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.

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