Mental multiplication and division (age 9+)

Activities to try

1. 1

   Scaling studio: watch the known fact grow

   Why it works: Multiplying/dividing by tens and hundreds scales a known fact by place value; tens×tens makes hundreds, so the number of zeros follows from the place-value reasoning, not from a memorised 'append a zero' rule.

   1. Take 40×60 and name the known fact hiding inside it (4×6=24).
   2. Place 24 on a place-value chart and scale it by ×100, watching the digits slide two columns to the left to make 2400 — because a ten times a ten makes a hundred, and both numbers contributed a ten.
   3. Explain in units: '4 tens × 6 tens = 24 hundreds = 2400', so the answer has two zeros, not one.
   4. Try the same scaling move on a division: link 360÷9 to the known fact 36÷9=4, then scale by ×10 to get 40.
   5. Contrast with a tricky case where a naive 'add a zero' rule would mislead (e.g. a decimal such as 0.4×60, which is 24, not 240) to feel why scaling by place value is the reliable idea.
2. 2

   Partition to make it friendly

   Why it works: The distributive law lets you break a hard factor into friendly parts and multiply each part, then recombine — so you only need facts you already know.

   1. Take 7×15 and split 15 into friendly parts: 10 and 5.
   2. Compute 7×10 = 70 and 7×5 = 35, then add: 70 + 35 = 105. State both multiplications explicitly so the second part is not dropped.
   3. Back the partition with an area model: a rectangle 7 wide split into a 7×10 tile and a 7×5 tile, whose areas sum to the whole.
   4. Say the full reasoning aloud and check it against an estimate (7×15 is a little over 100).
3. 3

   Two routes, same answer: choosing the friendliest

   Why it works: Mental maths is about selecting the friendliest of several valid routes to the same answer, not following one prescribed procedure.

   1. Take 7×15 and solve it two different ways: (a) partition into 7×10 + 7×5, and (b) factor-scaling (15 = 3×5, so 7×15 = (7×5)×3 = 35×3, or 7×3×5 = 21×5).
   2. Confirm both routes land on 105.
   3. Discuss which route felt quicker for this calculation and why.
   4. Repeat the compare-methods habit on a mixed set and justify the most efficient choice for each item.
   5. Optionally present the same calculation in more than one worked form side by side so the alternatives can be compared directly.