Transcript Compare And Order Non-rational numbers - Math GR. 6-8

Compare And Order
Rational Numbers
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Participant Objective
The purpose of the lesson is to determine
participants understanding of reading, comparing,
and ordering of fractions, decimals and percents.
Benchmark MA.6.A.5.3
Estimate the results of
computations with fractions,
decimals, and percents and judge
the reasonableness of the results
Reasonableness
Defined as:
•having good sense and sound judgment
•being prudent and sensible
•being plausible or acceptable
Reasonableness
examples
Proper Fractions
A proper fraction is a fraction
that is less than 1 and greater than
zero.
Decimals
A decimal is the representation of
a real number using the base 10
and decimal notation, such as
201.4, 3.89, or 0.0006.
Decimals vs Fractions
A "decimal" is a fraction whose denominator we do
not write but which we understand to be a power
of ten.
For example,
0.4 is read as four tenths or 4/10
0.74 is read as seventy four hundredths or 74/100
Percents
A percent is a way of expressing a
number as a fraction of 100.
Per cent means "per hundred”.
For example,
20% is read as twenty percent or 20/100
3% is read as three percent or 3/100
Fractions, Decimals, Percents
All represent a part of a whole.
•A fraction is based on the number into which the
whole is divided (the denominator). The numerator
(the top) is the PART, the denominator (the bottom)
is the WHOLE.
•A decimal is based on the number in terms of tenths,
hundredths, thousandths, etc.
•A percent is based on the number in terms of 100.
Judging Reasonableness
Example #1
Benchmark MA.6.A.5.2
Compare and order fractions,
decimals, and percents, including
finding their approximate
location on a number line.
What are rational numbers?
Rational numbers are parts of a whole. They can be
expressed as a fraction (1/4), a decimal (0.25) or a
percent (25%).
Rational numbers can be plotted on a number line.
Egyptian Fractions
• The ancient Egyptians only used fractions
of the form 1/n .
• All fractions had to be represented as a sum
of such unit fractions.
• This makes it easier to compare fractions.
Egyptian Fractions
How it worked:
Egyptians had a notation for 1/2 and 1/3 and
1/ and so on (these were called reciprocals
4
or unit fractions since they are 1/n for some
number n).
Egyptian Fractions
How did they write 2/5 or 3/4?
They were able to write any fraction as a
sum of unit fractions
2/ = 1/ + 1/
5
5
5
3/ = 1/ + 1/
4
2
4
Egyptian Fractions
So suppose Faith has 5 loaves of bread to
share among the 8 people.
Egyptian Fractions
Faith sees that she can give each person half
a loaf, with one loaf left over.
1
2
7
3
5
4
8
?
6
Egyptian Fractions
Faith takes the one loaf that is left and
divides it into 8 pieces, so each person gets
half a loaf and an eighth of a loaf.
1
5
2
6
3
4
7
8
Egyptian Fractions
Using Egyptian Fractions we see that
5/
8
= 1/2 + 1/8
Egyptian Fractions
Suppose Faith had 3 loaves to share between
4 people. How would she do it?
1
2
3
4
1
3
2
4
each person gets half a loaf and a fourth of
1/ + 1/ = 3/
a loaf.
2
4
4
Egyptian Fractions
What if she had 2 loaves to share among 5
people?
12345
12345
each person gets a fifth a loaf and a fifth of
1/ + 1/ = 2/
a loaf.
5
5
5
Egyptian Fractions
What if she had 4 loaves to share between 5
people?
1
2
3
4
5
each person gets half a loaf
?
?
Egyptian Fractions
What if she had 4 loaves to share between 5 people?
There is ½ of a loaf and 1 loaf left.
1 2
3 4 5 ?
each person gets a fourth a loaf in addition
1/ + 1/ + ?
to their a half of a loaf.
2
4
Egyptian Fractions
What if she had 4 loaves to share between 5 people?
There is 1/5 of a loaf left.
1
2
3
4
5
each person gets a fifth of the fourth that was
left in addition to their a half of a loaf and a
1/ + 1/ + 1/ = 16/
quarter of a loaf.
2
4
20
20
Using Egyptian Fractions to
Compare Fractions
Which is larger:
3/ or 4/ ?
4
5
Using Egyptian Fractions to
Compare Fractions
Using Egyptian fractions we write each as a
sum of unit fractions:
3/ = 2/ + 1/ = 1/ + 1/
4
4
4
2
4
4/ = 1/ + 3/
1/ + 6/
=
5
2
10
2
20 =
1/ + 1/
5/
1/ + 1/ + 1/
+
=
2
20
20
2
4
20
Using Egyptian Fractions to
Compare Fractions
3/
4
4/
5
4/
5
= 1/2 + 1/4
= 1/2 + 1/4 + 1/20
is the larger than 3/4 by exactly 1/20
Comparing Fractions using Decimals
Convert the fractions to decimals:
3/ =75/
4
100 or 0.75
4/ = 80/
5
100 or 0.80
80 (hundredths) is bigger than 75
(hundredths) therefore 4/5 is bigger than 3/4
Ordering Rational Numbers
One way to order rational numbers is
graphing them on a number line.
