Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 38004
Area Models: Multiplying Fractions
In this lesson students will investigate relationship between area models and the concept of multiplying fractions. Students will use area model to
develop understanding of the concept of multiplying fractions as well as to find the product of two common fraction. The teacher will use the free
application GeoGebra (see download link under Suggested Technology) to provide students with a visual representation of how area models can be
used at the time of multiplying fractions.
Subject(s): Mathematics
Grade Level(s): 5
Intended Audience: Educators
Suggested Technology: Computer for Presenter,
Internet Connection, LCD Projector, Java Plugin,
GeoGebra Free Software (Download the Free
GeoGebra Software)
Instructional Time: 50 Minute(s)
Freely Available: Yes
Keywords: fraction multiplication, area model, denominator, numerator
Instructional Design Framework(s): Direct Instruction
Resource Collection: CPALMS Lesson Plan Development Initiative
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Using GeoGebra and one other method (graph paper and colored pencils) student will use area models to multiply fractions and achieve 100% in the independent
practice.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students must have prior knowledge and understanding of multiplying whole numbers. They must also be familiar with arrays and part-whole thinking as applied to
multiplication.
Guiding Questions: What are the guiding questions for this lesson?
1. What is an area model? (The area model for multiplication is a pictorial way of representing multiplication. In the area model, the length and width of a rectangle
represent factors, and the area of the rectangle represents their product.)
2. When should we use models to solve problems with fractions? (When we multiply or divide fractions)
3. How can you determine the product of two fractions by looking at an area model? (The product is the area that is double-shaded (show below in purple)
4. What is the rule to multiply fractions? (Numerators are multiplied to get the numerator, and denominators are multiplied to get the denominator)
Teaching Phase: How will the teacher present the concept or skill to students?
During the Teaching Phase, teacher uses the GeoGebraTube applet "Area Model: Multiplying Fractions" to give visual definitions and demonstration of the use of area
model to multiply common fractions.
Definition - Pictorial way of representing multiplication. In the area model, the length and width of a rectangle represent factors, and the area of the rectangle
represents their product.
page 1 of 6 Demonstration
Move the upper and lower blue slider to represent the first fraction.
Move the upper and lower red slider to represent the second fraction.
Drag the red point to overlap the blue point.
Ask: "What do the length and the height of the rectangle represent?". (the factors)
Ask: "What does the double shaded area represent?". (the product)
How can you determine the product of two fractions by looking at an area model?
The product is the area that is double-shaded (show below on purple)
Related Terminology
After presenting the problem above, the teacher proceeds with definitions and demonstrations of important terms. The terms includes:
array
factors
product
Array
Definition - An array is a systematic arrangement of objects, usually in rows and columns.
Factors
Definition - Numbers that are multiplied together
Product
Definition - The result to a multiplication problem
page 2 of 6 During the Teaching Phase, the teacher will present the following problem: One half of Mrs. Jimenez's new room is an office. One third of her new office is covered with
tiles. What fraction of Mrs. Jimenez's new room is covered with tiles?
Ask students: "What do you need to find?"(the fraction of Mrs. Jimenez's new room covered by tiles)
Example #1 Using Area model to solve word problem
Teacher:
Ask students:
What does the lower half of the square represent? (The part of Mrs. Jimenez new room that is an office)
What does the third of the lower half of the square represent? (The part of the office covered with tiles)
Teacher will model using GeoGebra. The teacher will drag the red point on the GeoGebra application to overlap the blue point.
Ask students to describe what they noticed.
Possible answers:
the area shaded with blue is 1/2 which is the area of the new room that is an office.
the area shaded with red is 1/3 which is the area of the office covered with tiles.
there is an area that is double shaded represented in purple.
At this part of the discussion teacher can address guiding question #3 and guiding question #4
The product is the area that is double-shaded (show below on purple)
Discuss Guiding Question #4: What is the rule to multiply fractions?(Numerators are multiplied to get the numerator, and denominators are multiplied to get
the denominator).
Possible Misconceptions:
1. Students may believe that multiplication always results in a larger number.(Using models when multiplying with fractions will enable students to
see that the results will be smaller).
2. Mixed up the factors and the product in the area model.
Mixed up the numerator with the denominator and vice-versa.
Teachers can use the following example to demonstrate that answers can also be found using other methods such as folding paper or with the virtual manipulative
website.
Additional Examples
a. Three-fourths of the class is boys. Two-thirds of the boys are wearing tennis shoes. What fraction of the class are boys with tennis shoes?
b. One-fourth of Mr. Caruso's garden is covered by flowers and 1/6 of the flowers are black-eyed Susans. How much of Mr. Caruso's garden is covered by black-eyed
Susans?
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The teacher will use GeoGebra to visually model the situation presented "Luis had 1/2 of the lawn left to mow. On Sunday, he mowed 2/3 of what was left. What
fraction of the entire lawn did Luis mow on Sunday?"
While the teacher models using GeoGebra, the students will model on paper. During this part of the lesson the teacher will be introduce the concept of area model and
will discuss guiding question #1.
page 3 of 6 The GeoGebra application can be found in the following link: Multiplying Fractions. Adjust red sliders to represent 1/2 of lawn left to mow and adjust blue sliders to
represent two-thirds, which is the part mowed on Sunday.
