Lesson 13: Expressions with Rational Numbers

Let’s develop our signed number sense.

13.1: True or False: Rational Numbers

Decide if each statement is true or false. Be prepared to explain your reasoning.

  1. $(\text-38.76)(\text-15.6)$ is negative
  2. $10,000 - 99,999 < 0$
  3. $\left( \frac34 \right) \left( \text- \frac43 \right) = 0$
  4. $(30)(\text- 80) - 50 = 50 - (30)(\text- 80)$

13.2: Card Sort: The Same But Different

Your teacher will give you a set of cards. Group them into pairs of expressions that have the same value.

13.3: Near and Far From Zero

  $a$ $b$ $\text-a$ $\text-4b$ $\text-a+b$ $a\div \text-b$ $a^2$ $b^3$
row 1 $\text-\frac12$ 6            
row 2 $\frac12$ -6            
row 3 -6 $\text-\frac12$            

  1. For each set of values for $a$ and $b$, evaluate the given expressions and record your answers in the table.

  2. When $a= \text-\frac12$ and $b= 6$, which expression:

    has the largest value?

    has the smallest value?

    is the closest to zero?

  3. When $a= \frac12$ and $b= \text-6$, which expression:

    has the largest value?

    has the smallest value?

    is the closest to zero?

  4. When $a= \text-6$ and $b= \text-\frac12$, which expression:

    has the largest value?

    has the smallest value?

    is the closest to zero?

13.4: Seagulls and Sharks Again

A seagull has a vertical position $a$, and a shark has a vertical position $b$.

A vertical number line with the number zero indicated is labeled “vertical position in meters." On the number line, "a" is high above zero, "b" is slightly below zero and a horizontal line extends to the right at zero. A seagull has the vertical position of “a” and a shark has the vertical position of "b."

In the applet, you may choose to start by clicking on the open circles on the seagull and shark to drag them to new vertical positions. Once you have them in place, drag each of the other animals to the vertical axis to show its position, determined by the expression next to it.

GeoGebra Applet fJPk7QNu

  1. A dragonfly at $d$, where $d=-b$

  2. A jellyfish at $j$, where $j=2b$

  3. An eagle at $e$, where $4e=a$.

  4. A clownfish at $c$, where $c=-\frac{a}{2}$

  5. A vulture at $v$, where $v=a+b$

  6. A goose at $g$, where $g=a-b$

Summary

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

Sums:

$\frac12 + (\text-9)$

$\text-8.5 + x$

Differences:

$\frac12 - (\text-9)$

$\text-8.5 - x$

Products:

$(\frac12)(\text-9)$

$\text-8.5x$

Quotients:

$(\frac12)\div(\text-9)$

$\frac{\text-8.5}{x}$

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 $\text-8.5 \div 2 = \text-4.25$.

Practice Problems ▶