Lesson Plan: Fraction Word Problems

Lesson Objectives

Understand how to interpret and solve fraction word problems
Apply strategies for solving fraction word problems
Compare and contrast different fraction representations in word problems

Common Core Standards

3.NF.A.1 - Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
3.NF.A.3.D - Compare two fractions with the same numerator or the same denominator by reasoning about their size.
4.NF.A.2 - Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.

Prerequisite Skills

All prior fraction concepts
Basic problem-solving

Key Vocabulary

Fraction: A number that represents a part of a whole. It consists of a numerator (top number) and a denominator (bottom number), such as \( \frac{3}{4} \).
Equivalent Fractions: Fractions that have different numerators and denominators but represent the same value. For example, \( \frac{1}{2} = \frac{2}{4} \).
Common Denominator: A shared denominator that allows for easy comparison of fractions. For example, the common denominator of \( \frac{1}{3} \) and \( \frac{1}{4} \) is 12.
Number Line: A visual representation of numbers on a straight line, which helps compare and order fractions.
Greater Than (>): A symbol used to compare two numbers when one is larger than the other. Example: \( \frac{3}{4} > \frac{1}{2} \).
Less Than (<): A symbol used to compare two numbers when one is smaller than the other. Example: \( \frac{1}{3} < \frac{2}{3} \).

Multimedia Resources

Math Definitions Collection: Fractions https://www.media4math.com/Definitions--Fractions
Math Video Definitions Collection: Fractions https://www.media4math.com/MathVideoCollection--FractionsVocabulary
Slideshow: Fraction Definitions https://www.media4math.com/library/slideshow/fraction-definitions

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Identifying Equivalent Fractions with Pizza

Objective: Students will use visual representations of pizza slices to explore equivalent fractions.

Materials:

Paper and colored pencils (or fraction circle cutouts)
Printed images of a pizza divided into equal parts

Steps:

Draw a large circle on the board and divide it into 4 equal slices. Shade 2 of the slices.
Ask students: "What fraction of the pizza is shaded?" (Answer: \( \frac{2}{4} \)).
Now, divide the same pizza into 8 slices and shade 4 of them.
Ask: "What fraction is shaded now?" (Answer: \( \frac{4}{8} \)).
Guide students to see that \( \frac{2}{4} \) and \( \frac{4}{8} \) are equivalent.
Have students try drawing their own pizza fractions and finding equivalent fractions.

Discussion Questions:

What happens when we divide a pizza into more slices but shade the same portion?
Can you find another fraction that is equivalent to \( \frac{2}{4} \)?

Activity 2: Comparing Fractions on a Number Line

Objective: Students will use number lines to visualize and compare fractions.

Materials:

Printed blank number lines or drawn number lines on the board
Markers or sticky notes

Steps:

Draw a number line from 0 to 1 on the board.
Mark \( \frac{1}{2} \) in the middle, then mark \( \frac{1}{4} \) and \( \frac{3}{4} \) appropriately.
Ask students to help you place \( \frac{2}{4} \) on the number line.
Discuss: Is \( \frac{2}{4} \) the same as \( \frac{1}{2} \)? Why?
Give students their own number lines and have them plot \( \frac{1}{3} \), \( \frac{2}{3} \), and \( \frac{3}{6} \).

Discussion Questions:

Which fraction is closer to 0: \( \frac{1}{3} \) or \( \frac{2}{3} \)?
How does plotting fractions on a number line help us compare them?

Activity 3: Fraction Sorting Challenge

Objective: Students will classify fractions based on size and equivalency.

Materials:

Index cards with different fractions written on them
Three labeled bins or areas on the board: "Less than \( \frac{1}{2} \)", "Equal to \( \frac{1}{2} \)", and "Greater than \( \frac{1}{2} \)"

Steps:

Shuffle the fraction index cards and hand them out to students.
One at a time, students place their fraction in the correct category.
After all fractions are placed, review as a class and discuss any incorrect placements.

Discussion Questions:

What strategies did you use to decide where to place each fraction?
Can you find an equivalent fraction for one of the ones we sorted?

