12 × 3, 12 × 1, 12 × 1/2
Can we tell if a product will be bigger or smaller than twelve — without doing any multiplication? Yes, we can!
Here is twelve on the number line. Now we'll multiply twelve by different numbers and watch where each product lands.
First, what about twelve times three? Notice that three is bigger than one.
Multiplying by a number greater than one makes the product bigger. Twelve times three lands all the way over here, to the right of twelve.
Next, what about twelve times one?
Multiplying by one keeps the number exactly the same. Twelve times one is still twelve.
Now what about twelve times one half? One half is less than one.
Multiplying by a number less than one makes the product smaller. Twelve times one half lands here, to the left of twelve.
So here's the rule. If the other factor is greater than one, the product grows. If it equals one, it stays the same. If it's less than one, it shrinks.
You can compare a product to a factor just by looking at the other factor. No multiplying needed. Nice work!