I teach 8th grade Pre-Algebra in a Special Education resource room. This year I have been blessed with an amazing group of students! They all try so hard every day to grasp the concepts I am teaching them.
This week I was particularly proud of them. We are coming to the end of our unit on linear equations, and I came up with an activity which would really let them demonstrate their knowledge of linear equations and the flexibility to represent the information in various forms. I knew it was a very challenging activity, but I had confidence that they would all give it 110% of their effort.
Before we started, I warned them this was going to be really challenging. They moaned and groaned. I explained to them that this activity combines everything they've learned so far in this unit and will really show that they understand linear equations. I first displayed a similar but blank problem up on the SmartBoard. I pointed out that there were three forms of equations, a table, a graph, and the slope and x- and y-intercepts that needed to be identified. I told them to all put their detective caps on because they were going to solve a puzzle. I was going to give them just 1 piece of information, and they have to fill in every other piece of missing information. You should have seen their faces! Looks of panic, disbelief, and total lack of confidence.
I get really math-geeky and excited at this point. I explain to them that this is soooo cool because everything interconnects with everything else. That there is more than one way to figure out each piece of the puzzle. That there is more than one path to get from start to finish. They look at me like I've totally lost it!
(Warning: I'm about to ramble!)
So, I put one piece of the puzzle up on the board -- a standard form equation. I ask them what we can do with this? What information can we get from the standard form of an equation? They look at me with blank stares and shrugging shoulders. I guided them to finding the intercepts. Then I asked them what the intercepts could do for us? I started to see some light bulbs go on. I graphed the line using the intercepts. Then I explained that they could have also transformed the standard form into slope-intercept form first in order to graph it. Then I asked them what we can get from the graph? Someone volunteered "the slope!" So we found the slope (and I reminded them that if we had found the slope-intercept form first, we would already have the slope), and I asked what can we do with the information we have so far? No responses... I asked if we could write one of the missing equations? "Oh yeah!" So we wrote the slope-intercept form of the equation. (OK, here I go again with more questions -- I really had to drag it out of them the first time through!) Can we fill in the table of values yet? I saw mostly blank stares and shrugging shoulders again. "Substitution" was my clue to them. One students remembered that they could substitute the x-values I had provided them into the slope-intercept form of the equation to find y. So we went through that process, painfully... Then I erased all the y-values and asked them if they could think of another way to fill in the table without substitution. No responses. Which order pair do we already know? What's the special ordered pair in the table? "Oh, the y-intercept is the one with the zero for x!" OK, so how can we fill in the rest of the table if we know this one ordered pair? Can we use the fact that we already know the 'change in y over the change in x'? "Oh yeah, we know the slope, so we can use that!" We finished filling in the table using the slope. Once we had the table complete, I asked if we could now write the point-slope form. After we did, I explained that they could have also done that without the table since they had the slope and intercepts already and could have used one of the intercepts for the point in the point-slope form. Phew! Finally done! It was exhausting...
But when we were done filling in all the information in, they said "Can we do another one?" I was so excited that they wanted to do more! I thought since they seemed so confused and frustrated by the whole process, that they would never in a million years want more! But they were so persistent and determined to figure out how to put all the pieces of the puzzle together. We did a few more "puzzles" where I gave them different pieces of information such as just the completed table of values, or just the graph. When the class was almost over, they asked if we could do this again tomorrow!
My hope is that learning to be flexible and complete all the missing pieces of the puzzle from any starting point will help them truly understand linear equations. They really demonstrated Mathematical Practice Standard #1 - Make sense of problems and persevere in solving them. I've never seen them work so hard and I was very proud!