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

1. Chapter 10Computation Methods: Calculators, Mental Computation, and Estimation Lisa Jolley, Jessi Yates, and Sarah Golubski

2. Introduction • Computation has been around for a long time, and has taken quite a few different forms. • Piles of Stone, or an abacus to write algorithms on. • Competence with using different methods of computations is not just useful, but ESSENTIAL!

3. Teachers need to balance instruction and need to learn when to use written computations. • 1. All computation begins with a problem and with the recognition that computation is needed to solve the problem. • 2.Certain decisions must be made when doing computations. • 3. Estimation is always used to check on the reasonableness of the result. • There are two essential decisions in ever computation: 1. deciding on the type of result needed, and 2. deciding on the best method for getting that result.

5. The goal of teaching computation is to help students with: • Developing competence with each computational method. • Choose a method that is appropriate for the computation at hand. • Apply the chosen method correctly. • Use estimation to determine the reasonableness of the result.

6. Calculators • More than 80% of all mathematical computations in daily life involve mental computation and estimation, rather than written computations. • In the United States, 70%-90% of the time, instruction is focused on written computation. • The National Research Council believes “child should use calculators throughout their school work, just as adults use calculator throughout their lives. More important, children must learn to use them and when not to do so. They must learn from experience with calculators when to estimate when to seek exact answer; how to estimate answers to verify the plausibility of calculator results; and how to solve modest problems mentally when neither pencil nor calculator is convenient.”

8. The NCTM believes “appropriate calculators should be available to all students at all times.” • Teachers need to : • Help students understand how to use calculators appropriately • Showing students that not all problems can be solved with a calculator • Need to give students time to explore the calculator on their own • Show students how to handle calculators correctly. • Communicate with parents that caculators are being used in the classroom and how and why you are using them.

9. Calculator Myths and Facts • Myth: Using Calculators does not require thinking. • Fact: Calculators do not think for themselves. Students must do the thinking. • Myth: Using calculators lowers achievement. • Fact: Calculators can raise students’ achievement. • Myth: Using calculators always makes computations faster. • Fact: It is sometimes faster to compute mentally. • Myth: Calculators are useful only for computations. • Fact: Calculators are also useful as instructional tools.

10. Using Calculators Requires Thinking • Teachers must make it clear, calculators do not think, they only follow instructions. • Calculators do not discourage thinking. • If properly used, encourage thinking because they free students from long problems, and give more time to the important problem solving processes that generally precede, and often follow the computation.

11. Using Calculators Can Raise Student Achievement • Research has shown that students who use calculators have a better attitude towards mathematics. • Research has also shown that calculators in addition to traditional instruction can improve students’ basic skills and problem solving.

12. Calculators Are Useful for More Than Just Doing Computations • Calculators can be used to see pattterns, gives students practice with important skills, and reinforces knowledge for example, basic multplication facts , and division facts. • Calculators should be considered for use whenever computational skills are not the main focus.

13. There are two main uses for the calculator in the classroom; as a computational tool and as an instructional tool. • A calculator should be used as a computational tool when it: • Facilitates problem solving • Eases the burden of doing tedious computations • Focuses attention on meaning • Removes anxiety about doing computations incorrectly • Provides motivation and confidence

14. A calculator should be used as an instructional tool when it: • Facilitates a search for patterns • Supports concept development • Promotes number sense • Encourages creativity and exploration

15. Mental Computation • Mental computation is done “all in the head” • Without tools such as calculator or pencil and paper • Research has documented numerous methods for computing mentally • Many methods developed by children and are techniques that “make sense to them”

16. Mental Computation • Builds on thinking strategies used to develop basic fact • Naturally extends children’s master of basic facts • Extending facts is early step in in learning additional skills for mental computation • Takes different forms • Extending • Utilizing Place Values • Decompose (break up)

