Estimation and Computational Procedures for Whole Numbers

Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk

©2011 Pearson Education, Inc. All Rights Reserved

Chapter Nine

Estimation and Computational Procedures for Whole Numbers


©2011 Pearson Education, Inc. All Rights Reserved 9-2

What is Computation? Solve this problem: 42 – 16 Use the following strategies:

Unifix Cubes as tens and onesTally MarksStandard AlgorithmAdding UpSubtracting Back100 Square MethodCompensation Round up the subtrahend, add the difference to the answer.



What is Computation? Now solve this addition problem: 349 + 175 Use the following strategies:

Adding by Place ValueAdding the Number on in PartsStandard AlgorithmCompensation Round up an addend, subtract the difference from the sum

or the other addend.

What would you do with this problem?Mark was born in 1947. How old is he?



What is Computation? Computation includes:

Estimation 32 + 41 30 + 40 = 70Mental computation

Use of a calculator

Sometimes an estimate is sufficient

3 + 4 + 7 = 14



What is Computation? Children can and should be allowed to

create their own algorithms

There is no one best way to solve a problem, nor is there only one correct algorithm



Standard and Alternative Computational Algorithms

What is an algorithm? A set of step-by-step procedures used in solving

a problem. What are standard and alternative

algorithms? Standard algorithms are those that are typically

used within our society. They are often used because they are more efficient.

Alternative algorithms are those that are not commonly used in our society or are invented by the student.



Reasons to Explore Different Algorithms Alternative algorithms may help children develop

more flexible mathematical thinking and “number sense.”

Alternative algorithms may serve reinforcement, enrichment, and remedial objectives.

Alternative algorithms provide variety in the mathematics class.

Awareness of different algorithms demonstrates the fact that algorithms are inventions and can change.



Estimation and Mental Computation Mental computation involves finding an exact

answer without the aid of pencil and paper, calculators or other devices.

Estimation involves finding an approximate answer. Both should be developed along with paper and

pencil computationHelp determine unreasonable resultsContribute to an understanding of paper and pencil

proceduresHelp develop computational creativity



Mental Computation Benefits of mental computation Can enhance an understanding of

numeration, number properties, and operations

Promotes problem-solving and flexible thinking

Develops strategiesDevelops good number sense Often employed when a calculator is used



Strategies for Computational Estimation

Front-end – focuses on the left-most or highest place-value digits



Using the Front-End Strategy for Addition

Front-end strategy – focuses on the left-most or highest place value digits

Use the front-end strategy to solve the following problem

4396 + 1827 + 5450 + 2980 =Children will begin to recognize that the front-end

strategy always results in an estimate that is often less than or equal to the actual problem. Consider using an adjustment to the original estimate.

How would you adjust your estimate to the above problem?




Rounding – used when rounding a number to a specific place value

Estimate 23 x 78 IT IS:

…about 20 x 80 or 1600

…more than 20 x 70 or …1400+

…about 25 x 80 or 2000…more than 20 x 78

or 1560+




Clustering – used when a set of numbers is close to each other in value

World's Fair Attendance (1-6 July)Monday 72 250Tuesday 63 819Wednesday 67 490Thursday 73 180Friday 74 918Saturday 68 490

Estimate Average:

All about 70 000

Multiply the “Average” by number of values:6 x 70 000 is 420 000



Using the Clustering Strategy for Addition

This strategy is used when the numbers in a set are close to each other in value.

What number do these cluster around?

32 28 34 26 29 The numbers cluster around 30, so a good

estimate would be 5 x 30, or 150 Determine an estimate for these dollar amounts

using the clustering strategy

$3.11, $3.39, $2.94, $2.70, $2.61, $3.20




Compatible numbers – the process of adjusting the numbers so that they are easier to work with




Special numbers – involves looking for numbers that are close to “special’ values so they are easier to work with

Problem: Think: Estimate:7/8 + 12/13 Each near 1 1 + 1 = 2 23/45 of 720 23/45 near ½ 1/2 of 720 = 360 9.84% of 816 9.84% near 10% 10% of 816 = 81.6 103.96 x 14.8 103.96 near 100 100 x 15 = 1500

