Question

an integer, n, and its opposite, -n, combine to make 0

4 and -4 are opposites, they combine to make 0

-7 -6 -5 -4 -3 -2

do you understand?

1. essential question how are integers and their opposites related?

1. reasoning in order for an atom to have a zero charge, every proton, which has a charge of +1, must be matched with an electron, which has a charge of −1. a helium atom has 2 protons and 2 electrons. explain why a helium atom has a zero charge.

1. model with math explain how to use a number line to show that opposite quantities combine to make 0.

Explanation:

Question 1

Integers and their opposites are related such that for any integer \( n \), its opposite is \( -n \). When an integer and its opposite are added together, the sum is \( 0 \) (e.g., \( 4 + (-4) = 0 \)). Also, they are equidistant from \( 0 \) on the number line, with the integer to the right of \( 0 \) and its opposite to the left (or vice versa for negative integers).

A proton has a charge of \( +1 \), and an electron has a charge of \( -1 \). A helium atom has 2 protons and 2 electrons. The total charge from protons is \( 2\times(+1) = +2 \), and the total charge from electrons is \( 2\times(-1)= -2 \). When we add these charges together (\( +2 + (-2) \)), we get \( 0 \), so the helium atom has a zero charge.

1. Step 1: Choose a quantity and its opposite

Let's take a positive quantity, say \( 3 \), and its opposite \( -3 \).

1. Step 2: Locate on the number line

Find \( 3 \) on the number line (to the right of \( 0 \), 3 units away). Find \( -3 \) on the number line (to the left of \( 0 \), 3 units away).

1. Step 3: Show combination to 0

To show their combination, we can think of moving from \( 0 \) to \( 3 \) (a distance of \( +3 \)) and then from \( 3 \) to \( 0 \) by moving \( -3 \) units (or vice versa: moving from \( 0 \) to \( -3 \) and then \( +3 \) units). Mathematically, \( 3+(-3) = 0 \), and on the number line, the two movements (or the two points) cancel each other out to return to \( 0 \).

Answer:

An integer \( n \) and its opposite \( -n \) sum to \( 0 \) (\( n + (-n)=0 \)) and are equidistant from \( 0 \) on the number line.

Question 2