17. Extending 4 + 5 = 9 so 40 + 50 = 90 and 400 + 500 = 900

18. Using Place Value 675 – 200 = 475 675 – 50 = 625 675 675 -200 - 50 475 625

19. Combining Basic Facts with Place Value Recognize that 42 + 16 = 42 + 10 + 6 And 42 + 10 = 52 And 52 + 6 = 58

20. Decomposing 18 + 17 = 35 Decompose 17 2 + 15 Thus 18 + 17 = 18 + 2 + 15 and 18 + 2 = 20 S0 20 + 15 = 35

21. Mental computation can be done by using “friendly” or “compatible” numbers • Friendly numbers are ones that can be combined to makes numbers that are easy to compute • 10 is a friendly numbers so having children use ten-frame will help them see how to link different numbers to make 10 8 + ___ = 10 ___ + 5 = 10

22. Compatible Numbers 8 + 7 + 22 + 5 + 13 Recognize 8 + 22 = 30 8 + 7 + 22 + 5 + 13 7 + 13 = 20 30 + 20 = 50 And 50 + 5 = 55

23. Developing Proficiency by Practice Find numbers that add up to ten (find 3 sets) 6 1 6 3 8 3 7 6 5 2 2 4 2 5 4 1 6 2 3 4 9 8 2 6 8 2 2 1 7 1 4 6

24. Mental Computation • Encourages flexible thinking • Promotes number sense • Encourages creativity and efficient number use • Should be tried before using paper and pencil or calculator • Use friendly numbers

25. Example of Mental Computation: 397 4 = ? Think 400 4 = 1,600 and 397 is 3 less than 400 so 3 4 = 12 therefore 1,600 – 12 = 1,588

26. Example of Mental Computation:Look for an easy way 2 3 7 5 = ? Think 2 3 = 6 6 5 = 30 30 7 = 210

27. Example of Mental Computation:Use logical reasoning 15 120 = 10 120 = 1,200 Plus ½ of 10 120 = 600 And 1,200 + 600 = 1,800

28. Example of Mental Computation:Use knowledge about the number system 56 – 24 = ? 54 – 24 = 30 So 56 – 24 = 32

29. Mental Computations vs. Written Algorithms 165 + 99 When asked to compute mentally students generally 165 + 99 264 They did this in their head by using written format. “I added 5 plus 9 and got 14. I carried the 1, and 5 plus 9 plus 1 is 15. I carried that 1 and got 2. It’s two-six-four, or 264.” This is much different than mentally computing 99 is one less than 100 so one less than 165 is 164 so 100 + 164 = 264

30. Guidelines for developing mental computation skills • Encourage students to do computations mentally • Learn which computations students prefer to do mentally • Find out if students are applying written algorithms mentally • Plan to include mental computation systematically and regularly as an integral part of instructions • Keep practice sessions short (10 minutes at a time) • Develop children’s confidence • Encourage inventiveness • Mental computations or estimations?

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

32. Estimation • When to use estimation… • Before starting exact computation • While doing computation • After completing a computation

33. Teacher’s Role • Move them away from finding the “exact” answer. • Give immediate feedback and ask how they got their answer. • Encourage them to be flexible when thinking about #’s. • Develop different strategies to estimate.

34. Front End Estimation • Involves checking the leading digit (or front end) and the place value of that digit. • *Helps to reach an estimation quickly • Example: • 2.19 + 1.29 + 1.17 -------> 2 + 1 + 1

35. Compatible Numbers • Involves rounding the numbers in the problem to numbers that are easier to work with • *Depends on the operation as well as the numbers used • Example: • 64 x 8 ----> 60 x 8 or 38+67+49--->35+65+50

36. Flexible Rounding • Involves rounding to the numbers that are close but also compatible. They should be encourage to adjust in order to compensate for how the numbers were rounded. • Example:

37. Clustering • Estimate the average value of the number that the numbers cluster around. Multiply by the number of numbers in the group. • *This is best used when adding a long list of front end digits. • Example: 3.42 2.12 ~ 3 x 6 = 18 2.98 3.78 2.50 3.79

38. How to encourage awareness of strategies • Give them messy numbers so they will want to estimate rather than compute an exact number. • Example: 78342 + 83289 rather than 78 + 83 • Make sure students are not computing exact answers and then rounding to produce estimates • Have students share with peers how they made their estimates

39. How to encourage awareness of strategies • Develop a range for right answers so they understand that their can be more than one right answer due to the various strategies of estimation. • Encourage students to think of real-world situations that involve making estimates.