Lesson 13
Expressions with Rational Numbers
Lesson Narrative
As students start to gain fluency with rational number arithmetic, they encounter complicated numerical expressions, and algebraic expressions with variables, and there is a danger that they might lose the connection between those expressions and numbers on the number line. The purpose of this lesson is to help students make sense of expressions, and reason about their position on the number line, for example whether the number is positive or negative, which of two numbers is larger, or whether two expressions represent the same number. They work through common misconceptions that can arise about expressions involving variables, for example the misconception that \(\textx\) must always be a negative number. (It is positive if \(x\) is negative.) In the last activity they reason about expressions in \(a\) and \(b\) given the positions of \(a\) and \(b\) on a number line without a given scale, in order to develop the idea that you can always think of the letters in an algebraic expression as numbers and deduce, for example, that \(\frac14a\) is a quarter of the way from 0 to \(a\) on the number line, even if you don't know the value of \(a\).
When students look at a numerical expression and see without calculation that it must be positive because it is a product of two negative numbers, they are making use of structure (MP7).
Learning Goals
Teacher Facing
 Evaluate an expression for given values of the variable, including negative values, and compare (orally) the resulting values of the expression.
 Generalize (orally) about the relationship between additive inverses and about the relationship between multiplicative inverses.
 Identify numerical expressions that are equal, and justify (orally) that they are equal.
Student Facing
Let’s develop our signed number sense.
Required Materials
Required Preparation
Print and cut up slips from the Card Sort: The Same But Different blackline master. Prepare 1 copy for every 2 students. If possible, copy each complete set on a different color of paper, so that a stray slip can quickly be put back.
Learning Targets
Student Facing
 I can add, subtract, multiply, and divide rational numbers.
 I can evaluate expressions that involve rational numbers.
Glossary Entries

rational number
A rational number is a fraction or the opposite of a fraction.
For example, 8 and 8 are rational numbers because they can be written as \(\frac81\) and \(\text\frac81\).
Also, 0.75 and 0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(\text\frac{75}{100}\).