Activity 4: Converting Verbal Statements

Practice converting verbal statements into fraction expressions:

"One out of three."	\[ \frac{1}{3} \]
"Three out of four items"	\[ \frac{3}{4} \]
"There are total of five items. Two of them are selected."	\[ \frac{2}{5} \]

For each verbal statement that can be written as a fraction, identify which word corresponds to the numerator and which corresponds to the denominator.

Teach

Analyze fraction word problem examples:

Example 1: Pizza Fractions

"A pizza is divided into 8 slices. You and your friends eat 5 slices. What fraction of the pizza is left?"

Use this slide show to show the complete pizza and the partially eaten pizza:

https://www.media4math.com/library/slideshow/pizza-fractions-example-1

Show the full pizza divided into eight slices. Reinforce that this is the whole and becomes the denominator of the fraction.
Then show the slide that shows the three slices. This is the part of the whole and becomes the numerator of the fraction.
Use this information to model the fraction of the pizza left: \( \frac{3}{8} \).
Ask students to determine what fraction of the pizza that was eaten: \( \frac{5}{8} \).

Example 2. Interpreting Fractional Amounts

"A pizza is divided into 6 slices. You and your friends eat \( \frac{2}{3} \) of the pizza. How many slices did you eat?"

Use this slide show to show the complete pizza and the partially eaten pizza:

https://www.media4math.com/library/slideshow/pizza-fractions-example-2

Show the full pizza divided into six slices. Point out to students that since \( \frac{2}{3} \) of the pizza has been eaten, that only \( \frac{1}{3} \) of it remains.
The next slide shows one third of a pizza remaining. Notice that this pizza is divided into thirds.
The next slide that shows one third of the pizza when the pizza is divided into six slices. Point out that this is equivalent to the one third. Go back and forth from one slide to the other.
Use this information to model the fraction of the pizza left: \( \frac{2}{6} \).
Finally show that \( \frac{1}{3} \) and \( \frac{2}{6} \) are equivalent fractions by factoring a two from the numerator and denominator of the second fraction.

Example 3: Comparing Fractions

"Maria has a bag of 20 candies and she eats five of them. Isaac has a bag of 25 candies and also eats five of them. Who ate the larger fraction of candies?"

Use the information from the problem to complete this table:

	Number of Candies Eaten	Total Number of Candies
Maria	5	20
Isaac	5	25

Use the information to find the fractional amount each child has eaten:

Maria: \( \frac{5}{20} \) = \( \frac{1}{4} \)

Isaac: \( \frac{5}{25} \) = \( \frac{1}{5} \)

Since these are unit fractions, the one with the smaller denominator is the larger fraction. This means Maria ate the larger fraction of candies.

Example 4: Using Equivalent Fractions to Compare Fractions

Emma and Jack each drank a different amount of juice. Emma drank \( \frac{3}{4} \) of a cup, and Jack drank \( \frac{5}{8} \) of a cup. Who drank more juice?

Solution:

To compare \( \frac{3}{4} \) and \( \frac{5}{8} \), we rewrite them with a common denominator. The least common denominator of 4 and 8 is 8.

Convert \( \frac{3}{4} \) to an equivalent fraction with a denominator of 8:

\[ \frac{3}{4} = \frac{6}{8} \]

Now compare:

\[ \frac{6}{8} > \frac{5}{8} \]

Answer: Since \( \frac{6}{8} \) is greater than \( \frac{5}{8} \), Emma drank more juice.

Example 5: Using a Number Line to Compare and Order Fractions

Liam, Ava, and Noah each ran different distances. Liam ran \( \frac{1}{2} \) of a mile, Ava ran \( \frac{3}{4} \) of a mile, and Noah ran \( \frac{2}{3} \) of a mile. Who ran the farthest?

Solution:

To compare these fractions, we place them on a number line. First, we find equivalent fractions with a common denominator. The least common denominator of 2, 3, and 4 is 12.

Convert \( \frac{1}{2} \) to \( \frac{6}{12} \)
Convert \( \frac{3}{4} \) to \( \frac{9}{12} \)
Convert \( \frac{2}{3} \) to \( \frac{8}{12} \)

Now order them:

\[ \frac{6}{12}, \quad \frac{8}{12}, \quad \frac{9}{12} \]

Since \( \frac{9}{12} \) (Ava’s distance) is the greatest, Ava ran the farthest.

