Computational Fluency

Flexible & Accessible Strategies for

Multi-digit Addition and Subtraction

Math Alliance March 30, 2010Beth Schefelker and DeAnn Huinker

WALT

We Are Learning To…Develop flexibility in using computational strategies for addition and subtraction.

We will know we are successful when…Solve multi-digit addition and subtraction problems using strategies other than the traditional algorithm.

Base-ten Number System: Place Value Learning about whole number computation

must be closely linked to learning about the base-ten number system

The heart of this work is relating the written numeral to the quantity and to how that quantity is composed and can be decomposed.

Teacher Note, Computational Fluency and Place Value, Investigations Grade K-5. TERC, 2007

The Hundreds Chart

A Powerful Tool for Students

Computational Fluency

Flexibility Comfortable with more than one approach. Chooses strategy appropriate for the numbers.

Efficiency Easily carries out the strategy, uses

intermediate results. Doesn’t get bogged down in too many steps or

lose track of the logic of the strategy. Accuracy

Can judge the reasonableness of results. Has a clear way to record and keep track. Concerned about double-checking results.

Source: Russell, S.J. (2000). Developing computational fluency with whole numbers. Teaching Children Mathematics, 7, 154 - 158.

Addition Strategies

1. Add each place from left to right

2. Add on the other number in parts

3. Use a nice number and compensate

4. Change to an easier equivalent problem

Video

Directions Consider the

numbers. From the list of four

strategies, select an appropriate strategy for the numbers.

Solve this problem using the strategy.

581 + 397 = ?

Strategies• Add each place

from left to right• Add on the other

number in parts• Use a nice

number and compensate

• Change to an easier equivalent problem

Add Each Place from Left to Right

Add Each Place from Left to Right

Change to an Easier

Equivalent Problem

Change

to an Easier

Equivalent

Problem

Add On the Other Number in Parts

Add On the Other Number in Parts

Use a Nice Number & Compensate

Use a Nice Number & Compensate

“Start With” Tasks

Fold a piece of paper into four parts.Take turns facilitating: Draw a card and present the task to the group.

Everyone solves the problem using the same “start.”

Facilitator leads a discussion comparing solution paths & naming of the strategy.

438 + 295 = ?Facilitator: Draw a card and present the “start” task to the group.

Everyone uses the same “start” to solve it.

Facilitator: How do our solution paths compare?What is the name of the strategy?

438 + 295 = ?

Start with 400 + 200.Start with 438 + 200.Start with 295 + 400.Start with 295 + 5.Start with 438 + 300.Start with 433 + 300.

Moving to Subtraction

What are some of your thoughts on why subtraction is so much harder for students?

No pencils allowed.

Solve the problem by reasoning in your mind.

35 – 19 = ?

Then turn and share your reasoning.

Typical Instructional Sequence

Modelwith

Objects

Jump tothe

Standard Algorith

m

?

What’s missing from this sequence?

Developmental Approach to Computation

Direct Modeling Model the situation or action step-by-step using

objects or pictures and count, usually by ones, the objects.

Counting Approach Visualize the quantities to count-on, count-up-to,

count-down, usually by ones; use fingers to keep track of the counts.

Numerical Reasoning Strategy Use number relationships to strategically work with

quantities; break numbers apart and find easier ways to put them back together or to find differences.

35 – 19 = ?

Did you . . . Use the standard algorithm and

“borrow” Begin with 35 – 20 = 15 Begin at 19, count up by ones or in

parts Begin by subtracting 30 – 10 and 5 – 9 Begin by subtracting 35 – 10 Begin by adding 1 to both numbers

Video

Solve the problem in two different ways.

674 – 328 = ?

Subtraction Strategies

1. Subtract each place

2. Subtract the number in parts

3. Add up from the subtracted number

4. Use a nice number then compensate

5. Change to an easier equivalent problem

Add Up from the

Subtracted Number

Add Up from the

Subtracted

Number

Subtract the Number in Parts

Subtract

the Number in Parts

Use a Nice Number then

Compensate

Use a

Nice

Number

then

Compensate

Change to Easier

Equivalent Problem

Change to an Easier Equivalent

Problem

Subtract Each Place

Subtract Each Place

You Choose!

Fold your paper into four parts.Given a problem, pick a strategy. Then solve it using that strategy.

WAIT . . . . . . . . . . . . . . . . .Pass your paper to another person.That person studies the work and identifies which strategy was used.

Pick a Strategy. Solve it.

71 – 28 = ?Exchange papers. Study the work.Name the strategy.

Strategies• Subtract each

place

• Subtract the number in parts

• Add up from the subtracted number

• Use a nice number then compensate

• Change to an easier equivalent problem

Pick a strategy. Solve it.

82 – 47 = ?Exchange papers. Study the work.Name the strategy.

Strategies• Subtract each

place

• Subtract the number in parts

• Add up from the subtracted number

• Use a nice number then compensate

• Change to an easier equivalent problem

Pick a Strategy. Solve it.

754 – 273 = ?Exchange papers. Study the work.Name the strategy.

Strategies• Subtract each

place

• Subtract the number in parts

• Add up from the subtracted number

• Use a nice number then compensate

• Change to an easier equivalent problem

Pick a problem. Pick a strategy.

576 – 239 = ? Strategies• Subtract each

place

• Subtract the number in parts

• Add up from the subtracted number

• Use a nice number then compensate

• Change to an easier equivalent problem

Exchange papers. Study the work.Name the strategy.

Pick a Question to Discuss

What are some things students need to know in order to develop computational fluency for addition and subtraction?

In what ways might the use of varied strategies benefit and make computation more accessible for more students?

Closing Thought . . . .

The most appropriate computation method can and should change flexibly as the numbers and the context change.

Traditional algorithms are “digit-oriented” and “rigid” and rely on memorizing rules without reasons, and can lead to common errors.

Alternative strategies are “number-oriented” and “flexible” and rely on making sense of working with numbers, and build confidence in students.

Homework

Read the article, “Subtraction Strategies from Children’s Thinking.” Reflect on how the information helped you think further about fluency with subtraction.

Practice the five subtraction strategies using either of these problems: 87 – 38 = ? or 574 – 289 = ?

Gather student reasoning on two strategies, either addition or subtraction. Show a work sample (e.g., from class, your own, or the article) to a student and see if he or she can make sense of it. Then give the student a problem to try using the same strategy. Repeat with another work sample. Summarize the student’s reactions and sense-making.