standard and alternative computational algorithms chapter 11 tina rye sloan to accompany helping...

Standard and Alternative Computational Algorithms

CHAPTER 11

Tina Rye SloanTo accompany Helping Children Learn Math9e, Reys et al.

©2009 John Wiley & Sons

Focus Questions• 1. What is a computational algorithm? How and why are

manipulative materials useful in helping children develop understanding of algorithms?

• 2. How can teachers help children develop the addition algorithm? Do all children need to use the same addition algorithm?

• 3. What are two standard subtraction algorithms and how did they develop?

• 4. How does the distributive property support the development of the multiplication algorithm?

• 5. What is the partial-products algorithm for multiplication, and how is it related to the traditional multiplication algorithm?

• 6. Why is the traditional division algorithm the most difficult for children to master?

Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009

Computational Algorithms

•A computational algorithms is a computational skill with paper-and-pencil procedures.

•Focus has shifted to more attention on what children construct or develop for themselves.

•Computation has become a problem-solving process, one in which children are encouraged to reason their way to answers, rather than merely memorizing procedures that the teacher says are correct.


Balancing Conceptual Understanding and Computational Proficiency

A balance between conceptual understanding and computational proficiency is essential for developing computational fluency. To develop such an enhanced ability, recommendations for the teaching of computation include the following:

• Fostering a solid understanding of and proficiency with simple calculations

• Abandoning the teaching of tedious calculations using paper-and-pencil algorithms in favor of exploring more mathematics

• Fostering the use of a wide variety of computation and estimation techniques—ranging from quick mental calculation, to paper-and-pencil work, to using calculators or computers—suited to different mathematical settings

• Developing the skills necessary to use appropriate technology and then translating computed results to the problem setting

• Providing students with ways to check the reasonableness of computations (number and algorithmic sense, estimation skills)


Modeling Written Algorithms with Concrete Materials

•Use base-ten blocks and/or other concrete materials to model the "common" algorithms for addition, subtraction, multiplication, and division in the following slides. Compare your models with those used in our text.

27+ 35



•There were 61 children who did not sign up for hot lunch. There were 22 of these who went home for lunch. The rest brought a cold lunch. How many brought a cold lunch?



4

15 x 19

135 150285


Addition Algorithms

•Standard Addition Algorithm•Partial-Sum Addition Algorithm•Higher-Decade Addition


Addition: Standard Algorithm• When students attempt to use the standard

algorithm without understanding why it works, they may be more prone to errors than when they do addition in ways that intuitively make sense to them.

• Example: 27 + 35 1 27 Write 2 in the ones column +35 and 1 in the tens column. 62 2 + 3 + 1 = 6 tens


Addition: Partial Sums

• As group 5 used it, they added from left to right (tens first, then ones), which is more natural than working right to left. The partial-sum algorithm can be used as an alternative algorithm for addition (an end goal for students) or it can be useful as a transitional algorithm (intermediate step on the way to learning the standard algorithm). Ideally, however, children should be encouraged to work with whichever procedure they find easiest to understand.

Group 5 27 + 35 50 +12 62


Addition: Higher-Decade

Combinations such as 17 + 4 or 47 + 8 or 3 + 28, called higher-decade combinations, are used in a strategy sometimes referred to as “adding by endings.” Note that the two-digit number may come either before or after the one-digit number. In the Classroom 11-3 focuses attention on the relationship of 9 + 5, 19 + 5, 29 + 5, and so on. As a result of this activity, children realize the following:

• In each example, the sum will have a 4 in the ones place because 9 + 5 = 14, and the tens place will always have 1 more ten.


Subtraction

•Standard Subtraction Algorithm•Partial-Difference Subtraction Algorithm


Subtraction: Standard Algorithm

• The standard subtraction algorithm taught in the United States for the past 50 or 60 years is the decomposition algorithm. It involves a logical process of decomposing or renaming the sum (the number you are subtracting from). In the following example, 9 tens and 1 one is renamed as 8 tens and 11 ones:

91 11 – 4 = 7 ones –24 8 tens –2 tens = 6 tens

67

8 11


Subtraction: Partial-Difference

Take a moment to examine this algorithm. How does it work?


