(This is part of a series on teaching functions in a high school equivalency class. Other posts can be found here.)
My outline for this class:
- Review quiz
- Warm-up: Helping students to belong
- Mary’s Function
- The Border Problem
- Reflection
For homefun, I gave out:
In our fourth class, I hoped to teach students to build different equations based on the relationship of two variables. The central activity in the class was based on ways of seeing a border of squares around a square grid.
Review quiz
I started class with a review of last week’s quiz. On Mark’s suggestion, I created an answer key by pulling correct answers from different students in the class. I asked students to mark their quizzes and talk to each other questions they got wrong.
Warm-up
For today’s warm-up, I used a writing activity from Behavioral Insights for Education called Helping Students to Belong:
We’re going to start with some writing today. Please put your name on the top of a piece of paper I can collect. I want to read something to you and then ask for your reaction.
Many students struggle to fit in when they first arrive at a new school. Some worry that they are somehow different to others; some are concerned that teachers and students will judge them. Still others feel anxious that they won’t be able to keep up in class. With time however, most students become more comfortable with their new surroundings – they make new friends, they realize that differences are what make us unique and they gain confidence in the material they are learning in class.
Step 1: Take a few moments now to write about why some students feel like they don’t belong when they first arrive in school. Reflect on your own experience. Did you (ever) feel worried that you wouldn’t fit in, or that you wouldn’t be able to keep up in class?
Step 2: Now, write about how your experience at school has changed. Do you feel more ‘at home’ now than you did at the beginning of the year? Why is that? Did you make friends, did you find out where to get help from if you need it?
When students finished writing, I asked for volunteers to share. Some students talked about their fear of not being able to keep up after many years of being out school. They shared appreciation for how they were welcomed by teachers and other students. And there was an agreement that it is important to get over a fear of being in school.
Mary’s Function from WhatsApp
A few months ago, a friend who teaches at Borough of Manhattan Community College told me about how she has been using WhatsApp with one of her classes that only meets one day a week. She created a group within WhatsApp and then invited all of her students to join. The group allows students to help each other with homework. The teacher also sends video to the list that would be helpful to students. I have been interested to try it out, so I set up a group for my class when we started the semester. I have been encouraging students to post challenges to each other on the list. Mary posted the following function table in between classes, so I asked her to share it with the class.
Based on The Function Game, I put the table on the board and Mary gave outputs to suggested inputs from the class. Mary’s function was challenging for the class, generally, though most of the class was eventually able to give correct outputs for additional inputs.
Students noticed something interesting:
- The first few OUT numbers go up by 4, but that stopped at 25. However, if you move the 8 and 21 up, the pattern would continue. This noticing will be useful when we get to rate of change, though the inputs are not consecutive.
Once students knew the rule for Mary’s function, I used it to reinforce the idea of a function, since I had missed the opportunity to give a definition of a function in previous class.
- What is the rule?
- Times 2, plus 5.
- Plus itself, plus 5.
- Does is it work for every input/output pair? Yes.
- 0 x 2 + 5 = 5
- 2 x 2 + 5 = 9
- 4 x 2 + 5 = 13
- 6 x 2 + 5 = 17
- What if I put a 6 in again? Could something other than 17 come out? Why not?
- Write the following in your notes: When we work with functions, each input will give one output. If it’s a function, it will never have two different outputs for the same input. (This is one concept we can guarantee will be on the HSE exam since it has shown up on every readiness test.)
- Equations
- 0 x 2 + 5 = 5 (This is an equation, because it has an equal sign.)
- When you look at all of these equations, what’s the same? What’s staying the same? Times 2 and plus 5.
- What changes in the different equations?
- 0, 2, 4, 6 ← The number that is going in.
- 5, 9, 13, 17 ← The number that is coming out.
- An in/out table like Mary’s is one way to show a function. Another way is an equation. Here’s an example: IN x 2 + 5 = OUT (This equation is a way of writing Mary’s function.) The equation means that for whatever number that goes in, you multiply by 2, then add 5 to get the number that comes out.
The Border Problem
As an introduction, I reminded the class of our discussion of algebra as the generalization of arithmetic. Today, we’re going to practice using algebra to generalize our solutions. What does generalize mean again? To make general, true for different examples.
10×10
I had the 10×10 grid of the Border Problem prepared on chart paper. I told students I was going to have them look at a pattern and ask them questions, but I was only going to show it for a few seconds to start. Are you ready? Are you sure? I revealed the grid for about 3 seconds then taped it up again.
Students saw:
- a square
- 4 sides were shaded
- 100 chart
- times table
- box with yellow
- 100 squares
- 10 on each side are yellow
- 100 squares, 40 shaded
- a large square with a bunch of little squares
- right angles
- length and width
- 10 x 10 = 100
I then revealed the image again and asked the group:
Figure out without counting, without talking, without writing – How many squares are in the border (the yellow row all around the edge). Give me a thumbs up when you’re ready.
What I’m interested in is how many squares you found around the border, and more importantly, how you figured that out. I want to see how many different ways we can find for figuring out the number of squares in the border.
As you’re reading, you might take a minute to see how you count the number of squares in the border.
Pair/share – Talk to each other about what you got.
Then we came back together for discussion. At this point, I intended to ask students to explain how someone might think there are 40 squares on the border, but I forgot to ask. It was my first time teaching the Border Problem and, honestly, I was nervous that I would mess it up.
Let’s see some different methods for getting 36. Thumbs up if you’re ready. Listen for an explanation. Come up and explain if necessary.
Jermaine volunteered first. I asked him to come to the board to show us how he counted the squares in the border. He counted 10 squares across the top and the bottom. He then counted 8 on each side, not including the corners, which were already counted.
