Unit 5 Rational Number Arithmetic — Unit Plan

We can use positive numbers and negative numbers to represent temperature and elevation.

When numbers represent temperatures, positive numbers indicate temperatures that are warmer than zero and negative numbers indicate temperatures that are colder than zero. This thermometer shows a temperature of -1 degree Celsius, which we write $\text{-}1^\circ \text{C}$.

A thermometer positioned horizontally and with the bottom of the thermometer on the left, is labeled “degrees Celsius.” There are 4 evenly spaced large tick marks. Starting on the left, the large tick marks are labeled negative 5, 0, 5, and 10. Between each large tick mark, there are 4 evenly spaced small tick marks. The thermometer is shaded, starting from the left, up to negative 1.

When numbers represent elevations, positive numbers indicate positions above sea level and negative numbers indicate positions below sea level.

We can see the order of signed numbers on a number line.

A number is always less than a number to its right. So $\text{-}7 < \text{-}3$.

We use absolute value to describe how far a number is from 0. The numbers 15 and -15 are both 15 units from 0, so $|15| = 15$ and $| \text{-}15| = 15$. We call 15 and -15 opposites. They are on opposite sides of 0 on the number line but the same distance from 0.

We can represent signed numbers with arrows on a number line. We can represent positive numbers with arrows that start at 0 and point to the right. For example, this arrow represents +10 because it is 10 units long and it points to the right.

We can represent negative numbers with arrows that start at 0 and point to the left. For example, this arrow represents -4 because it is 4 units long and it points to the left.

To represent addition, we put the arrows “tip to tail.” So this diagram represents $3+5$:

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at 3. A second arrow starts at 3, points to the right, and ends at 8. there is a solid dot indicated at 8.

And this diagram represents $3 + (\text-5)$:

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, and ends at three. A second arrow starts at 3, points to the left, and ends at negative 2. There is a solid dot indicated at negative.

We can use the relationship between addition and subtraction to reason about subtracting signed numbers. For example, the equation $7 - 5 = {?}$ is equivalent to $5 + {?} = 7$. Here is a diagram that represents the addition equation.

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 5, and is labeled "plus 5." A second arrow starts at 5, points to the right, ends at 7, and is labeled with a question mark. There is a solid dot indicated at 7.

To get to the sum of 7, the second arrow must be 2 units long, pointing to the right. This tells us that positive 2 is the number that completes each equation: $5 + 2 = 7$ and $7 - 5 = 2$.

Notice that the addition expression $7 + (\text-5)$ also equals 2.

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 7, and is labeled "plus 7". A second arrow starts at 7, points to the left, ends at 2, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.

So we can see that $7 - 5 = 7 + (\text-5)$.

Here's another example. The equation $3 - 5 = {?}$ is equivalent to $5 + {?} = 3$.

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the right, ends at 5, and is labeled "plus 5." A second arrow starts at 5, points to the left, ends at 3, and is labeled with a question mark. There is a solid dot indicated at 3.

To get the to the sum of 3, the second arrow must be 2 units long, pointing to the left. This tells us that -2 is the number that completes each equation: $5 + \text-2 = 3$ and $3 - 5 = \text-2$.

Notice that the addition expression $3 + (\text-5)$ also equals -2.

A number line with the numbers negative 10 through 10 indicated. An arrow starts at 0, points to the right, ends at 3, and is labeled "plus 3". A second arrow starts at 3, points to the left, ends at negative two, and is labeled "minus 5". There is a solid dot and a question mark labeled at 2.

So we can see that $3 - 5 = 3 + (\text-5)$.

This pattern always works. In general:

$a - b = a + (\text-b)$

To find the difference between two numbers, we subtract them. Usually, we subtract them in the order they are named. For example, “the difference of +8 and -6” means $8 - (\text-6)$. We can find the value of $8 - (\text-6)$ by thinking $\text-6 + {?} = 8$. Representing this on a number line, we can see that the second arrow must be 14 units long, pointing to the right.