14.8 near 15



Reasons and Dangers for Having and Using Algorithms

Reasons for having and using algorithms include: PowerReliabilityAccuracySpeed

Dangers that are inherent in all algorithms include: Blind acceptance of resultsOverzealous application of algorithmsA belief that algorithms train the mindHelplessness if the technology for the algorithm is not

available. (Usikin, 1998)



Prerequisites for Developing Paper and Pencil Algorithms Demonstrate computational understanding of the

different operations Knowledge of some basic facts Good understanding of the place-value

numeration systemAn understanding of some mathematical

properties of whole numbers Understanding of the distributive propertyAn attitude of estimation



Teaching Computational Procedures

The following components are important to keep in mind when teaching computational procedures: Pose story problems in real-world contextsUse models for computationDevelop bridging algorithms to connect problems,

models, estimation, and symbolsDevelop the traditional algorithmExamine children’s workDetermine reasonableness of solutions



Developing Computational Fluency

Beginning work should focus on using proportional materials versus nonproportional

Children should be allowed to use their own language to describe computational processes

Teachers should model the mathematical language used with each operation

Use the calculator in lessons that help children think about the algorithms, develop estimation skills, and solve computational problems.

Teachers need to examine student’s work to look for error patterns or lack of understanding



Developing the Addition Algorithm

Pose story problems that are set in real-world contexts Instruction should use real problems which may

require children to regroupUse models for computation

With relevant numeration experiences children can work with larger numbers

Encourage estimation Transition to the recording phase by giving children

an organizational matEncourage children to use place-value language

as they describe their manipulations



Developing Bridging Algorithms for Addition

Partial sums algorithm – process of recording each partial sum individually before combining the partial sums to find the sum

The following addition problems are solved using the partial sums algorithm

56 +35

80 11 91

423 + 72 400 90 5 495



Developing the Subtraction Algorithm

Pose story problems set in real-world contexts Real problem contexts may require regrouping Encourage children to use the terms for regrouping and

renaming TradeGroupBreak apartBreak a tenMake a group

Encourage the following terms to be used meaningfully in a sentenceSubtractSubtractionDifference



Developing the Subtraction Algorithm (cont’d)

Use models and an organizational mat Move through a sequence that is unstructured

to a more systematic approachUse estimation and mental computation

Helps children to verify and feel confident about their concrete solutions



Developing Bridging Algorithms in Subtraction

Paper-and-pencil algorithms should follow the methods used by children when they subtract with base-ten blocks. Two children could work together with one child

manipulating the blocks on the organizational mat and the other child recording the results on paper.

Traditional Algorithm Numerous experiences with connecting concrete with

symbolic representations and explaining the subtraction procedure.

Encourage expanded notation.Use concrete simulations with problems containing

zeros.



Other Subtraction AlgorithmsComparison interpretation of subtraction

Can be modeled using matching techniques Decomposition algorithm (regrouping)

Known as the traditional algorithm that is commonly used today

Equal additions algorithm (“same change”) Based on compensation – what is added to the top

number (minuend) must also be added to the bottom number (subtrahend) to keep the difference the same.



Developing the Multiplication Algorithm

Pose story problems set in real-world contexts Equal groups and array interpretations may be the

most powerful Models for computation

Allow ample time to solve numerous problems concretely while recording, in their own way, what they did.

Once comfortable with place value language, they can record products and regroup them in a place value chart.

Allow children to use informal language



Developing Bridging Algorithms in Multiplication

Partial Products algorithm Uses place value language to make sense of the

results Use arrays to help support children’s understandingExpanded notation Distributive property

Build an array for 13 x 4 Using the array make a comparison to the partial

products algorithmUse the same strategy for 26 x 17



Developing the Division Algorithm

Difficult for children to learn as children need to…Know the division basic factsMultiply and subtract efficiently

Instruction shouldEmphasize one-digit divisors Some experience with two-digit divisors Experience with divisors up to four-digits

Use story problems in real-world contexts Use models for computationUse estimation and mental computation



Developing Bridging Algorithms for Division

Paper and pencil algorithms are a means for recording what has been done concretely

Introduce children to another way of writing 53 ÷ 7, using the traditional division box.

Avoid the phrase “7 goes into 53!”Use the listed algorithms to solve the problem

listed above Ladder algorithm (repeated subtraction)Pyramid algorithm (partial quotient) Traditional algorithm

How are these algorithms different or the same?

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Estimation and Computational Procedures for Whole Numbers