# 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 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Number of Ounces of Popcorn Kernels | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |