Here is a puzzle: x + 5 = 12. The beam is level, so both sides are worth the same. What could the mystery number x be? And more importantly, how could you be sure without just guessing?

Four equations on the balance beam. For each one, ask yourself what is being done to x, then watch the opposite move done to both pans. The last two are different from the first two, watch closely for how they are undone.

Next comes a division equation, shown step by step on the board (the beam interactive shows add, take-away and equal shares, not ÷). Watch x ÷ 3 = 5.

Ask what is happening to x: it is divided by 3. The opposite of dividing by 3 is multiplying by 3. Do that same move to both sides:

x ÷ 3 × 3 = 5 × 3

So x = 15. Check by putting 15 back into the original: 15 ÷ 3 = 5. Both sides match, so it is correct. Division is undone by multiplication, just as equal-share undoes multiplication.

Now we solve some together on the beam. We take them one at a time, and only the equation we are working on shows on the beam. First we do x + 6 = 10. Then x − 4 = 9. Then 4x = 12. Last comes x + 8 = 8, which has a surprise in it. Say what you think x is before we check that one.

Solve each one on the beam. Start with x + 3 = 7. Then x + 5 = 12. Then 2x = 10. Then x − 4 = 9. Then 3x = 12. Last comes the stretch, 4x = 18. Its answer lands between two whole numbers, so a mystery number here is allowed to be a decimal — that is not a mistake.