Explain two ways you can use a number line to order the integers $-3$, $+1$, and $-10$.

1. Plot each integer on the number line and arrange them from left to right. The integers will naturally fall in order from least to greatest.

2. Identify the relative positions of each integer based on their values. Since [tex]$-10$[/tex] is the smallest, it will be the furthest left, followed by [tex]$-3$[/tex], and then [tex]$+1$[/tex] which is the largest.

Absolutely! Ordering integers using a number line is a very visual and clear way to understand the concept of comparing and sorting numbers. Let's take the integers $-3$, $+1$, and $-10$ and explore two ways to order them.

### Method 1: Using a Horizontal Number Line

1. Draw the Number Line:
- Draw a horizontal line and mark a point in the middle as zero ($0$).
- Going to the right, mark positive integers ($+1$, $+2$, $+3$, and so on).
- Going to the left, mark negative integers ($-1$, $-2$, $-3$, and so on).

2. Plot the Integers on the Number Line:
- Locate and mark $-3$ on the number line: This will be three steps to the left of zero.
- Locate and mark $+1$ on the number line: This will be one step to the right of zero.
- Locate and mark $-10$ on the number line: This will be ten steps to the left of zero.

3. Order the Integers:
- On the number line, the further left a number is, the smaller it is.
- $-10$ is to the left of $-3$.
- $-3$ is to the left of $+1$.

Therefore, in ascending order, the integers are: $-10, -3, +1$.

### Method 2: Using Distance from Zero

Another method involves comparing their distances from zero:

1. Understand Negative and Positive Distances:
- The further a number is to the left of zero, the smaller it is.
- The further a number is to the right of zero, the larger it is.

2. Compare Integer Distances from Zero:
- $-10$ is 10 units to the left of zero.
- $-3$ is 3 units to the left of zero.
- $+1$ is 1 unit to the right of zero.

3. Order Based on Position Relative to Zero:
- Since $-10$ is the furthest to the left, it is the smallest.
- Followed by $-3$, which is closer to zero but still on the left side, making it larger than $-10$ but smaller than $+1$.
- $+1$ is to the right of zero, making it the largest among the three integers.

Thus, the order from smallest to largest is: $-10, -3, +1$.

Using these two methods, we confirm that the integers [tex]$-3, +1$[/tex], and [tex]$-10$[/tex] are ordered as [tex]$-10, -3, +1$[/tex]. Both approaches visually and conceptually help to understand the ordering of integers.