Distances on the Number Line 2
Alignments to Content Standards:
7.NS.A.1
Task
On the number line above, the numbers $a$ and $b$ are the same distance from $0$. What is $a+b$? Explain how you know.
IM Commentary
The purpose of this task is meant to reinforce students' understanding of rational numbers as points on the number line and to provide them with a visual way of understanding that the sum of a number and its additive inverse (usually called its "opposite") is zero. Students should have lots of opportunities to represent adding specific rational numbers before they work on answering this one.
Solution
We are given that $a$ and $b$ are the same distance from zero. However, from the above number line we can see that $a$ and $b$ are on different sides of zero. We can visualize this by representing $a$ and $b$ as directed distances on the number line:
If we start at zero and move $a$ units to the right, and then move the same number of units to the left, we will be back at 0.
We can also represent this symbolically. Since $a$ and $b$ are the same distance from zero but are on opposite sides of zero, we know that they are opposites, so $b = -a$. The sum of a number and its opposite is always zero.
$$
\begin{align}
a+b=& a+(-a) \\
=& a-a \\
=& 0
\end{align}
$$
Distances on the Number Line 2
On the number line above, the numbers $a$ and $b$ are the same distance from $0$. What is $a+b$? Explain how you know.