Lesson 6
Subtracting Rational Numbers
Lesson Narrative
In this lesson, students see that the difference between two numbers can be positive or negative, but the distance between two numbers is always positive. Using the geometry of the number line (MP7), they see that if you switch the order in which you subtract two numbers, the difference becomes its opposite.
For example, to find the difference in temperature between +70\(^\circ \text{C}\) and +32\(^\circ \text{C}\) we calculate \(70  32 = 38\), so the difference is 38\(^\circ \text{C}\). The distance between these two is also 38\(^\circ \text{C}\). On the other hand, to find the difference in temperature between +32\(^\circ \text{C}\) and \(+70^\circ \text{C}\) we calculate \(32  70 = 38\), so the difference is \(38^\circ \text{C}\). The distance is still 38\(^\circ \text{C}\). In general, if \(a  b = x\), then \(b  a = x\). By observing the outcome of several examples, students may conjecture that this is always true (MP8).
Learning Goals
Teacher Facing
 Compare and contrast (orally) subtraction expressions that have the same numbers in the opposite order.
 Recognize that the “difference” of two numbers can be positive or negative, depending on the order they are listed, while the “distance” between two numbers is always positive.
 Subtract signed numbers, and explain (orally) the reasoning.
Student Facing
Let's bring addition and subtraction together.
Required Materials
Required Preparation
Use of calculators is optional. In this lesson, the important insights come from observing the outcome of evaluating expressions. Practice evaluating the expressions is of secondary importance.
Learning Targets
Student Facing
 I can find the difference between two rational numbers.
 I understand how to subtract positive and negative numbers in general.
CCSS Standards
Glossary Entries

deposit
When you put money into an account, it is called a deposit.
For example, a person added \$60 to their bank account. Before the deposit, they had \$435. After the deposit, they had \$495, because \(435+60=495\).

withdrawal
When you take money out of an account, it is called a withdrawal.
For example, a person removed \$25 from their bank account. Before the withdrawal, they had \$350. After the withdrawal, they had \$325, because \(35025=325\).