Compare And Order Rational Numbers

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 Example

Reasonableness: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

Can you eliminate any answer choices? Why?

If you were to take the whole number portion of each fraction:

Add those whole numbers (2 + 4 = 6). She started with 10.Subtract 10 – 6 = 4. Choices B & D are unreasonable.

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 1/4 and so on (these were called reciprocals or unit fractions since they are 1/n for some number n).

Egyptian Fractions

How did they write 3/4?

They were able to write any fraction as a sum of unit fractions

3/4 = 1/2 + 1/4

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.

21 43 65

87 ?

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.

21 43

65 87

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?

21 43 21

43

each person gets half a loaf and a fourth of a loaf. 1/2 + 1/4 = 3/4

Egyptian Fractions

What if she had 4 loaves to share between 5 people?

each person gets half a loaf

?21 43 5 ?

Egyptian Fractions

What if she had 4 loaves to share between 5 people?

There is ½ of a loaf and 1 loaf left.

each person gets a fourth a loaf in addition to their a half of a loaf. 1/2 + 1/4 + ?

51 2 3 4 ?

Egyptian Fractions

What if she had 4 loaves to share between 5 people?

There is 1/5 of a loaf left.

each person gets a fifth of the fourth that was left in addition to their a half of a loaf and a quarter of a loaf. 1/2 + 1/4 + 1/20 = 16/20

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Using Egyptian Fractions to Compare Fractions

Which is larger: 3/4 or 4/5?

Using Egyptian Fractions to Compare Fractions

Using Egyptian fractions we write 3/4 as a sum of unit fractions:

3/4 = 2/4 + 1/4 = 1/2 + 1/4

Using Egyptian Fractions to Compare Fractions

Using Egyptian fractions we write 4/5 as a sum of unit fractions:

4/5 = 1/2 + 3/10 = 1/2 + 6/20

= 1/2 + 4/20 + 1/20 = 1/2 + 1/4 + 1/20

Using Egyptian Fractions to Compare Fractions

3/4 = 1/2 + 1/4

4/5 = 1/2 + 1/4 + 1/20

4/5 is the larger than 3/4 by exactly 1/20

Comparing Fractions using Decimals

Convert the fractions to decimals:3/4 =75/100 or 0.75

4/5 = 80/100 or 0.80

80 (hundredths) is bigger than 75 (hundredths) therefore 4/5 is bigger than 3/4

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.

Ordering Rational Numbers

greatestleast

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

Ordering Rational Numbers

1 . 1 1 2

0 . 2 3 4

1 . 0 5 6

0 . 4 5

Place zero in empty spots

0

Place the following numbers in order largest to smallest: 1.112, 0.234, 1.056, 0.45

Ordering Decimal Numbers

1 . 1 1 2

0 . 2 3 4

1 . 0 5 6

0 . 4 5 0

largestlargest

smallestsmallest

Place the following numbers in order largest to smallest: 1.112, 0.234, 1.056, 0.45

Ordering Decimal Numbers

1 . 1 1 2

1 . 0 5 6

0 . 4 5 0

0 . 2 3 4

largestlargest

smallestsmallest

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:2/5 , 11/2 , 3/4 , 15/6

Ordering Rational Numbers

Place the following numbers in order from smallest to largest: 2/5 , 11/2 , 3/4 , 15/6

2/5 = 24/60

11/2 = 3/2 = 90/60

3/4 = 45/60

15/6 = 11/6 = 110/60

Ordering Rational Numbers

largest

smallest

The order from smallest to largest is

2/5 , 3/4 , 11/2 , 15/6

2/5 = 24/60

11/2 = 3/2 = 90/60

3/4 = 45/60

15/6 = 11/6 = 110/60

Ordering Rational Numbers

largest

smallest

Which is the greater number, 23% or 2.5

Guided Practice #1

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%

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.

Step 2: Plot each point asked for in the problem: -1.5, 2, 21/2 , -2 5/6 , 1.4, 25%

Guided Practice #2

-25/6 -1.5 2 21/2 25% 1.4

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, 2, 21/2

Guided Practice #2

-25/6 -1.5 2 21/2 25% 1.4

Practice Activity #1

Silence is Golden

Practice Activity #2

Hexagon Domino Puzzle