How We Teach Integer Operations

Teaching integer operations can be frustrating, for both the instructor and the student. Memorizing a seemingly endless list of “rules” that can easily be misremembered or misapplied is a recipe for wrong answers and low confidence. Jason and Hallie made this video to demonstrate some potential problems with this approach.

Addition & Subtraction

• Addition does not inherently mean getting bigger, and subtraction does not inherently mean getting smaller - once we start operating on negative numbers, this is obviously no longer true. Trying to explain 3 + -4 as “start at 3 and get bigger by -4” is nonsensical and likely to confuse a student. 

• Teaching integer addition and subtraction as two completely different things with different sets of “rules” is not actually a good idea - if students learn addition and subtraction in isolation, they are likely to confuse the different “rules” for the different operations when presented with problems outside the context of a PK that only includes one type of problem. Unfortunately, our curriculum splits the two operations into two separate PKs. We can use Extending Knowledge to introduce students to integer subtraction when they are working on integer addition, and vice versa. 

• Actively discourage students from changing the signs of integer addition and subtraction problems. For one thing, if they are doing this without understanding why they are able to, they are likely to change signs even when it isn’t correct. More importantly, if their only way of understanding a problem is to turn it into a different problem, they likely don’t have a good conceptual understanding of the original problem.

• As numbers go up or to the right on a number line, they are getting bigger.

• As numbers go down or to the left on a number line, they are getting smaller.

• Negative numbers exist and are the opposite of positive numbers.

• The ability to correctly compare the size of negative numbers, e.g. -3 is bigger than -5.

• When we add a positive number, we get bigger.

• When we subtract a positive number, we get smaller.

• The ability to explain that “8 + 5” means “start at 8 and get bigger by 5.”

• The ability to explain that “8 - 5” means “start at 8 and get smaller by 5.”

• The ability to decompose numbers, which will be important for getting to and going past 0.

• Since adding a positive number means we get bigger, adding a negative number is exactly the opposite: we must get smaller.

  • • Therefore, “8 + -5” means “start at 8 and get smaller by 5.”

• Since subtracting a positive number means we get smaller, subtracting a negative number is exactly the opposite: we must get bigger.

  • • Therefore, “8 - -5” means “start at 8 and get bigger by 5.”

It is not necessary to change the signs in the problem in order to reason through them conceptually. Many students will have learned that “subtracting a negative is the same as adding a positive” without learning why. It is more helpful to emphasize that subtracting a negative and adding a positive "have the same result," since they both make the starting number bigger, but they are not literally the same problem. Use real world examples to help students understand this idea. "Filling in a hole in the ground" or "returning something to a store and getting a refund" are good examples to illustrate how subtracting a negative results in a real-world quantity getting bigger.

A sample conversation for the problem -3 - -5 might go like this:

Instructor: Where are we starting?

Student: At -3.

Instructor: Are we getting bigger or smaller?

Student: Bigger.

Instructor: That’s right! Why?

Student: Because subtracting a negative is the opposite of subtracting a positive.

Instructor: How much are we getting bigger by?

Student: By 5.

Instructor: So will we cross over 0?

Student: Yes!

Instructor: How do you know?

Student: Because we’re only 3 less than 0, and 5 is bigger than 3.

Instructor: So how far is it to 0? How much further will we still need to go?

Student: 3 to get to 0, and we need to go 5, so 2 more.

Instructor: So where will we end up?

Student: At positive 2.

Instructor: So what is -3 - -5?

Student: 2!

Other than the issues stated above, PK 3179 (Integer Addition) lines up with our approach until the bottom of page 6, when it introduces an entirely new method and set of questions.

Question 16 asks students to consider whether we should add or subtract the absolute values of the numbers in the problem, and whether the answer will be positive or negative. It is important to distinguish that we are not asking whether to add or subtract negative 18 and 9 - the problem clearly says to add them - but that we are talking about only their absolute values. In order to prepare students for this problem, it is important to start asking Extending Knowledge questions on previous pages, such as:

• Do you notice any patterns about your answers? 

• How will we know whether the answer will be positive or negative?

• Can you come up with a different problem using similar numbers that would have the same answer?

  • • For example, with -3 + 10 = 7, can they come up with 10 – 3 = 7? Think of it like an extension of fact families.

It is also helpful to talk about distances, since absolute value is the distance of a number from 0. In the example problem, -18 + 9, if we approach it with our usual framework, we would know that we are starting at -18 and getting bigger by 9. In other words, we are starting a distance of 18 away from 0, and then going a distance of 9 toward 0, so at the end of the problem, we will be a distance of 9 from 0. We actually subtracted those distances to find our answer. Since we started at a negative and didn’t cross over 0, we know that our answer will still be negative.

