Teaching the Framework 2


(This is my second blog post on my experience teaching functions in a high school equivalency class. My other post can be found here.)

Here’s an outline for my second class:

 For homefun (explanation below), I gave out:

In my second class this semester, my plan was to introduce functions with Maxine’s Rules for Love and related activities from the CUNY HSE Math Curriculum Framework, while looking for possible connections to our work on the Upside Down T visual pattern last week. Since it was early in the semester, I wanted to start with an icebreaker activity that would get people using each other’s names. In the last class, we had a race saying our names in order as quickly as we could. This week, I asked students to imagine that the value of their name could be calculated if a=$.01, b=$.02, c=$.03, etc. As they came in the room, I asked them to write their name on the board. Later we predicted who had the most valuable name and then got to work calculating to see if we were right.

The two Elizabeths in the class shared the prize with names worth $.98!

We then moved into the Algebra and Patterns brainstorm activity. I started by asking students, on their own, write down the first words they think of when they hear the word algebra. I then asked them to talk with a partner and finally, we shared out as a large group. To take notes, I used Mark’s phrase, You’re the hand and I’m the brain, and wrote down everything they said, organizing the responses in two columns, the first of math words and the second of non-math words.

Notice anything about their responses?

A few things I noticed:

  • There are a lot of negative feelings and experiences related to math in this class.
  • Many students don’t seem to know what algebra is. After a while, I was wondering if they were just telling me all the math words they knew. Though, of course, all of these topics could relate to algebra…
  • The algebra conversation prompted a question from Mauricio. It took a while for him to get around to his point and he was interrupted, but I think his question was something like, You know how we learn all this math in school, but then we don’t actually use it in life? I had a teacher who told me that it’s like lifting weights. Learning this kind of math can help your brain. (Mental note to share You Can Grow Your Intelligence.) This prompted a discussion from other students: Yeah, I never use math. We never use math, at least not in the jobs I’ve had. Which then prompted other students…
    • You know, I once really liked math. Simone seemed to come to this realization in the moment. It seemed like a bittersweet realization that something happened (maybe not her fault?) and maybe there is a chance that it could change in the future.
    • What makes us run away from math? This was also unprompted. Liz seemed to be asking if it is something in math or something in us? I would say neither, but really something in our education system.

Then we went through the same activity with the word patterns: Think individually, pair/share, share as a group. After students shared a number of different ideas related to patterns, I asked what all of these words have in common? What is a pattern?

Eventually, students talked about making observations (seeing what is happening in the world, being observant of our surroundings), collecting data (thinking about weather patterns and what meteorologists do) and making predictions about based on events in the past (we were talking about people who exhibit a pattern of a certain behavior). The idea I was trying to draw out is that we will be using patterns in this class to understand mathematical relationships, using practices that relate to science and our life in the world in general.

Then we jumped into the launch for Maxine’s Rules for Love. I asked students to discuss: Does age matter when people are dating? Some said yes, some said no. As long as both parties are of age. One student said she wouldn’t date anyone older than her. Another student said it was only a matter of maturity and that isn’t related to age.

I then told them about “my friend,” Maxine, who has a rule for the youngest people she will date:

half of her age, plus 7

Students worked through a couple questions:

Following Maxine’s rule, would a couple made up of a 44 year old and a 27 year old work? Explain your answer.
According to Maxine’s rule, what is the age of the youngest person you should date?

Students then considered Maxine’s rule for the oldest people she will date:

her age minus 7, times 2

Then I asked students to add their information to two tables on chart paper. I’m always a little nervous to ask people’s age, but no one seemed to have a problem. I told them that if they didn’t want to use their real age, they could use the age they feel in their hearts. A few of the older students said they didn’t care if people knew how old they are.

Unfortunately, I didn’t take a photograph until after we had talked through a number of the calculations and I had written notes on the chart.

What do you notice?

A few things I noticed:

  • Many of my students didn’t want to calculate half a year, so they rounded down. Shalisa and Andrew used half years.
  • There were a number of computation mistakes. Taking half of a number was challenging for a number of students, especially if their age is an odd number.
  • There’s a mistake that I didn’t catch. Do you see it?

We didn’t spend as much time on the oldest person chart, and I didn’t go through all the calculations with them.

