1. Number Sense

Lesson

The integers are made up of the positive and negative whole numbers, as well as the number $0$0. Here is a partial list of the set of integers, which continues up toward $+\infty$+∞ and down toward $-\infty$−∞.

$\left\{\dots,-3,-2,-1,0,1,2,3,\dots\right\}${…,−3,−2,−1,0,1,2,3,…}

We know that $1$1 is less than $2$2, that $2$2 is less than $3$3, and so on. We also know that $-11$−11 is closer to $-10$−10 than $4$4. All of this information can be represented visually using a number line, shown below. The arrows at each end indicate that the line extends infinitely in the positive and negative directions.

We can identify an integer on the number line by plotting a point at that integer. On the number line below we can see that the point is at the mark labeled with the integer $-3$−3. This means that the point on the number line is at $-3$−3.

However, not every number line has labels for all the marks. On the number line below, the location of the point on the number line has not been labeled.

In order to find the missing label we need to find the distance between each mark. We can do this by comparing the number of gaps between the existing labels with the distance between the integer labels.

In this case, we can see that there are three gaps between the labels of $0$0 and $3$3. This means that there is a distance of $3$3 units shared between three gaps, so each gap will be equal to $1$1 unit. Since the point is located one mark to the right of $3$3, the integer we are looking for is $1$1 unit greater than $3$3. So the location of the point is $4$4.

Let's try a similar problem.

Where is the point plotted on the number line?

**Think:** We can find the integer at the point by finding the distance between each mark.

**Do:** We can see that there are two gaps between $4$4 and $8$8. This tells us that one gap is equal to a distance of $2$2 units. Since the integer at the point is one mark to the right of $4$4, the integer we are looking for is $2$2 units greater than $4$4.

What integer is two units greater than $4$4? The answer is $6$6.

**Reflect:** When the number line is missing labels, we want to find the distance between marks in order to find the integer where the point is located. We can do this by comparing the number of gaps to the number of units between adjacent labels.

Notice that the point was plotted at the mark halfway between $4$4 and $8$8. This tells us that the integer represented by the point will be halfway between $4$4 and $8$8. This is another way to find that the point is plotted at $6$6.

When looking for the opposite of a meaning we usually try to reverse it. For example, the opposite of left is right because we can reverse moving to the left by moving to the right. When trying to find opposites on a number line, we can use the same approach.

Consider the integer $3$3. On this number line, the integer $3$3 represents "the location $3$3 units to the right of $0$0", shown in green. The opposite of this would involve reversing the direction. In other words, the opposite would be "the location $3$3 units to the left of $0$0", shown in blue.

This example shows that the opposite of the integer $3$3 is the integer $-3$−3.

We can use the same method to find the opposite of a negative integer. Consider the integer $-3$−3. This number represents "the location $3$3 units to the left of $0$0", shown in green, so its opposite will be "the location $3$3 units to the right of $0$0", shown in blue.

Opposite integers

Two integers are opposite if their locations on the number line are the same distance from $0$0, but on different sides of $0$0.

What about $0$0 itself? We can think about the opposite of $0$0 as being the number $-0$−0. But since $-0$−0 is the same as $0$0, the opposite of $0$0 is again $0$0. That is, the integer $0$0 is its own opposite.

This applet lets you visualize the idea of opposites. Slide point $A$`A` and see its opposite (point $B$`B`) move.

The number line isn't the only way to represent integers. Another common way is using two different colored counters. One color for positives and one color for negatives. If we have the same number of positive and negative counters, we call this a zero pair as each counter is canceled out resulting in zero. For example, below we have $2$2 positive counters ($+2$+2) and $2$2 negative counters ($-2$−2), so we have a zero pair. This tells us that $+2+(-2)=0$+2+(−2)=0.

The applet below allows us to use counters to explore integers. To use this applet drag over a positive (purple) counter and if desired, click on it to change it to a negative (red) counter. See the text at the top for the value of all the counters together.

Use the guiding questions below:

- Drag over a desired of counters and make them all negative. How many positive counters do you need to make a zero pair?
- Why do you think we use different colors for positive and negative?
- How might zero pairs help us in mathematics?
- Where might we see zero pairs in the real world?

When we use the number line to understand a real-world situation, we need to decide two things: where shall we put $0$0, and what direction shall be positive? In this way, the integers on the number line become signed numbers that we identify with locations in the real world. The size of the integer tells us the distance from $0$0, and the sign (either positive or negative) of the integer tells us the direction from $0$0.

Let the location of a city be represented by the integer $0$0, and let a point $7$7 km to the east of the city be represented by the integer $7$7. What integer represents the point $4$4 km to the west of the city?

**Think**: We can use a number line to represent this information. Since west is the opposite of east, the negative integers will represent points to the west of the city.

**Do**: The number line below shows the city at $0$0 and the point $7$7 km to the east at $7$7. One unit on the number line represents a distance of $1$1 km in the real world.

The point $4$4 km to the west of the city will be represented by the integer $-4$−4 on the number line.

Is the following number an integer?

$+19$+19

No

AYes

B

Where is the point plotted on the number line?

Think about the following statement:

"Arriving $14$14 minutes late."

Pick the statement that describes the opposite of "Arriving $14$14 minutes late".

Arriving $15$15 minutes late.

AArriving on time.

BArriving $14$14 minutes early.

CArriving $15$15 minutes early.

DSuppose "Arriving $14$14 minutes late" is represented by the number $14$14.

What directed number should represent "Arriving $14$14 minutes early"?