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WAJ3105 Numerical Literacy
31
TOPIC 2
OPERATION AND COMPUTATION
2.1 Synopsis In this chapter, you will develop techniques for mental computation and
estimation and explore paper-and-pencil procedures for adding, subtracting,
multiplying, and dividing whole numbers. Mental computation and estimation
require a solid understanding of numeration, a mastery of the basic facts, good
number sense, and an ability to utilize mathematical reasoning. This chapter also
gives outline the calculator and computer are tools for doing mathematical
computation. Appropriate uses of calculator and computer are away of increasing
the amount and the quality of learning afforded students doing the course of their
mathematics education.
2.2 Learning Outcomes
1. Perform calculation on pencil and paper, calculator and computer,
mental computation, and manipulative materials.
2. List and describe appropriate and inappropriate uses of calculator and
computers in teaching and learning of primary school mathematics.
2.3 Overview of Content
OPERATION AND
COMPUTATION
Pencil and Paper
Calculator and Computer
Mental Computation and
Estimation
Manipulative Materials
Appropriate
Inappropriate
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2.4 Teaching of Addition and Subtraction Besides learning about whole numbers and basic number concepts, children in
primary schools also need to acquire basic computational skills. The four basic
operations that children need to know and master include addition, subtraction,
multiplication and division. This topic focuses on two basic operations, that is .
addition and subtraction. Addition and subtraction are introduced in Kindergarten
and Primary One. These two operations are taught in school every year by
reviewing operations introduced previously and extending algorithms for work
with larger numbers.
2.4.1 Algorithms of Addition and Subtraction In this section, we examine the step-by-step procedures the algorithms for
adding and subtracting whole numbers. We focus on using models and logic to
make sense of the computational procedures for finding sums and differences,
regardless of the algorithm used. Mini-Investigation 2.1 asks you to analyze the
paper-and-pencil computational procedures you ordinarily use.
Essential Understandings for Section 2.4.1
There is more than one algorithm for adding whole numbers and more than
one algorithm for subtracting whole numbers.
Most common algorithms for addition and subtraction of whole numbers use
notions of place value, properties, and equivalence to break calculations into
simpler ones. The simpler ones are then used to give the final sum or
difference.
Properties of whole numbers can be used to verify the procedures used in
addition and subtraction algorithms.
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There are different concrete interpretations for addition and subtraction of
whole numbers, and certain ones are helpful in developing addition and
subtraction algorithms.
M I N I - I N V E S T I G AT I O N 2.4.1 : Communicating How would you complete the following subtraction calculation, using
paper-and- pencil?
2,004
- 1,278
2.4.2 Developing Algorithms for Addition
A model is a useful tool for explaining an algorithm. For example, the use of
base-ten blocks as a model to find the sum of two numbers involves actions with
the blocks that can later help illustrate the procedures used in a paper-and-pencil
algorithm for addition. In this subsection, we first look at an example that models
addition. Then we develop the related paper-and-pencil algorithm. Finally, we
use the properties of whole-number operations to verify that the steps in an
addition algorithm are logical.
Using Models as a Foundation for Addition Algorithms Example 2.1 shows
how base-ten blocks can be used to find a sum and thus provide models that
help explain the addition algorithms. In Example 2.1, you can think of the base-
ten blocks representing 369 and the base-ten blocks representing 244 as the
elements in two disjoint sets. The union of these sets is then found by joining the
two sets of blocks.
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Example 2.4.1 : Using the Base-Ten Blocks Model for Addition The two numbers shown are modeled with the base-ten blocks:
Use the base-ten blocks to find the sum of the two numbers and then write an equation to record the addition. Solution : Method 1 Step 1: Started by putting together all blocks of the same type:
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Step 2 : Regrouped 10 tens to make 1 hundred:
Step3: Regrouped 10 ones to make 1 ten:
The sum is 613, and recorded the addition with a vertical equation:
369
+ 244
613
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Method 2 : Step 1: Started by putting together the ones and then regrouped 10
ones to make 1 ten and was left with 3 ones:
Step 2: Next, put together the tens and then regrouped 10 tens to make 1
hundred and was left with 1 ten:
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Step 3: Finally, put together the hundreds, which gave me 6 hundreds, 1 ten,
and 3 ones:
The sum is 613, and recorded the addition with the equation 369 + 244 = 613
Practice: Use base-ten blocks to show 374 + 128. Write an
equation to record the addition. 2.4.3 Developing and Using Paper-and-Pencil Algorithms for Addition.
Let’s now look at two paper-and-pencil algorithms for addition that follow directly
from the models in Example 2.4.1. We use the same calculation, 369 + 244, to
show both of these algorithms. Along with the models in Example 2.4.1, these
algorithms again emphasize that in mathematics even a routine task often may
be done more than one way.
The first algorithm, which relates to Method 1 in Example 2.4.1, is the expanded
algorithm in which the values of each place are added first and later combined. Expanded Algorithm for Addition Think Write
369 + 244
Add hundreds: 300 + 200 = 500 500
Add tens: 60 + 40 = 100 100
Add ones: 9 + 4 = 13 + 13
Add the hundreds, tens, and ones: 613
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In the expanded algorithm, the order in which the numbers with a given place
value are added doesn’t matter because all partial sums are recorded.
The second algorithm, called the standard algorithm, relates to Method 2 in
Example 2.4.1 and involves starting with the ones and proceeding to add, with
regrouping, from right to left.
