6, 13, 20, 27, ...
The difference between each number is +7. This is called the first difference.
To find the equation: (first diff)(n) + (first term - first diff)
For this example it would be (7)(n) + (6 - 7) OR 7n - 1
Me: "Here is a sequence of numbers 5, 7, 9, 11, ... take a few moments on your own to write an equation to represent this pattern."
Many of the students look at me like I have two heads. Mind you, quite of few of the students have taken the test on this AND passed. I see a few students moving in the right direction, so I take a picture of their work with my iPad and display it on the Apple TV. Then a few more students have some recognition in their eyes.
Next I ask the students if there's a way we can get to the equation without remembering this ridiculous formula. **Insert crickets chirping** We discuss about how the pattern is changing each time, that the change is constant, and that's why it's the number multiplied by the variable. But what about the other number? We decide that we could plug in a 1 to try to find the 1st term then add or subtract something so that we really do get the first term.
Finally I get to the objective of the day. That I would like the students to be able to make connections between five different representation of a linear pattern; table, equation, graph, visual, and verbal. Before I told them what the 5 representation were I asked if they could figure it out. Would you believe that one of my classes did. I was so happy I could cry. As a class we came up with all representations for the pattern above. As we moved from one representation to another I asked the student how it related to the previous. The students were very insightful and you could see how confident they felt with having this opportunity to show their stuff.
I gave the entire class the equation y = 3x + 1 and asked them to work with their partner to create the other 4 representations. Each pair got poster paper, markers, graph paper, and glue. I was please with the results and I wasn't the only one. As the students were walking out of the room at the end of class they told me how much they enjoyed class today. They told me how smart they feel. They told me we should do this more often. Even the paraprofessional said she enjoyed class and loved how we make connections between all 5 representations.
Here are a few posers the students made.
I created index cards each with different information on each one. The pairs of students are given a card randomly and asked to create the other 4 representation.
This didn't go too bad. The major issue for most students was finding the equation (the only thing I actually taught them). But we reasoned our way through it rather than try to remember an algorithm.
From here I just need to clean up a few things with the students. Like to make sure the labeling on the axes is consistent:
Or just labeling the axes for scale.