# 6.2d and *6.2b - Compare/Order Fractions, Decimals, Percents (No Calc)

6.2    The student will

d)  compare and order fractions, decimals, and percents.  (NO CALCULATOR)

6.2  The student will(with NO CALCULATOR)

a)  compare and order positive rational numbers.*

Bloom's Level:  Understand, Analyze

### BIG IDEAS

• I can determine the best buy when items are one-third off, 40% off or \$50 off, find sale costs while shopping, create a budget, dilute mixtures, understand scales on maps, interpret probabilities and odds, and convert among metric units.
• I will be able to interchange fraction, decimal, and percent values to interpret real world situations.

### UNDERSTANDING THE STANDARD

• Percent means “per 100” or how many “out of 100”; percent is another name for hundredths.
• A number followed by a percent symbol (%) is equivalent to that number with a denominator of 100 (e.g., 30% = 30/100 = 3/10 = 0.3).
• Percents can be expressed as fractions with a denominator of 100 (e.g., 75% = 75/100 = 3/4).
• Percents can be expressed as decimal (e.g., 38% = 38/100 = 0.38).
• Some fractions can be rewritten as equivalent fractions with denominators of powers of 10, and can be represented as decimals or percents (e.g., 3/5 = 6/10 = 60/100 = 0.60 = 60%).
• Decimals, fractions, and percents can be represented using concrete materials (e.g., Base-10 blocks, number lines, decimal squares, or grid paper).
• Percents can be represented by drawing shaded regions on grids or by finding a location on number lines.
• Percents are used in real life for taxes, sales, data description, and data comparison.
• Fractions, decimals and percents are equivalent forms representing a given number.
• The decimal point is a symbol that separates the whole number part from the fractional part of a number.
• The decimal point separates the whole number amount from the part of a number that is less than one.
• The symbol  can be used in Grade 6 in place of “x” to indicate multiplication.
• Strategies using 0, 1/2 and 1 as benchmarks can be used to compare fractions.
• When comparing two fractions, use 1/2 as a benchmark. Example: Which is greater, 4/7 or 3/9?
4/7 is greater than 1/2 because 4, the numerator, represents more than half of 7, the denominator. The denominator tells the number of parts that make the whole. 3/9 is less than 1/2 because 3, the numerator, is less than half of 9, the denominator, which tells the number of parts that make the whole. Therefore, 4/7 > 3/9.
• When comparing two fractions close to 1, use distance from 1 as your benchmark. Example: Which is greater, 6/7 or 8/9? 6/7 is 1/7 away from 1 whole. 8/9 is 1/9 away from 1 whole. Since 1/7 > 1/9,  then 6/7 is a greater distance away from 1 whole than 8/9 so 8/9 > 6/7.
• Students should have experience with fractions such as 1/8, whose decimal representation is a terminating decimal (e. g., 1/8 = 0.125) and with fractions such as 2/9, whose decimal representation does not end but continues to repeat (e. g., 2/9 = 0.222…).  The repeating decimal can be written with ellipses (three dots) as in 0.222… or denoted with a bar above the digits that repeat as in 0.2

· Fractions, decimals and percents can be used to represent part-to-whole ratios.

- Example: The ratio of dogs to the total number of pets at a grooming salon is 5:8.  This implies that 5 out of every 8 pets being groomed is a dog.  This part-to-whole ratio could be represented as the fraction ( of all pets are dogs), the decimal 0.625 (0.625 of the number of pets are dogs), or as the percent 62.5% (62.5% of the pets are dogs).

· Fractions, decimals, and percents are three different ways to express the same number. Any number that can be written as a fraction can be expressed as a terminating or repeating decimal or a percent.

· Equivalent relationships among fractions, decimals, and percents may be determined by using concrete materials and pictorial representations (e.g., fraction bars, base ten blocks, fraction circles, number lines, colored counters, cubes, decimal squares, shaded figures, shaded grids, or calculators).

· Percent means “per 100” or how many “out of 100”; percent is another name for hundredths.

· A number followed by a percent symbol (%) is equivalent to a fraction with that number as the numerator and with 100 as the denominator (e.g., 30% = = ; 139% = ).

· Percents can be expressed as decimals (e.g., 38% = = 0.38; 139% = = 1.39).

· Some fractions can be rewritten as equivalent fractions with denominators of powers of 10, and can be represented as decimals or percents (e.g., = = = 0.60 = 60%). Fractions, decimals, and percents can be represented by using an area model, a set model, or a measurement model. For example, the fraction is shown below using each of the three models.

· Percents are used to solve practical problems including sales, data description, and data comparison.

· The set of rational numbers includes the set of all numbers that can be expressed as fractions in the form where a and b are integers and b does not equal zero.  The decimal form of a rational number can be expressed as a terminating or repeating decimal. A few examples of positive rational numbers are, , 82, 75%, .

· Students are not expected to know the names of the subsets of the real numbers until grade eight.

· Proper fractions, improper fractions, and mixed numbers are terms often used to describe fractions.  A proper fraction is a fraction whose numerator is less than the denominator.  An improper fraction is a fraction whose numerator is equal to or greater than the denominator.  An improper fraction may be expressed as a mixed number. A mixed number is written with two parts: a whole number and a proper fraction (e.g., 3 ).

· Strategies using 0, and 1 as benchmarks can be used to compare fractions.

- Example: Which is greater, or ? is greater than because 4, the numerator, represents more than half of 7, the denominator. The denominator tells the number of parts that make the whole.  is less than  because 3, the numerator, is less than half of 9, the denominator, which tells the number of parts that make the whole. Therefore, > .

· When comparing two fractions close to 1, use the distance from 1 as your benchmark.

- Example: Which is greater,  or ? is away from 1 whole. is away from 1 whole. Since,
, then is a greater distance away from 1 whole than . Therefore, .

· Some fractions such as , have a decimal representation that is a terminating decimal
(e. g., ) and some fractions such as , have a decimal representation that does not terminate but continues to repeat (e. g., = 0.222…).  The repeating decimal can be written with ellipses (three dots) as in 0.222… or denoted with a bar above the digits that repeat as in .

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

6.2d1  Compare 2 decimals through thousandths (No Calc) using manipulatives, pictorial representations, number lines, and symbols (≤, ≥, >, <, =).

6.2d2  Compare 2 fractions with denominators of (No Calc) 12 or less using manipulatives, pictorial representations, number line, and symbols, (≤ ≥, >, <, =).

6.2d3  Compare 2 percents using pictorial (No Calc) representations and symbols (≤, ≥, >, <, =).

·  6.2b1  Compare two percents using pictorial representations and symbols (<, ≤, ≥, >, =).

6.2d4  Order no more than 3 fractions, decimals, (No Calc) and percents (decimals through thousandths, fractions with denominators of 12 or less), in ascending or descending order.

·  6.2b2  Order no more than four positive rational numbers expressed as fractions (proper or improper), mixed numbers, decimals, and percents (decimals through thousandths, fractions with denominators of 12 or less or factors of 100). Ordering may be in ascending or descending order.

### KEY VOCABULARY

fraction, decimal, percent, numerator, denominator, equivalent, represent, compare, repeating decimal, terminating decimal, ascending, descending

Updated: Oct 27, 2017