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# Integers on the Number Line 2

Alignments to Content Standards: 6.NS.C.6.a

## Task

Below is a number line with 0 and 1 labeled:

1. Find and label the numbers $-2$ and $-4$ on the number line. Explain.
2. Find and label the numbers $-(-2)$ and $-(-4)$ on the number line. Explain
3. Find and label the number $-0$ on the number line. Explain.

## IM Commentary

The goal of this task is to study, with a number line, why it makes sense for a whole number $a$ that  $-(-a) = a$. The negative sign preserves the distance from 0 but switches the direction from 0. There are only two directions to go on the number line: left and right.  Therefore, if we switch directions twice this will bring us back to our original direction from 0.  Since the distance from 0 does not change, this means that taking the opposite of a negative number will produce a positive number with equal distance from 0.  This reasoning will be foundational for future 7th grade work where students must reason about the opposite of quantities such as (-2 + 8) and -(-2 + 8).

This task complements https://www.illustrativemathematics.org/illustrations/283 which investigates plotting negative integers on the number line.

## Solution

1. The numbers -2 and -4 are both negative and so they are both located to the left of 0. To place -2 on the number line, we need to move two units to the left of 0 and for -4 we move four units to the left of 0. The unit has been marked on the number line (in the positive direction) so we can count 2 and 4 equal tick marks to place -2 and -4 as shown below:

2. For -(-2) we are looking for the negative or opposite of -2. Since -2 is two units to the left of 0 this means that -(-2) will be two units to the right of 0. Similarly -(-4) is 4 units to the right of 0. These are plotted on the number line below:

Notice that -(-2) = 2 and -(-4) = 4. The first negative sign switches us from the right of 0 to the left of 0 and the second negative sign puts us back where we started.

3. Since 0 is neither to the right nor to the left of 0, taking the opposite of 0 will give 0, that is -0=0. For a positive number, such as 2, the opposite or -2 is a negative number. For a negative number such as -4, the opposite or -(-4) is positive. The number 0 is neither positive nor negative so we must have -0=0.

## Comments

#### Julie says:

over 1 year

We did this immediately after Integers on the Number Line 1 in the same class period, and I made some changes that worked out really well as a followup. I made the other number at the mark just to the right of 0's a 2 instead of a 1. That helped reinforce the idea of the negative of a number being the same distance from 0, but in the opposite direction (they had to think about it a little instead of just counting marks from 0).

Also, I added a part d to ask something like, "What distance is -4 from 0?" to see if they would answer 2 (counting marks) or -4 (not understanding distance as a non-negative number) or 4.

I am sooooo grateful to IM for the 6.NS.C tasks here. My district is using an ancient curriculum with publisher's supplements for CC, and the supplements are too horrendous to contemplate using. These tasks are incredibly helpful.