I have students work in pairs on the first part of the add_polynomials2_warmup. Students will have to understand what this problem is asking them to find MP1. They will also need to examine the structure needed to have the given polynomial represent the perimeter MP7. I give students about 3-4 minutes to make their sketches. Next, I choose several pairs of students to justify their sketches to the class using either the board or a document camera (MP3). As they present, I give other members of the class an opportunity to ask questions.
When we transition to the second slide, I have students think about the difference between this question and the first question (MP1). I begin by having students share ideas with a partner and as they develop a plan to solve the problem. Then, I allow each partnership to carry out their plan. Afterward, I ask 1 or 2 pairs share their responses and I allow the class to ask questions. Students should try to determine if the solution could represent the sides of a rectangle with the given perimeter.
This activity is called Numbered Heads Together. Each student is given their own dry erase board. Each student in each group is assigned a number 1-4. The objective is to have all four students work together on each of the nine problems in this section. The problems are given one at a time.
I give students approximately 2 minutes to work on each problem. At the end of two minutes I choose a number: 1, 2, 3, or 4. (For this I have used index cards numbered 1-4 or you can use an online random number spinner - students tend to like this better!). Whichever number comes up is the student who needs to show their work for the group. This students work is the the only work that counts for the group. I award points to groups with correct answers, or for making a lot of progress, or showing the most perseverence, etc.
I like this activity because:
The add_polynomials2 ticket out the door requires students to reason abstractly about polynomials (MP2). I encourage students to explain their ideas for Question 2 as thoroughly as possible. They can use examples to explain their thoughts to make their argument more easily understood (MP3). If time permits, I allow students to swap papers with their partners so that partners can give each other feedback before handing in their exit ticket.
This ended up being a really interesting way to assess student understanding. Students did really well with the first question. Many students ended up giving examples like (x+y)+(w+z). They were able to tell right away that all four terms had to be different if they were going to end up with four terms in the solution. The second question gave students more to think about. It was interesting to see how they rationalized that there could not be more than four terms when combining two binomials. Several students summed it up very well saying something along the lines of, "there could be four terms or less but never more than four terms." This type of thinking is abstract and often uncomfortable for students. However, this type of generalization is very important to the discipline of mathematics.