Pre-Algebra Out Loud: Learning Mathematics Through Reading and Writing Activities (2013)

Chapter 5. Fractions, Decimals, and Percents

WHAT? Description

Many students have difficulty learning and understanding how to work with fractions. Research shows that students who have a poor comprehension of what a fraction is will struggle with such operations as addition and subtraction problems that include fractions or fractional coefficients. The goal of this chapter is to help students understand that a fraction is a part of a whole or a part of a set. In addition, the lessons in this chapter explore the relationship of fractions, decimals, and percentages.

WHY? Objectives

Using the activities in this chapter, pre-algebra students:

· Repeat exercises that illustrate a rule and discover that rule

· Respond to writing prompts that target certain features about fractions, decimals, and percentages

· Compare and contrast various features about fractions

· Build semantic word maps that demonstrate and display fractional forms and concepts

Mini-Lesson 5.1 Pizza Math

CCSS Standard 6.RP: Ratios and Proportional Relationships

Understand ratio concepts and use ratio reasoning to solve problems.

A fraction has two parts: a numerator and a denominator. The denominator gives the size of the fraction as in a circle (represented as a pizza) divided into one-fourths (see below). The numerator gives the number of parts. Another way to think of this is in terms of the top number and bottom number:

· The top number (numerator) counts.

· The bottom number (denominator) gives what is being counted.

· We can see that .

· 4 of the .

We always write fractions in their simplest form.

Just about everyone has some experience dividing pizza or a pie into pieces. Let's assume each circle's pieces are all the same size. Consider the pizzas (circles) below divided into fourths, halves, and eighths.

is of the whole pizza! Two s is one-half of the pie.

Teaching Tip

Sketching these circles and the fraction bars on the board or using manipulatives or cut-outs will reinforce students' understanding of fractions as parts of a whole. Be sure to write the fractions inside the circles.

It is clear to see many relationships (equivalences) between the pieces of the circles such as these:

Three

Three

How many equivalent fractions can be found between the parts of the pizzas at the left?

To gain a deeper comprehension of these concepts, students can observe equivalencies by sketching fraction bars. Compare the following bars to see the same fractions given above:

Another way to think about fractions is to consider the ratio of two amounts of objects—for example, compared to to .

If we think of these five stars as a whole set, then we can consider two of the stars to be of the set.

We explore in this chapter several other ways to express fractions.

Activity 5.1: What's My Rule?

WHAT? Description

This is a guided discovery activity. Working alone or in groups, students are presented with several examples of a certain mathematical concept or rule. Then they brainstorm and write up the rule or conjecture, that is, what they think the rule might be. Students then solve problems using their conjecture. This activity is best completed with a teacher or teacher's aide available to check for accuracy.

There are many facts and rules regarding fractions and their forms. Worksheets 5.1, 5.2, 5.3, and 5.4. illustrate some of these features.

WHY? Objectives

During this activity, pre-algebra students:

· Discover mathematical rules while observing and completing several problems that use the rule

· Practice writing and using certain mathematical conjectures

· Work cooperatively to arrive at a well-expressed and accurate rule

HOW? Example

The goal here is to determine the rule from this set of equalities:

1. %

2. %

3. %

4. %

5. %

My rule: To convert decimal fractions into percentages, move the decimal point two places to make the number look larger. Then add a % sign on the right of the number.

Worksheet 5.1: To Terminate or Not

Name _____ Date _____

Directions: Consider the examples below, and fill in the rest of the rows. Then give your rule.

Activity 5.2: Writing Prompts

WHAT? Description

Math teachers regularly use writing prompts as a form of authentic assessment or to gauge their students' understanding of a concept. Writing prompts, such as those that follow, allow students to think about their own learning and how well they understand specific mathematical concepts. Writing prompts may be used in students' math journals or as a beginning or an ending to a lesson.

Consider the following prompts:

· Today I learned that …

· Something we did in math class today reminded me of …

· I am still not sure about …

· Outside class, I can use what we learned today when …

· I am confident when I do …

· I get confused when I …

· It would help me in class if …

Other prompts may be more focused on the content of the lesson:

· Explain what simplest form means.

· Compare a decimal to a fraction. How are they alike, and how are they different?

· Explain the role of the numerator and the denominator in a fraction.

WHY? Objectives

During this activity, pre-algebra students:

· Think about their own learning and practice metacognition, the study and understanding of how we learn and process information

· Practice writing and communicating about mathematics

· Compare and contrast certain mathematical or algebraic concepts

HOW? Example

Writing prompts may be as simple as “Fractions are …” or as complex as “The reason I struggle with adding and subtracting fractions is…”

Worksheet 5.2: Writing Prompts for a Lesson in Fractions

What I like about learning fractions is:

Something about fractions that I need to practice more is:

Fractions should be written in simplified form because:

The difference between the numerator and the denominator is:

Mini-Lesson 5.2 Conversions

CCSS Standard 6.NS: Number System

Apply and extend previous understandings of numbers to the system of rational numbers.

