- In this visualization video, we re-introduce the zero principle to subtracting positive and negative integers.
- By showing black and red squares combining and disappearing as well as black and red squares re-appearing in an attempt to show that we can add more squares of one colour as long as we add the same amount of the other colour as well.
- Following the introductory revisiting of the zero principle, we introduce the problem:
7 black squares take away 2 red squares
- As you can imagine, this is problematic for students as there are no red squares to take away!
- What must we do?
- If we recall the introduction showing that a black and red square can combine to disappear and also can “re-appear” when we want them to, we could simply add 2 red squares (so we can take them away) and 2 black squares.
Now, we replace the words with the math symbolic representation as well as re-arrange the squares to look a bit more “pretty” and easier to subitize:
Math Is Visual Prompts
Students are then given the following symbolically to represent concretely and then symbolically, on their own:
6 black – 4 red = ?
- Check out the animated video here:
- You can also consider using the following visual prompt images below:
It might be worth reviewing the initial video to ensure that students are picking up on the “rules” of the zero principle. Here is the initial video again, to avoid having to scroll up:
Hopefully, students will recognize that by using the zero principle, we can make 4 red squares appear as long as we also make 4 black squares appear.
Then, students can “take away” or subtract the 4 red squares and will be left with 10 black.
Rearranging the squares might make conceptually subitizing the total a bit easier.
Hope you and your students enjoy this exploration.
Math is visual. Let’s teach it that way!
Subtracting Positive and Negative Numbers
Subtracting positive numbers, such as 4 – 2, is easy. When we subtract negative numbers or subtract negative numbers to positive numbers, it gets more complicated.Here are some simple rules to follow when subtracting negative numbers.
Rule 1: Subtracting a positive number from a positive number – it’s just normal subtraction
For example: this is what you have learned before. 6 – 3 are two positive numbers. So solve this equation the way you always have: 6 – 3 = 3.
Rule 2: Subtracting a positive number from a negative number – start at the negative number and count backwards
For example: Say, we have the problem -2 – 3. Using the number line, let’s start at -2.
Now count backwards 3 units. So keep counting back three spaces from -2 on the number line.
The answer is -2 – 3 = -5.
Rule 3: Subtracting a negative number from a negative number – a minus sign followed by a negative sign, turns the two signs into a plus sign
So, instead of subtracting a negative, you are adding a positive. Basically, – (-4) becomes +4, and then you add the numbers.
For example, say we have the problem -2 – –4. This would read “negative two minus negative 4”. So we’re changing the two negative signs into a positive, so the equation now becomes -2 + 4.
On the number line, it starts at -2.
Then we move forward 4 units: +4.
The answer is -2 – (-4) = 2.
Rule 4: Subtracting a negative number from a positive number – turn the subtraction sign followed by a negative sign into a plus sign
So, instead of subtracting a negative, you’re adding a positive. So the equation turns into a simple addition problem.
For example: let’s say we have the problem 2 – (-3). This reads “two minus negative three”. The – (-3) turns into +3.
On the number line we start at 2.
Then we move forward three units: 2 + 3.
The answer is 2 – (-3) = 5.
Solution: This problem is quite simple: just add $3 and $6 and the result is $9.
The problem above can be solved using addition of integers. Owing $3 can be represented by -3 and owing $6 can be represented by -6. The problem becomes:-3 + -6 = -9
Look at the number line below. If we start at 0, and move 3 to the left, we land on -3. If we then move another 6 to the left, we end up at -9.
Rule: The sum of two negative integers is a negative integer.
Example 1: Find the sum of each pair of integers. You may draw a number line to help you solve this problem.
|Adding Negative Integers|
|-2 + -9 =||-11|
|-5 + -8 =||-13|
|-13 + -7 =||-20|
Do not confuse the sign of the integer with the operation being performed. Remember that:-2 + -9 = -11 is read as Negative 2 plus negative 9 equals negative 11.