Answer: Ava ran the farthest.

Example 6: Using Equivalent Fractions in a Recipe Context

A cake recipe calls for \( \frac{3}{4} \) cup of sugar. Mia only has a \( \frac{1}{4} \) cup measuring scoop. How many scoops does she need to measure \( \frac{3}{4} \) cup?

Solution:

Each scoop holds \( \frac{1}{4} \) cup. To find out how many scoops make \( \frac{3}{4} \) cup, we count by fourths:

\[ \frac{1}{4}, \quad \frac{2}{4}, \quad \frac{3}{4} \]

Answer: Mia needs 3 scoops.

Review

In this lesson, students learned how to compare and order fractions using equivalent fractions and number lines. They also applied these concepts to real-world situations, such as recipes and distance comparisons. Let's review the key ideas and vocabulary before working through some additional examples.

Key Vocabulary

Fraction: A number that represents a part of a whole. It consists of a numerator (top number) and a denominator (bottom number), such as \( \frac{3}{4} \).
Equivalent Fractions: Fractions that have different numerators and denominators but represent the same value. For example, \( \frac{1}{2} = \frac{2}{4} \).
Common Denominator: A shared denominator that allows for easy comparison of fractions. For example, the common denominator of \( \frac{1}{3} \) and \( \frac{1}{4} \) is 12.
Number Line: A visual representation of numbers on a straight line, which helps compare and order fractions.
Greater Than (>): A symbol used to compare two numbers when one is larger than the other. Example: \( \frac{3}{4} > \frac{1}{2} \).
Less Than (<): A symbol used to compare two numbers when one is smaller than the other. Example: \( \frac{1}{3} < \frac{2}{3} \).

Now, let's practice with some review examples to reinforce these concepts.

Provide additional fraction word problems for students to solve in pairs or small groups, such as:

"A pizza is divided into 10 slices. You and your friends eat 7 slices. What fraction of the pizza is left?"
"A case of sodas has 24 cans. What is the minimum number of sodas that need to be consumed before you can say that more than half of the sodas have been drunk?"

Encourage them to discuss their thought processes and strategies with each other.

Example 1: Comparing Fractions Using Equivalent Fractions

Tom and Jerry each have a pizza. Tom ate \( \frac{5}{6} \) of his pizza, while Jerry ate \( \frac{3}{4} \) of his pizza. Who ate more pizza?

Solution:

To compare \( \frac{5}{6} \) and \( \frac{3}{4} \), we first find a common denominator. The least common denominator of 6 and 4 is 12.

Convert \( \frac{5}{6} \) to an equivalent fraction with a denominator of 12:

\[ \frac{5}{6} = \frac{10}{12} \]

Convert \( \frac{3}{4} \) to an equivalent fraction with a denominator of 12:

\[ \frac{3}{4} = \frac{9}{12} \]

Now compare:

\[ \frac{10}{12} > \frac{9}{12} \]

Answer: Since \( \frac{10}{12} \) is greater than \( \frac{9}{12} \), Tom ate more pizza.

Example 2: Ordering Fractions Using a Number Line

Grace, Noah, and Lily each walked different distances. Grace walked \( \frac{2}{5} \) of a mile, Noah walked \( \frac{3}{10} \) of a mile, and Lily walked \( \frac{4}{5} \) of a mile. Order these distances from shortest to longest.

Solution:

To order the fractions, we first find a common denominator. The least common denominator of 5 and 10 is 10.

Convert \( \frac{2}{5} \) to \( \frac{4}{10} \)
Leave \( \frac{3}{10} \) as is
Convert \( \frac{4}{5} \) to \( \frac{8}{10} \)

Now order them:

\[ \frac{3}{10}, \quad \frac{4}{10}, \quad \frac{8}{10} \]

Answer: The distances in order from shortest to longest are \( \frac{3}{10} \), \( \frac{2}{5} \), and \( \frac{4}{5} \).

