by Josh Rappaport
In my March article, I described an alternative approach for doing subtraction problems, working them in a Left-to-Right (LTR) way rather than the traditional Right-to-Left (RTL) way. The method I described uses an innovative approach that encourages children to work with negative numbers to get the answer.
While that approach is conceptually expansive and exciting, I promised that for any homeschoolers who don't take to it, I would describe a third way to do subtraction in my next article. And that's exactly what I'll be doing right here and now.
To help us get started, let's think about the traditional abstract framework for both addition and subtraction problems. In the most general sense, simple addition problems, such as 15 + 18 = 33, rest on the framework: Part 1 + Part 2 = Whole (with the Whole being the answer we're looking for). On the other hand, basic subtraction problems, such as 33 – 15 = 18, rest on a different framework: Whole – Part 1 = Part 2 (with Part 2 being the answer we're searching for). In the subtraction problem just above, the Whole is the 33, Part 1 is the 15, and we are solving for Part 2, which is of course 18 in this particular problem.
As it happens, though, by juggling terms around a bit, we discover that there's a different way to conceive of subtraction. This alternative approach involves re-casting a subtraction problem — in a sense disguising it — to make it look like an addition problem. To understand this, we'll return for a moment to addition.
As noted above, the general framework for addition is this:
Part 1 + Part 2 = Whole
To transform this addition framework into a subtraction framework, we simply change Part 2 to What? (making What? the quantity we're trying to find), so that the problem's format now reads:
Part 1 + What? = Whole
and our goal is to find this What?
With all of these abstract terms being lobbed about, it's possible to get a bit confused, so let's re-ground this in the concrete by relating our new subtraction format back to the original problem.
The original addition problem, we recall, is:
15 + 18 = 33
What I'm saying is that instead of having children solve for the 33 (which they do if it's an addition problem), we can instead have children instead solve for the 18. When we set it up this way, we get the problem:
15 + What? = 33
Notice that when we work this problem by solving for the What?, we are essentially doing subtraction. We're doing the problem: 33 – 15 = What? However, and this is a big however, we're doing it in a form that "looks like" addition.
Doing this "switcheroo" benefits children in at least two ways. First, it's easier to teach children a "twist" on something they already know (addition, in this case) than to teach them a whole new process (subtraction, as it is traditionally taught). By re-casting the subtraction problem as an addition problem, you return children to their "terra firma" of addition, and that puts them at greater ease.
Secondly, showing children how to make this switch reinforces the idea that math involves flexibility and creativity. Solution methods are not chiseled in marble; rather they are fluid techniques that can and should be altered to suit a person's needs. By presenting various ways to do subtraction in these articles, I'm suggesting that we educators offer children a "menu" of ways to do subtraction. And then, we can let them choose the method that works best for them. That helps children feel empowered, and empowered learners are more confident learners.
Returning to our newly minted subtraction problem — 15 + What? = 33 — we see that we now have an addition problem with a twist. In typical addition problems, we solve for the Whole. But in this "addition" problem, we solve for one of the two parts: Part 2, to be exact. You might be wondering: how exactly do we teach this process? It turns out that it's not nearly so hard as it might seem. What's more, there's a lovely metaphor that helps children get a handle on the process.
The metaphor I use — an extended metaphor, actually — is that of a road trip. I tell children that in the problem 15 + What? = 33, they can imagine that they're taking a road trip to a desired destination. If they love the beach, let the destination be the beach; if it's the zoo where they always beg you to take them, let the trip end where the monkeys, zebras and giraffes cavort.
But here's the overall idea. The number that the problem starts with — 15 in our example — represents the journey's Starting Point. I usually make this the child's home. Then, make clear that we are traveling along a road that has Milepost marking every mile. And in addition, every Milepost that is a multiple of 10 (10, 20, 30, etc.) represents a Rest Stop where we make an imaginary stop as we take our trip (think: stop for gas, snacks, etc.).
To start figuring out 15 + What? = 33, the child first figures out the Rest Stops she will pass, and counts those up. For our trip, going from 15 to 33, we will stop at Rest Stops 20 and 30, so we'll use 2 Rest Stops. Add 1 to the number of Rest Stops, and we get the number of "legs" that the trip entails. So our trip entails 2 + 1 = 3 "legs."
