Unit 7 Rational Numbers — Unit Plan

Numbers can be positive or negative. On a horizontal number line, positive numbers are to the right of 0 and negative numbers are to the left of 0. We read "-3" as "negative 3". A positive number can be written with a "+" sign (like +4) or without one (like 4).

The number line tells us how numbers compare. The greater number is the one farther to the right, and the less number is the one farther to the left. For example, -2 is greater than -5 because -2 is to the right of -5, and -5 is less than -2 because -5 is to the left of -2. Any positive number is greater than any negative number.

Some numbers are closer to zero than others. To find out, we count how many units each number is away from 0. For example, -10 is closer to zero than -35 because -10 is only 10 units from 0, while -35 is 35 units from 0.

Number lines do not have to go left to right. In real life, many number lines go up and down. A thermometer is a vertical number line for temperature: higher on the thermometer means warmer. Below 0° is cold (negative), above 0° is warm (positive).

Elevation is another vertical number line. Sea level is 0 meters. A bird in the sky has a positive elevation (above 0). A fish swimming below the ocean surface has a negative elevation (below 0). An elevation closer to 0 means closer to sea level.

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. 

We can zoom into the number line to see decimal numbers between the integers. When each tick is a tenth, the point at 2.1 is one tenth to the right of 2, and the point at 2.5 is halfway between 2 and 3. On the left side of zero, the point at -2.4 is four tenths to the left of -2.

When the tenths are not shown, we can estimate where a decimal goes. For example, 3.2 lands a little to the right of 3, about two tenths of the way toward 4.

To compare two numbers, we look at where they sit on the number line. The greater number is the one farther to the right. Since -2.0 is to the right of -2.4, we say $\text-2.0 > \text-2.4$.

Two numbers that are the same distance from 0 and on different sides of the number line are opposites. For example, $\text-\frac{1}{2}$ and $\frac{1}{2}$ are opposites because they are both $\frac{1}{2}$ unit from 0, but on opposite sides.

To find the opposite of a number, we flip its sign. The opposite of 2.2 is -2.2, and the opposite of $\text-\frac{3}{2}$ is $\frac{3}{2}$. The opposite of 0 is itself — zero is the only number that is its own opposite.

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.

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

For example, -5 is to the right of -8, so -5 is greater than -8. And -8 is to the left of -5, so -8 is less than -5. The same rule works for any two numbers: -3 is greater than -7, because -3 is farther to the right; -9 is less than -1, because -9 is farther to the left.

Even without drawing a number line, we can picture it in our heads. For two negative numbers, the number with the smaller distance from 0 is greater. For example, -3.2 is greater than -3.4, because -3.2 is closer to 0 and therefore farther to the right.

We can write comparisons quickly using two symbols:

• > means greater than
• < means less than

So instead of writing "3 is greater than -3" we can write $3 > \text-3$. And "-35 is less than -10" becomes $\text-35 < \text-10$.

A helpful trick: the symbol always opens toward the greater number (like an alligator's mouth opening toward the bigger meal). Any number to the right of 0 is greater than 0, and any number to the left of 0 is less than 0.

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.

The absolute value of a number is its distance from 0 on the number line. Because distance is always at least 0, absolute value is always zero or positive — never negative.

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.

To write “the absolute value of a number”, we use two straight vertical bars around the number. So the absolute value of -5 is written $|\text-5|$, and we say $|\text-5| = 5$. Similarly, $|\text-12.9| = 12.9$ and $|\text-3.2| = 3.2$.

Opposites always have the same absolute value. The numbers -7 and 7 are opposites because they are both 7 units from 0, so $|\text-7| = |7| = 7$. The distance from 0 is the same no matter which side of 0 the number is on.

The only number whose distance from 0 is 0 is 0 itself, so $|0| = 0$.

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 "$|\text-8|=8$."

An inequality line is a green ray on a number line with an open circle at one end and an arrow pointing either left (less than) or right (greater than). The open circle marks the boundary, and every point on the shaded ray is a value that makes the inequality true.

Inequalities can also be written as sentences using $>$ (greater than) and $<$ (less than). For example, if $a$ is a traveler's age and the rule is "you have to be at least 16," the inequality is $a > 16$.

A solution to an inequality is any value that makes it true. For example, 10 is a solution to $n > 4$ because $10 > 4$ is a true statement. Inequalities usually have many (often infinitely many) solutions.

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.

The coordinate plane is a grid made from two perpendicular number lines: the horizontal $x$-axis and the vertical $y$-axis. Every point on the plane can be named by an ordered pair $(x, y)$ — the first number tells you how far to move along the x-axis, and the second number tells you how far to move along the y-axis.

The x-axis and y-axis can both include negative numbers. The point $(\text-3, 2)$ is 3 units left of the origin and 2 units up.

When two points share the same row (same $y$-coordinate) or the same column (same $x$-coordinate), the distance between them is the number of hops between the two different coordinates. For example, the distance from $(\text-6, 3)$ to $(1, 3)$ is 7 because the points share $y = 3$ and the $x$-coordinates are 7 units apart.

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.