Chapter 10 Computation Methods: Calculators, Mental Computation, and Estimation

1. Chapter 10 Computation Methods: Calculators, Mental Computation, and Estimation Krista Smith, Courtney Miller, CortneyYount

2. Calculators Computation has evolved from counting to using stones, but more recently (>20 years), the use of calculators has come into wide use, and has a greater impact in mathematics.

3. Calculators Balancing instruction: • All computation begins with a problem and with the recognition that computation is needed to solve the problem. • Certain decisions must be made when doing computation. • Estimation is always used to check on the reasonableness of the result.

4. Calculators “As a teacher, you need to help students understand how to use calculators appropriately, which involves showing students that NOT all problems can be solved with a calculator.” “Students using calculators posses a better attitude toward mathematics and an especially better self-concept in mathematics.”

5. Calculators Facts: • Calculators do not think for themselves. Students must still do the thinking. • Calculators can raise students’ achievement. • It is sometimes faster to compute mentally. • Calculators are also useful as instructional tools.

6. Calculators When to use a calculator: • Facilitates problem solving. • Eases the burden of doing tedious computations. • Focuses attention on meaning • Removes anxiety. • Provides motivation and confidence. • Promotes number sense. • Encourages creativity and exploration.

7. Calculators Inappropriate calculator use: • Students turning the calculators upside down to spell words. • Playing games. • Using the calculator to solve simple problems that they can do mentally. • Using calculators for cheating.

8. Mental Computation • Mental Computation is computation done • “all in the head”

9. Mental Computation • This builds on the thinking strategies used to • develop basic facts and naturally extends • children’s mastery of basic facts • It is often done by using compatible, or • “friendly” numbers, such as 10

10. Mental Computation The first step in developing proficiency with compatible numbers is learning to recognize them. Example: Find 2 or more adjacent numbers in a row or column with a sum of 10

11. Mental Computation Mental computation encourages flexible thinking, promotes number sense, and encourages creative and efficient work with numbers

12. Different strategies using Mental Computation • Solving a problem in whole-number addition • Ex) 165 + 99 • “I added 165 plus 100 and got 265, then I • subtracted 1 and got 264” • 4 x 6000 • “Extend the basic multiplication fact 4 x 6 = 24 • and combine with understanding of place value • 6000 • = 24,000

13. Why emphasize mental computation? • It is very useful • Mental computation is the most direct and • efficient way of doing many calculations • It is an excellent way to help develop critical- • thinking skills and number sense and to • reward creative problem solving • Proficiency in mental computation contributes • to increased skill in estimation

14. Guidelines for developing mental computation • Encourage students to do computations mentally • Learn which computations students prefer to do mentally • Find out if student are applying written algorithms mentally • Plan to include mental computation in instruction • Keep practice sessions short (about 10 minutes) • Develop children’s confidence • Encourage inventiveness

15. Estimation • Computational estimation is a process of producing answers that are close enough to allow for good decisions without performing elaborate or exact computations. • Typically done mentally

16. The Estimation Process • Before starting exact computation, students can use estimation to get a general sense of what to expect. • While doing computation, students can use estimation as a check to determine if the computation is moving in the right direction. • After completing a computation, students can use estimation to reflect on their answer and decide if it makes sense.

17. Factual Knowledge of Estimation • As awareness for the use of estimation students develop a greater respect for its power and view it as an essential part of the computational process. • Students who are proficient at written computations are not necessarily good estimators. • Research suggest that memorizing rules and procedures discourages students from using estimation.

18. Factual Knowledge of Estimation cont’d • Good mental computation skills and number sense provide the foundation for the successful development of estimation skills. • Development of good estimation skills is a process that takes years. • First opportunity to not focus on exact answers and yet is a natural aspect of mathematics.

19. The Start to Estimation • Talk about estimation includes the following vocabulary: about, almost, just over, and nearly. • As the students get older the vocabulary extends to the following words: approximate, reasonable, and unreasonable. • Estimation involves getting away from the mindset that only one answer is correct.

20. Examples of Estimation • If I have $2.00 and the sandwich I want is $0.99 and the soda I want is $0.79. Do I have enough money to cover the bill? What strategy should I use to figure this out? • Strategies to Estimation: • Front-End Estimation • Adjusting • Flexible Rounding • Clustering

21. Front-End Estimation Front-End Estimation involves The leading, or front-end, digit in a number The place value or that digit Example: Eggs: $2.13 Milk: $2.39 I have $5.00 do I have enough money? I will take the first digits and I see that they equal $4.00 $2.13 $2.39

22. AdjustingEstimation Strategy • A strategy in which the recognition of the estimation is more or less than that of the exact answer would be • Becomes natural to refine the first estimate by adjusting it. • Example: Used Front-End Estimation of 4000 for the sum of 1213+2926+1578. The student realizes that in order for it to be 4000 the front end digits would have had zeros behind. Knowing the numbers aren’t zeros then the estimate would need to be adjusted. Making the number higher.

23. Compatible Numbers • Compatible numbers for estimation is to chose numbers that are easier to work with and making the given operations needed to solve the problem. • Example: 64/8 is hard to compute mentally but, 64x8 is easier to compute mentally. When using compatible numbers one must remember that it also affects the operations used as well as the numbers.

24. Flexible RoundingAn Estimation Strategy • Rounding is set out to get numbers easier to work with making it an extremely useful estimation strategy. • Students should use flexible rounding when applying to estimation rather than rigid rounding. • Flexible rounding means to round numbers that are close but, also compatible.

25. ClusteringAn Estimation Strategy • Clustering, or averaging, strategy uses an average for estimating the sum. • Two-Step Process for Clustering: • Estimate the average value of the numbers that the value of numbers cluster around. • Multiply by the number of numbers in the group. • Clustering is a limited strategy, since it is appropriate only for quick estimating • the sum of a group of numbers that aren’t too different from one another.

26. When to chose a strategy? • When choosing the strategy it is ultimately up to the student. • Also the numbers and operations involved. • The key is to build the student’s confidence in the ability to chose the right strategy to use in trying to find the answer to a problem. • Use the following guidelines: • Give your students problems that encourage and reward estimation. • Make sure students are not computing exact answers and then rounding to produce estimates. • Ask students to tell how they made their estimates. • Fight the one-right answer syndrome from the start.