Algorithms for Addition and Subtraction

Children’s first methods are admittedly inefficient. However, if they are free to do their own thinking, they invent increasingly efficient procedures just as our ancestors did. By trying to bypass the constructive process, we prevent them from making sense of arithmetic.

Kamii & Livingston

If we don’t teach them the standard way, how will they learn to compute?

Algorithms in Arithmetic

An algorithm is a set of rules for solving a math problemgenerally involve repeating a series of

steps over and over

A variety of algorithms exist for addition, subtraction, multiplication and division.

Arithmetic Today The learning of the algorithms of arithmetic

has been the core of mathematics programs in elementary schools

Today’s society demands more from its citizens than knowledge of basic arithmetic skills.

There is general agreement among mathematics educators that more emphasis be placed on areas like geometry, measurement, data analysis, probability and problem solving

Current Traditional Algorithm

Addition1

47+28 75

“7 + 8 = 15. Put down the 5 and carry the 1. 4 + 2 + 1 = 7”

Subtraction 7 13

83- 37 46

“I can’t do 3 – 7. So I borrow from the 8 and make it a 7. The 3 becomes 13. 13 – 7 = 6. 7 – 3 = 4.”

Time to do some computing!

Solve the following problems. Here are the rules: You may NOT use a calculator You may NOT use the traditional algorithm Record your thinking and be prepared to share You may solve the problems in any order you choose. Try to

solve at least two of them.

658 + 253 = 297 + 366 =

76 + 27 = 314 + 428 =

Sharing Strategies

Think about how you solved the equations and the strategies that others in the group shared.

Did you use the same strategy for each equation?Are some strategies more efficient for certain

problems than others?How did you decide what to do to find a solution?Did you think about the numbers or digits?

Expanded Column Method

Number Line Method

Add on Tens, Then Add Ones

46 + 38

46 + 30 = 76 76 + 8 = 76 + 4 + 4

76 + 4 = 8080 + 4 = 84

Partitioning Using Tens Method

Nice Numbers Method

Lattice Method

First arrange the numbers in a column-like fashion.

Next, create squares directly under each column of numbers.

Then split each box diagonally from the bottom-left corner to the top-right corner. This is called the lattice.

Now add down the columns and place the sum in the respective box, making the tens place in the upper box and the ones place in the lower box.

Lastly, add the diagonals, carrying when necessary.

Strategies

In contrast to the traditional algorithm, these alternative algorithms are:Number oriented rather than digit oriented

Place value is enhanced, not obscured

Often are left handed rather than right handedFlexible rather than rigid

Try 465 + 230 and 526 + 98

Did you use the same strategy?

Teacher’s Role

Traditional Algorithm

Use manipulatives to model the steps

Clearly explain and model the steps without manipulatives

Provide lots of drill for students to practice the steps

Monitor students and reteach as necessary

Alternative Algorithms Provide manipulatives and guide

student thinking Provide multiple opportunities for

students to share strategies Help students complete their

approximations Model ways of recording strategies Press students toward more efficient

strategies

The reason that one problem can be solved in multiple ways is that…

mathematics does NOT consist of isolated rules, but of

CONNECTED IDEAS!

Time to do some more computing!

Solve the following problems. Here are the rules: You may NOT use a calculator You may NOT use the traditional algorithm Record your thinking and be prepared to share You may solve the problems in any order you choose. Try to

solve at least two of them.

636 - 397 = 221 - 183 =

502 - 256 = 892 - 486 =

Sharing Strategies

Think about how you solved the equations and the strategies that others in the group shared.

Did you use the same strategy for each equation?Are some strategies more efficient for certain

problems than others?How did you decide what to do to find a solution?Did you think about the numbers or digits?

Counting Down Using Tens Method

Partitioning Using Tens Method

Nice Numbers Method

The Counting-Up Method

The Counting-Up Method

Nines Complement

827 → 827

- 259 → 740 (nines complement)

+ 1 (to get the ten's complement)

1568

568 (Drop the leading digit)

Another Look at the Subtraction Problems

636 - 397 = 221 - 183 =

502 - 256 = 892 - 486 = Now that we have discussed some alternative methods

for solving subtraction equations, let’s return to the problems we solved earlier. Go back and try to solve one or more of the problems using some of the ways on the subtraction handout. Try using a strategy that is different from what you used earlier.

Summing Up Subtraction

Subtraction can be thought of in different ways: Finding the difference between two numbers Finding how far apart two numbers are Finding how much you have to “add on” to get from the smaller

number to the larger number.

Students need to understand a variety of methods for subtraction and be able to use them flexibly with different types of problems. To encourage this: Write subtraction problems horizontally & vertically Have students make an estimate first, solve problems in more

than one way, and explain why their strategies work.

Benefits of Alternative Algorithms

Place value concepts are enhancedThey are built on student understandingStudents make fewer errors

Suggestions for Using/Teaching

Traditional Algorithms We are not saying that the traditional

algorithms are bad. The problems occur when they are

introduced too early, before students have developed adequate number concepts and place value concepts to fully understand the algorithm.

Then they become isolated processes that stop students from thinking.

More and more, people need to apply algorithmic and procedural thinking in

order to operate technologically advanced devices. Algorithms

beyond arithmetic areincreasingly important in theoretical mathematics, in

applications of mathematics, in computer

science, and in many areas outside of mathematics.