0 Algebra 1 - Introduction to Relations and Functions

Unit 5 for my Algebra 1 kiddos is an introduction to relations and functions.  My goal with this unit is to review pre-algebra concepts such as ordered pairs and graphing points on the coordinate plane and emphasize vocabulary.  The EOI my students will take in the spring will have questions that specifically ask them to identify which relation is or is not a function, identify the domain or range of a given relation, and classify variables as dependent or independent.

After we get back from Christmas Break, we will finish up this unit with notes on function notation and determining the rule that a function follows.  I can't wait!  I'm soooo ready to get to Unit 6 - Rate of Change and Linear Functions.  I think I love to teach about slope and graphing more than anything else.  With Common Core, that will be shifting to eighth grade which makes me extremely sad.  When that happens, I guess I will have to find a new favorite unit to teach.

Here is our table of contents for this unit so far...

I already posted pictures and templates for the ordered pair foldable and coordinate plane foldable.

We made a frayer model to discuss an important vocab word: relation. 

Relation Frayer Model

Then, we discussed various ways to represent a relation.  I focused on the four types that my students will see on their end-of-year exam: ordered pairs, input/output table, coordinate plane, and mapping diagram.  My students were convinced that an input/output table was the same as a stem and leaf plot.  At least they have heard of a stem and leaf plot...

Ways to Represent a Relation Foldable - Outside

Ways to Represent a Relation Foldable - Inside

Once we had defined a relation, we could now define a specific type of relation: a function.  

Function INB Pages

Can you tell I love the Frayer Model for vocabulary?  We made one to summarize what we learned about functions.  I gave them the definition of a function and nothing else.  Then, I went through all of my released EOI test questions and copied and pasted all of the relations and/or functions into a Smart Board file.  I asked for a volunteer.  I showed them and the entire class a picture of a relation.  I instructed the students who had not volunteered to write down function or not a function.  Once everybody had written down their answer, the student who volunteered said their answer aloud.  Next, I had all of my students hold up their dry erase board to show the student.  After seeing the responses of their classmates, the student had to decide whether they should keep their original answer or change their answer.  Only after going through this process did I reveal whether it was a function or not.

This was a new way of doing things for me, and I kinda liked it.  I have students write their answers down and show them to me a lot.  But, I rarely have students show their boards to other students.  It reminds me of the "Poll the Audience" feature of Who Wants to Be a Millionaire.  I guess you could say that the person who has to voice their answer aloud is in the "Hot Seat."

Function Frayer Model

My students loved this.  I had lots and lots of students wanting to volunteer.  The conversations that ensued between students were AWESOME.  I also did something new this year.  I didn't teach my kids about the vertical line test to begin with.  I sort of let them discover it for themselves.  Of course, this worked better in some classes than others.

We ended this day's lesson by completing the function/not a function card sort that I love and have used a lot.  I used it last year with both my Algebra 1 and Algebra 2 classes.  And, I used it with both classes again this year.  This card sort is available from Math Tales From the Spring.

Function / Not a Function Card Sort

The next day, I had students create examples and non-examples of functions.  And, we summarized the vertical line test strategy that we had arrived during the previous lesson.  To give my students further practice, I assigned an error analysis task from Algebra's Friend.  My students are terrible at following directions.  Even though I told them that they were grading the answers given on the paper, I still had students who answered function or not a function instead of correct or incorrect.  My IEP students were especially bad about making this mistake.  I just don't know how to get them to slow down and read the instructions.

Domain and Range INB Pages

Next, we moved into our discussion of domain and range.  I used the DIXROY mnemonic device with my students.  I don't really know how many students even use this to help them remember the meaning of domain and range.  DIX: Domain, Inputs, X-Coordinates. ROY: Range, Outputs, Y-Coordinates.  I did see a student write the acronym on their index card to use on their semester test, so I guess at least one student has found it helpful.

