In this section of the lesson I will be teaching while projecting the resource: polynomials_add_subtract_launch.
This graphic organizer is designed as shown in slide 1. If you add the first two numbers in each row you get the third number in each row. If you add vertically, you get the bottom number in each column. The cell in the bottom right can serve as a check. This is because the bottom right cell represents the sum of both the two cells above it and the two cells to the left of it. Allow students to examine the table and verify for themselves that each row and column does sum to the appropriate value.
I allow students to work with their partner as they find the 5 missing values in this table. Students will need to reason using the values given to find the missing sums. I will call on several students to give a missing value and explain how they arrived at their solution.
In order to complete the missing cells, students have to reason quantitatively about the values (MP2). At some points they will be adding and at other points subtracting in order to find the missing values. I have students try to find the missing values by themselves first. Then, I have them compare their results with a partner. I ask students to take turns explaining what a value is and how they found it (MP3). For some values the partners may agree in their reasoning and others they may approach from a different perspective. The opportunity for discussion helps students to prepare for the investigation in the next phase of the lesson.
Now that students understand the structure of the organizer, they can solve for missing polynomial values. I like this activity because it allows students to get practice with additive reasoning dealing with polynomials (MP2). In order to find the missing values, students will need to examine the structure of each polynomial and work forwards and backwards to find the missing values in each of the cells (MP7).
Students should work with their partner on the investigation. I encourage students to "think out loud" while they are working so that they can convey their reasoning to their partner (MP3). I remind students to use the bottom right cell as a check to make sure that the two cells above and the two cells to the left add up to that value.
If time permits, I project one or two of these tables and I have students explain to the class how they found the missing values. As they present, I call on members of the class to ensure they are following along with their peer's explanation. I also encourage questions if a student cannot make sense of an explanation. All of these simple talk moves enable a more fluid exchange of ideas today, and in lessons to come.
For this ticket out the door, I ask students to make their own 3x3 puzzle. I give students the option of using numerical values, monomial values or polynomial values. I encourage them to make a choice that challenges their current level of understanding. Then, the student's choice provides some insight into how he/she is grasping the concepts associated with adding and subtracting polynomials.
Once students have designed their puzzles (put four values into the cells in appropriate places), they can trade puzzles with their partner. The partner can then try to find the other 5 missing values.
Caution: Make sure that students know that there are some puzzles that cannot be solved. Some examples are given in this document: polynomial_add_subtract_no_solve.pdf.
As a Bonus Question, I ask students to think about which four cells make the puzzle easier to solve or more difficult to solve. Ask students to concentrate on which cells need to be filled in, not on the value in the cells (MP1).
Explanation: If the four cells in the top left of the puzzle are filled in the puzzle is easier to solve because students only need to add. If the four cells on the bottom right of the puzzle are filled in the puzzle is more difficult because students will need to mostly subtract.
This lesson was adapted from "Polynomial Puzzler" on the NCTM Illuminations website.