Math = Love: Algebra 1 - Introduction to Relations and Functions

Tuesday, December 17, 2013

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!


44 comments:

  1. You asked for advice on your card sort. I've learned that more isn't always better for card sorts. Perhaps you could have divided the cards into two sets of 10 cards. For the first set, you could have left the cards side by side and just let the students choose independent or dependent. With that skill built up, you could have given them the second half of the cards to pair up and then classify. Kids who catch on more easily could make their own pairs, have a few distractors added to their cards, or write about the correlation between the variables.

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    1. Thank you so much for the advice! This was the first card sort I ever created on my own. I've always stolen already created ones from others before. There's a definite learning curve. I love the idea of breaking them into sets! That's brilliant. And, having the students who catch on make their own pairs is something I didn't think about at all. I think I tried to make this one card sort accomplish too much.

      Thanks so much for the feedback! It was immensely helpful!

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    2. Kathryn,

      This is a great way to differentiate the activity. Thanks for the idea.

      -Pamela

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  2. Love the question about whether a potluck is where people get together and smoke pot!

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    1. You can just never know what is going to come out of a student's mouth! :)

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  3. I teach independent and dependent as cause and effect. I make the kids say the phrases backwards and they see just how silly they sound. Love the INB! Always wanted to try one myself... never brave enough.

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    1. You should definitely just take the INB plunge. I did, and I haven't looked back since. If you need any help, I'd be happy to answer questions!

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  4. One of my students just suggested the idea of a table of contents for each unit, where we could put it, and why she feels it would be useful. I told her it was a great idea, and I think I know just the place to find a template! It's a perfect fit in our journals!! Thank you for sharing your documents. I'm finally integrating more typed foldables and lessons into my journals. I am such a hands on written person; it's hard for me to do. The kids love typed notes and I can tweek them when needed and not have to rewrite the whole thing! If I wanted to post what I have, what's your advice for the best method? Hope your second year is going great. Have a Merry Christmas!!

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  5. I'm saving this entire post! I'm a long term sub teaching remedial math and I've noticed this odd gap in the kids understanding that points right at this. They can do the mechanics but don't seem to get the point that one variable depends on the other so they are at a loss setting up problems. Spending some time working through this a second time might help them get it. Thanks, Merry Christmas

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    1. I agree! I noticed this gap last year with my students, so I've made it my goal this year to emphasize dependent/independent every single chance I get.

      Thanks for reading my blog!

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  6. Hi Sarah,

    I'm preparing for my unit on functions that I will start after break and going through your resources. I have not taught this lesson yet, but I don't think you should scrap your independent/dependent idea. I can see where the students became confused, because I tried pairing them up myself before looking at the key. For this type of activity, I think there may be a few too many examples. I teach 8th grade, so I will probably use 6 but I think 8 would be a good amount for high school. It is just a lot of reading, and kids tend to shut down when they see a lot of words.

    I can see how the examples would have worked when they were presented in ixl.com, because there was only one independent and one dependent variable on each page. When they are compiled as a group, there are too many similarities in the bunch. There are a lot of examples about food, which made it confusing. For example, the number of people they are able to serve could depend on the number of cookies that were baked...

    I plan on using the exact same activity with about 6 examples (since I teach Jr. High) and having them be a bit more concrete. One will pertain to weather, one to clothes, one to food, etc. Because the students are trying to decipher if the statement is independent or dependent, not which food group it belongs to. (I know that wasn't your intent)

    Thank you for all of these resources!! Your students are lucky to have you!

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    1. Thank you so much for the feedback!

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  7. On the independent dependent sorting activity I labeled the events that go together with the same letter. Students still had to decide which was the independent variable and which was the dependent variable. It worked wonderfully!

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    1. I love this tweak! Thanks for sharing!

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  8. Greetings I teach math and I am from Puerto Rico and discovered your site and it has helped me a lot, has great ideas and I'm using in my classroom. In my case it was so hard to take some of the issues but was unable to explain their activities better. Thank you very much for your ideas. Excuse my English but I'm not very good at that language, my language is Spanish.

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    1. How awesome! I'm sure your students are enjoying the activities you are creating and modifying for them!

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  9. This is awesome! I a excited to try out some of these ideas! Thank you!

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  10. We have a theatre company that works with different classes in our high school, teaching concepts kinesthetically. One day Ryan did an activity w/ independent/dependent. At one point he had the students pair up & one followed the other around the room. From that came the discussion of ind/dep. There was much more to it but my memory fails me. Another idea I stole from an 8th gr teacher is X walks, y jumps. It helps w/ graphing but also in that children become independent when they learn to walk. I thoroughly enjoy your creativity & wish I was gifted with half of it! Keep up the good work!

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  11. I love this entire post. I enjoy reading your posts and seeing all of your great ideas. I am interested to know how you post the Box with all of the document downloads. Is that an app? I would like to offer materials on my site, but I do not know how to do that. I use Box, so I would love to learn how to do it too. Thanks again!

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    1. Hey Denise! Thanks for reading my blog. It means a lot to me. Box probably has an app, but I've never used it. I used the web version of box.com. After I upload my files to the site, it gives me the option to "share my files." I choose the "embed" option which gives me html code to copy and paste in my blog post to make the files appear. Hope this helps!

