MSDE Mathematics Lesson Planning...

MSDE Mathematics Lesson Planning Organizer

Background InformationContent/Grade Level Multiplying Fractions or Whole Numbers by Fractions/Grade 5

Unit/Cluster 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.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. Lesson Topic Multiplication of Fractions

Relevance/Connections 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 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 3.NF.1

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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.

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.1

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. 4.NF.2

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 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

Component Details Which Standards for Mathematical Practice does this component address?

Materials Attachment #1 – Show What You KnowAttachment #2 - Elementary Recycling Results ChartAttachment #3- Grade Level Recycling ContestAttachment #4 – Exit Ticket, Activity 1Attachment #5 – Eco Park Planning ChartAttachment #6 – Eco Park Section PlannerAttachment #7 – Exit Ticket, Activity 2Attachment #8 – Grid optionsAttachment #9 – Exit Ticket, Activity 3

Warm Up/Drill Use what you know about whole numbers and fractions to answer the Make sense of problems and persevere in

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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?”

solving them

Reason abstractly and quantitatively

Construct viable arguments and critique the reasoning of others.

Model with Mathematics

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

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

Make sense of problems and persevere in solving them



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Expression Multiple Means

for EngagementKey QuestionsFormative AssessmentSummary

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 back 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 a different 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


Look for and make use of structure.

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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” ) of 24

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.”

Activity 2


of Representation

Multiple Means for Action and Expression

Multiple Means for Engagement

Key QuestionsFormative 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






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whole number (fraction of a number…i.e. 12 of 6).

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 3

1045

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.

Teacher circulates to address misconceptions and/or guide students toward deeper understanding.

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Solution Path Assessing Questions

Advancing Questions

Students are not finding the common denominator of 24.

If you continued, how many square yards would be used? Does it equal 24? Why or why

not? Is 38 the same

as 3

24 ? How do you

know?

How could you

show 38 of a set of

24? How much of the park would it cover?

Students are finding common denominators but the various fractions are not adding up to the whole park (24 square yards)

If you place the Eco Park sections on the grid of 24, would it cover all of the spaces or too many?

How could you show the sections of the park using exactly 24 squares?

Allow student groups to share their strategies for finding the total square units of each section of the park. Ask probing questions throughout the discussion in order to allow students to describe their thinking strategies. Some of the questions might include: “Which of the park sections is the largest/smallest? What

process did you use to arrive at this solution?” “Are any of the sections the same size? How is that possible

with different fractions of a whole?” “Which operation does this activity show? Explain your

thinking.”

“As a group, let’s review our findings so that we can move on to the next step of our planning.” See Attachment #6 – “Eco Park Section Planner” (As a group, insert equivalent fraction with a denominator of 24 into the second column.)

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To connect the word “of” to multiplication, pose the following questions to students:

To find the square yards for the Vegetable Garden, you found 38

of 24 square feet. How many yards did that equal? Is there a mathematical equation to show how you solved this? (38

of 24=38

× 24)

Continue with the remaining sections of the park to show multiplication equations.

If you have time, students can add the sections to check that it equals 24 square yards.

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.”

Exit Ticket: Attachment # 7 “Suppose the principal now has given you 36 square yards for the Eco

Park. How many square yards would the Butterfly Garden ( 16 )Water

Ecosystem ( 14 ), and Reflection Bench ( 1

12 ) be?”

Activity 3


of Representation

Multiple Means for Action and Expression

UDL Components: Principle I: Representation is present in the activity. Prior

knowledge of words and multiplication are activated in this lesson. Patterns and visual diagrams are used in this section to help students with multiplication of fractions.

Principle II: Expression is present in the activity. As students work in groups, physical interaction with each other and the concept presented are encouraged. Scaffolding enables students to break the objectives of the activity into manageable tasks that build on





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Multiple Means for Engagement

Key QuestionsFormative AssessmentSummary

prior knowledge. Principle III: Engagement is present in the activity. The task allows

for collaboration and active participation that permits students to explore, experiment, and act as peer mentors.

This activity addresses understanding of multiplying a fraction by a

fraction (i . e . 12

of 34 )

Pose to students the following “The principal decided to give us the original 24 square yards. It is now time to take the next step in our garden planning. What will be in each section? How will each section be divided? We will start by using a grid to divide up each section of the park.”

Distribute Attachment #8 (grids with 24 units – feel free to use which grid you wish to use to model with students for this activity.)

A. “Let’s begin by planning the Vegetable Garden so that we can

split it up for different vegetables. The Vegetable Garden is 9

24

square yards. The committee decided to devote 13 to carrots

and 23 to tomatoes.

On our grid, let’s choose 9 of the 24 square yards for our Vegetable Garden.

