My Algebra 1 students recently started what is probably my most favorite unit of the year: introduction to relations and functions. Our first skill is to be able to generate equivalent representations of a relation and determine if the relation is a function. Today's post focuses on the first part of that skill: generating equivalent representations of a relation.
In the past, I have made a 4-door foldable for this. It worked okay. Lots of writing which meant some students' notes were nearly illegible. My classes are bigger this year than they have ever been before, so I decided to pre-type as much of the foldable as possible.
Here's what the outside of the foldable looked like circa 2013:
You can see more of my notes from 2013 here.
Here's my new, pre-typed version.
We began our discussion by creating a set of ordered pairs. I had six students each volunteer an ordered pair. There was one restriction. The ordered pairs had to fit on the coordinate plane below it.
After writing our six ordered pairs, we moved to the input/output section of the table. We discussed that they would need to be able to organize ordered pairs into a horizontal table or a vertical table. I asked them which variable was used to represent input and which variable was used to represent output. They easily supplied x and y for these blanks. Next, we discussed where x and y belong on a vertical table vs. a horizontal table.
In the past, my students have sometimes struggled with calculating slope from a horizontal table because they aren't sure which values are x and which values are y. This is my motivating factor for including both horizontal and vertical tables here.
We took our ordered pairs from the previous section and turned them into both types of tables.
After making our tables, we graphed our six points on the coordinate plane.
In the past, I've spent some time reviewing graphing on the coordinate plane. This year, I'm going to assume that my students have mastered that concept in middle school. We'll see how that goes...
Finally, I introduced my students to my most favorite representation of a relation--mapping diagrams. I didn't tell my students why they are my favorite, but I know they'll be able to tell why soon when we start classifying relations as functions or non-functions.
We took our six ordered pairs and turned them into a mapping diagram.
First, I had them write their x values in the input oval without repeats. Then, I had them write their y values in the output oval without repeats. Some students began to be very concerned about this. How in the world do we know which input matches up with which output? I am so very glad you asked. That's why we need arrows to connect each input value to its corresponding output value.
And, our notes are done! Going through these four representations took about twenty minutes of our fifty minute class period. We spent the next twenty or twenty-five minutes working on a representations of relations telephone game.
I was inspired to make this telephone activity by a chemistry tweet.
Have you ever played the game "Telephone?" One person whispers a message to another person. That person whispers the message to yet another person. This continues until the message has been relayed through an entire line of people. Then, the person at the end shares the message they heard. This is compared the starting message, and usually quite a few laughs are had.
This works just like that, but we're dealing with math.
Give each student a sheet of paper, and ask them to accordion fold it on the lines between each section. This will save a lot of time later and throughout the activity!
Notice that the sheet of paper both begins and ends with a set of ordered pairs. If all things work as planned, the set of ordered pairs at the top and bottom SHOULD match. Of course, we know things don't always work exactly as we intend them to in our classroom.
All students were instructed to write their own set of six ordered pairs in the BOTTOM section of the page. Just like with our foldable, they needed to make sense that their ordered pairs would fit on the coordinate plane found on the page.
After writing their six ordered pairs, students traded papers with another student. They took the six ordered pairs written at the bottom and turned them into a table. Before passing the paper on to the next student, they had to fold up the paper so that the original six ordered pairs were hidden.
The next student to get the paper had to take the horizontal table and turn it into a coordinate plane. Before passing the paper on to the next student, he/she had to fold up the table to hide it from view.
The next student to get the paper will take the coordinate plane and turn it into a mapping diagram. Before passing it to the next person, he/she will fold up the paper to hide the mapping diagram from view.
The last person uses the mapping diagram to create a set of ordered pairs. After these ordered pairs are written, the paper is returned to the original owner. (We wrote our names next to our original set of ordered pairs to make returning the papers to their owners much easier!)
When students got their papers back, the first thing they had to do was check and see if the set of ordered pairs at the top of the page matched the set of ordered pairs at the bottom of the page. Many were surprised to see that the order didn't match, so we had to talk about why that would be the case.
For some students, their ordered pairs matched perfectly. For others, they didn't. This means they had to look through the different representations and figure out where things went wrong. So many awesome conversations came out of this. And, I hope this activity drove home the fact that precision is super important when translating from one representation to another! As my students found out, one tiny mistake can change the entire problem.
I loved the telephone structure of this activity, and I look forward to finding other ways to use this practice structure in class. It fits in tons of practice and error analysis without seeming like work!
Files for the foldable and activity can be found here.