Operations with Decimals

Addition and Subtraction

- Since many decimal numbers can be written as using fractions, we add them and subtract them in a manner similar to adding and subtracting fractions: add/subtract "like" fractions (same denominators and same decimal places)
Example: 
  24.076
+  3.19
=27.266

Subtraction with decimal squares, one square is the unit.

Pencil and paper algorithm is the traditional way of doing addition and subtraction
  2.4      5.6
+3.2    -3.4
=5.6   =2.2

Multiplication
Whole number x decimal
3 x 0.2
= 0.2 + 0.2 + 0.2
= 0.6

Area Model:

Pencil and paper algorithm:

Powers of 10:
Example:
0.3 (or 3/10) x 10^2 = 30

Example:
4.012 x 10^7
[4(1)+0(10^-1)+1(10^-2)+2(10^-3)] 10^7

Example:
28.7 x 10^-3
287/100 x 1/1000 = 287/10000 = 0.0287

Division:
Measurement/Subtractive
.9 / .3 = 3
---I---I---   

Sharing/Partitive
.9 / 3= 3
(---) (---) (---)
This shows that 3 friends are each given 3 peices of candy

Paper and pencil algorithm:

Division and converting decimals to fractions

Division powers of ten

Example: 4.07 / 10^2 = 0.0407
               Check by doing this: 4.07 x 10^-2 [4(1) + 0(10^1) + 7(10^-2)] (10^-2)
Example: 78.392 / 10^7 = 0.0000078392
               A positive exponent moves the decimal to the left that many places, and moves the decimal to the right if the exponent is negative.

Converting Decimals to Fractions

Helpful Hints
-  0.11111... = 1/9
-  0.01010... = 1/99
-  0.001001... = 1/999

A number over the fraction of 9 is repeating.
Examples:
2/9 = 0.2222...
3/9 = 1/3 = 0.3333...
4/9 = 0.4444...
5/9 = 0.5555...
6/9 = 2/3 = 0.6666...
7/9 = 0.7777...
8/9 0.8888...
25/99 = 0.25252525

To write a repeating decimal into a fraction, there are two ways.

One way is:

Another way is:

More ways of converting decimal examples are:
0.47 = 47/100
0.028 = 28/1000
0.777... = 7/9
0.2121... = 21/99

Deciamls and Rational Numbers

• The number 12.345 has 3 decimal places because of 345. the number 12 in front of the decimals is called the integer and the number 345 behind the decimal is in fact the actual decimal.
• Writing out a number in expanded form is as follows:                                                                  123.45= 1(100) + 2(10) + 3(1) + 4(1/10) + 5(1/100)
• Given the number 573.9634 what place value does the 9, 6, 3 and 4 have?

           9: Tenths
           6: Hundredths
           3: Ones and thousandths
           4: Ten thousandths

• 245.639 in written form is: Two hundred forty-five and six hundred thirty-nine thousandths
• 6.4193 in written form is: Six and four thousand one hundred, ninety-three ten thousandths
• The and is the way to separate the integer from the decimal

Here are three models for decimals: Decimal squares, base ten blocks, and a number line

                                                             Tenths- This represents 4/10 or 0.4

Hundredths- This represents 27/100 or 0.27

Thousandths- This represents 463/1000 or 0.463

Base Ten Blocks:

This picture shows 254
1.) If the long is the unit, what number is represented by the drawing? 25.4
2.) If the flat is the unit, what number is represented in the drawing? 2.54

Number Line:

Converting numbers:

• 42/100 = 0.42
• 64193/10000 = 6.4193
• 9/1000 = 0.009
• 436/10 = 43.6

Equality of Decimals: Two decimals are equal if they have the same numbers in the same place values not including extra zeros on the end of the number  Ex:  0.7 = 0.70

Inequality of decimals: To write these numbers in increasing order, start with the smallest number in the tenths place and keep going. 

• 0.036,  0.19, 0.195, 0.2  is in creasing order.
• 0.190 is less than 0.195

A rational number is any number that can be written in the form a/b where b is not a zero and a and b are integers is called a rational number.

Which of these has a repeating decimal representation?

