Expressing Relationships Day 6 of 6

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SWBAT represent patterns using numeric and variable expressions by comparing data from different sized checkerboards..

Big Idea

Students begin to generalize the patterns in the data.

Warm Up

10 minutes

As students enter they find posters at their math family's table group. Each group should have a poster representing each of the different sized checkerboards (4by4, 6by6, 8by8, and 9by9). I selected the posters with no mistakes in the data so that students could make generalizations from the comparisons. I start by asking them to share with their math family group any patterns they have noticed in the data over the past few days of the project and to use the posters to show. Then I have a few share out.

This is the culmination of a 6 day project that began in earlier lessons (checkerboard squares, squares in a row, checkerboard poster planning, describing checkerboard patterns, finding more patterns in the data).



20 minutes

On the board I have a large table drawn out listing different sized checkerboards and different sized squares that could be found on checkerboards. On the other side of the board I have a space for expressions.

At this point I ask them to look at their posters and tell me how many different sized squares are possible for a 4by4 checkerboard. When they tell me "four" I ask them to tell me what the sizes are and I put a box around those cells on my board. I do the same for the other sizes represented by their posters. To help them navigate some of the patterns we continue the discussion.

  • How does the number of different sizes relate to the size of the checkerboard?
  • How many different size squares would be possible on a 5 by 5 checkerboard? 10 by 10?
  • How do you know? What makes you say that?
  • How could we figure out how many different size squares would be on any size board?
  • What is the relationship between the number of different sizes and the size of the board?
  • Why does that make sense?

Next I tell them to use their posters again and tell me how many of the smallest squares are found on a 4by4 checkerboard. They tell me 16. I repeat the process for the other known checkerboards and then ask if they see a relationship between the size of the checkerboard and the number of smallest squares on it. Rather than have them share for the whole class I have them discuss it just in their groups for a moment. Then we continue the whole group discussion:

  • "Can you predict how many will be on a 5 by 5?" (25)
  • "How did you get that?" (5 squared)
  • "Where did the 5 come from?"
  • "Why does that make sense?"
  • "Could we use that to figure out how many for a 10 by 10?" "...n by n?"

I point out that even though we don't know the actual number for an n by n checkerboard, we can express what it would look like and how we would get it by using a variable in place of the unknown. We fill in the cell with n^2. They should have already predicted the term, but if they don't then go back and ask again where they found the number to square.

Now I want to focus on writing expressions.  I go to the "expressions" side of the board and I put ___+___+___+___ next to the table for the 4by4 four checkerboard and fill in the first blank with 4^2. Then I ask what the next term would be. I am hoping that someone will suggest 3^2, but not all of them have noticed the descending squares pattern and they may say 9 if they are looking at their posters. I will continue to write down whatever form they give me even if I have to go back and ask:

  • "is there another way to write that?"
  • "how did we get that?"
  • "what did we do to get that number?"

I don't want them to have any indication that what they are telling me is not acceptable, because the fact that they are participating and sharing ideas is great and new for many of them. If they haven't already mentioned the descending squares (4^2 + 3^2 + 2^2 + 1^2) I ask them to look at our first expression and discuss any patterns they may notice.

When we start the same process with the 6by6  checkerboard I want them to start seeing the relationships between each term being added in the expression so that we can express it like 6^2 + (6-1)^2 + (6-2)^2 + (6-3)^2 + (6-4)^2 + (6-5)^2 and I may add (6-6)^2. Then I want them to use the pattern to predict the number of squares on a 10 by 10 checkerboard and then an n by n.

The point that I want them to walk away with is twofold. Even if they aren't completely grasping the variable expression they have already gone beyond the original problem which they thought was too hard for them and now they are treating as the easy part, so I want them to learn that with perseverence and teamwork they can do more than they think. Mathematically I want them to begin to see variables as a way of representing patterns in a general way and expressing what happens no matter what the numbers are, which is so important to developing algebraic thinking.

Reflection assessment

24 minutes

This is sort of a combination of reflection, assessment, and exit ticket. Students work independently on this assessment using any of the posters at their group as well as their previous homework in which they described patterns they noticed themselves.

While students are working I circulate to ensure that they understand what the questions are asking them. They will often skip problem 1 because they mistake it as an example. In question 2 they may try to draw a checkerboard instead of a table, especially those students who had trouble making an organized table. I point out that in problem 1 I have given them a table of data for one checkerboard and problem 2 is asking them to show what the table looks like for the 9by9 checkerboard. They may also get confused by the directions to write an expression. I show them that in problem 3 I have given them the expression, which I point out, for the total squares on a 4by4 checkerboard and I'm asking them to show what the expression would look like for a 9by9. I remind them to be as detailed as they can in their description about how they find the numbers that go in the table. Many of them will not get the last question requiring a variable expression. Many of them put "n" or "n^2". Some may have misunderstood the question as asking for the number of different sized squares since we talked about that earlier.

The first two questions involve recognizing the patterns and demonstrating that they see the relationship between the data in the tables and the dimensions of the checkerboard. It also forces them to use an organized table. A handful of students were unable to set up an organized table on their posters and just had numbers written haphazardly about. (They will get a chance to do poster repair during lunch later in the week). The assessment also has them practice writing a numeric expression, describing a pattern, and writing a variable expression. However, I don't expect them all to be able to correctly represent the variable expression in its entirety. My students are also very low when it comes to verbal articulation of ideas or describing. This is something I work into lessons as much as possible throughout the year. When we go over this "assessment" I will highlight the connection between detailed description and the generalized variable expression. They will hopefully come to understand that the description needs to help others visualize and understand clearly, which helps lead to the generalized variable expressions.

When we go over this I focus on what they needed to do in order to demonstrate proficiency, because they will have the opportunity to improve this skill throughout the year.