Some things I noticed here:

  • Another mistake I missed. How did Carthian come up with 30 as the youngest or oldest person she could date? Carthian actually came up a second time and entered 50. I’m not sure where that number came from. Did she subtract 7 twice and then double?
  • Patrick just added 7 to his age instead of subtracting 7 and doubling.
  • Jermaine and Mauricio have different outputs. I can’t believe I missed that. This is the perfect example I needed to ask students if this situation is possible. If two people are 29 years old and we apply the same rule (subtract 7, multiply by 2), is it possible that one person can date an older person than the other? The realization that both outputs have to be the same is connected to a formal definition of a function: a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output. I missed a good opportunity there. Maybe I should clip that section of the chart and show it to the class later.

Looking back at these charts is a little humbling because I see that I probably should have spent more time making sure everyone understood the two rules before putting their data up. It’s okay that there were some computation mistakes, but it’s important that everyone understands the rules they’re applying. I felt some pressure to move on because I wanted to get to the next activity. There is also quite a range in math skills and knowledge in the class, so some students were finished fairly quickly while others were still calculating the age of the youngest person they could date.  When I teach it again, I might have students add data to the youngest person chart first and process that chart before moving to the oldest person chart. This would give all students some support so that they are more successful with the second chart. I also need to give myself a moment to double-check each calculation on the chart before discussion, so that I know which examples to bring to students’ attention. I don’t think it’s necessary that I personally correct each one, but someone should. Ideally, I would have been able quietly ask a student to double-check their work before we went over the chart together.

Some goals from Maxine’s Rules for Love:

  • Use a rule, inputs and outputs
  • Build a function table
  • Understand that with functions, each input can have only one output

The last activity of the night was My Teacher is a Computer. I told the class that I was going to pretend to be a computer. I had a rule in my head. I would ask them to feed me numbers (inputs). I was going to do something to each number and then spit out another number (output). Their job was to guess what I was doing to the inputs to produce the outputs. If they thought they knew the rule, they should give me a discrete thumbs up. I would then ask them to prove it by giving the correct output for a new input.

Before we started, I asked the group, So as soon as you know the rule… you should shout it out, right?

No! They said back.  We should give you a thumbs up.

I told the class I was ready. Someone give me an input, please.

Someone yelled, 4.

I put the 4 in my right hand, pretended to stick in my right ear and then washed my hair with my hands as I thought. I pulled the number out of my left ear… If a 4 goes in,  a 13 comes out. Then I recorded the input and the output in a table.

Add 9, said Maureen. The class groaned.

I laughed and said, If you know the rule, should you yell it out? No!!

Let’s try again. Let me reprogram the computer. Okay, I have a new rule. Someone, please give me an input.

3.

If a 3 goes in, a 10 comes out.

Add 7! Someone else said.

The class groaned. DON’T SAY THE RULE! a few students said.

Third time’s a charm. I came up with a new rule, asked for inputs and recorded outputs as we went.

When students gave me a thumbs up, I asked if they could supply the next output based on an input. When most of the class had their thumbs up, I asked the class to reveal the rule: +3.

We played a couple more times with one-step rules using addition, subtraction and multiplication.

After class, I took some notes on what went well and what could have been better:

We made it through some review of sequences and the first visual pattern, then a discussion of algebra and patterns, with the two key points: Studying patterns lets us make observations, collect data and make predictions & algebra is the generalization of arithmetic. We will need to come back to both of these ideas.

I had hoped to get to function machines tonight, but ran out of time. Something went at least 30 minutes longer than I thought it would. We didn’t get to a writing reflection at the end or any sort of closing activity.

I’m concerned that there are some stronger voices that may dominate the class. There was also a lot of talking over each other. I’ll need to work on the discourse. I’d like to make a list of student names for the next class and check people off as they speak, so that I can keep track of whose voices I should try to encourage and who I need to ask to step back a bit.

Would a quiz be useful at the beginning of class? Yes. But designed to help them learn, not so much to assess.


One thought on “Teaching the Framework 2

  1. Hi Eric, I used both “The Value of Your Name” and “Maxine’s Rules For Love” as warm-ups to working on functions in my intermediate HSE class. Both exercises are energizing and fun, and eased the class into working on in-out tables. Your lesson plan is great and detailed – the class obviously got into it and got a lot out of it. What I take away from seeing your lesson play is “slow down” and, “let them struggle” “Let THEM work it out, and don’t be too helpful”.

    PS, I’ve liked other posts – “problems” in Math Memos. My approach is “jump into a problem and try to swim” – so it’s great to have access to some doozies. Thank you and keep ’em coming!

    Cheers,

    Richard

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