Erica Burnison C. A. JACOBS INTERMEDIATE, DIXON, CA
7th Grade Math : Unit #2 - Writing expressions : Lesson #1

Checkerboard Squares Day 1 of 6

Objective: SWBAT use the formula for the area of a square to find the number of single squares on a checkerboard and to look for patterns of regularity to find additional possible squares.
Standards: 6.EE.A.2c MP1 MP4 MP7 MP8
Subject(s): Math
60 minutes
1 Warmup - 10 minutes

Projected on the screen are several WRONG answers to their homework from last night. Students are asked to check for any of these and also check for agreement with others in their math family group. If they find or suspect an error they need to use their "conversation moves" sentence starters that are taped to their desks (created in earlier lesson in this unit called "conversation moves")  to help them find and correct the error(s). This is a process first introduced to students in an earlier lesson in this unit ("out of order").

The wrong answers are ones that I expect from common mistakes that they would benefit from catching and correcting themselves or with the help of a peer. My goal is for them to learn from their mistakes, which has a better chance of happening if they notice and correct the mistakes themselves. I give them the following wrong answers to look for:

1.  900, 81  2.  14, -14  3.  2, 25  4.  49  5.  7.2, 7.5  6. 35, 81, 1947  7.  41, 55.01, 39.2 

8.  36, -35  9.  225, 72, 20  10. 3

As they are working together to correct their mistakes I am checking their planners as well as problems 5 and 6. I am checking these to make sure they have not mistakenly used the left to right rule with addition and division in (246 + 36/9) and that they haven't forgotten that 2(11) means to multiply in [35 - 2(11) + 48]. I am also listening for some of the statements and questions from their "conversation moves" that are taped to their desks. I am specifically listening for: "how did you get that?", "what did you get?", and any of the statements for returning to the task. When I hear these I draw the classes attention to it by saying "I really like what I'm hearing at this group" or "I just heard 'what did you get?' at this math family".

After we go over the right answers to the homework I give them another 2 minutes to give and get help to correct an answer. They can ask me too, especially if all members got it wrong.

2 Exploration - 30 minutes

This is the first day of what will probably be a week long lesson in which students will be expected to determine the number of squares on an 8 by 8 checkerboard, a 6 by 6, a 4 by 4, and a square checkerboard of any dimension. They will need to determine what different sizes of squares are possible and record in a table the number of each size on the checkerboard and then find the sum. They will find patterns in the sums of the different sized checkerboards and generalize a rule in a variable expression to determine the number of squares on any sized checkerboard. For most of them this is the first time they have been asked a question like this so I wanted one with a pretty simple entry point and one in which there were easy points of scaffolding.

I pass out one checkerboard per group (I picked them up at thrift stores for $0.49 - 4 each). It is a must to give them the real thing! I tell them I want to know how many squares are on the checkerboard. They immediately start counting. Everybody is in! None of them see the complexity and they all have confidence they can solve this problem so they engage immediately!

 I circulate to see if they are counting all of the squares or if they are using the area formula. Several students will shoot their hands up really fast and say they have an answer. I respond that this is a really tricky problem and they need to make sure everyone in the group understands how to proceed. I check in with each group to show me what they think and what they tried. Then I have them share with the group. I project the checkerboard onto the white board so we can draw on it as they share what they did to find that there are 64 squares on the board. You can also use slides 1-3 in the powerpoint to count with them and reinforce the area formula after a student has explained it.

Hopefully, a student will come up with the idea that there are different sized squares. If they do I will ask these students what they mean by that and have them point out other squares on the board. If they don't, I will outline a 2 by 2 square on the board. They usually respond with surprise and dive right back in! 

Tech Break Down
Flexibility

If the tech breaks down, as mine did, your best bet is a wet erase marker on a white board. I drew an 8 by 8 checkerboard on the white board with a vis-a-vis pen. Then we can use dry erase pens to highlight different sizes and counting strategies on the bo

Meaning vs. memory
Developing a Conceptual Understanding

Most of my students do not really understand what area means or represents. If they have learned to memorize what procedure to carry out for area and perimeter problems I have found that their learning is very fragile. Those who don't understand the concept or definition of what area is will not understand why the formula works and, subsequently remember it or apply it incorrectly. Students often come with the misconception that area IS multiplication.

This lesson helps provide a conceptual rationale for how the area formula connects to addition, because we are adding up multiple rows of the same number. The checkerboard makes it really easy to see, so I can point out that we have 8 squares in the bottom row and another 8 stacked on top of them, etc. in order to see that we have 8 squares eight times. 

Although this lesson seems like a really simple one, it provides so many opportunities for extension. But I also really like it because it helps provide intervention for so many of my students that didn't understand where the area formula comes from. I think it is important to meet students at their level of understanding. My students know there is a formula for calculating area, but they don't know what the numbers represent or why the formula works.

When I asked how many of the smallest squares on the checkerboard most of them added the 8 rows of 8 and didn't think of multiplying. This was fine, because it helped them connect meaning to the formula and helped them understand why multiplying works.

3 Closer - 5 minutes

I like to close by summarizing what we have done so far and clarifying the next task. I like to go through the first 5 slides of the powerpoint again to summarize. It really reinforces the area formula, which many of my students don't fully understand. In addition, some kids need to see the overlapping squares again, because this is going to be hard for them to keep track of. For the patterns to really emerge in this long term lesson students need to be able to count these squares.

I tell them that their homework is to try to find a good way to count those larger squares. I also tell them if they make a mess of their homework paper (many of them have already taken out a fistful of color markers) that I will have a new one for them tomorrow. I remind them to get their folks at home to help.

Student Confidence
Safety

I launched this series of lessons because most of my students are far below grade level. Not only are their skills low, but many have a negative self image when it comes to math. They can't imagine that they or many of the students in class are capable of .