The purpose of this introduction is to present the essential question and then to make sure students know how to model the problems using counters. The modeling method is presented so that students see the relationship between addition and subtraction; it is not the standard way to model integer sums using counters (MP5). As students are working on the four problems presented, it will be important to emphasize step 3 - removing counters from the pile. Using problem 1 as an example, I may ask "What did you remove from the sum of 5 and 0 in order to make the sum 5 + 3? " [answer: 3 negatives or -3]. I'll then ask "What operation do we use when removing items from a set" [answer: subtraction]. I'll make a big deal out of this connection during the introduction, but students will have time to explore this more in the problem solving section. It is worth noting with students that the modeling of the first addend is an example of the identity property of addition. (MP7). There is a prompt to ask that question in the work. If students are having difficulty with the modeling, I will make up additional problems that are similar to the ones presented.
In this section, students work with their partners to complete the table. The purpose of the table is to collect data that will help students answer the questions. This section is an exercise in MP7 and MP8 as students see that adding a value is the same as subtracting its opposite. As students are working to complete the 8 rows of the table, I will walk around to make sure that information is being filled in correctly. If after the introduction, there are a few students who still don't understand the modeling, I will go to them first. I will also try to interact with as many groups as possible on 1 problem. I'll ask them to model the problem and to tell me what they removed from their set in order to make the second addend appear in the right pile. As I am checking in with groups, I will make a note of different people to call on during the questions discussions. I will be looking for a few distinct responses to bring to the class. Some that maybe lack clarity and some that are clearly written.
Once the majority of groups have finished, we will discuss the answers to the questions. Questions 1 and 2 are a chance for students to practice MP3. As groups present, I will ask "Does that make sense?" and take volunteers to respond. It is really imporant that students are okay with having their responses critiqued and to emphasize that this is part of the learning process.
Question 3 and 4 are to help build the transition from concrete models to numeric representations without a model.
This section now gives students a chance to apply what has been learned in the previous sections. It is okay for students to use counters as needed, but hopefully students can apply the conclusion from the problem solving section that adding a number is the same as subtracting its opposite.
Question 9-12 emphasizes this relationship since only variables are used. These 4 problems are a chance for students to reason abstractly (MP2). If students struggle here, I will bring them back to the conclusion from earlier. Many students may feel uncomfortable not having specific numbers. I will encourage them to initially asign a value to each variable.
Problem 13-17 are there so that student identify not only the equivalent subtraction problem, but also apply the commutative property (MP7). I want student to see the multiple representations for an expression.
Extension problems 18-21 have students solve equations using counter models and problem 22-25 students solve actual equations. It is okay if not every student makes it to the extension during this lesson, though I will find time at some point to complete it whether in a large group or small group. I want students to do really well with the first 17 questions. The extension problems are meant to present a challenge for those who quickly mastered the lesson.
The exit ticket is a step-by-step recreation of the modeling process. Problems 1 and 2 are meant as a guide to help students solve question number 3. In problem 1 students identify the value as 6 or 6 + 0. In problem 2, students are asked what they can remove (subtract) to increase the value of the set. Finally in problem 3, students are asked to write a sum and difference.