Lesson Plan: 5.NF.B.4 Multiplication of Fractions
(This lesson should be adapted, including instructional time, to meet the needs of your students.)
Background Information /Content/Grade Level / Multiplying Fractions or Whole Numbers by Fractions/Grade 5
Unit / Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Essential Questions/Enduring Understandings Addressed in the Lesson / Essential Questions:
If a fraction is multiplied by a whole number or another fraction, will the product increase or decrease? Why?
How are multiplying whole numbers and multiplying fractions the same and/or different?
Enduring Understandings:
Multiplication of a whole number by a fraction yields a product less than the whole number.
Multiplication of a fraction by a fraction yields a product less than either factor.
Standards Addressed in This Lesson / 5.NF.B.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product ab× 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.
It is critical that the Standards for Mathematical Practice are incorporated in ALL lesson activities throughout the unit as appropriate. It is not the expectation that all eight Mathematical Practices will be evident in every lesson. The Standards for Mathematical Practice make an excellent framework on which to plan your instruction. Look for the infusion of the Mathematical Practices throughout this unit.
Lesson Topic / Multiplication of Fractions
Relevance/Connections / 5.NF.B.4b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of 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.
· Multiplication of a whole number by a fraction (i.e. 6 × 12 ) is essentially repeated addition with like denominators.
· Multiplication of a fraction and a whole number (i.e. 6 × 12 ) results in portioning the whole number into equal parts.
Student Outcomes / Students will be able to multiply a fraction by a whole number or fraction and interpret the product to show evidence of understanding through multiple representations (i.e. visual model, concrete models, written explanations, equations, etc.)
Prior Knowledge Needed to Support This Learning / 3.NF.A.1 Understand a fraction 1b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction ab as the quantity formed by a parts of size 1b
4.NF.A.1 Explain why a fraction ab is equivalent to a fraction n × an × b by using visual fraction models, with attention to how the number and size o the parts differ even though the two fractions themselves are the same size. Use this principal to recognize and generate equivalent fractions.
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 12. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols <, = , or > and justify the conclusions, e.g. by using a visual fraction model.
· Understanding that a fraction is the numerator divided by the denominator
· Understanding that a fraction can be part of a whole or set which can be shown by partitioning the whole or set into equal parts
· Understanding that multiplication can be repeated addition as well as showing groups of a number/fraction
· Understanding that multiplication can be shown as an array
· Understanding that multiplication is equal groups of numbers of objects (i.e. 2 × 3 = 2 groups of 3)
· Understanding that when multiplying two whole numbers, the product is larger than both factors.
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Method for determining student readiness for the lesson / Use questions from Show What You Know! (Attachment #1) to assess students’ understanding of fractions and multiplication (from prior knowledge listed above.)
We will assess:
· Understanding that multiplication can be shown as an array
· Understanding that multiplication is equal groups of numbers of objects (i.e. 2 × 3 = 2 groups of 3.)
· Understanding that when multiplying two whole numbers, the product is larger than both factors
Learning Experience /
Component / Details / Which Standards for Mathematical Practice does this address? How is each Practice used to help students develop proficiency? /
Warm Up / Use what you know about whole numbers and fractions to answer the following question:
“At Styrofoam Elementary School, only 14 of 12 classes recycle. How many classes at that school are recycling?” (use numbers, pictures or words)
Have multiple students show their representation of the problem. Then ask:
What you notice about all of the representations?”
“What do they all have in common? How are they different?”
Motivation / Tell students that they are now part of an exclusive “MSDE Eco Club” which will be working to promote environmental awareness within the school building and grounds. They will be part of a planning team to promote recycling within the building as well as plan an Eco Park on school property. An Eco Park is a designated area within school grounds with the goal of providing real life applications to environmental concepts. These may include a butterfly garden, a vegetable garden, habitats and ecosystems.
To start these activities, here are a few options:
1. Have a gallery walk with poster titles including “What to Include in our Eco Park”, “Ways to Promote Recycling”
2. Have students research what can be included in the Eco Park and share their ideas.
Activity 1
UDL Components
· Multiple Means of Representation
· Multiple Means for Action and Expression
· Multiple Means for Engagement
Key Questions
Formative Assessment
Summary / UDL Components:
· Principle I: Representation is present in the activity. Through the Warm-up and Motivation, prior knowledge is activated. Students are encouraged to display their solutions in a variety of ways, including diagrams, text, images, graphs, etc. Key items in the text were emphasized by providing different scenarios so the students can clarify their thinking about the questions being asked.
· Principle II: Expression is present in the activity. The activity provides alternatives in the requirements for rate, timing, and range of motor actions necessary to interact with the instructional materials/physical manipulatives, such as fraction bars.
· Principle III: Engagement is present in the activity. It is based on an authentic situation that occurs in their school, making the outcomes real, student-centered, and purposeful. The tasks were designed to allow the students to have active participation in the lesson.
This activity addresses understanding of multiplying a whole number by a fraction.
