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 - PowerPoint PPT PresentationTRANSCRIPT
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 .
25 15
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 .
16
37
SHOWDOWN!!! Which of the following rational
numbers is not between and ?
a) b)
c) d)
29
112
17
221
13
15
SHOWDOWN!!! How many rational numbers are
between 3.76 and 3.77?
SHOWDOWN!!! Write three rational numbers
between: &
27
37