Lesson 18Expressions with Rational Numbers

Learning Goal

Let’s develop our signed number sense.

Learning Targets

  • I can add, subtract, multiply, and divide rational numbers.

  • I can evaluate expressions that involve rational numbers.

Lesson Terms

  • rational number

Warm Up: True or False: Rational Numbers

Problem 1

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

  1. is negative

Activity 1: Card Sort: The Same But Different

Problem 1

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

Activity 2: Near and Far From Zero

Problem 1

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

Problem 2

When and , which expression:

  1. has the largest value?

  2. has the smallest value?

  3. is the closest to zero?

Problem 3

When and , which expression:

  1. has the largest value?

  2. has the smallest value?

  3. is the closest to zero?

Problem 4

When and , which expression:

  1. has the largest value?

  2. has the smallest value?

  3. is the closest to zero?

Are you ready for more?

Problem 1

Are there any values could you use for and that would make all of these expressions have the same value? Explain your reasoning.

Activity 3: Seagulls and Sharks Again

Problem 1

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

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.

  1. A dragonfly at , where

  2. A jellyfish at , where

  3. An eagle at , where .

  4. A clownfish at , where

  5. A vulture at , where

  6. A goose at , where

open applet in presentation mode
Print Version

A seagull has a vertical position , and a shark has a vertical position . Draw and label a point on the vertical axis to show the vertical position of each new animal.

  1. A dragonfly at , where

  2. A jellyfish at , where

  3. An eagle at , where .

  4. A clownfish at , where

  5. A vulture at , where

  6. A goose at , where

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."

Lesson Summary

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

Sums:

Differences:

Products:

Quotients:

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

  • By putting a little dot between the factors, like this: .

  • By putting the factors next to each other without any symbol between them at all, like this: .

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

  • By writing the division symbol between the numbers, like this: .

  • By writing a fraction bar between the numbers like this: .

When we have an algebraic expression like and are given a value for the variable, we can find the value of the expression. For example, if is , then the value of the expression is , because .