by Josh Rappaport
Back in August we learned some math tricks that boost computation speed with MULTIPLICATION; now let's take our mental math journey's second leg. This month we'll learn tricks for ADDING and SUBTRACTING whole numbers with speed and understanding.
ADDITION first. If the only way children learn to add whole numbers is the "good-old," right-to-left (RTL) approach that requires "carrying" or "regrouping," they'd probably have trouble doing a problem like 496 + 275 in their mind, doing the "carrying" without writing anything down.
So I'd like to introduce an approach that lets children work this same kind of problem more easily, working the sums from left to right (LTR) instead.
Note: By RTL I mean working the problem in the traditionally taught way, starting with the furthest-right place value (the 1s digits) and methodically working through the digits to the left of it (next the 10s digits, then the 100s digits, etc.). By LTR I mean the opposite approach: working through the largest place values first, then methodically working through the digits to the right of it.
Let's start with an easy problem: 49 + 26. With the LTR approach, you add the digits on the left first: the 4 (which stands for 40) plus the 2 (which stands for 20). Adding those, and you get 60. Then adding the digits in the 1s place value — 9 + 6 — you get 15. Just add your partial answers — 60 + 15 — and you get the final answer, 75. It's that simple.
There are two main benefits to this LTR approach:
Benefit #1) It's quicker than the RTL technique. In fact, this approach is so quick and easy (once children get used to it) that children can actually do the math in their head. And boy, do they get excited when they can do this and consistently get the answers right. Try it and see for yourselves.
Benefit #2) Using the LTR approach forces children to use (and therefore think about) the place values of the digits. In other words, you have to view the 4 of 49 for what it really is: 4 x 10, or 40. And of course the same is true for the 2 of 26, which is really 2 x 10, or 20. If you think about it, when kids use the traditional RTL procedure, they actually don't need to realize that the 4 means 40, or that the 2 means 20. As a result, students using the RTL approach exclusively often view "place value" as a minor topic they learned "once upon a time," rather than what it truly is: a key structural element, part of the warp and woof of our entire number system.
Here are a few practice problems, from easier to harder, to get children started using the LTR approach to addition:
a) 16 + 18
b) 27 + 44
c) 75 + 38
d) 146 + 139
e) 257 + 586
Answers: a) 34, b) 71, c) 113, d) 285, e) 843
Now I can almost hear a few of you folks muttering to yourselves, saying things like: Wait, is this Rappaport guy saying that the way I learned to add is "wrong"? Rest assured, I'm not saying that anything you learned is wrong. I'm just saying that the way that we (I'm in this bunch, too) learned to add is not the best way, necessarily. It's simply not the best way if you want children to deeply understand what they're doing, and if you also want them to become highly proficient at doing math in their heads. So if you like those two goals, check out this LTR approach. It's "blowing in the wind."
Now let's turn our attention to SUBTRACTION: Like the RTL approach to addition, the time-worn RTL approach to subtraction leads neither to computational speed nor to deep understanding of what's happening in the process.
The good news is that once again, switching to an LTR approach brings the same twin benefits of speed and understanding. Consider computational speed with the problem 43 – 16. With the RTL approach, you'd have to "borrow" or "re-group." While there's nothing wrong with that conceptually, it's hard for children to do this in their minds; it simply involves too many steps for most children. (If you are unsure, ask your kids if they'd like to — or even can — work such a problem mentally.)
Now let's see how we can do this same problem, 43 – 16, using an LTR approach. Starting in the 10s place, children take away the 10 (the 1 in the 10s place) from the 40 (the 4 in the 10s place). That right there leaves 30. Next children take away the 6 from the 3 in the 1s place. Impossible? Not at all! With just a little practice learning how negative numbers work (see below for a fast way to teach this), children will recognize that 3 – 6 = – 3. Then children combine their two answers: 30 and – 3 just means 3 less than 30, which is 27, the answer.
Let's take a similar problem, but we'll view it in a more streamlined way: 82 - 27
80 - 20 = 60
2 - 7 = – 5
Putting together 60 and - 5, you get 55, the answer.
Now that we're "on a roll," let's look at a subtraction problem with three digits: 333 – 158
Here is how the answers flow out using the LTR approach:
300 – 100 = 200
30 – 50 = – 20 (At this point you have 180, since 200 and – 20 = 180)
3 – 8 = – 5 (So the final answer is 175, since 180 and – 5 = 175)
In my tutoring I've found that parents often worry about their children using the LTR approach to subtraction, fearing their children will get confused by negative numbers. In fact, though, most children not only enjoy the LTR approach to subtraction, they also have an intuitive "feel" for negative numbers (where they get it, I'm not sure). In any case, below I show a quick way to teach the concept of negative numbers, a prerequisite to the LTR approach.
