Quick challenge, no pencils: what is 25 × 16 in your head? Before you reach for the column method, ask yourself if there's a cleverer route. Would you ever actually write this one down, or is there a way to make the numbers do the hard work for you?

Three tricky multiplications, three different strategies. Watch each pair of panels and see how a hard multiplication turns into an easy one. As you watch, ask yourself why the answer doesn't change.

144 ÷ 8 by halving

Dividing by 8 is the same as halving three times, because 2 × 2 × 2 = 8. Read the three snapshots in order. The first shows 144 halved to 72. The second shows 72 halved to 36. Before you look at the third, predict where we'll land. Three halvings and we're there, no long division needed.

Now we work some together on one machine. The machine is already set to the friendly × 50 rule for our first problem, 24 × 25, which we reshape by double-and-halve to 12 × 50 — so 12 is the deliberately-halved input to send through. Watch first, then predict.

Learning the hidden-rule trick, ready for the next step. Before we start, here is how you work out a rule you cannot see. You probe the machine: send in a friendly number like 1 or 10, read what comes out, and reason backwards. If you send 1 in and 50 comes out, the rule must be × 50. If you send 10 in and 500 comes out, that confirms × 50. Once you are sure, you build that rule and press Check. We will use exactly this probe-and-build move in the Class Challenge, so get comfortable with it here.

Together, name the strategy for each before we reveal: 24 × 25, then 7 × 49, then 96 ÷ 4, and finally 45 × 11. Say which strategy fits before you send anything through.