All That and a Bag of Chips! Using Counters to Combine Integers
Lesson 8 of 20
Objective: SWBAT combine and simplify integers by making zero pairs with counters
Students enter according to the Daily Entrance Routine. When they arrive at their desks they find a Do Now paper to complete silently.
When students have answered all questions they must bring their paper to me. I stand on the right side of the room where I have diner-like booths built against the walls. I check all answers and if the student is correct, they get to form a group in the booths. There are three available booths for 4 students each. The purpose for this separation is two-fold: I will be able to weed out the students still struggling with ordering integers so that I can help them in the main part of the classroom and it is also a motivator to get students to work quickly. Some of my students have not been using urgency to enter the classroom and get right to work, so I am hoping that this is an opportunity to reward students who are completing their work at a decent pace.
Students in booths review the answers together and draw number lines for each problem to illustrate their answers. I review the answers with the students left in the center of the classroom. I use number lines to illustrate the number that is out of order. I encourage them to continue working with urgency because there will be other opportunities for earning booths in the future.
Students are given 1.5 minutes to write their heading on their paper and copy the aim. During this time I will distribute 20 blue and 20 red counters to each pair of students (including those in booths).
Next, I ask one student to read the “notes” written on their paper for integers as a review of the basic definition. Then we fill in the blanks for the colors of the counters that will be used for the rest of the week (red + and blue -). We also have a brief discussion about the concept of zero pair. I ask students to tell me what integer represents one red counter and what integer represents one blue counter. Then I ask students to tell me what 1 + -1 equals. I explain that the red and the blue counter make a "zero pair". I also make sure to pose the example of two blue counters and ask if this is a zero pair. This will allows students to think about problems where two negatives will be added. The use of counters today is also a great way to practice MP5 where students are using these tools to reason about integer operations in a concrete and hands-on way.
I then draw a number line on the board that shows zero and the location of 5. I ask the students to show me 5 with their chips. I stop to make sure student have 5 red chips on their desk. If students have blue chips to show 5, I ask them to review their notes for the appropriate color to be used for positive 5. I go back to the number line and I plot -5 on the number line. I ask students to show this with counters as well, and to place the blue counters next to the red counters. I ask, “what happens when you combine -5 and 5, what do we get?” (zero) as well as, “what is the vocabulary word we learned this week for two opposites that combine for a result of zero?” (additive inverse). I push the understanding of this definition with the following questions:
- Can someone explain further: what are the additive inverses in this example?
- Deconstruct the term “additive inverse”. Why use the word “additive” and why use the word “inverse”?
Students remaining in the center of the classroom (not in booths) will be asked to reseat themselves so that they have a partner(s) and are closer to the front of the room. I instruct students to use their counters to model each of the problems on their paper. They must draw the counters for the first 4 problems using the symbols ⊕ and ⊖. For the problems on the back of their task paper, they are instructed to use the counters to model each expression, but they do not have to draw it. Once students finish the task, they are to check off one of the boxes at the end of the paper to indicate their level of understanding for the lesson.
While students are practicing during this time they are still separated according to their answer to the do now. Although the do now does not involve integer operations, it does involve a more basic understanding of integers and their order. By having students in the center of the room that either did not understand the “Do Now” or were not working quickly during that time, I am able to focus more attention on improving their sense of urgency and also answering any questions to clarify ordering and comparing integers.
During the last three minutes of the previous section I ask students to return to their desks, pack up, and leave their “Task” paper out. Once these expectations have been met and students are waiting silently, I put up three problems on the board with larger numbers:
- 108 + (-59)
- 36+ (-76)
- -13 + (-15) + 16
I ask three students to volunteer to complete the problems at the board so that other students can observe. Because the numbers are larger, it does not make sense to draw counters. So I ask for my volunteers to show the “work” necessary to complete these problems. I ask students to discuss the strategy or “rule” used when combining numbers with alternate signs and combining integers with the same sign. For questions like #2 above, I ask the question “How do you determine the sign of an answer for these types of problems”? If a student says something like, “I take the sign of the biggest number” I respond by asking “but is negative 76 bigger than 36?” This leads to a discussion where I encourage students to think of a vocabulary term we learned this week that they can use to talk about the “distance from zero” (MP6). If they still cannot figure out the correct term, I tell them it is absolute value and I ask someone to volunteer to restate the answer to my question using this word.
I provide students with examples of number lines and plotted points and ask them to model this integer with their counters. Then I ask them to model with chips the idea that I can go to an integer and return to zero. These are zero pairs.