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.

Thank You #MTBoS

In the past couple weeks, it’s really struck me what a unique and special community the MTBoS truly is.  The things the people in this community have done and created are amazing in and of themselves, but that’s just a piece of it.  The way the people in this community support each other is incredible.

I will be scrolling through Twitter and can’t help but smile when I see a simple Tweet turn into a lengthy conversation between people from all over the world.  It’s SO cool to see the collaboration that takes place through the #MTBoS.  How can you not smile when you see people who may have never met in real life encourage others after a rough day, offer advice or constructive feedback on an idea, and take the time to look up a resource for someone.

It recently struck me how much of an impact the MTBoS has had on my teaching.  Most days, I would not have one complete lesson  without the MTBoS -things I have taken from people in this community are woven throughout my entire day nearly every day.  One day recently when it really hit me,  I had implemented differentiated review stations I learned about at TMC 16 from Michelle Naidu and part of those stations included open middle problems and quarter the cross.  I used Notice/Wonder from Annie Fetter that I learned about via Sara Van Der Werf to introduce the idea of a function with Hannah Mesick‘s birthday analogy.  I used Estimation 180 and Desmos as a warm-up.

I could stop there with the thank you for everything the MTBoS does to share resources and support teachers, but I can’t.  Because that’s only part of the greatness that is the MTBoS.  What’s even more awesome is your name doesn’t have to be Dan Meyer or Fawn Nyugen to be part of it.  If you’re passionate about math education, you’re in.  You’re part of the group.  They even include an unknown girl like me!

Not too long ago, I threw a question out there, and  I know I shouldn’t be, but I was honestly surprised that people actually replied!

Enough people were willing to share their input that the feed actually had a place to “show more” replies.  That made me excited.  🙂


I had something funny happen not too long ago in one of my classes that I didn’t think any of my colleagues would fully appreciate, but I knew Casey would.  And she did.

And then there’s this.  🙂

So thank you MTBoS not only for all the amazing resources you provide to teachers but for being there and getting it -after the good, the bad, and the ugly days you understand what it means to hold the challenging and exhausting, yet rewarding and amazing title of teacher.

Soft Skills: Choices and Consequences

Soft skills is a word that’s thrown around pretty regularly at my school, so when I saw this week’s (ok last week’s) #MTBoSblogsplosion theme was soft skills I didn’t think I would have too much trouble coming up with something to write about.  Once I started thinking about it though, I struggled to think of things I thought were “worthy” of writing about because I know it’s nothing new.  I also doubted myself for doing many of them.  Am I just being a crab about some of those things? Do I just need to let it go?

  • It’s not uncommon for me to say at the start of a class before the bell rings to a student or group of students when they walk in the room extremely loud, “Why don’t you leave the room and come back in an appropriate way.”
  • As students are transitioning from their desks to groups, I’m known to say, “Alright, everyone back to their desks.” and before they even sit down typically at least one student already has their hand raised because they know I will ask, “Now why did I have you go back to your desks?”
  • When a student says to me something like, “I don’t have a pencil.”  My response is always, “Ok, so what are you going to do about it?” rather than responding to the unasked question.
  • As often as I can, I try not to let a student interrupt me when I’m working with a another student or talking to a teacher.  Even if the conversation was essentially over when the student interrupted, I will have the student return to their seat and purposefully continue the conversation with the teacher -even if that means explaining why I’m continuing to talk when we were done talking.

There have been so many times this year that I have been frustrated over some of those non-math things that come up.   I know those things are important, but I’ve asked myself at times if it’s worth it to continue to pour my energy into those things when they don’t even directly involve math.  One time I was talking with a colleague about some of my frustrations, and he made the comment that I’m teaching them how to be people.  That struck me.  I needed to hear someone else say out loud what I knew was true -that those things are important and worth putting time and energy into.  I love how Liz Mastalio words it, “Honestly, the math is secondary”.  (I also love how she does Friday questions with her students.  It’s similar to Sara Van Der Werf’s name tents.  I might have to try that out sometime!)

Choices and Consequences.  That is the underlying idea in many of the soft-skill related things I do with my students.

