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Distances on the Number Line 2


Alignments to Content Standards: 7.NS.A.1

Task

Numberline_with_oppo_544004b9c22e5f8d252f6059b0821c8c

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:

A_and_b_56a3be764696c6200f98919663799706

Ab_7028cdc105ff6781ad7e1a7a4eee2a2b

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.

$$ \begin{align} a+b=& a+(-a) \\ =& a-a \\ =& 0 \end{align} $$

Peter Cincotta says:

about 5 years

I like this task. I think it is a good way to help a teacher to determine a student's level of understanding of abstractness on the number line as well as directed distances. I particularly like that there is no mention and no hint of a suggestion that the number represented by "b" is negative or less than zero.