🔥 Flame and Ice: Working with Integers

In this episode we add, subtract and multiply integers. Our helpers are the flame and the ice!

Every flame is worth +1 and every ice is worth −1. Three flames make +3, and three ice make −3.

When a flame meets an ice, they cancel each other out: (+1) + (−1) = 0. This is called a zero pair. +1 and −1 are opposites.

To work out (+5) + (−3), put down 5 flames and 3 ice. Match each ice with a flame: 3 zero pairs vanish. That leaves 2 flames, so the answer is +2.

(+2) − (−3): we must take 3 ice away from 2 flames, but there is no ice! First add 3 zero pairs, so the value is still +2. Then take away the 3 ice. That leaves 5 flames, so the answer is +5.

Subtracting a number is the same as adding its opposite. Turn the subtraction into an addition and change the sign of the second number: (−4) − (+1) = (−4) + (−1) = −5.

A common mistake: −3 − 5 is not −2! Taking 5 away from −3 means adding −5 to −3: 3 ice and 5 more ice make 8 ice. The correct answer is −8.

For multiplying, look at the pattern: 3 × 2 = 6, 3 × 1 = 3, 3 × 0 = 0. The answer goes down by 3 each time, so 3 × (−1) = −3. Positive × negative = negative.

Now let’s multiply by −3: −6, −3, 0. This time the answer goes up by 3 each step, so (−3) × (−1) = +3. Negative × negative = positive.

Summary: a flame is +1, an ice is −1, and a pair makes 0. Subtracting means adding the opposite. When you multiply, the same signs give a positive answer and different signs give a negative one. Now try it in Frost & Flame!