Whenever we use the standard algorithm and there are 10 or more ones, we
regroup 10 ones as 1 ten and then add the tens. If there are 10 or more tens, we
regroup 10 tens to make 1 hundred and then add the hundreds, regrouping as
needed. This process continues for as many digits as there are in the addends. Example 2.4.2: Using the Expanded and Standard Algorithms for Addition Use either the expanded or the standard algorithm to find the sum Solution Method 1 : Add the ones, then the tens, and finally the hundreds. Each
time write the partial sum. Then find the total of the partial sums.
562 + 783 5 140 1200
1345
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Method 2: The first add the ones. Then add the tens and regroup.
Finally, add the hundreds. So that 13 hundreds are 1 thousand and
3 hundreds.
562
+783
1345
2.4.4 Developing Algorithms for Subtraction
Models can be used to explain algorithms for subtraction in much the same way
as they are used to explain addition algorithms. We first use models to illustrate
the procedures for subtraction. Then, we use those procedures to develop pencil-
and-paper subtraction algorithms. Finally, we use mathematical reasoning to
justify the subtraction algorithm.
Using Models as a Foundation for Subtraction Algorithms. Our use of base-
ten blocks in addition demonstrated that the procedures for finding a sum may be
modeled in various ways. We also saw that the process used to join and regroup
the base-ten blocks connects closely with a model of and the definition of
addition. Similarly, using base-ten blocks to find differences demonstrates that
many different procedures for modeling subtraction are also available. The
procedures shown below for using base-ten blocks to model a procedure for
subtraction.
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Example 2.4.3 : Modeling a Procedure for Subtraction The larger number in the subtraction calculation shown is modeled with base-ten blocks:
Find the difference by using the base-ten blocks and write an equation to record the subtraction. Solution
Method 1: To have enough ones to take away 8, I begin by trading 1 ten for 10
ones. I then take away 8 ones from the 15 ones, leaving 7 ones:
Next, I take away 1 ten from the 3 tens remaining and am left with 2 tens:
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With no hundreds to take away, I now find the difference, 227, and record
the subtraction:
245
-18
227
Method 2: Started at the hundreds place and note that there are 0 hundreds
to take away. And then take away 1 ten from the 4 tens, leaving 3
tens:
Now need to take away 8 ones but have only 5 ones. And then take away
the 5 ones, leaving 2 hundreds and 3 tens:
Now trade 1 ten for 10 ones and take away 3 ones, leaving 2 hundreds, 2
tens, and 7 ones:
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Finally, the difference is 227 and record the subtraction with the equation 245 –
18 = 227 2.4.5 Developing and Using Paper-and-Pencil Algorithms for Subtraction. Let’s now look at two paper-and-pencil algorithms for subtraction. We use the
calculation task modeled in Example 2.4.3 to develop these algorithms. The first
algorithm is based on Method 2, whereby he subtracted the values of each place
beginning on the left. In this algorithm, called the expanded algorithm, we start
with the greatest number and repeatedly take away as much as is possible to do
mentally before moving from left to right.
In the expanded algorithm, subtracting could begin at any place because the
order of subtracting will not change the difference. The second algorithm, based
on Method 1 in Example 2.4.3, is called the standard algorithm and involves
starting with the ones and proceeding to subtract, with regrouping, from right to
left.
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If not enough ones are available to subtract when we use the standard algorithm,
we regroup 1 ten as 10 ones and then subtract the ones. If there are not enough
tens to subtract, we regroup 1 hundred as 10 tens and subtract. 2.5 Teaching of Multiplication and Division In this section, we look at algorithms for multiplication and division of whole
numbers. We begin by using models to help explain these algorithms and then
use the properties of whole numbers to justify the algorithms. 2.5.1 Developing Algorithms for Multiplication
As with addition and subtraction algorithms, models provide a physical basis for
explaining algorithms for multiplication. The models used here include base-ten
blocks and pictorial models that represent multiplication as finding the area of a
rectangle. Using the processes suggested by the models, we develop the related
paper-and-pencil algorithms for multiplication. Finally, we use mathematical
reasoning along with properties of whole numbers to verify that the steps in a
multiplication algorithm are logically correct. Developing and Using Paper-and-Pencil Algorithms for Multiplication.
We now use the multiplication calculation modeled in Example 2.4.4 to examine
two paper-and-pencil algorithms for multiplication. Partial products play an
important role in each. The first algorithm based on that model involves breaking
apart the numbers according to the place value of each digit and multiplying each
digit according to its place value to obtain the partial products. In this algorithm,
called the expanded algorithm, the partial products are added to find the final
product.
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Example 2.4.4
The second algorithm, called the standard algorithm, involves forming only
two partial products.
In this case, the first factor is multiplied by the ones digit of the second factor and
the numbers are regrouped to form the first partial product. Then the first factor is
multiplied by the tens digit of the second factor.
Example 2.4.5 further illustrates the use of these two algorithms.
Example 2.4.5 : Using the Expanded and Standard Algorithms for
Multiplication Choose either the expanded or standard algorithm to calculate the product
345 x 666
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Solution Caleb’s thinking: First, I multiplied the ones by 6, then the tens, and then the
hundreds. Then, I added all the partial products. Here is what I came up with:
345
X 666
30
240
1,800
2,070 Makenzie’s thinking: First, I multiplied the ones by 6 and regrouped. Then, I
multiplied the tens by 6, added the extra tens, and regrouped. Finally, I multiplied
the hundreds and added the extra hundreds. Here is what I came up with:
2 3
345
X 6
2,070
2.5.2 Developing Algorithms for Division
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