More Fun with Algorithms
By Michael Naylor

In April we took a look at algorithms for addition and multiplication. Students who learn a variety of algorithms and possibly who are even given a chance to invent their own will develop into powerful users of numbers.

Subtraction algorithms (Grades 1-3)
When introducing two-digit subtraction, use base-ten blocks and start with a problem in context, perhaps: "Your chickens laid 32 eggs. You gave 15 of them to your teacher. How many are left?"

Ask your students to model the problem with the blocks on a place-value mat showing tens and ones. Have students write the digits of the number to subtract on two small squares of paper and place them beneath the base-ten blocks as a reminder of how many pieces they need to remove from each position. What happens when there aren't enough ones?

After students have had many experiences with the manipulatives, provide them with charts divided into labeled columns of tens and ones. Have them repeat the actions with the blocks, using the charts to record their moves. They should cross out and rewrite numbers as needed to mirror the same actions with the blocks. As students become more comfortable with regrouping, use of the models will become less and less necessary.

Equal addends method of subtraction (Grades 3-8)
In many parts of the world, students learn a subtraction algorithm different from our own. This algorithm makes a great puzzle for students and reinforces place value concepts. Here's an example of how the algorithm is used to compute 520 - 348:

Starting at the right, the 0 is changed to a 10, but instead of subtracting 1 from the 2 in the tens place, instead 1 is added to the 4 in the subtrahend:

10 minus 8 may now be computed, and our attention turns to the tens column. We can't take 5 tens from 2 tens, so again the 2 is changed to a 12 and 1 is added to the 3 in the hundreds column. All columns can then be computed.

Students will be amazed when they first see this algorithm. After they've tried it and are convinced that it works, then challenge them to explain it!

Why does it work? When you add 10 to the ones column, you compensate for it by subtracting an extra ten in the next column, and so on. In effect, you add 10 to both the top number and the bottom number, so the difference is not changed at all!

Scaffolding method of division (Grades 4-8)
Long division is the most feared algorithm – and with good reason. No other basic algorithm is as confusing and offers so many opportunities for errors as the standard division algorithm.

Fortunately there's an alternative, one that I have used with great success with children as young as third graders. It's called scaffolding division, and not only is it fun and easy, it also makes sense and develops number sense as well.

Suppose we wanted to divide $138 among six people – how would you do it? You might think, "I'll give $20 to each person, that makes $120, there's $18 left over, so that's $3 more per person, or a total of $23 for each person." Scaffolding works exactly the same way.

Here's that same example worked by a student:

Notice that there are many ways to arrive at the answer – the "right" choice of which numbers to choose in each step depends on how much number sense the individual student has.

For example, students who are computing 117 divided by 4 may think of it in different ways:

Student A: "I know that 4 x 25 is 100, so I'm going to try 25 per group first. I'll write 25 to the right, that's a total of 100 taken from 117, which leaves 17 remaining. Ah! That's an easy one. 4 fours is 16, So I'll just write the 4 to the right and then take away 16, that leaves 1 remaining. 25 + 4 = 29, so 117 divided by 4 is 29 remainder 1."

Student B: "I'm going to try 20 first so I'll just write 20 to the side. 20 times 4 is 80, subtract from 117 to leave 37. I know that 4 x 8 is 32, so I'll write 8 next and then I'll subtract 32 to leave 5. I can then take one more 4 away, so I'll just write 1 and then subtract 4 to leave a remainder of 1. 20 + 8 + 1 = 29. My answer to this problem then is 29 remainder 1."

The advantages of scaffolding
I like to introduce this algorithm within the context of money. Pretend the school has given $1,000 to share among all the students in the class. Suppose there are 23 students in your class. Write "1000 ÷ 23" on the board. Pretend to give each student $1, and have them jot down this amount. On the board, subtract from $1,000 the amount you distributed. (1000 - 23 = 977). Repeat this, then ask if there's a faster way.

Try $10 next, and then subtract $230 from the amount. Want to try $20 next? How much is 20 x 23? (twice the $230 from last time, so $460). Keep going until there is less than $23 left; circle this and write "remainder" on the board. Ask the students how much they each have (they have been keeping track of this). You've just divided 1000 by 23 to get 43 remainder 11.

There are many advantages of using scaffolding:

• It's fun and it makes sense.
• It develops estimation skills.
• Students are engaged in mental arithmetic – they are thinking throughout the process, not just following an algorithm.
• Students develop number sense.
• The more number sense that students possess, the more efficient the process.
• There are many correct ways to arrive at a solution.
• There are fewer opportunities for error than with long division.
• Students who practice scaffolding are better able to divide mentally.

Hint: I found that scaffolding works well when I've picked easy numbers that can be multiplied mentally. Recall with 138/6 you could choose 20 or 10 as your first try – it didn't matter. Choosing 18 would also have worked, but multiplying 18 and 6 mentally is a lot tougher than multiplying 10 and 6.

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Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

May, 2007, Vol.37, No.8