# 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 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.

### Student Facing

• I can find the difference between two rational numbers.
• I understand how to subtract positive and negative numbers in general.

Building On