Estimation is a very valuable real-life skill to learn--I use it daily when shopping, traveling, and allotting time for work projects. Every 10th question on a number sense test deals with estimation (approximation), where the answers do not have to be exact. They are considered correct as long as they are within 5% (above or below) the exact answer. The simplest types of these problems involve only addition, but middle school tests this year had none of this type of question. So here we will cover the problems that involve both addition and subtraction. This concept showed up 12 times this year--always at question # 10.
Number Dojo Level: 47
Points to Consider:
- In spite of the fact that the answers don't have to be exact, estimation problems are still the most commonly missed concepts on number sense tests. I think this is for two main reasons: 1) there are more of them, and 2) they typically look harder than they really are. Students often will skip these questions when they don't immediately see a pattern in the numbers.
- The rules of number sense tests dictate that each answer to an estimation problem must be an integer, even if the exact answer is a fraction, decimal, or irrational number.
- Whenever possible, it is best to round before performing the calculations in the problem.
- When adding or multiplying, the answer is larger than the original numbers. As such, the margin of error is greater (in other words, the acceptable range of answers is larger). When dividing or subtracting, the margin of error is smaller. So when rounding, take this into consideration.
- Write the answer in its rounded form. In other words, do not try to "un-round" and estimate the exact answer at this point. I have seen many students miss these questions by filling in random digits to replace the zeros.
How to Solve:
- Look for the largest number in the question. Round to the leftmost two digits in that number. For example, if that number is a 5-digit number like 79366, round to the nearest thousand to get 79000. Typically, you are safe rounding every other number in the question to that same place value, even if the number has fewer digits.
- Perform the additions and subtractions with these (easier-to-use) round numbers. Combine terms (even out of order) if they make the operations easier. For example, if we have 530 + 790 - 230, knock out the 30s first (530 - 230) and then add the 790.
- Remember that when the numbers are smaller (fewer digits), your acceptable range of answers is smaller. Round more precisely with these numbers.
Example 1: 3271 - 1428 + 6312
- Notice that each number has 4 digits. Round to the nearest hundred for each one. Think: 3300 - 1400 + 6300.
- 3300 - 1400 + 6300 = 3300 + 6300 - 1400 = 8200. Write this down.
- The exact answer is 8155, so the acceptable (+/- 5%) range would be between 7748 & 8562. We are well within range.
Example 2: 918 - 2951 + 4368
- Notice that the largest number has 4 digits, and round each to the nearest hundred. Think: 900 - 3000 + 4400.
- Change the order to make it easier: 4400 - 3000 + 900 = 2300. Write this down.
- The exact answer is 2335; the acceptable (+/- 5%) range would be between 2219 & 2451. We are well within range.
Example 3: 123 - 456 + 789 - 911
- Each of the numbers has 3 digits, so we will round to the leftmost 2 (the 10s place) on each. Think: 120 - 460 + 790 - 910.
- Change the order to make it easier: 790 + 120 - 460 - 910. 790 + 120 = 910. 910 - 910 = 0. We are left with -460; write this down.
- The exact answer is -455, so the acceptable (+/- 5%) range would be between -477 & -433. We are within range.