Rule: The sum of two positive integers is a positive integer.
Example 2: Find the sum of each pair of integers. You may draw a number line to help you solve this problem.
|Adding Positive Integers|
|+2 + +9 =||+11|
|+17 + +5 =||+22|
|+29 + +16 =||+45|
Do not confuse the sign of the integer with the operation being performed. Remember that:
+29 + +16 = +45 is read as Positive 29 plus positive 16 equals positive 45.
How to Add and Subtract
Positive and Negative Numbers
This is the Number Line:
|Negative Numbers (−)||Positive Numbers (+)|
|“−” is the negative sign.||“+” is the positive sign|
If a number has no sign it usually means that it is a positive number.
Balloons and Weights
Let us think about numbers as balloons (positive) and weights (negative):
This basket has balloons and weights tied to it:
Adding a Positive Number
Adding positive numbers is just simple addition.
|We can add balloons (we are adding positive value) the basket gets pulled upwards (positive)|
- is really saying
- “Positive 2 plus Positive 3 equals Positive 5”
- We could write it as (+2) + (+3) = (+5)
Subtracting A Positive Number
Subtracting positive numbers is just simple subtraction.
|We can take away balloons (we are subtracting positive value) the basket gets pulled downwards (negative)|
- is really saying
- “Positive 6 minus Positive 3 equals Positive 3”
- We could write it as (+6) − (+3) = (+3)
Adding A Negative Number
Now let's see what adding and subtracting negative
Quick Insight: Subtracting Negative Numbers
A math teacher recently asked how to explain the concept of subtracting negative numbers to her class. Why is 8 – (-6) = 14 the same as 8 + 6 = 14?
I've long internalized negatives as “opposite” and subtraction as “opposite of addition” so in my head, I had a notion of “opposite of opposite of addition” which simplifies down to “addition”.
But that inner verbalization was still pretty abstract. After thinking of a better intuition, here was my reply:
Great question! I had to think about it for a bit. Addition and subtraction are related, but slightly different, than positive and negative numbers.
Imagine going on a walk. You're facing forward, and take 8 steps forward. This is really:
0 + 8
0 is your starting point. The “+” means “facing forward” and “8” means “8 steps in the direction you're facing”. Ok.
Now, let's say we want to keep facing forward and take 6 more steps. That'd be:
8 + 6 = 14
Which gives us 14 steps from our starting point. What if we had faced backwards and took 6 steps?
8 – 6 = 2
Which means we're pretty close to our starting point, just 2 steps away. What if we had faced backwards but walked backwards 6 steps?
8 – (-6) = 14
Ah! The addition/subtraction tells us which way to face, and the positive/negative tells us if our steps will be forward or backward (regardless of the way we're facing).
In a sense, the addition/subtraction acts as a verb (“face forward” or “face backward”), and the positive/negative acts as an adjective (“regular steps” or “backwards steps”). Or maybe it's an adverb, modifying how we walk (walk forwardly, walk backwardly). You get the idea.
For older students, “subtracting a negative” can be seen as “cancelling a debt”. If I have a debt of $30, and someone “subtracts it”, I've effectively gained $30. In general, if you remove a disadvantage, you have improved your situation — a positive.
These explanations are a bit abstract, the walking one is more fun to try directly. I actually walked around while thinking through the intuition. (If you're adventurous, you might start thinking about taking side steps, or jumping, and how that would be represented.)
When doing simple arithmetic, we only track the final location, not orientation. Facing backwards and walking backwards might have us looking at 0 while we advance forward. But mathematically, our endpoint is the same: 8 – (-6) = 8 + 6 = 14.
If we care about the way we're facing, we need a more complex math object (a vector) to keep track of our orientation as well as position (“14, facing forward” vs. “14, facing backward”). Perhaps we'd use a line integral, moving along a path and tracking the direction we face as we go.
A fitting analogy leads to questions about what else is possible.