Example 3: Using Equivalent Fractions in a Recipe Context

A recipe calls for \( \frac{5}{6} \) cup of oil, but the measuring cup only shows \( \frac{1}{3} \) cup. How many \( \frac{1}{3} \) cup scoops does the recipe require?

Solution:

We need to find out how many \( \frac{1}{3} \) cup scoops make \( \frac{5}{6} \) cup. First, we rewrite both fractions with a common denominator. The least common denominator of 3 and 6 is 6.

Convert \( \frac{1}{3} \) to \( \frac{2}{6} \)

Now divide \( \frac{5}{6} \) by \( \frac{2}{6} \):

\[ \frac{5}{6} \div \frac{2}{6} = \frac{5}{6} \times \frac{6}{2} = \frac{30}{12} = 2.5 \]

Answer: The recipe requires 2 and a half \( \frac{1}{3} \) cup scoops of oil.

Quiz

Answer the following questions.

Lisa has a ribbon that is \( \frac{3}{4} \) of a meter long. Jake has a ribbon that is \( \frac{5}{8} \) of a meter long. Who has the longer ribbon?
A classroom has 24 students. \( \frac{2}{3} \) of the students are wearing sneakers. What fraction of students are not wearing sneakers?
Mark and Zoe each built a tower out of blocks. Mark’s tower is \( \frac{5}{6} \) of a meter tall, and Zoe’s tower is \( \frac{3}{4} \) of a meter tall. Whose tower is taller?
A recipe calls for \( \frac{2}{3} \) of a cup of milk. Thomas only has a \( \frac{1}{3} \) cup measuring scoop. How many scoops does he need to measure exactly \( \frac{2}{3} \) cup?
Olivia is reading a book. She has read \( \frac{3}{5} \) of the book, while Ethan has read \( \frac{4}{6} \) of his book. Who has read a larger portion of their book?
A sports field is divided into 8 equal sections. If Maria’s team is using \( \frac{6}{8} \) of the field and Luis’ team is using \( \frac{3}{4} \) of the field, who is using more of the field?
Which fraction is greater: \( \frac{5}{8} \) or \( \frac{2}{3} \)? Use equivalent fractions to compare them.
A lemonade stand sold \( \frac{5}{10} \) of its drinks in the morning and \( \frac{3}{6} \) of its drinks in the afternoon. Which part of the day did they sell more drinks?
Arrange the following fractions in order from least to greatest: \( \frac{2}{3} \), \( \frac{3}{4} \), and \( \frac{5}{6} \).
Emma and Leo each ran a race. Emma ran \( \frac{7}{12} \) of a mile, and Leo ran \( \frac{2}{3} \) of a mile. Who ran farther?

Answer Key

1. Lisa’s ribbon is longer because \( \frac{3}{4} = \frac{6}{8} \), which is greater than \( \frac{5}{8} \).
2. \( \frac{1}{3} \) of the students are not wearing sneakers.
3. Mark’s tower is taller because \( \frac{5}{6} > \frac{3}{4} = \frac{9}{12} \), and \( \frac{5}{6} = \frac{10}{12} \).
4. Thomas needs 2 scoops of \( \frac{1}{3} \) cup to measure \( \frac{2}{3} \) cup.
5. Ethan has read more because \( \frac{4}{6} = \frac{2}{3} \), which is greater than \( \frac{3}{5} \).
6. Both teams are using the same amount of the field because \( \frac{3}{4} = \frac{6}{8} \).
7. \( \frac{2}{3} \) is greater than \( \frac{5}{8} \) because \( \frac{2}{3} = \frac{16}{24} \) and \( \frac{5}{8} = \frac{15}{24} \).
8. The lemonade stand sold the same amount in the morning and afternoon because \( \frac{5}{10} = \frac{3}{6} \).
9. Least to greatest: \( \frac{2}{3}, \frac{3}{4}, \frac{5}{6} \) (or \( \frac{8}{12}, \frac{9}{12}, \frac{10}{12} \)).
10. Leo ran farther because \( \frac{2}{3} = \frac{8}{12} \), which is greater than \( \frac{7}{12} \).