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  12. I'm introducing functions, relations, domain, range, etc. to a very intelligent group of sixth graders and have been reading and reading and reading, looking for the best way to do it. THANK YOU for all the materials you have posted and shared! What an awesome resource you have provided! Due to the age of my students, I doubt I'll use anything else from your blog, but wow...I'm speechless about the help this will give me.

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    1. Thanks! I hope your lessons went well!

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  13. Thank you so much for sharing the resources you made - they're a big help!

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  14. So, I didn't read all of the comments people posted, so if this is a repeat I apologize. First, I love your creativity and I've borrowed several of your ideas for my own classroom. Second, I found that to get around students not understanding certain lingo I used a few scenarios from things they liked. For example, The number of the Doctor would be paired with how many regenerations he's went through. (assuming you know about Doctor Who :) ) So the IV would be the regenerations since that's what inflicts the change in the number that he is (DV) i.e. the 12th Doctor has went through 12 regenerations. I've used sports points or record sales as well. Anything that they're into that you can put in this type of scenario. Just a thought. Thanks for your helpful blogs!

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  15. Sarah, I have saved your blog to my favorites bar. I am new to teaching Algebra this year and you have seriously been a life-saver. Thank you for your amazing work! Keep it coming!!!

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    1. Thanks, Laura! Have fun teaching algebra this year!

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  16. I love your blog...absolutely love it. I feel like your ideas for INB are so applicable to my classroom teaching. While I no longer teach algebra, I do teach 8th grade math and I feel like I can pre-teach students for algebra with this year's material using many of your ideas. Thank you!

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    1. Hey Sarah !! this is so nicely organized and easy to teach. I would love to have these functions cutouts and foldable cutouts in email if you can email me on kparikh@theburlingtonschool.org. I can not save and print it in color. thanks a lot

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  17. Thank you so much for all your hard work and then sharing it with us! I plan on using this unit as my pre-teaching/review for my 8th grade Algebra 1 students.

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  18. I love your ideas! I used the independent and dependent cards as an opener for my lesson. I gave each student one of the statements and had them try to find the corresponding statement. Then the pair decided which one depended on the other. This made a great introduction to independent and dependent variables. My students love to move around and collaborate, so this was the perfect way to get them moving and thinking.

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  19. how long does this unite 5 take? i mean how many 45 min period? is there any lesson here that u did not mention, like homework checking, solving problems, test, quiz etc. and i didnt see the page 44(ordered pairs) and 45(part of the coordinate plane), it wasn't posted?
    PS: i can download the pdf files, however .pub spurred files cant.

    thanks two millions

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    1. Here's a direct link to all my files so you can find what's missing: https://app.box.com/s/abl3ycndkyzb7jcujpdf

      I'm not sure how many periods this took. I would say we spent 5 days on notes. Then, add in time for quizzes, etc. But, that's just a guess. It's going to depend on your students, their maturity levels, and their previous math backgrounds.

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  20. Thanks for post. People who are really concerned about their website may take help from the after effects templates so as to make your work more impressive and eye-catching.

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  21. Helpful ideas . I Appreciate the points ! Does someone know if my assistant would be able to get access to a blank TN LB-0441 copy to work with ?

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  22. My friend edited a template FL DoR F-1120 example at this place http://goo.gl/dZo8q3

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  23. Thank you for sharing. Very insightful for someone using interactive notebooks for the first time. Be Bless and Stay Encouraged

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  24. Hi Sarah, you tweeted asking for help with ideas for independent/dependent variables, and I reckon here is the best place to respond because there's more space!

    First, I feel you. My UNIVERSITY students also have trouble with this sort of thing, though I am talking here about statistics rather than algebra. I long ago ceased to be surprised by how the language impedes this sort of activity! But it's still frustrating!

    Anyway, one problem is the very words we're using here. They are two very similar-sounding words and it's easy to mix up which one is called which if the words themselves mean little to you. Like the names of twins, you're always mixing up which name goes with which twin.

    Most of them probably do know what "depends" means and this is good. The problem with many of my students is actually the word "independent". In natural English it means "not related to anything", which is not at all the situation we're in here since we're explicitly relating these things together! Alternatively it might mean "happening without intervention", which often doesn't apply to these situations either, or can be interpreted in the opposite way. For example, if you have some number of cucumbers, the number of pickles happens by itself with no input from you.

    I like to tell them that the word independent has its own special meaning here, and in this context it simply means NOT dependent. That is, the dependent variable DEPENDS on the other variable, and the independent variable is NOT the dependent variable.

    There is a more serious problem in my mind, though, to do with causation and relationship. In some of those stories, the causation could actually be the opposite way than you are suggesting. For example, someone planning a concert might have a fixed duration in mind, which would CAUSE them to choose a certain number of songs. In this context, the number of songs is the dependent variable and the length of the concert is the independent variable. Without knowing the full story, you actually can't decide!

    Even worse, in statistics and science, there's no particular reason that there has to be a causal relationship, but you still might be interested in predicting one variable from the other. Even if there is a causal relationship, the thing doing the causing doesn't have to be the output variable! When a car breaks at high speed, it is clear that the speed the car is going will cause the skid marks on the road to be longer. But if you are a crash scene investigator, you want to calculate the speed from the length of the skid marks.

    So it seems to me that in order to choose the dependent and independent variable, you actually need the full story including the information the person has and the information they want to find out. Without that, I can't be completely sure.

    I'd recommend having a three-way match up: story-dependent variable-independent variable. It will be easy to match the variables to their story, and the story will give more info to help choose the dependent and independent variable.

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