Breaking products apart

Idea: Cut one factor

Keep one factor whole and cut the other into pieces that end in zero.

• $67\times 7=60\times 7+7\times 7=469$
• Cut from above too: $43\times 27=43\times 30-43\times 3$
• Pick the cut with the smaller leftover

Formula: The law behind it

$a\times (b+c)=a\times b+a\times c$

Split one factor into a sum. The round piece may lie above the factor or below it.

Example: Cut from above

Compute $43\times 27$ in your head.

1. $27$ sits close to $30$
2. $43\times 30=1290$ and $43\times 3=129$
3. $1290-129=1161$

Common mistake: Cutting both factors

$40\times 20+3\times 7=821$ is not $43\times 27$: the cross terms are missing.

Instead Cut one factor only, or track all $4$ pieces: $821+340=1161$.

Idea: Multipliers with a reflex

Some multipliers are a short step from $10$ or $100$.

• By $5$: times $10$, then halve: $78\times 5=\frac{780}{2}=390$
• By $9$ and $11$: $76\times 9=760-76$, $52\times 11=520+52$
• By $25$: times $100$, then quarter: $36\times 25=\frac{3600}{4}=900$

Quick check: Times 15

Compute $64\times 15$.

Idea: Add from the left

Keep one running total and feed the second number in, largest piece first.

• $487+300=787$, then $787+50=837$, then $837+6=843$
• Every stage is close to the answer, so a cut-off still leaves an estimate

Idea: With the clock running

Look at both factors and spend a second choosing the cut.

• $5$, $9$, $11$ or $25$: use the reflex
• Otherwise cut the factor closer to a round number
• Say the running total aloud