## Learning Math: Number and Operations

# Fractions, Percents, and Ratios

## Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.

**In This Session**

Part A:Models for the Multiplication and Division of Fractions

Part A:

**Part B:**Decimals and Percents

**Part C:**Fibonacci Numbers

**Homework**

In this session, we’ll look at several topics related to fractions, percents, and ratios. As in earlier sessions, we’ll look at graphical and geometric representations of these topics, as well as some of their applications in the physical world. As you work through the activities in this session, reflect on how mathematics is reasonable and logical, and how it is helpful to look for the logical patterns that emerge when you think about a mathematical situation.

For information on required and/or optional material, see Note 1 below.

**LEARNING OBJECTIVES
**In this session, you will do the following:

• Understand how to use area models for computation with fractions and decimals

• Use benchmarks to estimate the “reasonableness” of answers to percent problems

• Understand the meaning of “percent”

• Solve percent problems with proportions

• Explore Fibonacci numbers

### Notes

**Note 1
Materials Needed:
**8 1/2-by-11 transparencies (you can cut them into halves or quarters)

**•**

**Magic markers in a variety of colors**

**•****A board with a meter stick or number line on it (optional)**

**•****An elastic band wide enough to stretch around a meter stick (optional)**

**•****Pineapple or pine cone**

••

**One or two toothpicks**

**•**

### Key Terms

**Previously Introduced**

quotative division

A quotative division problem is one where you know the number of items in each group and are trying to find the number of groups. If you have 30 popsicles and want to give 5 popsicles to each person, figuring out the total number of people is a quotative division problem.

**New in This Session**

**area model for multiplication**

The area model for multiplication is a method of multiplying fractions (between 0 and 1) by representing the multiplied fractions as areas of a whole. The same model can be used to divide fractions that are between 0 and 1.

**common denominator model for division**

The common denominator model for division is a method of dividing fractions by finding a common denominator and then dividing the numerators.

**Fibonacci sequence**

The Fibonacci sequence is a series of numbers in which the first two elements are 1, and each additional element is the sum of the previous two. The sequence is 1, 1, 2, 3, 5, 8, 13, 21, . . . .

**golden mean**

The golden mean is the limit of the ratio between two consecutive Fibonacci numbers. It is exactly (1 + ) 2 and approximately 1.618. Often, the Greek letter phi (ø) is used to represent the golden mean.

**golden rectangle**

A golden rectangle is a rectangle whose sides are in the ratio of 1 to ø, where ø is the golden mean. A golden rectangle can be cut into a square and a smaller golden rectangle.

**percent**

Percent means some part out of 100. It can also be represented as a fraction or decimal. For example, 45% means 45 out of 100, 0.45, and 45/100.

**proportion**

Proportion is an equation that states that two ratios are equal, for example 2:1 = 6:3.

**ratio
**

A ratio indicates the relative magnitude of two numbers. The ratio 3:1 means that the first quantity is equivalent to three times the second quantity. The ratio 2:3 means that twice the first quantity is equivalent to three times the second quantity. This relationship may be written 2:3 or as an indicated quotient (2/3).