Arithmetic Sequences with Visual Patterns + VNPS + Desmos

I haven’t blogged in a while, like years, so this feels a little strange and hopefully some of you find this post useful….

This blog is going to talk about one of my favorite lessons, that I have spent time refining over the past couples of year. It’s about arithmetic sequences, which is the culminating activity in a chapter about linear functions. It’s a nice way to create a need and an understanding about the equation for find the nth term of an arithmetic sequence.

When students entered the room, I created random groups of 3 using https://www.flippity.net/ . Once the students were in groups at VNPS (I have a combination of dry-erase boards and chalk boards where students work throughout my classroom) I projected the first task on for them to work on.

I gave students time to do this task in their groups and monitored their work. I reminded groups to take with who was writing. It is always fun seeing ideas rotate around the classroom and the debates that start to happen, definitely a buzz going on in the classroom. Here is some of their work, sorry so many…

When they got back together as a class we ignored all the work on the boards and started off by TALKING about the patterns and what students saw and how they saw the pattern progressing, I used to skip this step but Fawn Nguyen who makes https://www.visualpatterns.org/ emphasized this to me. Different groups talked about their perspective. Then we quickly went over the answers to the questions. I then asked….”How many trees are there in stage n of the pattern?” and put the groups back to work. Below you will see some more work that showcases that.

So, if you take a peak at the last 2 images you will see two different equations. A student from one of the last groups ask “who’s equation is right?” Of course, he thought his own groups was. Another student from the other groups said, ”maybe they are both right?” Of course I replied with “is that possible? What would that mean?” Somebody mentioned they were the same equation and I said “how could you check that?” and then I walked away (power move for student-centered philosophy is sometime students learn and debate more when you walk away and leave students to figure out some amongst themselves).

We then used the Desmos graphing calculator as a tool to verify our thinking and test out the expressions and equations the students came up with. Using Desmos as a tool for verifying is severely underused in my opinion. We used the TABLE function in Desmos and entered the information we had from the first part, then we wrote expression that help break down the pattern into parts, then we tested our expressions to see if they matched. 

The discussions we had as a group and how the students verbalized it all was really awesome, most students were really engaged and it was great to see the different ideas. Students actually begged to do another one, which I had planned for but was going to skip because this took way more time than I had planned. Here was the next visual pattern.

I particularly like this pattern due to the use of 2 shapes but only asking questions about the circles. Of course, especially for anyone teaching middle school, emphasizing that we are only addressing the number of circles is important. Although I didn’t ask students to draw this time, several groups did. Here is the quick check of the expressions that came up with with stage n.

Several groups we pretty excited that if they just jumped to the equation or expression it would give all the other results for them, that the power of expressions. 

Like I mentioned, this was a lead in to teaching arithmetic sequences. Hope you enjoyed reading, feel free to ask any questions.