Domain and Range (DIXROY) Notes

Domain and Range / Representations of a Relation Practice

This half-sheet was stolen, ahem, borrowed from Mrs. Hester.  She posted a picture of this organizer from her 8th grade interactive notebook, and I fell in love.  So, I typed up one at the last minute.  I uploaded the template for you below.  Everything isn't perfectly aligned and centered, but it was the best I could do with the limited time I had.  If I wasn't such a procrastinator, my notebooks would be better.

Independent and Dependent Variables INB Pages

Last year, some of my students really struggled with the difference between dependent and independent variables.  So, this year, I set out to teach this topic better.  I'm pretty sure I failed, but I'll share with you what I did anyway.  

Independent and Dependent Variables Notes

We took notes over the difference between independent and dependent variables.  I filled out the arrows above three different ways for the three different sections of Algebra 1 that I teach.  None of the ways led to universal understanding.  When I look at a scenario, I can easily tell you which variable depends on the other.  This is the dependent variable.  Thus, the other variable is independent.  Easy.  

Many of my kids cannot do this.  I will think I have come up with the perfect example that will make all things clear to all students.  Ms. Hagan's Outfit.  The Weather.  Which one is dependent?  Which one is independent?  They will tell me something like this: "The weather depends on Ms. Hagan's outfit."  This leads to me saying, "So, if I change my outfit, the weather will change?"  It sounds crazy.  But, they will answer yes to this and truly mean it.  I don't think my kids understand the word "depend."  Or, maybe they don't understand cause and effect.  

There were basically two groups of students in each section where I taught this lesson.  The students who thought this was the most obvious thing we had ever done in Algebra 1.  Why in the world would we spend an entire 50-minute period on this lesson?  Then, there were the students who missed every single question.  They would flip the dependent and independent variable.  Every single time.  I was almost tempted to tell them to just write the opposite of what they thought the answer should be.  Okay, that's terrible.  And, I would never actually do that.  But, why can't my students get this?  

I created a card sort that I thought would take 10 minutes tops.  We spent at least 35-40 minutes on it.  And, some students still didn't finish.  

Independent and Dependent Variable Card Sort

* I accidentally glued one pair on backwards.  The number of songs performed should be independent, and the duration of the concert should be dependent.  Oops.  I was in a hurry, and it shows!  

I did a quick google search for sets of independent and dependent variable scenarios.  I found such a set on www.ixl.com.  I copied and pasted these scenarios and edited them to remove the names.  I thought this activity would be too easy for my students if I left the names in the scenarios.  I think I could have left the names in, and my students STILL would have struggled.

I passed out a sheet containing the 20 statements.  I gave students these instructions:

1. Cut out the 20 rectangles.

2. Recycle your trash.

3. Pair up the statements.

4. Have Ms. Hagan check to make sure your statements are paired up correctly.

5. Classify each statement as dependent or independent.  Arrange these in two columns.  

6. Have Ms. Hagan check your classifications.

7. Glue these in your notebook on page 52 and 53.  Be sure to label the columns as dependent and independent.  

Students were okay with steps 1-2.  But, Step 3.  Oh my goodness.  "How am I supposed to know how to pair these up?"  "This is so hard."  "What's a potluck?"  "Is that where people get together and smoke pot together?"  "This is impossible."  "What does the word duration mean?"  "You mean pickles come from cucumbers?!?"  

I would go around and check my students' statements to make sure they were paired up correctly.  It was not out of the ordinary for students to only have 3/10 pairs made correctly.  They were pairing up statements that had NOTHING in common.  It was a nightmare.  

A few of my high-achieving students caught on early and were able to complete this activity with little assistance.  The rest of my students.  I just don't know.  This isn't supposed to be that hard.  They couldn't get them paired up let alone classify them as dependent or independent.  I was frazzled.  My students were frazzled.  

If you have any insight on how to teach independent and dependent variables, please leave a comment.  I'm begging you.      

I have uploaded the PDF templates that I created for my students to use below.  If you can't get them to load, please make sure that you have Flash/Shockwave installed.  If problems persist, feel free to send me an e-mail.  I will be happy to attach the files and send them to you!