Ways to Multiply and Divide - Estimation of Products and Quotients

Estimation is the process of finding an approximate value that is 'close enough' to the actual answer. It helps in checking if a calculated answer is reasonable. On a number line, estimation looks like moving a point to the nearest landmark number (like a multiple of $10$ or $100$) to simplify mental math.

To round a number to the nearest ten, look at the digit in the ones place. If the digit is $5, 6, 7, 8,$ or $9$, round up by adding $1$ to the tens digit and making the ones digit $0$. If the digit is $0, 1, 2, 3,$ or $4$, round down by keeping the tens digit the same and making the ones digit $0$. For example, $47$ is closer to $50$ than $40$ on a visual scale, so it rounds to $50$.

To round a number to the nearest hundred, look at the digit in the tens place. If the tens digit is $5$ or more, round up (e.g., $672$ becomes $700$). If it is less than $5$, round down (e.g., $629$ becomes $600$). Visually, this is like identifying which century mark a number is nearest to on a long ruler.

Estimating Products involves rounding each factor to its highest place value before multiplying. For instance, to estimate $38 imes 52$, we round $38$ to $40$ and $52$ to $50$. The estimated product is $40 imes 50 = 2000$. This can be visualized as an area model where the dimensions are adjusted to the nearest tens for easier calculation.

Estimating Quotients is most effective when using 'Compatible Numbers' rather than strict rounding. Compatible numbers are numbers that are easy to divide mentally. For example, to estimate $254 \div 6$, we change $254$ to $240$ because $24$ is a multiple of $6$. This makes the mental division $240 \div 6 = 40$ much faster.

The approximation symbol $\approx$ is used to represent an estimate. Unlike the equal sign ($=$), the wavy lines of $\approx$ indicate that the numbers are not exactly equal but are approximately equivalent for the purpose of the calculation.

When multiplying rounded numbers that end in zeros, multiply the non-zero digits first and then append the total count of zeros. For example, in $700 \times 30$, multiply $7 \times 3 = 21$ and then add the three zeros to get $21,000$.