Estimation and Computational Procedures for Whole Numbers

1. Chapter Nine Estimation and Computational Procedures for Whole Numbers Is it reasonable?

2. 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 • 100 Square Method • Compensation Round up the subtrahend, add the difference to the answer.

3. What is Computation? • Now solve this addition problem: 349 + 175 • Use the following strategies: • Adding by Place Value • Adding the Number on in Parts • Standard Algorithm • Compensation 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?

4. What is Computation? 3 + 4 + 7 = 14 • Computation includes: • Estimation 32 + 41 30 + 40 = 70 • Mental computation • Use of a calculator • Sometimes an estimate is sufficient

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

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.

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.

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 computation • Help determine unreasonable results • Contribute to an understanding of paper and pencil procedures • Help develop computational creativity

9. Mental Computation Benefits of mental computation Can enhance an understanding of numeration, number properties, and operations Promotes problem-solving and flexible thinking Develops strategies Develops good number sense Often employed when a calculator is used

10. Strategies for Computational Estimation Front-end – focuses on the left-most or highest place-value digits

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?

12. Strategies for Computational Estimation …more than 20 x 70 or …1400+ …about 20 x 80 or 1600 …about 25 x 80 or 2000 …more than 20 x 78 or 1560+ Rounding – used when rounding a number to a specific place value Estimate 23 x 78 IT IS:

13. Strategies for Computational Estimation Estimate Average: All about 70 000 Multiply the “Average” by number of values: 6 x 70 000 is 420 000 Clustering – used when a set of numbers is close to each other in value World's Fair Attendance (1-6 July) Monday 72 250 Tuesday 63 819 Wednesday 67 490 Thursday 73 180 Friday 74 918 Saturday 68 490

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

15. Strategies for Computational Estimation Compatible numbers – the process of adjusting the numbers so that they are easier to work with

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

17. Reasons and Dangers for Having and Using Algorithms Reasons for having and using algorithms include: • Power • Reliability • Accuracy • Speed Dangers that are inherent in all algorithms include: • Blind acceptance of results • Overzealous application of algorithms • A belief that algorithms train the mind • Helplessness if the technology for the algorithm is not available. (Usikin, 1998)

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 system An understanding of some mathematical properties of whole numbers Understanding of the distributive property An attitude of estimation

19. Teaching Computational Procedures • The following components are important to keep in mind when teaching computational procedures: • Pose story problems in real-world contexts • Use models for computation • Develop bridging algorithms to connect problems, models, estimation, and symbols • Develop the traditional algorithm • Examine children’s work • Determine reasonableness of solutions

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

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 regroup • Use 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 mat • Encourage children to use place-value language as they describe their manipulations

22. Developing Bridging Algorithms for Addition 56 +35 80 11 91 423 + 72 400 90 5 495 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

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 • Trade • Group • Break apart • Break a ten • Make a group • Encourage the following terms to be used meaningfully in a sentence • Subtract • Subtraction • Difference

24. Developing the Subtraction Algorithm (cont’d) • Use models and an organizational mat • Move through a sequence that is unstructured to a more systematic approach • Use estimation and mental computation • Helps children to verify and feel confident about their concrete solutions

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.

26. Other Subtraction Algorithms • Comparison 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.

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

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 understanding • Expanded notation • Distributive property • Build an array for 13 x 4 • Using the array make a comparison to the partial products algorithm • Use the same strategy for 26 x 17

29. Developing the Division Algorithm • Difficult for children to learn as children need to… • Know the division basic facts • Multiply and subtract efficiently • Instruction should • Emphasize 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 computation • Use estimation and mental computation

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?