Constructing an Understanding of Integer Addition

Making sense of integer addition and subtraction is hard.   The algorithms are complicated and hard to remember.   On their face the results sometimes seem counter intuitive.   Sometimes the answer is positive, sometimes it is negative.   Sometimes the answer gets “bigger” when you subtract and sometimes it gets “smaller” when you add.   Because of all this messiness, students have a really hard time knowing whether an answer makes sense if they haven’t had the chance to build some conceptual understanding before jumping to an algorithm.

I start helping my students make sense of the process using a chip model.    They use black chips to represent positive numbers and red chips to represent negative numbers.    They combine the chips, making zero pairs (one red and one black chip) to find the sum.    I start with the chip model because most students think of addition as combining sets of objects.

After students have mastered the chip model, I move on to a number line model.   The curriculum that I use (Connected Math) does a really nice job introducing the number line model.    It talks about addition as combining sets by providing a context in which two kids each have a number of video games and then talking about the combined number of video games.    It illustrates the problem using a chip model.   Then, it goes on to talk about addition also being representative of a context in which one “adds on”.   It provides a context in which there is a temperature at sunrise and that is “added on” to as the temperature rises over the course of a day.    They model this problem using a number line.

After introducing students to the number line model, I take them out into the hall to walk the number line.   Prior to this, I have had my student aides create a number line for each table group.    Each number line is created using painters tape (for ease of removal when the time comes).   The numbers on each number line range from -10 to 10.    I begin by modeling a couple of integer addition problems on the line.    I walk forward for positive numbers and backward for negative numbers.

After modeling several problems, I have each group complete a set of problems by walking on the number line.    Each group gets a laminated index card with the problems to be completed and a dry erase marker to record their answer.    The first group member walks the number line to solve the first problem.    The other members of the group check his or her work.   The next person in the group walks the number line for the second problem and the others check the work.    Each problem is completed by a new student until everyone in the group has walked the number line.   At that point, the group rotates through again until all the problems  have been completed.

Once students have mastered integer addition on the number line, they use models to construct an algorithm for integer addition.    They begin with a group of four problems.   They solve these problems using a model.    Then, they identify what is the same about the problems and create two more problems to fit the group.   (All of the problems in the group have addends in which the signs match.)    Finally, they come up with an algorithm for the adding problems within the group.   They repeat this process for a second group of problems in which the signs of the addend pairs do not match.    If students have trouble figuring out the algorithm, I remind them that each number has a magnitude (size) and a direction (sign) and suggest that they consider the two parts separately.   This usually helps them get to the algorithm.

After creating the algorithm, students test it on rational numbers (mixed numbers) and verify its efficacy using a number line model.    Finally, they test rational number addition for commutativity.

After students have created an algorithm, I use the foldable shown below to summarize the lesson.

I follow up with this Always, Sometimes, Never exit card.