The first "leg" always goes from the Starting Point to the First Rest Stop. For our trip, the first leg runs from our Starting Point at Milepost 15 to the Milepost 20 Rest Stop. That short "leg" takes just 5 miles. The next "leg" goes from Rest Stop 20 to Rest Stop 30. That "leg," obviously, takes exactly 10 miles. The final "leg" always takes us from the last Rest Stop to our Destination. In our example, we travel from Rest Stop 30 to Milepost 33: a distance of just 3 miles.
To get the answer to our subtraction problem, we simply sum the lengths of our journey's three "legs": 5 + 10 + 3 = 18. So the answer is 18. In other words: 33 – 15 = 18
Some people call this approach Subtraction by Adding Up, or Subtracting the Make-Change Way, since it's the way people count back your change after a purchase. But the main point is that this is a perfectly legitimate way to teach subtraction, and in fact, it is one that children generally understand intuitively. I'd venture to say that children innately grasp this approach better than they grasp the traditional approach with borrowing or "re-grouping." That approach is, in my experience, the hardest one for children to grasp.
Now that you understand this approach, here are a few problems to help you start teaching this technique. These three problems all have two Rest Stops: a) 42 – 25, b) 53 – 38, and c) 61 – 47
Next, some problems with three Rest Stops: d) 41 – 17, e) 63 – 39, f) 95 – 62
Finally, some problems with more than three Rest Stops: g) 51 – 9, h) 72 – 19, j) 93 – 29
Teaching Tip: I find that when first teaching this method, it helps to make a diagram of the road trip. To do so, you (or your children) can draw out a number line, and on it clearly mark the Starting Point, the Rest Stops, and the Destination Point. Small tick marks on the line can represent the individual Mileposts.
Finally, I'd like to mention one big benefit to teaching this approach. I've found that children find it much easier to do subtraction problems mentally (in their head, without pencil or paper) when they use this approach. Next time you take a real road trip, check out my claim. Give your children some subtraction problems and encourage them to use this approach. If your kids are at all like the children I work with daily, you and they will be pleasantly amazed by their ability to do problems like 53 – 29 in their heads: getting these right time and again. And not only that, but also deeply understanding what they're doing.
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.
In my March article, I described an alternative approach for doing subtraction problems, working them in a Left-to-Right (LTR) way rather than the traditional Right-to-Left (RTL) way. The method I described uses an innovative approach that encourages children to work with negative numbers to get the answer.
While that approach is conceptually expansive and exciting, I promised that for any homeschoolers who don't take to it, I would describe a third way to do subtraction in my next article. And that's exactly what I'll be doing right here and now.
To help us get started, let's think about the traditional abstract framework for both addition and subtraction problems. In the most general sense, simple addition problems, such as 15 + 18 = 33, rest on the framework: Part 1 + Part 2 = Whole (with the Whole being the answer we're looking for). On the other hand, basic subtraction problems, such as 33 – 15 = 18, rest on a different framework: Whole – Part 1 = Part 2 (with Part 2 being the answer we're searching for). In the subtraction problem just above, the Whole is the 33, Part 1 is the 15, and we are solving for Part 2, which is of course 18 in this particular problem.
As it happens, though, by juggling terms around a bit, we discover that there's a different way to conceive of subtraction. This alternative approach involves re-casting a subtraction problem — in a sense disguising it — to make it look like an addition problem. To understand this, we'll return for a moment to addition.
As noted above, the general framework for addition is this:
Part 1 + Part 2 = Whole
To transform this addition framework into a subtraction framework, we simply change Part 2 to What? (making What? the quantity we're trying to find), so that the problem's format now reads:
Part 1 + What? = Whole
and our goal is to find this What?
With all of these abstract terms being lobbed about, it's possible to get a bit confused, so let's re-ground this in the concrete by relating our new subtraction format back to the original problem.
The original addition problem, we recall, is:
15 + 18 = 33
What I'm saying is that instead of having children solve for the 33 (which they do if it's an addition problem), we can instead have children instead solve for the 18. When we set it up this way, we get the problem:
15 + What? = 33
Notice that when we work this problem by solving for the What?, we are essentially doing subtraction. We're doing the problem: 33 – 15 = What? However, and this is a big however, we're doing it in a form that "looks like" addition.
