Transcript Reys.ch11.Computational.Algorithms

CHAPTER
11
Standard and Alternative
Computational Algorithms
Tina Rye Sloan
To accompany Helping Children Learn Math10e, Reys et al.
©2012 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,
10th Edition, © 2012
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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-andpencil 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)
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Modeling Written Algorithms with
Concrete Materials
• 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?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Modeling Written Algorithms with
Concrete Materials
4
15
x
19
135
150
285
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Addition Algorithms
• Standard Addition Algorithm
• Partial-Sum Addition Algorithm
• Higher-Decade Addition
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009
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
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Subtraction
• Standard Subtraction Algorithm
• Partial-Difference Subtraction Algorithm
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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:
8 11
91
–24
67
11 – 4 = 7 ones
8 tens –2 tens = 6 tens
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Subtraction: Partial-Difference
Take a moment
to examine this
algorithm. How
does it work?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Multiplication with One-Digit
Multipliers
2 × 14 = 2 × (10 + 4) = (2 × 10) + (2 × 4)
= 20 + 8 = 28
Array of Distributive
Property of Multiplication
Array of 2 x 14
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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 = ______
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Lattice Multiplication
• Use the lattice method to solve the problem
275 x 92
2
7
5
9
2
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Division
• Division with One-Digit Divisors
• Division with Two-Digit Divisors
• Making Sense of Division and Remainders
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Division with One-Digit
Divisors
Can you explain the
Subtractive Division
Algorithm? How does
it compare to the
Distributive Division
Algorithm?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9th Edition, © 2009
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Building Computational Proficienc
• 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).
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012
Problems
Jill and Jeff both collected
baseball cards. Jill had 27
cards and Jeff had 35. How
many did they have
together?
x
14
2
4 52
-
74
58
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
10th Edition, © 2012