The purpost of the introduction is to make sure students know two things: 1) The essential question; 2) How to find sums using counters.
I will make sure that every students has 10 positive counters and 10 negative counters. I like to use green for positive and red for negative, but I have often seen people use black for positive as a link to the phrase "in the black". I do not have any black counters.
Before students work on the first 8 problems, I will teach them how to use the counters. I will describe the value of each color and then explain a zero pair. Then using the document camera, I will place various combinations of positive and negative chips on display. I will ask, what is the sum? I will give students a few seconds to process and then I will cold call a student. Depending on my mood that day and the dynamic of the class, I may have them write their responses on a whiteboard. Sometimes whiteboard work magically elevates their mood!
After we have done several problems and I see that nearly all students can easily find sums, I will have them do the first 8 problems. I will ask that they model using their chips, but I would like them to write plusses (+) and minuses (-) to model the problems as well. This will give them a tool to use if they need verification of an integer sum (MP5). It will also make it easy for me to diagnose any issues as I am walking around the room.
Students may need to be reminded that the parentheses around the second addend are visual only. They help to visually separate the negative value from the addition operation but they have no effect on the computation. Also, look out for students that may confuse the parentheses with absolute value! The difference is easy for a math teacher to see but not always a young student who is trying to make sense of so many new (or fairly new) symbols and concepts!
This narrative refers to page 1:DetermineIfSumIsPosNegOrZeroUsingCounters_Intro.docx
Now that students have a handle on how to use the chip model, we can address the essential question of the lesson. Students will work with their pairs or trios to fill in a table of 11 sums to evaluate if a sum is positive, negative or zero. The first question should lead students to see that positive sums can result from a positive and a negative or two positives. The second question should lead students to see that different signs or two negatives can lead to a negative sum. Question 3 is where students make the connection between absolute value and the resulting sign. I think MP2 and MP7 are the primary practices that students must put to use to make sense of the questions. Questions 5 and 6 then lead students to create a rule for sums (MP8). As students work through the tasks, I will be looking to make sure that the tables are being filled in correctly first, so that students can then correctly answer the questions. For question 3, I will push students to be as specific as possible and to use correct mathematical language. For example, if a student says a sum of different signs will be negative when there are more negatives, I will ask them what they mean by more negatives. What mathematical term could they use instead of more negative?
Problems 1-3 of the independent practice give students a chance to apply their understanding of the day's essential question on integers that are not as practical to model. Here is a chance to make sense of structure to determine the sign of the sums (MP7).
Problems 4-7 present an even more abstract application of the the essential understanding. Students must make sense of the relative magnitude of the two addends in order to find the sign of the sum. (MP2).
Problems 8-9 ask students to test the truth of statements and explain their reasoning. (MP3).
Problems 10-11 are the easiest problems of the set. If I have did my job during the introduction section of the lesson, no student should struggle with these two problems. Yet, it is a reminder that the counter model is a useful way ot understand the sums of integers. (MP5)
Problems 14-16, ask students to use the commuatative property to evaluate the sum of three addends. Two of the sums make a zero pair. Attending to structure (MP7) in this manner, is a useful method for evaluating sums.
The first 3 extension problems ask students to determine the sign of sums of non-integer rational numbers.
The next 3 problems ask students to solve simple one-step equations. If a student struggles with one of these questions, I will use language from the current lesson. Using problem 22 as an example I may ask: What is the sign of the sum? Based on the sign of the sum and the sign and magnitude of the known addend, what must be the sign of the unknown addend? Will the unknown addend have a magnitude that is greater than, less than, or equal to the known addend. Again, this is an application of MP2.
The last 4 problems of the extension are to let students see that the counter model can be applied to non-integer rational numbers. I want to empower students (before teaching a formal lesson) to solve such sums on their own.
This narrative refers to pages 3-4 of:
The exit ticket asks students to explain cases where a sum will be negative, positive and zero. I have required students to use the counter model in their explanations. I want to get them in the habit of making drawings as a way to support their reasoning. (MP4)