Writing Fractions as Decimals

To find the decimal equivalent of a certain fraction, we divide. Example: 3 divided by 4 is , which gives .75. We can read as 15 divided by .

Simplified fractions “terminate,” or have a finite number of terms, only if their denominator is a 2 or 5 or both. Example: The fraction has a denominator of 40, which has a prime factorization of . The decimal for the fraction is .025.

A fraction such as will not terminate but will have a repeating pattern. Another example is . The denominator has a prime factorization: . Also, the decimal fraction is .2083333 … and does not terminate.

All simplified fractions have a decimal that either terminates or a pattern that repeats forever.

Teaching Tip

Fractions are always written in their simplified form, often called lowest terms. A fraction is in lowest terms when both numerator and denominator have no factors in common. Another way to say this is the top and bottom of a fraction are “relatively prime.”

Writing Decimals as Fractions

Reading a decimal correctly provides an excellent way to write a fraction.

A decimal greater than 1 can be written as a mixed number or as an improper fraction, that is, a fraction whose numerator is greater than its denominator.

A special case is a decimal that has a repeated pattern, for example, .25252525 …

To find its equivalent simplified fraction, set

Remind students that they may need to simplify fractions, because solutions must always be in lowest terms:

Writing Decimals as Percents

To write decimals as percents, move the decimal point two places to make the decimal look larger and add the percent sign: % and , for example. Another way to explain the rule is this: “To change a decimal to a percent, move the decimal point to the right.” Having to memorize which direction the decimal moves, left or right, may cause confusion. It's best to remember the percent always looks larger than the decimal and the decimal always looks smaller than the percent. In fact, they are equivalent to each other, as is their fractional representation. This means we multiply by 100. Percent means “per one hundred” so 25% is 25 parts out of 100.

Examples

Convert .056 to a percent: %

Convert 125% to a decimal:

Convert .05 to a percent and then to a simplified fraction: per

Teaching Tip

Always write fractions in simplified form. If you do this from the beginning of discussing fractions, students will realize it is part of the process and will always do it this way.

Activity 5.3: Comparison-and-Contrast Matrix

WHAT? Description

Students can use the comparison-and-contrast matrix (Vacca & Vacca, 1999) to compare and contrast related features of related concepts. Choose the concept and features, and ask students to fill in the blank squares. The completed matrix may then be used as a study guide or turned in as an assignment. Students' answers may be objective or subjective depending on the concepts or features to be compared.

WHY? Objectives

During this activity, pre-algebra students:

· Research and reflect on the similarities and differences of fractions, decimals, and percents

· Complete a comparison-and-contrast matrix and use the matrix as a study guide

· Receive feedback about any misconceptions they have of the meaning of the concepts in their matrices

HOW? Example

Worksheets 5.3 and 5.4 will give students practice in this activities.

Worksheet 5.3: Comparison-and-Contrast Matrix: Rational Numbers

Name _____ Date _____

Directions: Fill in the cells in the following matrix. Each row will have four forms for each fractional number. The different forms are given in the first row, at the top of each column.

Worksheet 5.4: Comparison-and-Contrast Matrix: Fractions

Name _____ Date _____

Directions: Draw a sketch of the given fraction, and write a complete sentence using the description of the fraction.

Activity 5.4: Semantic Word Maps

WHAT? Description

Semantic word maps are used to depict and display the relationships between key concepts and terms. These word maps often resemble flowcharts or webs connecting mathematical terms. Arrows connect related concepts and often display a hierarchy of the term, where one term is a set and the others are subsets of that set. For example, if algebra is the main set, then equations, variables, and constants are all parts of the set or subsets of the set “algebra.” After students have completed their semantic word maps, pose critical-thinking questions—questions that go beyond the calculations and use higher-order thinking skills regarding the relationships of the concepts.

WHY? Objectives

During this activity, pre-algebra students:

· Explore relationships between mathematical concepts and terms

· See the hierarchy of key concepts

· Create study guides displaying key concepts

HOW? Example

Consider the following related concepts:

Text boxes and arrows can be used to display the connections or hierarchy of terms:

Worksheet 5.5: Semantic Word Map

Name _____ Date _____

Directions: Use the following terms in the given word map to show the relationship of the terms. You do not need to use all of the terms.

· fraction, decimal, percent

· rational numbers, real numbers

· numerator, denominator

· division, quotient

· ratio, proportion

Worksheet 5.6: Semantic Word Map

Name _____ Date _____

Directions: Place fraction words in the circle. Choose at least six terms that are related to fractions. Use your textbook to find and choose the words that will help you create a semantic word map inside the circle.