According to the Standards published by the National Council of Teachers of Mathematics, designed to guide math education, “Continual emphasis on computational estimation helps children develop creative and flexible thought processes and fosters in them a sense of mathematical power.” It isn’t just for getting a decent approximation; it’s a crucial tool for helping children strengthen their fundamental understanding of, and facility with, numbers.
In a previous post I made the case for the importance of developing estimation skills in students. Here is a collection of specific strategies you can use to do this. It is by no means meant to be exhaustive; hopefully, in addition to some specific tools that you can teach your children now, this post also sparks some ideas for similar strategies that you can help your children develop. I strongly suggest that you frame these as mental math games or puzzles and not as important skills to master for a successful future in mathematics, or some such lofty goal! Truth be told, however, estimation activities can be a lot of fun AND are quite important to one’s developing facility with numbers.
Front-End Estimation: Addition
Estimate the sum of the following three numbers:
283 + 512 + 637
Traditionally, students in the US are taught to line these up by place value and add from the right side, carrying (regrouping) when needed. Instead, for a quick estimate, start on the left and add the digits in the hundreds place: 200 + 500 + 600 = 1300. Now ‘eyeball’ what’s left, and you can see something in the eighties, something in the teens, and something in the thirties. The numbers in the eighties and the teens add up to close to 100, and add to that the remaining number and you end up with about 150 from the tens and one places. 1300 + 150 = 1450, which isn’t far from the exact sum of 1432. If you just need an estimate, this is a very effective way of doing it, and in many situations an estimate is all we really need.
Grouping Friendly Numbers
Estimate the sum of the following five numbers:
32 + 54 + 82 + 71 + 15
Glancing quickly at the tens place, look for digits that add to 10 (or even 20). For example, I see that 3 + 7 = 10, so I could mentally replace 32 and 71 with 100. I could also replace 82 and 15 with another 100. Overall, a reasonable estimate of the sum would be 100 + 100 + 50, or 250. The precise sum is 254.
Front-End Estimation: Multiplication
The technique of focusing on the left-most digit works well for multiplication as well as for addition.
Estimate the product of 312 x 7.
Mentally calculate 300 x 7. This is done by removing the two zeros, multiplying 3 x 7, and putting the two zeros back.
300 x 7 = 2100
To obtain a closer estimate, if one so desires, keep 2100 in your head, round 12 down to 10, and multiply 7 by 10. Then add that to the 2100 you already have.
300 x 7 = 2100 and 10 x 7 = 70 à 2100 + 70 = 2170
The precise product is 2184.
Adding fractions with different denominators can be tricky for many students. Here’s an example of one of the most common errors students make when adding fractions:
When a student doesn’t have a good sense for the relative size of fractions, let alone performing operations on them, this seems to be a logical way to add them: Add the numbers on top and add the numbers on the bottom. Easy enough! When one has a basic understanding of fractions, however, one can easily see that adding the numerators and adding the denominators generally does not yield a sensible answer.
Let’s replace each fraction with a ‘landmark’ fraction, a common fraction that we can more easily get our minds around, to which it is close. 6 is about half of 11, so the first fraction is close to one-half. 12 is about half of 25, so the second fraction is also close to one-half. That means that the sum should be close to one 1. The fraction 18/36, however, is exactly one-half, which makes no sense at all.
One way to approximate the sum of fractions is to replace them with landmark fractions and add those. The more practice one gets working with fractions, the more landmark fractions one can use, and therefore the more accurate one’s estimate will be.
It’s easy to make a quick little game out of practicing these estimation skills, and this is a great way to pass the time in a waiting room, on line in a supermarket, or in traffic. Keep in mind that if this is approached with a sense of fun, a sense of challenge and puzzlement, a child is much more likely to enjoy it (and to ask to do it again!) than if you frame it as important skill-building. It just so happens that it’s important skill building as well!!!