Doing this "switcheroo" benefits children in at least two ways. First, it's easier to teach children a "twist" on something they already know (addition, in this case) than to teach them a whole new process (subtraction, as it is traditionally taught). By re-casting the subtraction problem as an addition problem, you return children to their "terra firma" of addition, and that puts them at greater ease.
Secondly, showing children how to make this switch reinforces the idea that math involves flexibility and creativity. Solution methods are not chiseled in marble; rather they are fluid techniques that can and should be altered to suit a person's needs. By presenting various ways to do subtraction in these articles, I'm suggesting that we educators offer children a "menu" of ways to do subtraction. And then, we can let them choose the method that works best for them. That helps children feel empowered, and empowered learners are more confident learners.
Returning to our newly minted subtraction problem — 15 + What? = 33 — we see that we now have an addition problem with a twist. In typical addition problems, we solve for the Whole. But in this "addition" problem, we solve for one of the two parts: Part 2, to be exact. You might be wondering: how exactly do we teach this process? It turns out that it's not nearly so hard as it might seem. What's more, there's a lovely metaphor that helps children get a handle on the process.
The metaphor I use — an extended metaphor, actually — is that of a road trip. I tell children that in the problem 15 + What? = 33, they can imagine that they're taking a road trip to a desired destination. If they love the beach, let the destination be the beach; if it's the zoo where they always beg you to take them, let the trip end where the monkeys, zebras and giraffes cavort.
But here's the overall idea. The number that the problem starts with — 15 in our example — represents the journey's Starting Point. I usually make this the child's home. Then, make clear that we are traveling along a road that has Milepost marking every mile. And in addition, every Milepost that is a multiple of 10 (10, 20, 30, etc.) represents a Rest Stop where we make an imaginary stop as we take our trip (think: stop for gas, snacks, etc.).
To start figuring out 15 + What? = 33, the child first figures out the Rest Stops she will pass, and counts those up. For our trip, going from 15 to 33, we will stop at Rest Stops 20 and 30, so we'll use 2 Rest Stops. Add 1 to the number of Rest Stops, and we get the number of "legs" that the trip entails. So our trip entails 2 + 1 = 3 "legs."
The first "leg" always goes from the Starting Point to the First Rest Stop. For our trip, the first leg runs from our Starting Point at Milepost 15 to the Milepost 20 Rest Stop. That short "leg" takes just 5 miles. The next "leg" goes from Rest Stop 20 to Rest Stop 30. That "leg," obviously, takes exactly 10 miles. The final "leg" always takes us from the last Rest Stop to our Destination. In our example, we travel from Rest Stop 30 to Milepost 33: a distance of just 3 miles.
To get the answer to our subtraction problem, we simply sum the lengths of our journey's three "legs": 5 + 10 + 3 = 18. So the answer is 18. In other words: 33 – 15 = 18
Some people call this approach Subtraction by Adding Up, or Subtracting the Make-Change Way, since it's the way people count back your change after a purchase. But the main point is that this is a perfectly legitimate way to teach subtraction, and in fact, it is one that children generally understand intuitively. I'd venture to say that children innately grasp this approach better than they grasp the traditional approach with borrowing or "re-grouping." That approach is, in my experience, the hardest one for children to grasp.
Now that you understand this approach, here are a few problems to help you start teaching this technique. These three problems all have two Rest Stops: a) 42 – 25, b) 53 – 38, and c) 61 – 47
Next, some problems with three Rest Stops: d) 41 – 17, e) 63 – 39, f) 95 – 62
Finally, some problems with more than three Rest Stops: g) 51 – 9, h) 72 – 19, j) 93 – 29
Teaching Tip: I find that when first teaching this method, it helps to make a diagram of the road trip. To do so, you (or your children) can draw out a number line, and on it clearly mark the Starting Point, the Rest Stops, and the Destination Point. Small tick marks on the line can represent the individual Mileposts.
Finally, I'd like to mention one big benefit to teaching this approach. I've found that children find it much easier to do subtraction problems mentally (in their head, without pencil or paper) when they use this approach. Next time you take a real road trip, check out my claim. Give your children some subtraction problems and encourage them to use this approach. If your kids are at all like the children I work with daily, you and they will be pleasantly amazed by their ability to do problems like 53 – 29 in their heads: getting these right time and again. And not only that, but also deeply understanding what they're doing.
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.
