How Does It Grow?
Lesson 7 of 7
Objective: SWBAT generalize patterns into a verbal description and a variable expression.
I give students directions in the classroom before they enter the classroom. They will each receive a card with an expression on it. They will recognize that each card shows one strategy for adding numbers mentally that the class came up with during our number talks in previous lessons (Let's talk addition, what were they thinking, and Delightful decimals ). Once they they enter the room they will find cards taped to each table group, one of which matches the card that they received before entering the room. They don't need to solve the problem. They just need to sit down at the correct group: the group that shows an equivalent expression to the one on their card. This will determine their family for the day.
Once students find their family, I tell the class that they are the experts on these strategies since they came up with them and they can get help from one another and offer help to those who look stuck. I expect that some of my students will be resistant. They will insist they don't understand their strategy. I just keep letting them know that their card shows a mental strategy for solving one of the problems at one of the tables. If a stronger intervention is necessary:
- I may take them to another student to get some help.
- I may recommend that they go look at each of the table problems to see if one of them looks like it would match.
- I may suggest that they ask someone who looks like they have found their group and ask how they did it.
- I may have them take a closer look at what the numbers seem to be doing in the expression card they have and what effect it might have on the solution or how we might approach the problem.
This lesson can be frustrating at the outset. But, I want to leverage this activity to help my students learn that they can work through their frustrations and figure something out if they stick with it and examine it further (MP1).
Whole Class Discussion
Our class discussion today is episodic. We will work back and forth between small group and whole group sharing as we discuss strategies. I will begin by writing a number sentence on the board:
48 +2 + 37 – 2
Then I ask students to find the person(s) in their group who have a card showing a similar strategy. I say, "point to that person or persons." They may not notice that this one, 48 – 3 + 37 + 3, is also similar until after the conversation.
When all groups have made their identifications I ask them to take 3 minutes with their group and decide how they know it is equal to their table problem. I say, "think about how it helps to solve the problem mentally."
As I circulate I record some of the comments I hear my students make as they are explaining/discussing. I am listening for the different ways they have of explaining the property/strategy:
- "since the same number is being added and subtracted, it is like adding nothing”
- “adding and subtracting the same number cancels out”
- “when you take away the same number being added you end up with the original numbers in the problem”
I have even had students cover up the number being added and subtracted to show the original problem. I am also listening for reasons that this property is helpful or useful like “it lets us change a problem into a friendlier problem.” After 3 minutes I ask students to share with the whole group and I write their responses on chart paper.
Then I tell them to point to the person who has a card showing a strategy that looks like this next one:
40 + 20 + 6 + 7
I tell them there is only one person in their group with this one so they don't confuse it with: (40 + 20) + (6 + 7) .
I ask them again to take 3 minutes with their group and decide how they know it is equal to their table problem and how it helps to solve it mentally. When we demonstrate how to do the addition mentally we add up in chunks 40+20 = 60; 60 + 6 = 66; 66 + 7 =? I might stop here and point out that when I add this in my head I break down the 7 even more: 66 + 4 + 3, in order to make the next friendly number.
Then I have them point out the person who has:
(40 + 20) + (6 + 7)
I purposely use the same numbers in this example as I did in the last example so that students reflect on the difference difference between chunking and adding up one number at a time. The question I have them discuss and share here is:
How is this strategy slightly different from the last one. What do we do differently?
I want to hear them say that the "parentheses change the order or operations." I ask students to look at their table problem again and ask them what number is grouped with the 40? the 20? I rewrite the problem to record their response: (40 + 6) + (20 + 7). I ask them to discuss in their groups why it is helpful to break them up and regroup them in a different way.
After our lengthy discussion, I give the class growth pattern assignment to work on in class. Most of my students will finish in class, but some will take it home for homework. I pass out linker cubes for students to model with as they work. I give each group 20 linker cubes that they have to share, so they can show what they are doing and help them explain to each other what they think.
Many of the students will jump straight to the drawing of what Figure 4 would look like and then set up the table and get stuck on extending and generalizing the pattern because they have not taken enough time to describe how the pattern is growing in detail. Some students will try to extend the table to the 10th figure and then get stuck on the 100th. I sit down with them and ask them to describe how the pattern is growing. Many of my students just say it is growing by one each time. So I ask how many cubes are in figure one. They usually ask if they should include the black one and I say it's up to them, but right now we're going to look at how the figure is changing. They will say the black one isn't changing, because there is always exactly one in each figure. I tell them if they want to set that one aside for a moment and focus on the white ones that are growing they can do that. In the end they tell me either that there are 3 or 4 cubes in figure one (depending on whether they are counting the black one for now). So I say that if it is changing by one each time I would expect there to be 4 or 5 in figure 2, but there are 6 white ones and 1 black one. Usually they will now say they meant that it was growing by 3 each time, but I back them up and say, but in a way it is growing by one, how is that. They point out that it is growing by one in each of three directions. At this point I can ask for a certain figure how many white ones are in one direction, in another, and in the top, etc. Then I tell them to talk about that in their group and start working on the other figures.(MP3)
The most common mistake on the last problem which asks for a variable expression for the nth figure is that they just put "n". I ask them to describe how they came up with 301 for the 100th. Usually they will say there will be 100 cubes in each direction plus the one black one, so that's 100 three times plus one. Then I ask how many white ones will be on the right side of the nth figure, they say n, and then I ask how many on the left... That's usually enough for them to see their error. If not, I go through the whole figure and then ask how many times we would have n cubes. Some students get 30 and 300 for the 10th and 100th, so I ask them to explain it for me just to see if they excluded the black one on purpose or not. If not, they usually catch their mistake. If they did we discuss why they thought it might not be included (usually because that part of the pattern isn't growing), but then I say the question is about how many cubes, so if it's a cube it should count.