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 \(x\) 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).
Lesson overview
 13.1 Warmup: True or False: Rational Numbers (5 minutes)
 13.2 Activity: Card Sort: The Same But Different (10 minutes)

13.3 Activity: Near and Far From Zero (15 minutes)
 Includes "Are you Ready for More?" extension problem

13.4 Optional Activity: Seagulls and Sharks Again (10 minutes)
 There is a digital applet in this activity.
 Lesson Synthesis
 13.5 Cooldown: Make Them True (5 minutes)
Learning goals:
 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.
Learning goals (student facing):
 Let’s develop our signed number sense.
Learning targets (student facing):
 I can add, subtract, multiply, and divide rational numbers.
 I can evaluate expressions that involve rational numbers.
Required materials:
 preprinted slips, cut from copies of the blackline master
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.
Glossary:
 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 \(\text\frac{75}{100}\).
 Access the complete Grade 7 glossary.
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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