First, remind your children than in ordinary subtraction, we always take LESS from MORE, and when we do so, we get an answer that's POSITIVE. i.e.: 8 – 3 = 5, meaning + 5. Then just explain that when, on the other hand, we subtract MORE from LESS, we get the same number that we'd get if the numbers were reversed, but now the answer is NEGATIVE. For example, since 8 – 3 = 5, when we do 3 – 8, we still get 5, only now it is – 5.
Want a quick way to help your children grasp negative numbers conceptually? Refer to money. Consider positive numbers as representing money we have, and negative numbers as representing money we owe. Using this framework, the problem 3 – 8 (the same as + 3 – 8) mirrors the story: you have $3, but you owe someone $8. After paying the person the $3 you have, you still owe $5, so your answer is – 5. You can explain that owed money is viewed as "negative" because in the real world, it's a "negative thing" to owe money.
One more benefit to the LTR approach. When children subtract using this approach, they learn how negative numbers work and get started using them. This early work with negatives gives children a big advantage and spares them conceptual confusion when they deal more extensively with negatives in PreAlgebra.
For those who don't take to this way of subtracting (not everyone does), I'll describe another way to subtract mentally in next month's column. But for now, why not give it a try?! I'd venture to bet that your children will enjoy it, even if you might find it a bit strange at first.
Here are a few practice problems to get you and your children started with the LTR approach to subtraction:
a) 31 – 17
b) 55 – 29
c) 87 – 58
d) 237 – 159
e) 362 – 193
Answers: a) 14, b) 26, c) 29, d) 78, e) 169
Back in August we learned some math tricks that boost computation speed with MULTIPLICATION; now let's take our mental math journey's second leg. This month we'll learn tricks for ADDING and SUBTRACTING whole numbers with speed and understanding.
ADDITION first. If the only way children learn to add whole numbers is the "good-old," right-to-left (RTL) approach that requires "carrying" or "regrouping," they'd probably have trouble doing a problem like 496 + 275 in their mind, doing the "carrying" without writing anything down.
So I'd like to introduce an approach that lets children work this same kind of problem more easily, working the sums from left to right (LTR) instead.
Note: By RTL I mean working the problem in the traditionally taught way, starting with the furthest-right place value (the 1s digits) and methodically working through the digits to the left of it (next the 10s digits, then the 100s digits, etc.). By LTR I mean the opposite approach: working through the largest place values first, then methodically working through the digits to the right of it.
Let's start with an easy problem: 49 + 26. With the LTR approach, you add the digits on the left first: the 4 (which stands for 40) plus the 2 (which stands for 20). Adding those, and you get 60. Then adding the digits in the 1s place value — 9 + 6 — you get 15. Just add your partial answers — 60 + 15 — and you get the final answer, 75. It's that simple.
There are two main benefits to this LTR approach:
Benefit #1) It's quicker than the RTL technique. In fact, this approach is so quick and easy (once children get used to it) that children can actually do the math in their head. And boy, do they get excited when they can do this and consistently get the answers right. Try it and see for yourselves.
Benefit #2) Using the LTR approach forces children to use (and therefore think about) the place values of the digits. In other words, you have to view the 4 of 49 for what it really is: 4 x 10, or 40. And of course the same is true for the 2 of 26, which is really 2 x 10, or 20. If you think about it, when kids use the traditional RTL procedure, they actually don't need to realize that the 4 means 40, or that the 2 means 20. As a result, students using the RTL approach exclusively often view "place value" as a minor topic they learned "once upon a time," rather than what it truly is: a key structural element, part of the warp and woof of our entire number system.
Here are a few practice problems, from easier to harder, to get children started using the LTR approach to addition:
a) 16 + 18
b) 27 + 44
c) 75 + 38
d) 146 + 139
e) 257 + 586
Answers: a) 34, b) 71, c) 113, d) 285, e) 843
Now I can almost hear a few of you folks muttering to yourselves, saying things like: Wait, is this Rappaport guy saying that the way I learned to add is "wrong"? Rest assured, I'm not saying that anything you learned is wrong. I'm just saying that the way that we (I'm in this bunch, too) learned to add is not the best way, necessarily. It's simply not the best way if you want children to deeply understand what they're doing, and if you also want them to become highly proficient at doing math in their heads. So if you like those two goals, check out this LTR approach. It's "blowing in the wind."
