(This is my third blog post on teaching functions in a high school equivalency class. Other posts can be found here.)
My outline for this class:
- Quiz – Intro to Visual Patterns and Functions
- 5 Things in Common
- Unit 1, Activity 2: My Teacher is a Computer (review)
- Unit 1, Activity 3: One Rule to Bind Them All
- Unit 1, Activity 4: Function Machines
- Exit slip: What do you know about functions? What is one question you have about functions?
In our third class of the semester, my goals were to complete Unit 1 of the CUNY HSE Math Curriculum Framework so that students would have a basic understanding of what a function is, along with the formal definition of a function, how they can be expressed in rules and tables, and how a function rule must work for each input/output pair in a function table. I wanted to do all this while not overwhelming my more beginning students and while keeping more advanced students engaged and challenged.
I started class with a quiz. After 20 minutes, I allowed students to talk to each other for about 10 minutes before I collected their work. The goals of the quiz were for students to see some of the work we had done again, so they would know what they had learned or still need to review, and I also wanted to get a sense of what students were able to do. I also hoped using some low-stake quizzes through the semester would help students with test anxiety (though I’m realizing now that this will only happen if I allow time for students to reflect on taking tests and share strategies). I let students talk to each other at the end because I wanted them to have some quick feedback and I wasn’t planning to go over the quiz immediately. My guess was that the discussion wouldn’t influence how students did on the quiz overall, though there might be some quick writing in of answers, which I was willing to risk.
Looking over the quiz later, I found that the results were mixed.
Most students were able to predict the number of squares in the 10th figure of the Upside Down T Pattern by drawing or seeing a pattern in the growth.
Many students were able to use Maxine’s rule to calculate the youngest age someone should date, but very few students evaluated the results correctly.
Others evaluated the ages, but didn’t use Maxine’s rule.
Some evaluated the output correctly with some minor mistakes.
A couple students showed a full understanding.
And most of the class was able to find rules for the function tables and complete a missing output:
After the quiz, we did an icebreaker, which I’ve been calling a class challenge. I dealt out playing cards to create groups of four (all the Kings get together, all the Jacks, all the Aces, etc.). In groups, students were challenged to find 5 things they all had in common with each other, with the extra challenge of finding commonalities that were unique to their group. I like using this activity towards the beginning of a semester since it gets students talking quickly and can break down some of the cliques that can form in a class. Celeste, who was starting as a volunteer, joined a group of students. After about 10 minutes, each group shared a few commonalities they found: Everyone here speaks a second language, we all have kids, we all like to travel…
Then we played a couple quick rounds of My Teacher is a Computer, as practice for determining the rule for a group of inputs in a function table.
At this point, I had intended to make a point about the definition of a function: If a number goes into a function more than once, the same number must come out each time. Each input can only have one output. My intention was to make this clear in the last class with Maxine’s rule (I missed the opportunity to compare Mauricio’s and Jermaine’s outputs for the oldest age chart). Somehow, I forgot again and didn’t use these function tables to make that point. After the activity, I should have asked something like, Is it possible for me to put a 4 into the first function a second time and get something other than 9? Could I get 6, for example? Why not? I need to remember to nail down this definition in the next class.
Then we moved on to main activity: One Rule to Bind Them All. I told the class that this activity would give them an opportunity to learn something new about tables and rules. In the five groups we formed earlier, I gave each group one input/output pair. I asked each group to keep their input/output pair secret so that they only saw their own. Their task was to come up with multiple rules that might produce their output from their input. The goal was to write on chart paper as many rules as possible that work for their input/output pair.
While students worked, Celeste and I walked around to help make sure the rules they generated worked for the input/output pair and to guide them towards generating new rules. I was looking to make sure each group came up with the rule, multiply by 2 then add 2. In order for the next part of the activity to work, I needed each group to have that rule.
After about 20 minutes, all the groups had filled chart paper with multiple rules. I put all five charts in a row on the board and asked students what they noticed:
- There are lots of different rules.
- We all used addition, subtraction, multiplication and division.
- We wrote the rules in different ways. Some people just wrote the steps. Other people used In and Out.
