# Exponent Unit

This was the first time I’ve taught exponents without explicitly telling students the “rules” at some point within the unit.  Many students still said things like, “Oh, so when you divide, you subtract the exponents.”  I have mixed feelings over this.  Yes, I want my students to notice patterns, but not at the expense of understanding the math they are doing.  This is one of the things I struggle ensuring as a teacher -that after my students have noticed patterns, they still understand what is actually happening.

I started the unit with a modified version of Andrew Stadel’s exponent mistakes worksheet.  (I know I found someone else’s version of this worksheet that I modified, but I can’t remember where I got it.)  This was something we came back to periodically throughout the unit.  On one of the last days of the unit, we went over the correct answers as a class for the first time.  After going over the sheet, I asked my students to think back to their reaction when I first gave them the worksheet.  Many sort of freaked out and several others were convinced that some of the problems were actually correct.  It was fun for me to see them realize they had learned something throughout the unit because they could now correctly do all of the problems.

The rest of the first day we focused on identifying the base and writing things in expanded form.  The next several days I spent at least one full day on the product rule, power rule, and quotient rule.  The link for the worksheets I used is at the end of this post.  Again, I know I modified those worksheets from ones I found somewhere online at one point, but I can’t remember where I found them.

I used this Which One Doesn’t Belong? as a warm-up one day.  I’ve really been loving using these as warm-ups this year.  I love how much vocab students use while doing these.

About this point in the unit, I was not in my morning class a few days in a row due to state testing with my 6th graders.  I was looking for self-checking practice for students on exponent problems.  The challenge for me was we hadn’t talked about the zero power yet or negative exponents.  Most everything I was finding online included those types of problems.  Here’s what I came up with.

I modified Kate Nowak’s row game to work for where my students were at.

I don’t know if “Two Truths and a Lie” is the correct name for the next worksheet I created, but I couldn’t think of another name and was running out of time, so I went with it.  Basically, students were to simplify 3 different problems.  Two of the problems would have the same answer (the two truths) and the other problem had a different answer (the lie).

I also had a sheet of Yohaku puzzles ready which I LOVED, but I didn’t end up using it then.  I did, however, use it later in a few of my classes.  I love that there are so many different solutions to these puzzles.  I definitely want to look at the other puzzles on that site for future use.

When I was finally back with all classes after state testing, we reviewed using this Desmos activity I created.

absolutely LOVE this Desmos activity from Mathy Cathy for an introduction to zero and negative exponents.

We ended the unit with some more practice combining all different types of problems.

# 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.

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, …

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.

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.

# 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.

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.

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.

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!

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.

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.

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.)

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.

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.

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.”

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.