Arithmetic and Geometric Sequences

Arithmetic and geometric sequences was another unit where this was my first time teaching this when I was the one introducing this to students.  There were parts of this unit that I really liked, but there are definitely things I want to improve for next year.  Looking back on the unit, I’m surprised at how many students struggled with arithmetic sequences after having just spent so much time on linear functions.

I started day 1 with this warm-up:


After that we did Notice/Wonder using the following image.  Next time, do I keep the sequences color coded but have them random on the page rather than sorted with the line down the middle?


We then took notes on arithmetic sequences, common differences, geometric sequences, and common ratios.  I used some of Sarah Hagan’s notes found here and made others similar to hers after reading this post.  I also used her half sheet as practice.  The link to my notes is at the bottom of this post.

The warm-up for day 2 was this Which one Doesn’t Belong?


On day 2, we talked about writing the rule for the arithmetic sequences.  We made the connection to common difference and the slope of a line as well as the zero term and the y-intercept.


Sara Van Der Werf’s “Sarified” standards have become my Bible for my state standards.  I LOVE how she has combined the vocab allowed along with the examples given for each benchmark.  However, I’ll admit I was a little bit perplexed by the standards and examples for arithmetic sequences, so if you’re a MN teacher reading this and can clarify for me, please do!

Here’s how the standards define an arithmetic sequence:

f(x) = mx b, where x = 0, 1, 2, 3, …

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But then in this example, unless I’m completely off track here -which could absolutely be the case, -1 would have to be the 1st term based on the answers given.

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So after debating and overthinking, I decided to give students some examples where the first number given was the zero term and others where the first number given was the first term.  I also told them in the directions which term the first number given was.  Was this the best solution?  I don’t know, but it’s what I did.  If you have thoughts on what I could do next time, please share!


Next up was writing rules for geometric sequences.  I didn’t want to just give the formula to students and have them use it, but I couldn’t think of anything I felt was great.  Last minute, I came up with the following and again had students notice and wonder things about it.  Was it the best thing ever?  No, but my students were able to come up with the formula for geometric sequences on their own after looking at this image.


Again, we took notes on geometric sequences and did a couple examples together as a class.


When students graphed the sequences, we also looked at the graph on Desmos and talked about how because we see a small portion of the graph, it may look linear, but when we look at more of the graph we see that it isn’t linear.


When students were working with arithmetic and geometric sequences separately, they did fine, but the wheels started falling off for some students when the sequences were mixed up.  Students mixed up where the zero term went and where the common difference/common ratio went.  I was frustrated that my students were frustrated because, to me, this wasn’t really that much new stuff.  Arithmetic sequences were like linear functions which we had just spent weeks working with.  After school that day I put on my “non-math teacher brain” and tried to look at the formulas from my students’ perspective.

Arithmetic:  f(x) = mx + b  (m = common difference;  b = zero term)

Geometric:  f(x) = ab^x (= zero term;  b = common ratio)

When I wrote the two formulas out like that, I understood why students were frustrated. Although there’s a “b” in both formulas, it doesn’t represent the same thing in both.  In arithmetic sequences it’s the zero term, but in geometric sequences it’s the common ratio.  I could see and understand why they felt the “rules” were changing on them.  Once I understood the confusion, I was able to address this better when working with students.  When students were confused, I pointed out that in both formulas the common ratio/common difference is the number “with the x“.  I don’t know that’s the best way to go about helping students who struggle with that though.

If anyone else has students who struggle with this and have found ways to introduce this so that students don’t get mixed up with this or have ways of explaining this so students understand, please share!

Here is the link to the notes I used in this unit as well as a few worksheets.

Much of the format in the notes is stuff I’ve borrowed or modified from Sarah Hagen.  I don’t do true interactive notebooks (INB).  I upload the notes as a pdf to Google Drive for my students at the start of a unit, and they use an app called Good Reader on their iPads so they can write or type in the notes.  Here is Sarah’s page of INB resources.

Pythagorean Theorem and Rational/Irrational Numbers

It’s been one of those roller coaster weeks.

I finished my Master’s presentation last weekend (yay!), and a couple people have asked me if I feel like a “Master” now.  I have confidently responded, “Yes, I am definitely a much better teacher now than I was two years ago, and my Master’s program has played a huge role in that.”

I went into the week excited to be a teacher and not a teacher going to grad school for the first time in two years, but I got a dose of humble pie on Monday when the majority of my lessons were flops.  The week has been a series of ups and downs since then.  So it feels weird to write this post, which is a rough skeleton outline of a unit I taught several weeks ago that I felt went pretty well, after a week like this, which felt like a major flop teaching wise.

This was my first time truly teaching the Pythagorean Theorem rather than just reviewing it with students.  I’ve really been working this year to find ways to get my students to notice and discover things in math rather than me telling them stuff.  I’ve been surprised that often times a small change I make to a lesson makes a huge impact on the overall lesson.  (I’ve come a long way with my 8th graders.  6th grade is a whole other story.  That’s my project this summer.)

