Developing number sense to check if answers make sense in context
Reserve & Extensions • K-12
Not every math problem requires an exact answer. In everyday life -- shopping, cooking, budgeting, science -- quick estimates are often more useful than precise calculations. More importantly, estimation helps you catch errors: if your calculator says a $3.50 item costs $350.00 after tax, your number sense should set off alarm bells.
The simplest estimation strategy: round numbers before computing.
Estimate: 487 + 312 + 196
Round to the nearest hundred: 500 + 300 + 200 = 1000
Exact answer: 995. The estimate is very close.
Estimate: 38 × 52
Round: 40 × 50 = 2000
Exact answer: 1976. Again, close enough for a quick check.
Front-end estimation uses only the leading (leftmost) digits, then adjusts with the remaining digits.
Estimate: $3.87 + $5.42 + $2.19
Front-end: $3 + $5 + $2 = $10
Adjust: $0.87 + $0.42 + $0.19 is roughly $1.50
Estimate: $10 + $1.50 = $11.50
Exact: $11.48. Front-end estimation with adjustment is very accurate.
Compatible numbers are numbers that work well together -- they divide evenly, add to round numbers, or are easy to compute mentally.
To estimate 7,342 / 8, think: "What is close to 7,342 that divides nicely by 8?" Try 7,200: 7,200 / 8 = 900. Or 8,000 / 8 = 1,000. The exact answer (917.75) is between these estimates.
An order of magnitude estimate tells you the general size -- is the answer in the tens, hundreds, thousands, or millions? This is the most basic reasonableness check.
The population of a city: order of magnitude 105 or 106. Your height in millimeters: order of magnitude 103.
Fermi problems (named after physicist Enrico Fermi) ask you to estimate quantities that seem impossible to know, by breaking them into pieces you can estimate.
How many piano tuners are there in Chicago?
The actual number is estimated at 100-200, so our order of magnitude (101 to 102) is right.
Students often think estimation means "guessing." It does not. Good estimation uses number sense, rounding, and logical reasoning. An estimate of 38 × 52 should be close to 2000 -- not "a few hundred" or "ten thousand." If your exact answer and your estimate are wildly different, one of them is wrong.
After solving any problem, ask: "Does this answer make sense?" If you calculated that a car traveling 60 mph covers 3 miles in 2 hours, something is wrong. If you found the area of a bedroom to be 50,000 square feet, that is a football field, not a bedroom. Always sanity-check your results.
1. Estimate: 4,821 + 3,197 + 1,988
Round: 5,000 + 3,000 + 2,000 = 10,000. (Exact: 10,006.)
2. Use compatible numbers to estimate: 6,391 / 7
6,300 / 7 = 900. (Exact: 913.)
3. A store receipt shows: $12.95 + $4.50 + $8.99 + $3.25. Estimate the total before looking at the register.
Round: $13 + $5 + $9 + $3 = $30. (Exact: $29.69.)
4. Fermi problem: About how many words are in a typical 300-page novel?
A page has roughly 250-300 words. Using 250: 300 × 250 = 75,000 words. (Typical novels range from 60,000 to 100,000 words, so this is reasonable.)
5. A student calculates the area of a rectangle 12 m by 8 m and gets 960 m2. Is this reasonable?
No. Quick estimate: 10 × 8 = 80 m2. The correct answer is 12 × 8 = 96 m2. The student likely made an error (perhaps multiplying 12 × 80 instead of 12 × 8). Estimation caught the mistake.