving forward is critical for students success. Especially since this type of math can get confusing and students may get frustrated and discouraged quickly.

I think if I were to teach this stream during my practicum I would use the Minds On portion of the class to have students recall previous learning about patterns and making T charts, and I would also include some big ideas at the beginning of the class so students can see how this math is used in the real world. I think one of the biggest challenges in math in grades 7/8 is getting students to realize it's not useless. I've had multiple students during my observations asking why they have to learn these concepts since 'they aren't ever used in the real world except by mathematicians.' For example. a summary of this essay about MC Escher's Circle Limit figures could be presented to the class as a 'hook' or a way of showing them pattering at a larger scale than the classroom, and this Yahoo Answers thread gives a lot of 'casual' algebra uses.