I started this unit off having students notice and wonder using the following image from this Desmos graph.


I wasn’t quite sure where the lesson would go after this.  I had notes ready to go, but before I got to that I put an example where students needed to find the length of the hypotenuse up on the board.  I was excited to see students drawing the squares to find the hypotenuse.  Even though I was the one to show them the picture above,  I don’t know if I would have thought to do a problem problem using that visual until our discussion in class and we started doing the example together.  I decided to forgo the notes that day and continue with more examples using their own way of thinking about these problems before I formally introduced the Pythagorean Theorem.

By the end of that first day, students’ work looked something like this.

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I can’t remember if I gave them a problem where they had to find the length of a leg the first day or the second day, but I’m pretty sure it was before I formally introduced the Pythagorean Theorem.

By the end of the first day some students got it.  Others felt completely lost, but that was a great opportunity to have the conversation about how if students were confused that was a good thing.  I tell my students it means they’re paying attention and are engaged in what’s going on enough to be confused.  I tell them to stick with it and don’t give up.  I tell them to trust me.  I’ll get them where they need to be.  It was only the first day of the unit.

And guess what?  Later that week, those that were confused that first day, got it, and I reminded them of how confused they felt the first day and that conversation.  That I told them to trust me and stick with it and that they’d get it.  And they did.

One day when reviewing the Pythagorean Theorem in one of my classes, I asked a student why it worked.  I wish I would have recorded his response.  I thought it was pretty perfect!  “Well, if you find the area of the square on that leg and the area of the square on the other leg and add them together it equals the area of the square on the hypotenuse.  Then to find the length of the side you take the square root.”

I made this Desmos activity for some in class practice later that week.

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I also used this unit to talk about rational vs. irrational numbers.  It wasn’t quite where I planned to introduce it, but my units got switched around a bit because of state testing.

We spent a day or so on rational and irrational numbers.  I expected to find a Desmos card sort on this, but couldn’t so I came up with this one.  Shortly after I created that one, I also saw that Joel Bezaire also created one.  You can find his here.

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The last big concept in this unit was finding the distance between two points using the Pythagorean Theorem.  I started that day by reviewing the Pythagorean Theorem.  Then I posed this question to students.


It was fun to see my students excited that they were able to figure this out on their own.  To me it seems like such a small difference from the types of problems we had been doing, but to my middle schoolers, this is an entirely different problem at first glance.  It’s fun to see many students who at the start of the year would shut down when any new type of problem was put in front of them now trying different things to attack a new problem.

Then I gave students two ordered pairs and asked them to find the distance between the two points.  As one group worked on this problem, one person started by graphing the points and another started by finding the difference between the y-values and the x-values like they had done earlier in the year to find slope.  Their conversation was another one I wish I had recorded.  As they compared their strategies I overheard, “No, it’s the same thing.  When you subtract the x‘s, it gives you the length of this leg, and when you subtract the y‘s, it gives you the length of this one.”

What I really liked about this unit was there was very little direct instruction -a little bit when I formally introduced the Pythagorean Theorem and a little bit when students took notes on rational and irrational numbers.  Other than that students were in groups working on problems the majority of the time.

Movement: Incorporating Math Concepts into Movement Activities

(I’ve written several other posts on movement:  Post 1 • Post 2 • Post 3)

Movement has been a huge part of my classroom since attending Sara Van Der Werf’s session at a conference my first year teaching.  When I first started being more intentional about incorporating movement into my classroom, I would often have students do 10 jumping jacks, 5 push ups, or skip around the room, etc.  Every once in a while I have an epiphany about a way to incorporate the concepts I’m teaching students into these movement activities and think to myself, “Why did it take me so long to come up with this?!”

Here’s the list of what I’ve found to work best so far, more for my future reference than anything.

  • Perimeter/Area – I have students skip around the perimeter of the room or hop through the area of the room.  (I will also often add clockwise or counterclockwise to the directions.  I’m always shocked at the number of students who don’t know these words!)
  • Exponents – When doing something like jumping jacks, I started giving students an exponent to evaluate, rather than just a number.  This way I can sneak in exponents all year long.  “Do 3 to the second power jumping jacks.”
  • Prime/Composite – I’m all about finding ways to expose students to vocab words all year long.  I sneak these vocab words in by having students “Do a prime number of push-ups.” or “a composite number of sit-ups.”
  • To get back to their desks when we’re doing with a movement activity, I will sometimes have students count the number of “hops” to get to their desk and then do something with that number such as find the prime factorization of the number or we’ll talk about who hopped a prime number of times or whose number is divisible by 3, etc.
  • Ratios – I pick 2 activities and have students do them at a specific ratio.  For example, “Do jumping Jacks and sit-ups at a ratio of 3:2.”  (This was my “Why haven’t you thought of this before now?” moment of this year.  6th grade is ALL about ratios.  Seriously, why did this one take me so long to do?!)

I would love to hear how you incorporate the skills you’re teaching into movement activities in your classroom!