Unit 7 Rational Numbers — Unit Plan

Positive numbers are numbers that are greater than zero. Negative numbers are numbers that are less than zero. The meaning of a negative number in a context depends on the meaning of zero in that context.

For example, if we measure temperatures in degrees Celsius, 0 degrees Celsius corresponds to the temperature at which water freezes.

In this context, positive temperatures are warmer than the freezing point, and negative temperatures are colder than the freezing point. A temperature of -6 degrees Celsius means that it is 6 degrees away from 0 and that it is less than 0. This thermometer shows a temperature of -6 degrees Celsius.

If the temperature rises a few degrees and gets very close to 0 degrees, without reaching it, the temperature is still a negative number.

Another example is elevation, which is a distance above or below sea level. An elevation of 0 refers to sea level. Positive elevations are higher than sea level, and negative elevations are lower than sea level.

In this context, a bird flying in the sky would have a positive elevation because it is higher than sea level. An octopus or a shark would have a negative elevation because it is swimming below sea level. 

Two numbers that are the same distance from 0 and on different sides of the number line are opposites. For example, points A and B are opposites because they are both 2.5 units away from 0 and on opposite sides of 0.

We can also say that the opposite of 8.3 is -8.3, and the opposite of $\text-\frac32$ is $\frac32$. The opposite of 0 is itself.

Here is another labeled number line with some rational numbers. A rational number is a number that can be written as a positive or negative fraction or zero.

The number 4 is positive, and its location is 4 units to the right of 0 on the number line. The number 4 can be written as $\frac41$ or $\frac{16}{4}$ or any other equivalent fraction. 

The number $\text-\frac23$ is negative, and its location is $\frac23$ units to the left of 0 on the number line. To locate $\text-\frac23$ on the number line, we can divide the distance between 0 and -1 into thirds and then count 2 thirds to the left of 0.

All fractions and their opposites are rational numbers.

The phrases “greater than” and “less than” can be used to compare numbers on the number line. For example, the numbers -2.7, 0.8, and -1.3, are shown on the number line.

Because -2.7 is to the left of -1.3, we say that -2.7 is less than -1.3. We write:

$\displaystyle \text-2.7 <\text -1.3$

In general, any number that is to the left of a number $n$ is less than $n$.

We can see that -1.3 is greater than -2.7 because -1.3 is to the right of -2.7. We write:

$\displaystyle \text-1.3 >\text -2.7$

In general, any number that is to the right of a number $n$ is greater than $n$.

We can also see that $0.8 > \text-1.3$ and $0.8 > \text-2.7$. In general, any positive number is greater than any negative number.

We can compare two numbers by looking at their positions on the number line: The number farther to the right is greater. The number farther to the left is less.

Sometimes we want to compare which one is closer to or farther from 0. For example, we may want to know how far away the temperature is from the freezing point of $0 ^\circ \text{C}$, regardless of whether it is above or below freezing. 

The absolute value of a number tells us its distance from 0.

For example, the absolute value of -4 is 4, because -4 is 4 units to the left of 0. The absolute value of 4 is also 4, because 4 is 4 units to the right of 0. Opposites always have the same absolute value because they are both the same distance from 0.

The distance from 0 to itself is 0, so the absolute value of 0 is 0. Zero is the only number whose distance to 0 is 0. For all other absolute values, there are always two numbers—one positive and one negative—that have that distance from 0.

To say, “the absolute value of 4,” we write "$|4|$."

To say, “the absolute value of -8 is 8,” we write "$|\text-8|=8$."

Let’s say a movie ticket costs less than $15. If $c$ represents the cost of a movie ticket, we can use $c < 15$ to express what we know about the cost of a ticket.

Any value of $c$ that makes the inequality true is called a solution to the inequality.

For example, 5 is a solution to the inequality $c < 15$ because $5<15$ (or “5 is less than 15”) is a true statement, but 17 is not a solution because $17<15$ (“17 is less than 15”) is not a true statement.

If a situation involves more than one boundary, or limit, we will need more than one inequality to express it.

For example, if we knew that it rained for more than 10 minutes but less than 30 minutes, we could describe the number of minutes that it rained ($r$) with the following inequalities and number lines: $\displaystyle r > 10$

$\displaystyle r < 30$

Any number of minutes greater than 10 is a solution to $r>10$, and any number less than 30 is a solution to $r<30$. But to meet the condition of “more than 10 but less than 30,” the solutions are limited to the numbers between 10 and 30 minutes, not including 10 and 30.

We can show the solutions visually by graphing the two inequalities on one number line.

Just as the number line can be extended to the left to include negative numbers, the $x$- and $y$-axes can also be extended to include negative values. This creates the coordinate plane, a system that can be used to describe the locations of points.

For example, point $B$ can be described by the ordered pair $(\text-4,1)$. The $x$-value of -4 tells us that the point is 4 units to the left of the $y$-axis. The $y$-value of 1 tells us that the point is 1 unit above the $x$-axis. Point $B$ is located in Quadrant II.

The same reasoning applies to the points $A$ and $C$. The $x$- and $y$-coordinates for point $A$ are positive, so $A$ is to the right of the $y$-axis and above the $x$-axis. Point $A$ is located in Quadrant I.

The $x$- and $y$-coordinates for point $C$ are negative, so $C$ is to the left of the $y$-axis and below the $x$-axis. Point $C$ is located in Quadrant III.

Quadrant IV contains points whose $x$-coordinates are positive and whose $y$-coordinates are negative.

Points in the coordinate plane can give us information about a situation. One common situation is about money.

For example, to open a bank account, money has to be added to the account. The account balance is the amount of money in the account at any given time. If we put in $350 when opening the account, then the account balance will be 350.

Sometimes we may have no money in the account and need to borrow money from the bank. In that situation, the account balance would have a negative value. If we borrow $200, then the account balance is -200.

A coordinate plane can be used to display both the balance and the day or time. This allows us to see how the balance changes over time or to compare the balances of different days. Similarly, if we plot data such as temperature over time in the coordinate plane, we can see how temperature changes over time or compare temperatures at different times.

A factor of a whole number is a whole number that divides evenly into that number, without a remainder. For example, 1, 2, 3, 4, 6, and 12 are all factors of 12 because each of them divides 12 evenly, without a remainder.

A common factor of two whole numbers is a factor that they have in common. For example, 1, 3, 5, and 15 are factors of 45. They are also factors of 60. We call 1, 3, 5, and 15 common factors of 45 and 60.

The greatest common factor (sometimes written as GCF) of two whole numbers is the greatest of all the common factors. For example, 15 is the greatest common factor for 45 and 60. 

One way to find the greatest common factor of two whole numbers is to list all of the factors for each and then look for the greatest factor they have in common. To find the greatest common factor of 18 and 24, first list all the factors of each number.

• Factors of 18: 1, 2, 3, 6, 9, 18

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

The common factors are 1, 2, 3, and 6. Of these common factors, 6 is the greatest one, so 6 is the greatest common factor of 18 and 24.