Activity #1 Using Area model to solve word problem Students:
1. Provide students with graph paper.
2. Ask students to draw a 2 by 3 rectangle.
3. Divide the upper segment of rectangle in half.
4. Divide the left segment of the rectangle in thirds.
Teacher:
Ask students:
What does the lower half of the square represent? (The part of the lawn left to be mowed)
What does thetwo third of the lower half of the square represent? (The part of the lawn that was left that was mowed on Sunday)
Students:
5. Provide students with colored pencils and ask them to use one color to represent the half of the lawn left to be mowed. Students must colored in horizontally.
6. Ask students to use a different color and ask them to color in vertically the fraction that was mowed on Sunday.
Teacher:
The teacher will modele using GeoGebra; at this time the teacher will drag the red point on the GeoGebra application to overlapthe blue point. The area model should
look similar to the area model that the students draw.
Ask students to describe what they noticed.
Possible answers:
the area shaded with red is 1/2 which is the part of the lawn which was mowed on Saturday.
the area shaded with blue is 2/3 which is the part of the remaining area which was mowed on Sunday.
there is an area that is double shaded represented on purple.
At this part of the discussion teacher can reinforce guiding question #3, which was discussed during the Teaching Phase.
Possible Misconceptions:
1. Students may believe that multiplication always results in a larger number.(Using models when multiplying with fractions will enable students to see that the results
will be smaller).
2. Mixed up the factors and the product on the area model.
Activity #2 Teacher:
Use GeoGebra application or thevirtual manipulative website to randomly generate between 4- 5 problems.
On GeoGebra, move the red and blue sliders to create a new fraction for students to model and solve.
On the virtual manipulative website, click on test meand to create a new problem click on new problem.
For this activity, have students to work with a partner and provide each student with graph paper and colored pencils.
Steps:
1. Teacher generates a random model.
2. Students draw the model and find the answer.
The teacher will use GeoGebra or the virtual manipulative website to visually discuss the problems randomly generated.
Management Tips:
Since problems will be randomly created, answer may vary.
Provide students enough time to model each problem.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Students will be given a Summative Assessment assignment. The assessment is designed to reinforce the concept of multiplying fractions. If students struggle on the
assignment, the teacher will be made aware of misunderstandings, and shortcomings of the lesson taught. When "weak" areas are identified on the assignment, the
teacher can address during the next available class period and will have students take a short online practice on IXL website.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Summarize lesson by asking students to debrief on what they learned on this lesson. Have students share what they discovered about the relationship between area
models and multiplying fractions. Discuss all guiding questions during the closure. Teacher will be sure to focus on reinforcing vocabulary with the class and will ask
students to explain what they learned as a result of the lesson.
Summative Assessment
The teacher will use the Summative Assessment assignment to determine if the students have reached the learning targets for this lesson. Students must show 100%
mastery in the summative assignment in order for the skill to be considered mastered.
page 4 of 6 Formative Assessment
Because students are expected to know how to multiply whole numbers using area models before beginning this lesson, the teacher should check for that knowledge.
This virtual manipulatives web site can assist in this check of student knowledge:Rectangle Multiplication
Another check on students' prior knowledge is this worksheet:Multiplying Fraction Practice.
Feedback to Students
Teacher will give feedback to students during Activities 1 and 2 of the guided practice. The teacher's feedback includes discussion of possible misconceptions.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations: During this lesson students are working on real-world problem to contextualize the concepts, this will help visual learners to comprehend better
the concept. Students will be working with hands-on activities, which will reinforce the skill. Students will be also working in small groups to brainstorm and complete
different tasks during the each activity, this think-share-pair strategy will allow students to learn from each other. Teacher will circulate continuously, monitor and check
in with each group on their progress and understanding of the tasks.
Extensions: The lesson itself can stand alone or can be extended to master other related skills. Teacher might like to extend to multiplying mixed numbers or to
divide fractions.
Suggested Technology: Computer for Presenter, Internet Connection, LCD Projector, Java Plugin, GeoGebra Free Software
Special Materials Needed:
For this lesson the following materials are needed
Pencil
Graph Paper
Ruler
Colored Pencil
Further Recommendations: Before giving the activities to the students, it is important that the teacher become familiar with the content and the GeoGebra
application.
Additional Information/Instructions
By Author/Submitter
This resource is likely to support student engagement in the following standards of mathematical practice
MAFS.K12.MP.2. Reason abstractly and quantitatively.
Rationale: This math practice includes students' reasoning and explanations of how they use models, pictures, and strategies. This math practice can be use during the
teaching phase while using GeoGebra to demonstrate and discuss the first problem.
MAFS.K12.MP.4. Model with mathematics.
Rationale: It requires students to extend the models and strategies they developed to multiply fractions. This math practice can be use during activity #1 of the guiding
practice.
MAFS.K12.MP.6. Attend to precision.
Rationale: They multiply fractions efficiently and accurately as well as solve problems in both contextual and non-contextual situations with the use of area model. This math
practice can be use during the entire lesson specially during discussions.
SOURCE AND ACCESS INFORMATION
Contributed by: celia segarra
Name of Author/Source: celia segarra
District/Organization of Contributor(s): Broward
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.5.NF.2.4:
Description
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a
sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create
a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction
side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply
fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
Remarks/Examples:
Examples of Opportunities for In-Depth Focus
page 5 of 6 When students meet this standard, they fully extend multiplication to fractions, making division of fractions in grade
6 (6.NS.1) a near target.
page 6 of 6