After Jermaine explained his method, I asked students to raise their hand if they understood what Jermaine explained. When the volunteer had trouble explaining, I asked Jermaine to explain his method again. Then I wrote down the following calculation on chart paper and asked, Does this calculation represent what you did? When he agreed, I wrote Jermaine’s name below.
10
10
8
+8
—
36
Jermaine
After writing the calculation, I asked, How many understand Jermaine’s method? How many people used that method?
Does anyone have another method? Give me a thumbs up. I then went through the same process with five more students.
I was looking for six possible ways of calculating the number of squares in the border. Students found all six! Take a look at the list below and see if you can show how each of them of them is connected to a different way of seeing the border in the 10×10 square grid.
I then asked students to translate these calculations into equations like we did with Mary’s function earlier. We talked through 10 + 10 + 8 + 8 = 36 as an example, then gave students time to do the rest in their group. I then invited students to write the other equations under the different ways of seeing the border.
- 10 x 4 – 4 = 36
- 10 + 9 + 9 + 8 = 36
- 9 x 4 = 36
- 100 – 64 = 36
- 8 x 4 + 4 = 36
Later, Celeste and Mark pointed out to me that it was really hard to read these notes from the back of the room. I have struggled with how to use a chalkboard and chart paper in this classroom. It’s embarrassing to see how messy my chart paper is. I definitely need to plan my notes out better so that students can follow and refer back to what we have done earlier.
Celeste told me about another difficulty students had when they wrote equations. Students struggled to turn a calculation like this into an equation:
Many students wrote this:
8 x 4 = 32 + 4 = 36
instead of
8 x 4 + 4 = 36
Clearly 8 x 4 is not equal to 32 + 4 or 36, but it’s understandable that students would make this mistake. They have been taught that the equal sign means the answer, so we get the answer to 8 x 4 then we add 4 to get 36. We will need to return to this misconception later.
6X6
I then asked the students, without the benefit of a picture, to visualize a 6 x 6 square grid.
What if I said I don’t care how many squares there are, but I want to know how you would figure it out? Give me a thumbs up when you have a way of figuring it out. You can use whichever method you like.
While students talked to each other about how they counted the squares, I posted a 6 x 6 grid on chart paper.
I then asked the class, How would could you use Jermaine’s method to figure out how many squares there are in the border of the 6×6 grid? Another student determined that Jermaine would count the squares on the top and the bottom (6 each) and add the sides (4 each): 6 + 6 + 4 + 4 = 20 squares in the border.
As a group we went through each method to count the number of squares in the border. In hindsight, this seems like a pretty tedious way of going through the methods. When I teach this lesson again, I think I will have each group choose the method they like and then have them explain their equation for the 6×6 grid.
Finally, I asked students to think about how they would figure out the number of squares in the border of any square grid. I then asked them to write an explanation for finding the number of squares in any square grid, using Jermaine’s method. I later invited Mauricio and Ismael to write their explanations on chart paper.
Interestingly, Jermaine objected to the Ismael’s description, saying that he added the two numbers and never multiplied.
For an impromptu homefun assignment, I asked the class to write written explanations for the other five methods. Almost immediately, I realized that it wasn’t a good assignment, since most students wouldn’t be able to do it on their own. In the end, students must have realized it wasn’t a good assignment either, because a week later only two students turned it in. Here’s an example, which doesn’t include generalizations, but includes clear explanations for how each of the 10×10 methods work.
In the spirit of reflecting on strengths and challenges of this class, I’ll start with some of the successes. In this class more than any other, many students were up at the board, either explaining their way of seeing, writing an explanation of someone else’s method or doing calculations. The activity was also really useful for helping students listen to each other carefully. They had to understand each other’s methods in order to apply them with a different sized grid. A number of students also reflected on how this lesson taught them that there are many ways to solve one problem. That is a valuable lesson, especially for students who aren’t confident in their own problem-solving skills.
After the class, however, I didn’t feel that it had been as successful as I’d wanted it to be. There was a lot of engagement, but I also felt that I could have guided students more easily towards generalizing their methods. I had hoped to get to algebraic equations by the end of class, underestimating how difficult writing generalizations would be. I don’t think I gave students enough support in writing these explanations. In the 6×6 part of the lesson, I kept too much control by going through each method one by one. Some of the advanced students were certainly bored, though it was definitely useful for some more beginning students. It felt like a long stretch of teacher-focused time that would have been more productive if I had planned a student activity to explore the 6×6 grid.
Or maybe it would have been more productive to use just one student’s method and generalize it. For example, we could have used Jermaine’s method to have a conversation similar to this:
How did Jermaine calculate the number of squares in the border of the 10×10 grid?
10 + 10 + 8 + 8 = 36
How would he calculate the number of squares in the border of the 9×9 grid?
9 + 9 + 7 + 7 = 32
An 8×8 grid?
8 + 8 + 6 + 6 = 28
Etc.
7 + 7 + 5 + 5 = 24
6 + 6 + 4 + 4 = 20
5 + 5 + 3 + 3 = 16
4 + 4 + 2 + 2 = 12
3 + 3 + 1 + 1 = 8
What’s changing and what’s staying the same?
What if the grid was 100 squares long on one side? 100 + 100 + 98 + 98.
What if the grid was s squares long on one side?
In talking to Celeste and Mark after class, I felt a little better realizing that just because you can teach almost anything in basic functions through this problem (use of variables, creating equations, rate of change, starting amount, combining like terms, simplifying equations, use of parentheses, etc.), it doesn’t mean you have to cover it all. There will be other classes and opportunities for students to generalize and learn to write symbolic equations.
