Lesson objective: Understand that for every expression that includes subtraction, there is an equivalent expression using addition of the opposite of the number being subtracted.

Students bring prior knowledge of finding sums of rational numbers. This prior knowledge is extended to subtraction of integers as students use distance and direction on the number line to understand the result of subtraction and its connection to adding an opposite. A conceptual challenge students may encounter is not being able to extend their thinking to subtracting negative numbers. It can be very confusing that the result of a difference may be larger than the number they start with.

The concept is developed through work with a number lines, which allows us to explore distance and direction. Students begin to think about how, when *a* and *b* are rational numbers, *a* - *b* and *a*+(-*b)* are equivalent expressions because they have the same value, and we can verify this with a number line.

This work helps students deepen their understanding of operations, because they explore and use a relationship between addition and subtraction. Students learn the new connection of these operations that subtracting gives the same result as adding an opposite, and that it is often more efficient to think about a subtraction that way.

Students engage in Mathematical Practice 3 (Construct viable arguments and critique the reasoning of others) as they explore adding and subtracting with signed numbers with an emphasis on distance and direction as they work to support of disprove Jack's claim and address Susan's concern. They also engage in Mathematical Practice 7 (Look for and make use of structure) as they use distance and direction to connect addition and subtraction with positive and negative numbers.

**Key vocabulary:**

**Special materials needed:**

- Blank number lines (optional)