A number line with the numbers negative 10 through 10, indicated. An arrow starts at 0, points to the left, ends at negative 6, and is labeled "minus 6". A second arrow starts at negative 6, points to the right, ends at 8, and is labeled "plus 14". There is a solid dot at 8.

The difference of two numbers tells us how far apart they are on the number line and in which direction. The difference of +8 and -6 is 14 because these numbers are 14 units apart, and 8 is to the right of -6.

A number line with the numbers negative 10 through 10 indicated. Two solid dots are on the number line located at, negative 6 and 8. An arrow starts at negative 6, points to the right, ends at 8, and is labeled "positive 14."

If we subtract the same numbers in the opposite order, we get the opposite number. For example, “the difference of -6 and +8” means $\text-6 - 8$. This difference is -14 because these numbers are 14 units apart, and -6 is to the left of +8.

A number line with the numbers negative 10 through 10 indicated. Two solid dots are on the number line located at, negative 6 and 8. An arrow starts at 8, points to the left, ends at negative 6, and is labeled "negative 14."

In general, the distance between two numbers $a$ and $b$ on the number line is $|a - b|$. Note that the distance between two numbers is always positive, no matter the order. But the difference can be positive or negative, depending on the order.

Sometimes we use positive and negative numbers to represent quantities in context. Here are some contexts we have studied that can be represented with positive and negative numbers:

• Temperature
• Elevation
• Money
• Inventory

Using positive and negative numbers (and operations on positive and negative numbers) helps us understand and analyze the situations in context. To solve problems in these situations, we just have to understand what it means when a quantity is positive, what it means when a quantity is negative, and what it means to add and subtract quantities.

When two points in the coordinate plane lie on a horizontal line, we can find the distance between them by subtracting their $x$-coordinates.

When two points in the coordinate plane lie on a vertical line, we can find the distance between them by subtracting their $y$-coordinates.

Five line segments on a coordinate plane with the origin labeled “O”. The numbers negative 5 through 5, are indicated on the horizontal axis and the numbers negative 4 through 5 are indicated on the vertical axis. Each line segment is either vertical or horizontal. The first line segment is vertical that begins at the point negative 5 comma 3 and ends at the point negative 5 comma negative 4; the line segment is labeled with the expression 3 minus negative 4. The second line segment is horizontal and begins at the point negative 5 comma negative 4 and ends at the point negative 2 comma negative 4; the line segment is labeled with the expression negative 2 minus negative 5. The third line segment is vertical and begins at the point negative 2 comma negative 4 and ends at the point negative 2 comma 2; the line segment is labeled with the expression 2 minus negative 4. The fourth line segment is horizontal and begins at the point negative 2 comma 2 and ends at the point three comma 2; the line segment is labeled with the expression 3 minus negative 2. The fifth line segment is vertical and begins at the point 3 comma 2 and ends at the point 3 comma five; the line segment is labeled with the expression 5 minus 2.

Remember: The distance between two numbers is independent of the order, whereas the difference depends on the order.

We can use signed numbers to represent the position of an object along a line. We pick a point to be the reference point and call it zero. Positions to the right of zero are positive. Positions to the left of zero are negative.

A number line with the numbers negative 10 through 10 indicated. A point is indicated at zero and is labeled "reference point." Another point is indicated at negative 4 and is labeled "4 units to the left of zero." A third point is indicated at 7 and is labeled "7 units to the right of zero."

When we combine speed with direction indicated by the sign of the number, it is called velocity. For example, if you are moving 5 meters per second to the right, then your velocity is +5 meters per second. If you are moving 5 meters per second to the left, then your velocity is -5 meters per second.

If you start at zero and move 5 meters per second for 10 seconds, you will be 50 meters to the right of zero. In other words,

$\displaystyle 5\boldcdot 10 = 50$

If you start at zero and move -5 meters per second for 10 seconds, you will be 50 meters to the left of zero. In other words,

$\displaystyle \text-5\boldcdot 10 = \text-50$

In general, a negative number times a positive number is a negative number.