A. 3/16

B. 3/15

C. 3/10

D. 3/9

E. All of the above

-Not A because it can be written as 3/(2x2x2x2) 0.1875

-Not B because it can be written as 3/(3x5) 0.2

-Not C because it can be written as 3/(2x5) 0.3

- It is D because it is written as 3/(3x3) 0.3 repeating

-Therefore it is not E

- A fraction or rational number can be written as a terminating decimals if the prime factorization of the denominator contains only 2's and or 5's with the fraction in lowest terms.

Multiplication and Division of Fractions

 Multiplication of fractions is the process of multiplying the numerators (top two numbers) by each other, and the same with the denominators. A visual of take the fraction is as follows...

Another visual of multiplication of fractions is as follows...



The multiplication of improper fractions is displayed below...

The division of fractions has one more step than multiplication. The second fraction needs to flip and the sign needs to be changed into a multiplication sign as follows...

Properties of Fractions:

Closure: fraction x fraction = fraction

Commutative: A/B x C/D = C/D x A/B

Associative: (A/B x C/D) x E/F = A/B x (C/D x E/F)

Identity: A/B x 1 = A/B x 1/1 = A/B

Distributive: A/B ( C/D + or - E/F) = A/B x C/D + or - A/B x E/F

Inverse: A/B  B does not equal zero. B/A so that A/B x B/A = 1

 Properties of Fractions Game

Decimal Place Values:

Repeating Decimals:

Conversions:

• Decimals into Fractions Extra Practice

Introduction to Fractions

• Fractions were first introduced in measurement problems, to express a quantity that is less than a whole unit. The word fraction comes from the Latin word “fractio” which means “the act of breaking into pieces”.
•  Fraction: A fraction is an ordered pair of integers a and b, with b not equal to 0, written . The integer a is called the numerator of the fraction, and the integer b is called the denominator of the fraction.

            3 Models for Fractions:  1.) Part to whole  2.) Divison  3.) Ratio

1.) Part to Whole-  To interpret the meaning of any fraction we must
agree on the unit (The "Whole") ex: a cup, an inch, the area of a hexagon, a whole pizza, etc. Understand that the unit is subdivided into b parts of equal size
understand that we are considering a parts of the unit.
            Area Models: are the shaded part of a shape

Set Model-  A finite set of objects U is chosen to represent the unit which is divided into n(U) = equal parts. A fraction is then visualized by circling the number of objects that represent the numerator of the fraction.
= 3/10
Number-Line Model: The unit is represented by the length of the line segment between 0 and 1 on the number line. That portion of the number line is subdivided into equally sized segments.

2.) Division- For any numbers A and B with b not equal to 0  For Example: to compute 3 divided by 4, we use the sharing concept of division

3.) Ratio 
-In this case, fractions are used to compare one amount to another.
-Using Cuisenaire Rods, we compare the lengths of two rods.
-For example, if we compare the dark green and red rods. How many red rods does it take to have the same length as the dark green rod? What fraction does the red rod represent if the dark green rod is considered the unit? How about comparing the red rod to the yellow?

 

Fraction bars with + - / x

Today we worked on computing with fraction bars with addition, subtraction, multiplication and division.

Piaget has charted the cognitive development of preadolescents, and his research indicates that even at the age of 12, most children deal only with symbols that are closely tied to their perceptions. For example, the symbolic representation of 1/6+2/3=5/6 has meaning for most elementary school children only if they can relate it directly to concrete or pictorial representations.

• Obtaining Common Denominators: If each part of the 5/6 bar is split into 2 equal parts and the 1/4 bar is split into 3 equal parts, both bars will have 12 parts of the same size. These new bars show that     5/6 + 1/4 = 13/12 (Improper) that changed into 1 1/12 (Mixed Number)  and then, 5/6 - 1/4 = 7/12

Subtraction of same fraction numbers

Multiplication: To determine 1/3 x 4/5, split each shaded part of the 4/5 bar into 3 equal parts. One of these split parts is 1/15 of the whole bar, and 4 of these split parts is 4/15 of the whole bar.

 (You can think of this as having 4/5 of an amount and taking 1/3 of each fifth).

Division: Using the measurement approach to division, we can write 1/2 / 1/6 = 3, becuase the shaded portion in the picture below of the 1/6 bar can be measured off (or fits into) exactly 3 times on the shaded part of the 1/2 bar.

Similarly, 5/6  /  1/3 = 2 1/2 because the shaded portion of the 1/3 bar can be measured off (or fits into) 2 1/2 times on the shaded part of the 5/6 bar.