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Chapter Nine. Estimation and Computational Procedures for Whole Numbers. Is it reasonable?. What is Computation?. Solve this problem: 42 – 16 Use the following strategies : Unifix Cubes as tens and ones Tally Marks Standard Algorithm Adding Up Subtracting Back - PowerPoint PPT Presentation
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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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©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.
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-3
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?
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-4
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-5
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-6
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.
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-7
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.
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-8
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-9
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-10
Strategies for Computational Estimation
Front-end – focuses on the left-most or highest place-value digits
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-11
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?
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-12
Strategies for Computational Estimation
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+
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-13
Strategies for Computational Estimation
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-14
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-15
Strategies for Computational Estimation
Compatible numbers – the process of adjusting the numbers so that they are easier to work with
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-16
Strategies for Computational Estimation
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-17
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)
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-18
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-19
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-20
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-21
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-22
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-23
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-24
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-25
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.
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-26
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.
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-27
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-28
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-29
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
Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk
©2011 Pearson Education, Inc. All Rights Reserved 9-30
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?