Multiplication

•Multiplication with One-Digit Multipliers•Multiplication by 10 and Multiples of 10•Multiplication with Zeros•Multiplication with Two-Digit Multipliers•Multiplication with Large Numbers


Multiplication with One-Digit Multipliers

2 × 14 = 2 × (10 + 4) = (2 × 10) + (2 × 4) = 20 + 8 = 28

Array of 2 x 14

Array of Distributive Property of Multiplication


Multiplication by 10 and Multiples of 10

• Multiplying by 10 comes easily to most children, and is readily extended to multiplying by 100 and 1000 as children gain an understanding of larger numbers.

• Multiplying by 20, 30, 200, 300, and so on is an extension of multiplying by 10 and 100. Emphasize what happens across examples and generalize from the pattern. For example, have children consider 3 x 50:

3 x 5 = 15 3 x 5 tens = 15 tens = 150

3 x 50 = 150 Then have them consider 4 x

50: 4 x 5 = 20

4 x 5 tens = ____ tens = _____ 4 x 50 = ______


Multiplication with Zeros

•When zeros appear in the factor being multiplied, particular attention needs to be given to the effect on the product or partial product. Many children are prone to ignore the zero.

•When an estimate is made first, children have a way of determining whether their answer is in the ballpark.


Multiplication with Two-Digit Multipliers

Arrays or grids offer one way to bridge the gap from concrete materials to symbols and they also help illustrate, once again, why the partial-products algorithm makes sense.


Lattice Multiplication

2 7 5

9

2


•Use the lattice method to solve the problem

275 x 92

Multiplication with Large Numbers

•As children experiment with using a calculator for multiplication, there will come a time when they overload the calculator. Sometimes the number to be entered contains more digits than the display will show. At other times the factors can be entered, but the product will be too big for the display.

•When this happens, children should be encouraged to estimate an answer and then use the distributive property plus mental computation along with the calculator.


Division

•Division with One-Digit Divisors•Division with Two-Digit Divisors•Making Sense of Division and Remainders


Division with One-Digit Divisors

Can you explain the Subtractive Division Algorithm? How does it compare to the Distributive Division Algorithm?


Division with Two-Digit Divisors

•Work with two-digit divisors should aim toward helping children understand what the procedure involves but not toward mastery of an algorithm.

• The calculator does the job of multi-digit division for most adults, so there is little reason to have children spend months or years mastering it. Other mathematics is of more importance for children to learn.


Making Sense of Division and Remainders

• Pass out 17 candies to 3 children. (Each child receives 5 candies with 2 candies left over. Or, if the candies can be cut into pieces, each child could have 5 candies plus 2/3 of a candy.)

• To make each Valentine’s card you need 3 pieces of lace. You have 17 pieces of lace. How many cards can you make? (You can make 5).

• If 17 children are going on the class trip and 3 children can ride in each car, how many cars are needed? (You will need 6 cars. With 5 cars you could seat only 15 children, with 2 children still waiting for a ride. With 6 cars, you can seat all 17 children, with 1 seat left over.)

• Note that the remainder is handled differently in each of these real-world problems.


Checking

•The calculator can serve many other functions, but its use in checking has not been overlooked by teachers. Nevertheless, the calculator should not be used primarily to check paper-and-pencil computation.

•Encourage estimation extensively, both as a means of identifying the ballpark for the answer and as a means of ascertaining the correctness of the calculator answer.


Choosing Appropriate Methods

• Children must learn to choose an appropriate means of calculating.

• Sometimes paper and pencil is better; sometimes mental computation is more efficient. Other times use of a calculator is better than either, and sometimes only an estimate is needed.

• Encouraging students to defend their answers often yields valuable insight into their thinking. Children need to discuss when each method or tool is appropriate, and they need practice in making the choice, followed by more discussion, so that a rationale for their choice is clear.


Building Computational Proficiency

•Computational fluency with addition, subtraction, multiplication, and division is an important part of mathematics education in the elementary grades; however, “developing fluency requires a balance and connection between conceptual understanding and computational proficiency” (NCTM, 2000, p. 35).


How Many Strategies?

•Examine the problems in the following slide:• How many different ways can you solve them using

mental computation and/or written computation? Compare your strategies with those used in our text.


74

- 58

14

x 2

4 52

Jill and Jeff both collected baseball cards. Jill had 27

cards and Jeff had 35. How many did they have

together?

Problems


standard and alternative computational algorithms chapter 11 tina rye sloan to accompany helping...

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