When discussing PK 3180 (Integer Subtraction), the first few pages are in line with our approach; for example, it talks about how subtracting a positive means getting smaller. On page 4, the same type of question as in the addition PK occurs about whether we add or subtract the absolute values of the numbers and what the sign will be. Again, we can address this by talking about distance. In the example below, we know that we are starting a distance of 17 below zero, and then going a distance of 12 more away from zero (because we are getting smaller), so altogether we are a distance of 29 below zero, which means the answer must be -29.

Page 5 (below) introduces a totally new approach – thinking of subtraction as “how far apart.” If students have been with us for a while, this will be a familiar way of thinking about subtraction; if not, you may need to start with examples using only positive numbers with positive answers to illustrate why this is a valid way of thinking about subtraction, e.g. 10 - 7 means “how far is it from 7 to get to 10.”

The page instructs the student to think about whether we are counting from left to right or from right to left. It may not be clear to the student why this determines whether the answer will be positive or negative. Switching to a vertical number line and emphasizing whether we are getting bigger (more positive) or smaller (more negative) as we go from the second number to the first will make it clearer, as getting bigger means the answer will be positive and getting smaller means it will be negative. No matter what, think of this as one more approach that might work well for certain students, but always go back to our consistent approach to emphasize why this technique and the answers we get from it make sense. As long as the student can correctly explain how they got their answer and why it makes sense, we don’t need to force them to use a specific method to solve.

Page 7 (below) introduces the idea of subtracting by adding the opposite (i.e. changing the signs). DO NOT gloss over the very small text in example 1 that says “because subtracting a negative and adding a positive both make the starting numbers bigger.” This is the only way it connects to our consistent approach. We want to emphasize that subtracting a negative and adding a positive have the same result, but (no matter what the page says) they are not literally the same problem – it’s not actually necessary to change the signs if we understand what the problem is asking, which we do using our consistent approach. Actively discourage students from changing the signs on their paper – if the only way they can understand the problem is by turning into a different problem, they don’t understand the original problem on a conceptual level.

In conclusion, we can make the PKs work for us by utilizing our own Visual Clues and Extending Knowledge questions and by making connections between what the page says and our consistent approach.

Multiplication

Our approach to multiplication builds on our understanding of adding and subtracting integers, as well as our multiplication construct (“this, that many times”). We will extend that to what it means to multiply by a negative number by relying on the fact that negatives are the opposite of positives.

Start with an example the student will be familiar with using positive numbers. Let’s take 3 x 2. What does that mean? Hopefully, the student will be able to say “3, 2 times.” We need to be a little more specific, so ask what we are doing with the 3s. Hopefully, the student will say “adding them.” So we have established that multiplying a number by positive 2 means “add the number twice.”

3 x 2 = + 3 + 3 = 6

-3 x 2 = + -3 + -3 = -6

Now we need to figure out what multiplying by negative 2 would mean. Since multiplying by 2 meant “add twice,” and -2 is the opposite of 2, the opposite of “add twice” would be “subtract twice.” So when we multiply a number by -2, it means “subtract the number twice.”

3 x -2 = - 3 - 3 = -6

-3 x -2 = - -3 - -3 = 6

Division

We can use our division construct language (“how many of these are in that?”) to explain why dividing two numbers with the same sign results in a positive answer. 10 ÷ 5 means “how many 5s are in 10?” There are 2 (5 + 5 = 10), so 10 ÷ 5 = 2. Similarly, -10 ÷ -5 means “how many -5s are in -10?” Again, there are 2 (-5 + -5 = -10), so -10 ÷ -5 = 2.

When dividing a positive number by a negative number (or vice versa), we can take a similar approach to multiplication, where we talked about a number, a negative number of times. 10 ÷ -5 means “how many -5s are in 10?” If we’re counting by -5s, we’ll never get to 10 - we’re going the wrong direction! -5, 2 times is -10, -5, 3 times is -15, etc. We need to go the opposite direction. Just like with multiplication, above, -5, negative 1 times is +5, -5, negative 2 times is +10, etc. So that means there are -2 -5s inside of 10.

An alternate strategy to explain why dividing numbers with different signs results in a negative answer relies on opposites. 10 ÷ -5 is the opposite of 10 ÷ 5, and the opposite of 2 is -2, so 10 ÷ -5 = -2. This matches the approach taken by PK 3274 (Multiplying and Dividing Integers).