Archive for May, 2012

A teacher asked me the following question. My response follows.

My students are having trouble adding and subtracting signed numbers correctly.   Many had the line “two negatives equals a positive” memorized and applied it universally, which would usually get them the right answers when dividing or multiplying, but clearly doesn’t work with addition and subtraction. How do I get them to remember the rules of basic arithmetic with signed numbers? And are there any resources that might help?

This is a problem that I have experienced frequently! Students struggle since they remember a rule but apply it in the wrong situation. This can be frustrating for students since they might think that they are doing well, remembering the mathematical rules, but then discover that they are applying the rule incorrectly.

The underlying issue in a situation like this is that the students are just remembering rules rather than working to make sense of the rule. Recall that we must focus on developing mathematical practices in our students. These mathematical practices include:

  • Making sense of problems and persevering in solving them.
  • Reasoning abstractly and quantitatively.
  • Constructing viable arguments and critiquing the reasoning of others.
  • Modeling with mathematics.
  • Using appropriate tools strategically.
  • Attending to precision.
  • Looking for and making use of structure.
  • Looking for and expressing regularity in repeated reasoning.

Working with integer operations is a wonderful opportunity for developing these mathematical practices with our students!

I encourage you to take some time to work with your students to help them to make sense of operations with integers using manipulatives. You can use two color chips or any similar objects that come in two different colors, one color to represent positive quantities and one to represent negative quantities. In addition, students will need to think and develop a definition for what it means to add and subtract.

In my experiences, I have found that students often come to the conclusion that addition means to combine quantities and subtraction means to take one quantity away from another. Starting with addition, work with your students to use the manipulatives to model problems like:

  • 5+3
  • 5+(–3)
  • –5+3
  • –5+(–3)

It is critical that students model these situations with the manipulatives so that mental imagery can begin to be formed. Later, when we transition away from the manipulatives, we want students to be able to close their eyes and imagine the situation at hand. This mental imagery will prove to be very powerful in the sense-making process and helping students to become proficient at working with integer operations.

Furthermore, students must realize that a positive quantity and the same negative quantity combine together to make zero. This idea can be developed using the context of money. Spending money can be considered “negative” and depositing money can be considered “positive.” If an equal quantity is deposited and spent, the net effect is a $0 change to the bank account. In my classes, students refer to this idea as “making zero.”

When all of these ideas are in place (meaning of addition, making zero), students are ready to tackle the problems presented. It will be necessary to work through them in sequence as you work to help your students build the mental imagery needed to fully make sense of integer operations. I will share how the last two problems will look as your students build the models with the manipulatives. In this case, the red circles represent negative quantities and the green circles represent positive quantities.


Notice that three of the negative circles (3 negatives) join with three of the positive circles (3 positives) to make zero. In terms of the value of this combinations, we have two negatives (2 red circles) so –5+3=–2. Let’s consider another example.


With the mental imagery formed with activities using manipulatives like these, students are much less likely to make the “two negatives always make a positive” error. It becomes clearer in the minds of our students when they can envision combining (focus on the definition of addition) a negative quantity with another negative quantity and seeing a combination that can be nothing but negative! Clearly, the combination of two negatives with addition cannot be positive!

With subtraction, a little additional work is needed. Recall the definition of subtraction as “taking one quantity away from another quantity.” Also recall that we can “make zero” without changing the overall value of the quantity of items. With this in mind, let’s consider two subtraction problems.

–5 – (–3)

This problem tells us to start with a quantity of negative 5 (5 red circles) and take away negative 3 (3 red circles).

In modeling it this way, it becomes much more clear that –5 – (–3) = –2. Imagine students manipulating objects so that they are physically taking away a quantity of red objects from a larger quantity of red objects. It becomes clear and the mental imagery can be formed.

Let’s look at another situation where we are to take away a quantity that doesn’t seem to be present.

–3 – (–5)

This problem tells us to start with a quantity of negative 3 (3 red circles) and to take away negative 5 (5 red circles). Since we cannot take away more than we have, we take advantage of the idea of “making zero.”

Note that we start with a quantity with a value of negative 3 (3 red circles). By adding two pairs of red/green circles (making zero), the value of the quantity remains negative 3. But now, we can take away the required negative 5 (5 red circles)!

This will help students to see that –3 – (–5) = 2.

If it’s not too late, it is important for students to explore, build mental imagery, make sense, reason, construct arguments, etc. before being handed the traditional rules for operating with integers. In fact, what I have found to be the case is that the students will eventually come to discover the rules for themselves. For example, students may say, “I notice that when I subtract a negative, it has the same effect as adding the opposite of the sign of the second quantity…does that always happen?” Celebrate those moments! Ask the student to share this with the class and ask the class to think about this observation! Discuss it, talk about it, and ask the students to work to confirm it! Once the students all agree to this claim, post it on the wall and give it a name like, “Susie’s Rule.”

As time goes by, students will begin to transition from using the manipulatives to using paper and pencil (they may use ‘+’ and  ‘–’  symbols instead of the color objects). Continue to push students to develop mental math as they imagine the manipulatives or these symbols.

I have a PowerPoint that animates the red and green circle idea that I can send to you if you email I also  have some classroom activities that connect to and extend the ideas presented in this post – again email if you are interested.

In conclusion, focus your classroom activities on the mathematical practices that should be developed in students. Use the context of integer operations to develop these mathematical practices. Make students aware that this is truly what doing mathematics is all about! Avoid any tricks, gimmicks, sayings, or songs that may or may not hold mathematical meaning for students. Rather, build mental imagery and allow students to make sense!