denseness of rational numbers
DESCRIPTION
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
![Page 1: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/1.jpg)
Denseness of Rational Numbers
Pre-Algebra
Mrs. Yow
![Page 2: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/2.jpg)
What does it mean to be DENSE?
![Page 3: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/3.jpg)
Which material is more DENSE here?
Why???????
![Page 4: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/4.jpg)
Which material is more DENSE here?
The Hair!!!
![Page 5: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/5.jpg)
Compare Rational Numbers
(Find numbers between) Using Models Using Common Denominators Using Place Value Using Definition of Less Than
![Page 6: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/6.jpg)
Using Models
Fraction Wall
![Page 7: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/7.jpg)
Using Models
Number Line
![Page 8: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/8.jpg)
Using Common Denominators
When the denominators of two fractions are the same,
the one with the greater numerator represents
the larger rational number.
![Page 9: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/9.jpg)
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.
![Page 10: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/10.jpg)
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.
![Page 11: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/11.jpg)
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
![Page 12: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/12.jpg)
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
![Page 13: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/13.jpg)
Example Find three rational numbers
between & .56
89
![Page 14: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/14.jpg)
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…..
![Page 15: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/15.jpg)
Examples Write each repeating decimal as a
simplified fraction.
1.) 0.11111…
2.) 0.2222…
![Page 16: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/16.jpg)
CLASSWORK FACE TIME (20-25 minutes
![Page 17: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/17.jpg)
SHOWDOWN!!!
![Page 18: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/18.jpg)
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
![Page 19: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/19.jpg)
SHOWDOWN!!! Using your calculator, find a rational
number between and .1378
740
![Page 20: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/20.jpg)
SHOWDOWN!!! Using your calculator, find a fraction
between the rational numbers and . (DOK 3)
1941
613
![Page 21: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/21.jpg)
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
![Page 22: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/22.jpg)
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
![Page 23: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/23.jpg)
SHOWDOWN Write three numbers between: -2.4
< x < -2.31
![Page 24: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/24.jpg)
SHOWDOWN!!! Write a number that is greater than
but less
than .
16
37
![Page 25: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/25.jpg)
SHOWDOWN!!! Which of the following rational
numbers is not between and ?
a) b)
c) d)
29
112
17
221
13
15
![Page 26: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/26.jpg)
SHOWDOWN!!! How many rational numbers are
between 3.76 and 3.77?
![Page 27: Denseness of Rational Numbers](https://reader033.vdocuments.site/reader033/viewer/2022051020/568161a6550346895dd1643b/html5/thumbnails/27.jpg)
SHOWDOWN!!! Write three rational numbers
between: &
27
37