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# 6-2 Comparing Bits and Pieces - Concepts and Explanations

## Fractions as Parts of a Whole

In the part-whole interpretation of fractions, students should determine what the whole is, divide the whole into equal-size parts (that are not necessarily the same shape), recognize the number of parts they need to represent the situation, and form a fraction by placing the parts needed over the number of parts into which they have divided the whole.

### Example

If there are 24 students in the class and 16 are girls, then you can represent the part of the whole that is girls as 16/24. You can also represent 16/24 as 2/3.

The denominator 3 tells into how many equal-size parts the whole has been divided, and the numerator 2 tells how many of the equal-size parts have been shaded.

## Fractions as Measures of Quantities

In this interpretation, students think of fractions as numbers.

### Example

A fraction can be a measurement that is “in between” two whole measures.

Students see this every day in references such as 2 1/2 brownies or 7 3/4 inches.

## Fractions as Decimals

Students need to understand decimals in two ways: as special fractions with denominators of 10 and powers of 10, and as a natural extension of the place-value system for representing quantities less than 1.

### Example

To find the decimal representation of the fraction 2/5, rewrite it with a power of 10 in the denominator.

2/5 - 4/10

The fraction has tenths in the denominator, so the decimal equivalent places the 4 in the tenths place.

4/10 = 0.4

## Ratio

Students build understanding of ratios as comparisons of numbers. Students express ratios in different ways: with the language of for every, using the word to, with colon notation (a : b), and using the word per.

### Example

When you say that 1/6 of a school is sixth graders, strictly speaking, this is not a number but a ratio. It compares a part to the whole; for every 6 students, 1 is a sixth grader.

The ratio of the sixth-grade fundraising goal to the seventh-grade fundraising goal is 60 : 90.

Mary runs at 5 miles per hour.

## Unit Rate

A unit rate is a comparison in which one of the numbers being compared is 1 unit. You can use unit rates to calculate equivalent ratios.

### Example

Finn runs 10 miles in 2 hours.

Finn runs 2.5 miles in a half hour (or 30 minutes).

Finn runs 1 mile in 1/5 hour (or 12 minutes).

The statement Finn runs 1 mile in 12 minutes expresses a unit rate.

## Rate Table

Rate tables are a way to express equivalent ratios. For example, if you know that 1 ounce of popcorn kernels yields 4 cups of popped corn, you can use a rate table to calculate other amounts

### Example

Cups of Popcorn From Ounces of Kernels
 Number of Cups of Popcorn Number of Ounces of Popcorn Kernels 4 8 12 16 20 24 28 32 36 40 44 48 1 2 3 4 5 6 7 8 9 10 11 12