Computation

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


32

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.


33

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.


34

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:


35

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


36

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:


37

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


38

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


39

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.


40

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:


41

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:


42

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.


43

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.


44

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


45

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

Explore in the internet and get the informations

Computation

Documents

Transcript of Computation