If you would have told my younger self that choices and consequences would be something I would repeat to my students, I would have rolled my eyes at you.  I heard that from my parents all. the. time.  And I now understand why.

That is not an idea that comes naturally to most middle school students.  They struggle to see the connection between their own actions and the consequences -whether those be positive or negative.  As often as I can when having conversations with students, I try to ask questions to help them realize and how their own actions connect to the consequence. I want them to be the ones to actually verbalize it rather than me.

Then there was this day.  This is still one of my most favorite days ever as a teacher.  I think all students left my room that understanding that if they made the choice to engage themselves in class, the consequence was that they might actually enjoy it.  🙂



Movement: A Game Changer in my Middle School Classroom

Much of my first year teaching was spent trying to figure out what on earth to do with 6th graders!  While I always knew I wanted to be a teacher, I always said I never wanted to teach middle school, and when I got my first job knowing I would be teaching 6th graders, I was really at a loss.  When I was a 6th grader, I was still in elementary school.

I felt that my job as their first math teacher in the middle school was to teach them how to be a middle school student.  I still fully agree with that statement; however, my view of what it looks like to teach them how to be a student is drastically different now than it was  as a first year teacher.

I went to two sessions at two different conferences my first year teaching that were game changers for me when it comes to teaching middle schoolers.  By my second year teaching, I was able to honestly say that I loved teaching my 6th graders, but without those two things, I think it would have taken me a lot longer to be able to say that I really truly love being a middle school teacher.  I am now teaching middle schoolers all day and love it.  I’m exhausted all. the. time. but I love it.

The first game changer was the idea of using games to get students practicing different skills.  I blogged about one of the things I took from that session here.  The second was a session on incorporating movement into the classroom.  One of the presenters was none other than Sara Van Der Werf.  (Sometimes I feel like all I write about are ideas I’ve stolen from Sara.  This entire blog could really just be an ode to Sara.  Thanks Annie for saying that so much better than I ever could!)

Anyway, I took SO many things from that session that I was able to implement immediately into my classroom and so many ideas that I was able to build upon to make work for me.  I mentioned one of them, Balance Points, here as well how I use her Stand and Talks here, but there are so many other things that I continue to use from her session.

When I started incorporating more movement activities in my classroom, I was worried they would take away from the precious time I have with my students.  However, I quickly realized something.  The times when I knew my students needed a brain break but tried to push through to finish what we were on were often way less productive than the times I would stop and give them a quick break.  There are still times that I literally stand in amazement in my classroom over how focused my students are after giving them a short break.  I also worried about the transition time from class work to brain break back to class work.  Yes, the first time I do a brain break with students it takes a bit longer to explain it, but after that, as long as I give it a name so students know what I’m talking about the next time, the transitions go pretty smoothly.  I also found that these brain breaks are a great way to review concepts we had previously covered.

My goal is to post semi-regularly on a different movement activity I do in my classroom.  We’ll see how consistent I can be, but today’s activity is what I call Divisibility Hop.

While writing this post, I looked up Sara’s version of this game from my notes from her session and realized that it has morphed into something a bit different in my classroom.  I think both versions are great.

Sara’s version is called Factor Hop.  She puts a number in each of the 4 corners of the room -these numbers are the factors.  She calls out a number, and if the number you are standing by is a factor of that number, you hop to another corner of the room.

I’ve been calling the game Divisibility Hop.  I put 4 numbers in the corners of the room.  They are generally 3-4 digit numbers.  I call out a number, and the number I call out is the factor.  If the number students are standing next to is divisible by the number I call out, they move to another number.  I will change up the numbers periodically.

The first time we play this I tell students something along the lines of, “The name of the game is Divisibility Hop, so you can’t walk to the other number.  I don’t care how you get to another number as long as you don’t walk, you’re safe, and everyone around you is safe.”

I’ve had students do the worm across my room, penguin walk, seal walk, you name it.  For many kiddos, this is the perfect sensory activity!

Today I had a student curl in a ball and start rolling across the room and say, “They see me rollin’.”

Yes, I truly love teaching middle schoolers.  They are the best.