- There’s one rule that is on all the pages. Really? What do you mean? Multiply by 2 and add 2. Yes? It’s on each pages. Could you show us what you mean? …
Latiffah noticed that x2, +2 was used by each group. I pressed her to explain so that other students could see the equivalence of these different ways of writing the rule:
- Multiply by 2, Add 2
- 1 + 1 + 2 = 4
- Multiply by 2, Plus 2
- In(2) + 2 = Out
- Multiply by 2 +2
I then put each of the original five input/output pairs into a table for students to consider.
Looking at the first pair (1, 4), I asked whether times 4 could work for this table. It seems fine to me. 1 times 4 is 4. It works. I waited for students to say, No, that doesn’t work for the next one. Oh… then how about plus 5? 3 plus 5 equals 8. No, that doesn’t work for the first one. It’s times 2, plus 2. What do you mean? Etc. I was trying to get a few students to clearly state that the rule for a function table has to work for every input/output pair in the table.
My final question for them to consider: How many input/output pairs would you need to come up with a rule for a function? Is one pair enough? Many voices said that you needed more than one, so I felt comfortable that they understood the main point of this activity. A few people said that two was enough. I said that we would come back to this question later in the semester to see if that was true. I’ll need to remember this when we get to Unit 5: Nonlinear Functions.
We then moved on to Function Machines. I introduced the activity by saying, by this point in the class, we have practiced a few different kinds of function skills. For Maxine’s Rule, you were given a rule and put numbers in to find the numbers that come out. For My Teacher is a Computer, you were given inputs and outputs and had to come up with the rule. With function machines, you are going to practice both of those skills.
I drew an image of a function machine with an input/output table to the right:
I explained that a function machine can be a helpful way to think about functions. Something goes in, a rule is applied, and something comes out. I spoke about a paper factory. What goes in? Wood. What comes out? Paper. What happens in the factory? We’re not really exactly sure, a bunch of processes, but it turns wood into paper…
We then looked at the table on the right. A 3 goes into the function, something happens, and a 6 comes out. A 5 goes in, something happens and a 10 comes out. Students came up with a number of versions of the rule: times 2, plus itself, double. Students then worked on the function machine worksheets to fill in missing rules, inputs and outputs.
In the last few minutes of class, I asked students to write about functions. I’m trying to incorporate a bit of writing into each of our math classes, for assessment, to help students remember what they have learned, to practice writing skills. Here are some excerpts from their reflections:
What do you know about functions?
- Function to me means “it works,” a rule put in place, a pattern that makes sense when the rule is applied.
- All functions have rules. These rules is what gives us our functions. Also the rules has to work for every input/output pair.
- Functions have a bunch of tricky patterns.
- Functions is a combination of processes (operations) that need an input to interact and get the output.
- Functions are not my thing because I don’t know much about them, but I’m willing to learn. Some people don’t know how to divide. Teach some of that please.
- I learned functions can be written in many different ways and everyone has different methods they use.
- I don’t really know how to explain or categorize functions.
- I know that functions are letters that replace numbers. Instead of writing the number, you can put a letter in its place.
What is one question you have about functions?
- I need to know more about how to get input from output (if the input is unknown).
- What is the easiest way to find an output?
- What is a quick or go to rule when it comes to looking into in/out?
- Is there any method we haven’t learned about how to figure out the function? Can we use variables too?
- Can functions get harder than what we are learning now?
- Can a function have more than 2 steps?
- Are functions used in all kinds of math?
- I would like to know what kind of functions will be exactly on the test.
- I would like to learn more about how functions work with decimals.
- Is it possible to teach me more about functions, fractions, and algebra?
- How can you make it more fun and much easier to understand, please?
As I write this, it’s been a few weeks since this class happened. I’m wondering how many of these questions we have answered since then and what I should do to respond. Would it be useful to bring these questions back to the class for general discussion? We will certainly answer some of these in the course of the semester. Others we probably won’t. For example, I don’t plan to get into a lot of work with fractions and algebra. It’s just too big of a quagmire for most students. I think we would get stuck there for the rest of the cycle, and lose the opportunity to go farther in functions. Making choices of where to focus our attention is really difficult, when we only have 13 classes in the semester.
