Denseness of Rational Numbers

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Denseness of Rational Numbers. Pre-Algebra Mrs. Yow. What does it mean to be DENSE?. Which material is more DENSE here?. Why???????. Which material is more DENSE here?. The Hair!!!. Compare Rational Numbers (Find numbers between). Using Models Using Common Denominators

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### Denseness of Rational Numbers

Pre-Algebra

Mrs. Yow

Compare Rational Numbers(Find numbers between)
• Using Models
• Using Common Denominators
• Using Place Value
• Using Definition of Less Than
Using Common Denominators

When the denominators of two fractions are the same,

the one with the greater numerator represents

the larger rational number.

If Denominators are Unlike

The Fundamental Law of Fractions can be used to

write equivalent fractions with the same denominator

if the denominators of the fractions to be compared are

different.

The Cross-Product can also be used to compare

fractions that have different denominators.

Using Place Value
• Same procedure for comparing whole numbers in that we start on the left with the place with the largest value and compare each place as we move to the right.
• Rationale for this process is based on the use of common denominators.
Using Definition of Less Than
• Whenever a positive rational number is added to a first rational number to get a second rational number, the first number is less than the second.
• For example, , so we know that .
Denseness of Rational Numbers
• Between any two rational number there exists an infinite number of other rational numbers.
• We can find rational numbers between any two rational numbers using common denominators and place value (much like we do when comparing rational numbers).
• A discussion of denseness is important in classrooms to help students understand, for example, that is NOT the only rational number between and .
Example
• Find three rational numbers between & .
Repeating Decimals and Fractions
• Recall that every rational number in fraction form can be written as a terminating or repeating decimal.
• If it is a repeating decimal, it has a denominator of “9”, “99”, “999”, etc…..depending on how many digits are repeating…..
Examples
• Write each repeating decimal as a simplified fraction.

1.) 0.11111…

2.) 0.2222…

CLASSWORK

FACE TIME (20-25 minutes

SHOWDOWN!!!

Determine the validity of the following statement.

“If x and y are rational numbers, then x < y < 0 guarantees that x2 < y2.”

a) Always true

b) Sometimes true

c) Sometimes false

d) Never true

SHOWDOWN!!!

Using your calculator, find a rational number between and .

SHOWDOWN!!!

Using your calculator, find a fraction between the rational numbers and . (DOK 3)

SHOWDOWN!!!

Find the product of and . Then

divide the product by 2. Will the answer yield

a rational number between and ?

SHOWDOWN!!!

3.45 is a solution to the inequality 3 < x < 3 .

Which statement justifies that 3.45 is a true value for x?

a) 3.45 is less than 3 .

b) 3.45 is greater than 3.5 and less than 3 .

c) 3.45 is greater than 3 and less than 3 .

d) 3 is greater than 3.45.

SHOWDOWN

Write three numbers between: -2.4 < x < -2.31

SHOWDOWN!!!

Write a number that is greater than but less

than .

SHOWDOWN!!!

Which of the following rational numbers is not

between and ?

a) b)

c) d)

SHOWDOWN!!!

How many rational numbers are between 3.76 and 3.77?

SHOWDOWN!!!
• Write three rational numbers between: &