My primary goal for 1st term in 8th grade math is to get my students comfortable with writing equations in slope-intercept form. I spend a good chunk of the term working on visual patterns – using The Pattern and Function Connection as my primary resource. As students build linear patterns and draw them, they easily can find the rate of change and identify it as the part that is being added over and over again. What they don’t catch onto so easily is why that shows up as multiplication in the equation. If you just have to add four, shouldn’t the equation be y= x+4.

In order to help them understand the connection between repeated addition and multiplication, I wrote this worksheet, entitled Adding, Adding, Adding…. It’s a bit difficult to describe, so I suggest you just click on it and take a look for yourself. The best news is that it really works! After working through this, almost all of my students understand where the multiplication comes from and why it works. If I were to teach exponential functions, I would teach them in the exact same way and call the worksheet Multiplying, Multiplying, Multiplying. In fact, I may have already written that and can dig it out of my files for a later post if anyone is interested.

FYI – the directions in the worksheet ask them to mark the rate of change (slope) in blue and the zero-step (y-intercept) in red all four representations (picture, graph, table, equation). This is what that looks like. I have these posters hanging on a bulletin board in my room.

A word of caution concerning this assignment, though. This is not an independent work assignment. With my Honors students, I divided them into groups of four and had them read it aloud and work while I wandered and answered questions. This seemed to work well. With my regular math class, I have many low readers. So they took turns reading sections aloud to their ‘sideways partner’ and then we discussed the directions as a class and then they worked on one section. When that section was completed, we repeated the process with the next section. Students will need help working through this, but it’s totally worth the time and effort to get them to make the connection between repeated addition and multiplication.

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katelpsnice worksheet!

I’ve noticed that even when students are savvy enough to know that 2+2+2+2 can be written as 4*2, they almost see the multiplication as a function of the fact that there are 4 of them (thus the important part of the multiplication being the 4). Then you give them a table of values of x & y, for example x = 1 y = 5; x = 2 y = 7; x = 3 y = 9, they can find that pattern of +2 every time, but they struggle to relate that back to multiplication, even if they understood the concept of repeated addition as multiplication. You help them get to that nicely.

I’ve noticed that even worse is repeated multiplication. Of course they can say 2*2*2 = 2^3, but if you ask them to write a pattern that results in the formula y = 2^x, they almost all really struggle. They’ll verbalize “it’s double every time,” but getting that into algebra is tough.