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:

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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?

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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?

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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.

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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!

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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.

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

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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.

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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.

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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.

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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.

Relations, Functions, and Function Notation

I’m teaching 8th grade for the first time this year.  I feel like there are so many huge, important things introduced in 8th grade, and I find myself way overanalyzing how to go about presenting the information to students in a way that they will truly understand what’s going on rather than memorize and follow a set of procedures.  I’m also learning how far I can push students to figure things out on their own without leaving them feeling completely lost and frustrated.

Relations and functions was one of those things that I was somewhat dreading having to teach because I wasn’t quite sure how to go about it, but I’m really happy with how much of those lessons have gone.

Day one of the unit started with Which One Doesn’t Belong to get students thinking about graphs on the coordinate plane.

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Then students did notice/wonder with the following image.  One student noticed that it looks like it came from a textbook.  Ha!  That may have been the first time all year they had seen anything from a textbook.  Whether it’s a good thing or not, I rarely, if ever, use a textbook with students.  When I use it, it’s for my own reference and isn’t shared with students.

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As students noticed other things about the image, they were able to tell me the similarities and differences among the different representations of a relation.  I didn’t have to teach it!  They taught it to each other.

Day two started with the following problem.  Two days in a row with images from the textbook.  That was definitely a first for the year!  I wasn’t anticipating liking how this would go as much as I did, but I loved how some students were stuck at first because the graphs have no numbers and eventually figured it out.

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Then I used Sarah Carter’s telephone activity.  I liked how this went, but next year I want to change it up so that students are participating more often.  If I put students in groups of 4, I may give each of them a sheet and have each person start by writing down ordered pairs. Then I would have everyone pass the paper around so that all students have a sheet at all times.  They would also get practice with each of the different ways of representing relations this way.

I also realized that at times I need some work on giving directions…  In my first class I had one student whispering the ordered pairs to their partner rather than using the piece of paper to pass the “message” along!  Oops!

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The next thing was magic I tell you.  Magic.  I put the following image up and had students notice/wonder about it.  Things had been going really well up to this point, and I worried adding the next thing would completely throw some students for a loop.

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I looked up some students’ actual birthdays through our school’s grade book website.  They were immediately engaged.  Thank you Hannah for this!!

Again, they were the ones to tell me “you can’t have two birthdays”, which lead into a conversation on functions and the similarities and differences between relations and functions where I helped fill in the correct vocab.  I honestly expected this to be a stumbling block for some students, but it really was a non-issue.

This was one of my slides from day three.  Students all had mini-whiteboards with coordinate grids on the back.  I started by picking 3 inputs and outputs and had students plot them on the coordinate plane and decide whether it was a function or not.  Then I had students create their own examples of functions and non-functions and had some write their answers on the SMART Board.

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Then I had students do a stand and talk and discuss the two sides of the chart they had just made.  At this point in the lesson, the green vertical lines weren’t up there yet, but in every class while students were working in their groups, I had someone come up to the board and point out the vertical line to their partner!  It was awesome!  Again, the students came up with the vertical line test on their own and taught it to each other.  I didn’t have to!

Day four started with a review of the vertical line test and this Desmos activity from Cathy Yenca.

All of my students have iPads, and when students were working in pairs on this, we had to revisit what it looks like to be working in pairs when both students have devices.  I told them how sometimes they look like toddlers playing.  Some gave me confused looks at first, but I explained how if you ever watch toddlers play together, they don’t actually play together.  They play next to each other and don’t interact.  They laughed, but I told them that’s what they look like sometimes.  They got the point and things improved after that.

I used another Desmos activity on day five from Rockstar MathTeacher followed by the introduction to function notation.

I started by putting a couple problems like y = 3x + 8 up and asked students what y equals when x = 5, etc.  Then I put the picture below on the board and had students try to figure out what the “stuff” on the right meant.  (And yet another image from a textbook!  I’m almost positive this unit more than doubled the textbook pictures students had seen all year in my class.)

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Students were frustrated at first.  They thought I was crazy for asking them to do this, but they got it.  You could see the lightbulbs go off and how proud they were of themselves.  During the stand and talk, I again had students coming up to the board to point things out to their partners.  I don’t ever remember that happening before this unit.  It was fun to see.