Now let's turn our attention to SUBTRACTION: Like the RTL approach to addition, the time-worn RTL approach to subtraction leads neither to computational speed nor to deep understanding of what's happening in the process.
The good news is that once again, switching to an LTR approach brings the same twin benefits of speed and understanding. Consider computational speed with the problem 43 – 16. With the RTL approach, you'd have to "borrow" or "re-group." While there's nothing wrong with that conceptually, it's hard for children to do this in their minds; it simply involves too many steps for most children. (If you are unsure, ask your kids if they'd like to — or even can — work such a problem mentally.)
Now let's see how we can do this same problem, 43 – 16, using an LTR approach. Starting in the 10s place, children take away the 10 (the 1 in the 10s place) from the 40 (the 4 in the 10s place). That right there leaves 30. Next children take away the 6 from the 3 in the 1s place. Impossible? Not at all! With just a little practice learning how negative numbers work (see below for a fast way to teach this), children will recognize that 3 – 6 = – 3. Then children combine their two answers: 30 and – 3 just means 3 less than 30, which is 27, the answer.
Let's take a similar problem, but we'll view it in a more streamlined way: 82 - 27
80 - 20 = 60
2 - 7 = – 5
Putting together 60 and - 5, you get 55, the answer.
Now that we're "on a roll," let's look at a subtraction problem with three digits: 333 – 158
Here is how the answers flow out using the LTR approach:
300 – 100 = 200
30 – 50 = – 20 (At this point you have 180, since 200 and – 20 = 180)
3 – 8 = – 5 (So the final answer is 175, since 180 and – 5 = 175)
In my tutoring I've found that parents often worry about their children using the LTR approach to subtraction, fearing their children will get confused by negative numbers. In fact, though, most children not only enjoy the LTR approach to subtraction, they also have an intuitive "feel" for negative numbers (where they get it, I'm not sure). In any case, below I show a quick way to teach the concept of negative numbers, a prerequisite to the LTR approach.
First, remind your children than in ordinary subtraction, we always take LESS from MORE, and when we do so, we get an answer that's POSITIVE. i.e.: 8 – 3 = 5, meaning + 5. Then just explain that when, on the other hand, we subtract MORE from LESS, we get the same number that we'd get if the numbers were reversed, but now the answer is NEGATIVE. For example, since 8 – 3 = 5, when we do 3 – 8, we still get 5, only now it is – 5.
Want a quick way to help your children grasp negative numbers conceptually? Refer to money. Consider positive numbers as representing money we have, and negative numbers as representing money we owe. Using this framework, the problem 3 – 8 (the same as + 3 – 8) mirrors the story: you have $3, but you owe someone $8. After paying the person the $3 you have, you still owe $5, so your answer is – 5. You can explain that owed money is viewed as "negative" because in the real world, it's a "negative thing" to owe money.
One more benefit to the LTR approach. When children subtract using this approach, they learn how negative numbers work and get started using them. This early work with negatives gives children a big advantage and spares them conceptual confusion when they deal more extensively with negatives in PreAlgebra.
For those who don't take to this way of subtracting (not everyone does), I'll describe another way to subtract mentally in next month's column. But for now, why not give it a try?! I'd venture to bet that your children will enjoy it, even if you might find it a bit strange at first.
Here are a few practice problems to get you and your children started with the LTR approach to subtraction:
a) 31 – 17
b) 55 – 29
c) 87 – 58
d) 237 – 159
e) 362 – 193
Answers: a) 14, b) 26, c) 29, d) 78, e) 169
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two teenage children. Josh is the author of the Algebra Survival Guide, and the companion Algebra Survival Guide Workbook, both of which will soon be available for homeschoolers as a computer-based Learning Management System, developed and run by Sleek Corp., of Austin, TX.
At his blog, Josh writes about math education, offering tips and tricks. Josh also authors Turtle Talk, a free monthly newsletter with an engaging "Problem of the Month." You can see a sample issue here or subscribe at this site. Josh also is co-author of the "learn-by-playing" Card Game Roundup books, and author of PreAlgebra Blastoff!, a "Sci-Fi" cartoon math book featuring a playful, hands-on approach to positive and negative numbers.
In the summer Josh leads workshops at homeschooling conferences and tutors homeschoolers nationwide using SKYPE. Contact Josh by email @ josh@SingingTurtle.com or follow him on Facebook, where he poses two fun math Qs/day.