Throughout this activity, refer to the Essentials Questions:
· If a fraction is multiplied by a whole number or another fraction, will the product increase or decrease? Why?
· How are multiplying whole numbers and multiplying fractions the same and/or different?
A. Pose the following problem to students:
A survey of 6 classes showed that each class filled 13 of its plastic bottle recycling bin. How many total bins were filled?” Have students work in groups to solve this problem. Allow students to explore using various representations such as manipulatives, visual models and written communications. Ask volunteers to show their thinking to the group.
· “Explain the process used to find the solution.”
· “Write an equation that shows the same relationship between the numbers”.
· (Try to elicit both a repeated addition and then multiplication equation to show their understanding of the connection between the two.)
B. Another part of the survey showed that the same 6 classes filled their paper recycling bins to 34 full.
How many total bins of paper were filled?” Allow students to explore using various representations such as manipulatives, visual models and written communications.
· “How is this problem different than the last?”
· “Why is this answer a mixed number, while your answer to the last question was a whole number?’ (Work to elicit that in the previous problem, 3 is a factor of 6 and hence will go in an equal number of times whereas 4 is not a factor of 6 so the remainder of pieces will become the fraction in the mixed number.)
C. (See Attachment #2 ---titled “Elementary Recycling Results Chart”) Post the chart for the class to use in this activity, with the strategy columns blank.
Work together as a class to complete the chart by various strategies used to find the total number of bins filled. Be sure to make a connection between the repeated addition and multiplication equations.
· How does repeated addition relate to the factors of a multiplication problem?
· 8 groups of 56 is equivalent to 8 × 56. Do you agree or disagree? (Guide students to the understanding that ‘of’ is another way to express multiplication.)
· Explain how the products of these problems are different from the products of an equation with two whole number factors?” (Connect to the enduring understanding ‘Multiplication of a whole number by a fraction yields a product less than the whole number.’)
D. Pose the following to students: As part of the MSDE Eco Club, you notice that the classes are recycling paper more than plastic. To promote more recycling, Wye Oak Elementary decided to have a recycling contest in grades 3 - 6. The following are the results:
Post Chart (Attachment #3- “Grade Level Recycling Contest” )
Divide the class into four groups, one to represent each of the grade levels. As a group, have students represent the total amount of bins filled using both a visual representation as well as the equation(s) that matches.
Ask the following questions:
1. “Which grade level won the contest?
2. How do you know?”
As this activity concludes, please be cognizant of the Enduring Understanding: “Multiplication of a whole number by a fraction yields a product less than the whole number.”
E. Distribute the following exit ticket to students (Attachment #4)
“The Kindergarten classes are using 3 recycling bins. Each bin was 25 full. How many recycling bins were filled? Explain the process used to find the solution by showing at least 2 representations for your solution.” / This activity will allow students to make sense of problems and persevere in solving them by choosing their own methods to solve each problem.
(SMP #1)
This activity allows students to begin to reason abstractly and quantitatively as have to represent the problems in different ways.
(SMP #2)
As the students choose their method for solving each problem they will have to construct viable arguments and critique the reasoning of others as they listen to the conversations of other students.
(SMP#3)
The students analyze each problem and model with Mathematics by using equations and diagrams as needed.
(SMP #4)
Activity 2
UDL Components
· Multiple Means of Representation
· Multiple Means for Action and Expression
· Multiple Means for Engagement
Key Questions
Formative Assessment
Summary / UDL Components:
· Principle I: Representation is present in the activity. It uses visual diagrams and charts throughout the tasks to help students plan and complete their activities. The activity presents an authentic problem that replicates community-minded projects in which elementary schools actually engage students.
· Principle II: Expression is present in the activity. Different ways of representing the scenario are presented to the students, and different physical models are provided for the students to use.
· Principle III: Engagement is present in the activity. The task allows for active participation as students explore the different possibilities for the Eco Park. Students are expected to self-reflect through the completion of the activity.
This activity addresses understanding of multiplying a fraction by a whole number (fraction of a number…i.e. 12 of 6) and a review of equivalent fractions.
A. Ask students to complete the following warm-up to review/assess previous skills needed for this activity.
Order the following park sizes in order from least to greatest: 14 310 45 12
B. Explain to students that after learning about the recycling of your school community, it has been decided that more awareness of the environmental issues could help to promote appreciation and action to help save the planet! As members of the MSDE Eco Club, we are going to plan an Eco Park. The principal has designated 24 square yards to the Eco Club for this project.
The area of the park has been divided into 5 sections to include: a butterfly garden, a vegetable garden, a reflection bench, an animal habitat and a water ecosystem.
Post the chart (Attachment #5 – “Eco Park Planning Chart”) for the class to explore how to partition the Park.
Divide students into groups of 3 or 4. Distribute grid paper to each group (and possibly scissors to cut up the grid paper), manipulatives (i.e. 24 counters) and any other tools to allow students to find the fraction of the Park. Remind students that each paper square or manipulative would equal 1 square yard.