Then this went up on the board next, and I asked students to figure out the pattern and to use it to complete the bottom two rows.

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Once again, students were the ones to figure this out and teach it to each other, and I just helped fill in the vocab words here and there or nudge them to use the correct vocab words.

Day six was a quiz and more practice with function notation.

Day seven started with Set.

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My last class was struggling to find the last set.  I asked if they were ready to call it good and move on.  And one students immediately said, “No!  We’re not stuck on the escalator!”   There was no way I was going to let them quit after that.

Then students worked on a Tarsia puzzle.  Most students were familiar with this puzzle from when they had me as sixth graders. The responses I got when I took them out were, “Yes!  I love these things!”  and “Oh yeah, I remember these.  These are fun.”

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Honestly, the plan was for students to work on this half the hour and then move on to linear function.   However, I decided to let students continue working for two reasons.  One, they were working!  In my first class, it was a little bit louder than usual, but as I looked around every group was on task talking about math!  In my other class, students were quietly working and focused during last hour of the day on a Friday!

The bigger reason I decided to let students keep working was because when we do activities like this I tend to underestimate how long it will take students to finish and as a result, maybe a couple groups will finish and the rest won’t, never getting that feeling of accomplishment and of having completed the puzzle.  I wanted as many students as possible to end the week feeling that way, and most did.

I wish I could remember where I got the file for that puzzle.  It was likely on Mr. Barton Maths website.  If anyone knows for sure, please let me know.  Here are the files I used.  Note:  the card that has a 12 on it has a typo.  The function is missing the equal sign.  It should read f(x) = -5x² – 3x + 14; f(-3)

It was a good week and a half or so!  I also had graphing stories ready to go when I had a few extra minutes in a class.

A Teacher’s Noticings and Wonderings from Day 1

These past few days have been somewhat of a blur with changing to a 7 period day, teaching all middle school for the first time ever, and even teaching in a new classroom where I’m not used to the set up yet.  I’d forgotten how exhausting teaching is with all of the decisions we make in a day!

But I am LOVING the changes!  I don’t know if it’s the fact that I’m teaching all middle schoolers now, that half of my classes are students I’ve taught before, or that I’ve been trying quite a few new things in my classroom, but it’s been a great start to the year!

I know I’m probably super late on Annie Fetter’s Notice/Wonder video, but it was new to me over the summer.  I’ve been asking students to notice and wonder quite a bit already this year and also ask students “What else? What else? What else?” thanks to Sara Van Der Werf.

During the first day of “Notice/Wonder” and “What else?”, I noticed and wondered things as the teacher.

The first happened when students were noticing and wondering things about Fawn Nguyen’s Noah’s Ark problem.

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At one point, I was sort of hiding out in the back of the room so students could focus on the picture on the board.  It didn’t take long for me to notice that a few students had turned around and looked to me.  I exclaimed, “You guys!  You aren’t noticing and wondering things about my face!  Look at the board!  You’re noticing and wondering things about the picture up there!”  Maybe not the best reaction I could have had, but that’s what happened.

Now, I’m wondering what I can do to change the culture in my classroom so that students don’t always look to me for the answers and trust themselves to be able to come up with their own mathematical ideas about problems.

 

The second instance occurred when we were talking about group work norms after using Sara Van Der Werf’s 100 numbers task.

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After seeing pictures of themselves doing the task, students were answering the question, “What does good group work in math class look like?”  As they were giving me responses, I kept saying “What else?”.  At one point a student said, “Can you give us a hint?  What does it start with?”  Another instance of a student assuming that I had a specific answer I was looking for.  The student was surprised when I said that there wasn’t anything particular I was looking for and that I wanted them to come up with responses.  Again, I wondered how I can change that culture in my classroom this year.

 

At first I was sort of hard on myself regarding both of these situations because they both happened in classes of students I’d had before.  However, when I taught them before, I didn’t know what I know now.  Now I know, and I’m glad I have the opportunity to work with these students again and teach them the things I now know.

Also, I shouldn’t jump over the fact that I noticed those two situations in my classroom and that they stood out to me.  I don’t know that I would have done that in the past.  I may have noticed them in the past and found them funny, but now I notice them and realize that they are examples of something I need to work on with students in my classroom.