denseness of rational numbers

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 Using Place Value Using Definition of Less Than

Using Models

Fraction Wall

Using Models

Number Line

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 denominatorif the denominators of the fractions to be compared aredifferent.

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 .

56

3 1 47 7 7

3 47 7

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 .

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35

Example Find three rational numbers

between & .56

89

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!!!

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 .1378

740

SHOWDOWN!!! Using your calculator, find a fraction

between the rational numbers and . (DOK 3)

1941

613

SHOWDOWN!!! Find the product of and .

Then

divide the product by 2. Will the answer yield

a rational number between and ?

56

910

56

910

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.

13

12

13 1

313

12

SHOWDOWN Write three numbers between: -2.4

< x < -2.31

SHOWDOWN!!! Write a number that is greater than

but less

than .

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37

SHOWDOWN!!! Which of the following rational

numbers is not between and ?

a) b)

c) d)

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112

17

221

13

15

SHOWDOWN!!! How many rational numbers are

between 3.76 and 3.77?

SHOWDOWN!!! Write three rational numbers

between: &

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