On a number line, the rational number to the
right of another rational number is greater.
least
greatest
Ordering Rational Numbers
A second method is to convert all rational numbers
to decimals. Place the following numbers in order
largest to smallest: 1.112, 0.234, 1.056, 0.45
1
0
1
0
.
.
.
.
1
2
0
4
1
3
5
5
2
4
6
0
Place zero in
empty spots
Ordering Decimal Numbers
Place the following numbers in order largest to
smallest: 1.112, 0.234, 1.056, 0.45
1
0
1
0
.
.
.
.
1
2
0
4
1
3
5
5
2
4
6
0
largest
smallest
Ordering Decimal Numbers
Place the following numbers in order largest to
smallest: 1.112, 0.234, 1.056, 0.45
1
1
0
0
.
.
.
.
1
0
4
2
1
5
5
3
2
6
0
4
largest
smallest
Ordering Rational Numbers
Besides using number line, and decimals, you
can use the common denominator method.
Convert all rational numbers to fractions with
common denominators. Place the following
numbers in order from smallest to largest:
2/ , 11/ , 3/ 15/
5
2
4,
6
Ordering Rational Numbers
Place the following numbers in order from
smallest to largest:
2/ , 11/ , 3/ 15/
5
2
4,
6
2/ = 24/
5
60
11/2 = 3/2 = 90/60
3/ = 45/
4
60
15/6 = 11/6 = 110/60
smallest
largest
Ordering Rational Numbers
The order from smallest to largest is
2/
5
,
3/ 11/
4,
2
,
smallest
2/ = 24/
5
60
11/2 = 3/2 = 90/60
3/ = 45/
4
60
15/6 = 11/6 = 110/60
largest
5
1 /6
Guided Practice #1
Which is the greater number, 23% or 2.5
Percent means per hundred. So, 23% is the
same as 23/100 or 0 .23
0 .23 is smaller than 2.5
2.5 > 23%
Guided Practice #2
Graph this set of numbers on a number line.
What is the order of the set of numbers from
least to greatest?
-1.5, 2, 21/2 , -2 5/6 , 1.4, 25%
Guided Practice #2
Step 1: Draw a number line from -3 to 3 with
equal intervals.
-25/6
-1.5
25%
1.4
2
21/2
Step 2: Plot each point asked for in the
problem: -1.5, 2, 21/2 , -2 5/6 , 1.4, 25%
Guided Practice #2
Step 3: Use the points plotted on the number
line to write the numbers in order from least
to greatest.
-25/6
-1.5
25%
1.4
-25/6 , -1.5, 25%, 1.4, 2, 21/2
2
21/2
Directions
1. two teams with a set of plates.
2. Each team member should have a plate.
3. This is a silent game, if a student talks
during play, a point will be deducted from
the team’s score.
4. Show the first problem.
Directions
Continue
5. Once the problem is uncovered, the team
members with the appropriate plates should
arrange themselves at the front of the class to
represent the number/problem displayed from
the transparency.
6. The students have 30 seconds to form the
number/problem at the front of the class. After
30 seconds, each team is given a point for a
correct answer.
7. An additional point is given to the first
team to “present” the correct answer.
8. Play continues until all of the
numbers/problems have been displayed.
9. The winning team is the one with the most
points at the end of the game.
Presentation Numbers
1. five thousand, two and one tenth
2. Seven hundred twenty-three and eight
hundredths
3. Eighty and four thousandths
4. Nine thousandths
5. One and fifty-three hundredths
6. Two hundred one thousand, thirty-six
Presentation numbers continue
7. Fifteen and two tenths.
8. One hundred twenty thousand, three
hundred seven and four tenths.
9. Four hundred twenty-five thousand, three
hundred seventeen and eight thousandths.
10.one hundred forty-six thousand, three
hundred ninety-seven
Guided Practice………..
Place the following numbers in order from greatest
to least.
0.75, 0.615, 0.58, 0.195
1. Line-up numbers and add zero(s)
0.750
0.615
0.580
0.195
2. Look at number in the tenths place, all the
numbers are different, so arrange that
number in order from greatest to least.
.615 ,.195, .580, .750, than reorder all
.750
.615
.580
.195
How do you compare and order nonnegative rational numbers?
To answer this question click on the
webpage. Do the practice exercise of
your choice, complete exercise
according to web directions.
http://www.aaamath.com
Conclusion
• Understanding place value can help you compare
and order numbers.
• Start at the far left place value of the numbers,
adding zero(s) as place holders when the numbers
don’t have the same number of place values.
• Compare the digit of each place value where the
digits are different.
• The way those digits compare is the way the
whole numbers compare.
• Use > (greater than), < (less than), or = (equal to)
when comparing numbers.