My Favorite Mistake

Last year sometime I came across this video from the Teaching Channel and have been incorporating my version of that into my classroom this year.  I post a warm-up problem on the board and have students complete it on a 1/4 sheet of paper, collect them, and I re-write an incorrect solution on the board and we discuss the mistake and why it was my favorite mistake.

Over the summer, I also read Sarah Carter’s post about having students evaluate their mistakes here. (And of course she has a poster you can download here.  Her posters are great!)  I loved the idea of having students evaluate their mistakes and write about it.  I haven’t talked with students about types of errors yet, so I created my own version of her form.

Here are the files to download.  I’ve included a PDF and a Word Doc version.

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I thought the warm-up routine would pair nicely with the reflection form.  After doing My Favorite No as a warm-up for a while to get students familiar with that process, I used that sheet for the first time last week when passing back tests.

I wasn’t quite sure how this would go over with students, but overall I was super impressed with their effort while filling this out as well as their responses.  Students did this on one of the first days back from Christmas break, and because of that I sort of expected students to push back a bit and not be willing to do something different.  I shouldn’t have doubted my students!  I don’t know if I’ve ever had students so engaged when going over a test before.  They were analyzing the various mistakes they made, writing about them, and fixing them.  It was great!

I’m looking forward to doing more of this with students, but I know this process could be improved.  I know I need to work more on connecting students’ responses on this sheet to their future work and discussing with them more why filling this out is important.  What ideas for improvement do you have?

Have you done something similar to this with your students?  What does it look like in your classroom?


My Favorites: 2 Equation Activities

I decided to give the #MTBoS #MtbosBlogsplosion blogging initiative a try, at least for this week.  This week’s topic is “My Favorite” and in typical me fashion, I couldn’t pick just one, so here are a couple of my current favorites related to solving equations.  Ask me in a couple weeks what “my favorites” are, and I’d likely give you a different answer depending on what topic I’m covering then.

My Favorite App:  SolveMe Mobiles

I’m probably super late in discovering SolveMe Mobiles, but I love it!  I’ve used it with 6th, 7th, and 8th graders and all students have been enjoyed it.  One of my 6th graders came to me this week, “Can you please help me with this puzzle?  I was working on it last night with my brother and we couldn’t figure it out.  We stayed up until 10:00 trying to figure it out.”  You know you’ve found a winner when a students voluntarily does math at home, gets his brother to play along, and they stay up late trying to figure it out!


If you haven’t checked this app out yet, I strongly encourage you to do so.  (There is also a website if your students don’t have iPads.)  I love that the app is simple and easy to use, but it also offers many different features.  Students can draw on the screen, zoom in and out, and you can drag down from the mobile and it creates an equation with the shapes.  I have only used the “play” mode with students, but they are also able to build their own.

If students type in an incorrect answer the mobiles tilt to reflect that one side is heavier than the other.


I’ve also had some great conversations with kids when they do something like I’ve got pictured below and they tell me that it’s balanced and wonder why it doesn’t work.


My Favorite Movement Activity:  Balance Points

My first year teaching I went to Minnesota’s annual math conference and attended Sara Van Der Werf’s session on movement.  The entire session was a game-changer for me.  Not only was it the first time I was able to experience first-hand Sara’s awesomeness, but I also took away a ton of stuff I instantly applied to my classroom.  Right away, I saw the impact movement had on my students and my lessons, AND realized that movement activities don’t have to take away from the time we’re doing math.

To play Balance Points, students are paired up, and I put an equation on the board such as x + 5 = 9.  While staying connected in some way, such as holding hands, students must show the correct answer by having that many body parts touching the ground.  I encourage them to be creative and won’t accept “boring” answers.  0609_wheelbarrowrace

I wish I had pictures to share of my students playing this.  They LOVE it.  There isn’t a person in the room who isn’t smiling and laughing, myself included!   There’s usually a lot of  “Come quick!  Check our answer!” because students are about ready to fall over from the crazy positions they’ve come up with.  So fun!!  I haven’t done this, but you could use this for order of operations as well.

What other ways could you use this with students?  Let me know what you try!