We can use signed numbers to represent time relative to a chosen point in time. We can think of this as starting a stopwatch. The positive times are after the watch starts, and negative times are times before the watch starts.

Three points are labeled on a number line. The numbers negative 10 through 10, in increments of 1, are indicated. A dot at negative 4 is labeled "4 seconds before the start time". A dot at 0 is labeled "start time". A dot at 7 is labeled "7 seconds after the start time".

If a car is at position 0 and is moving in a positive direction, then for times after that (positive times), it will have a positive position.

Number line. 7 evenly spaced tick marks. Scale negative 15 to 15, by 5's. Three equal sized arrows pointing to the right, one from 0 to 5, 5 to 10, and 10 to 15.

$5 \boldcdot 3 = 15$

For times before that (negative times), it must have had a negative position.

A blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing right and a dot are plotted. The first arrow points from negative 15 to negative 10. The second arrow points from negative 10 to negative 5. The third arrow points from negative 5 to 0. A dot is above 0.

$5 \boldcdot \text-3 = \text-15$

If a car is at position 0 and is moving in a negative direction, then for times after that (positive times), it will have a negative position.

A blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing left and a dot are plotted. The first arrow points from 0 to negative 5. The second arrow points from negative 5 to negative 10. The third arrow points from negative 10 to negative 15. A dot is above 0.

$\text-5 \boldcdot 3 = \text-15$

For times before that (negative times), it must have had a positive position.

A blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing left and a dot are plotted. The first arrow points from 15 to 10. The second arrow points from 10 to 5. The third arrow points from 5 to 0. A dot is above 0. 

$\text-5 \boldcdot \text-3 = 15$

Here is another way of seeing this:

• A positive number times a positive number always results in a positive number.
• A negative number times a positive number or a positive number times a negative number always results in a negative number.
• A negative number times a negative number always results in a positive number.

Any division problem is actually a multiplication problem:

• $6 \div 2 = 3$ because $2 \boldcdot 3 = 6$.
• $6 \div \text- 2 = \text-3$ because $\text-2 \boldcdot \text-3 = 6$.
• $\text-6 \div 2 = \text-3$ because $2 \boldcdot \text-3 = \text-6$.
• $\text-6 \div \text-2 = 3$ because $\text-2 \boldcdot 3 = \text-6$.

Because we know how to multiply signed numbers, that means we know how to divide them.

• A positive number divided by a negative number always results in a negative number.
• A negative number divided by a positive number always results in a negative number.
• A negative number divided by a negative number always results in a positive number.

A number that can be used in place of the variable that makes the equation true is called a solution to the equation. For example, for the equation $x \div \text-2 = 5$, the solution is -10 because it is true that $\text-10 \div \text-2 = 5$.

We can represent sums, differences, products, and quotients of rational numbers (and combinations of these) with numerical and algebraic expressions. 

We can write the product of two numbers in different ways.

• By putting a little dot between the factors, like this: $\text-8.5\boldcdot x$.
• By putting the factors next to each other without any symbol between them at all, like this: $\text-8.5x$.

We can write the quotient of two numbers in different ways as well.

• By writing the division symbol between the numbers, like this: ${\text-8.5}\div{x}$.
• By writing a fraction bar between the numbers, like this: $\frac{\text-8.5}{x}$.

When we have an algebraic expression like $\frac{\text-8.5}{x}$ and are given a value for the variable, we can find the value of the expression. For example, if $x$ is 2, then the value of the expression is -4.25, because $\text-8.5 \div 2 = \text-4.25$.

We can apply the rules for arithmetic with rational numbers to solve problems.

In general, $a - b = a + \text- b$.

If $a - b = x$, then $x + b = a$. We can add $\text- b$ to both sides of this second equation to get that $x = a + \text- b$.

Remember: The distance between two numbers is independent of the order, while the difference depends on the order.

And when multiplying or dividing:

• A positive number multiplied or divided by a negative number always has a negative result.

• A negative number multiplied or divided by a positive number always has a negative result.

• A negative number multiplied or divided by a negative number always has a positive result.