How can we show the 13 of the Vegetable Garden ( 9

24 )on our

grid? (note to teacher: use the steps below to model for students how to show a fraction of a fraction on the grid.)

1. Show the 9 out of 24 on the grid. Shade the 9 out of 24 one color (such as yellow.)


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2. Use a different color (such as blue) to shade 13 of the

924 .

(When you color on your chart, the overlapped section 13 of the

vegetable garden, should be a combination of the colors (green.)

3. “How many square yards would the carrot section be?” ( 324 )

4. Show the equation 13 of

924

= 324 Remind students that ‘of’ can

be the same thing as multiplication so that the equation could

read “13 ×

924

= 324

5. Now that we have the carrot section portioned, let’s continue

with our planning. If the tomato section is 23 of

924 , what

fraction of the Eco Park would the tomato section be?

B. Continue partitioning the various sections of the Eco Park in the same manner. Students will decide how big each section will be. Some examples could be:

Butterfly Garden: 14 of the Butterfly Garden should be the

Butterfly Bush: Dark Knight, 34 should be Beebalm Bush

Animal Habitat: 13 of the Animal Habitat should be toads,

23

should be native insects. of 24

Water Ecosystem: 56 of the Water Ecosystem should be the

pond, 16 waterfall

Herb Garden: 12 of the Herb Garden should be basil,

12 oregano.

Here is an example of how the garden can be partitioned.

VGCarrot

VGTomato

VGTomato

AHToad

AHInsect

AHInsect

VGCarrot

VGTomato

VGTomato

WEPond

WEPond

WEPond

VGCarrot

VGTomato

VGTomato

WEPond

WEPond

WEWaterfal

l

BGDark

Knight

BGBeebalm

Bush

BGBeebalm Bush

BGBeebalm

Bush

HGBasil

HGOregano

C. As you have finished partitioning the different sections, it is time to build student understanding of how to multiply fractions. You can ask questions such as:

If 23 of the Animal Habitat is native insects, ( 2

3of 3

24 ) remembering that ‘of’ can mean multiplication, write a mathematical equation to show the section of the Animal

Habitat that will be devoted to insects. 23

× 324 =

672 =

112 .

What do you notice about the equation? (After discussion, elicit that you could multiply the numerators and

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denominators across to find the product.) “Write an equation to determine what fraction of the Animal

Habitat is devoted to toads.” (Based on student understanding, continue with the other

sections to model multiplying the fractions, or move on if students are ready.)

What do you notice about the product compared to each of the factors in these equations? Do they increase or decrease? Why?

D. Break students into groups of 3 or 4 students. Allow the groups to solve the following problems together.

For the tomato section (23 of the vegetable garden), students

planted tomato seeds of which 16 were Cherry Tomato

seeds, 13 were Beefsteak Tomato seeds, and

12 were Hybrid

Tomato seeds. What fraction of the vegetable garden will each these types of tomatoes cover? Work together to solve these problems and be prepared to share your solutions. Choose visual models, words, manipulatives, or other tools to assist in understanding and explanation (including the original grid.)

Have groups share their strategies and solutions with the class. Make sure to highlight the various strategies from different groups.

As an extension, have students create a map of their own Eco Park showing all sections and the label the fractional parts.

Exit Ticket: See Attachment #9“When a fraction is multiplied by another fraction, will the product increase or decrease? Why? Prove your answer using words, numbers, models and/or symbols in your explanation.”

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Closure In order to connect the 3 activities from above, write the following 3 expressions on the board:

8 ×14

34 ×6

34 ×

28

Pose the following to students: 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? How are multiplying fractions with whole numbers or another

fraction used in the real world? What would be some

examples? (i.e. 14 off of a $20 sweater.)

Write a word problem for each of the expressions above. Take time to share the various word problems.

Supporting InformationInterventions/Enrichments

Special Education/Struggling Learners

ELL Gifted and Talented

Vocabulary – ELL and special education students need to know and understand the following terms to be successful with these activities: product, factors, partition, multiplication (including repeated addition,) expression, equation

Use of grid paper and calculators as needed Use of manipulatives including counters, fraction towers, etc. as needed to model multiplying

fractions by a whole number or fractions of a fraction. Understanding of basic fractions and multiplication concepts by providing visual models and use of

manipulative as mentioned above.

Technology Computer with internet access (for researching Eco Parks) LCD projector (optional) Document camera Calculators Overhead (transparency could be used for multiplying fractions using a grid.)

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Resources(must be available to all stakeholders)

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Attachment #1

Show What You Know!

1. What are at least 2 ways to illustrate 4 × 2 = 8?

2.What are at least 2 ways to illustrate 4 × 2 = 8? Draw as many arrays as you can to show the product of 20.

3. When multiplying 2 whole numbers, will the product of those factors be greater than or less than each of the factors? Explain your thinking using words, numbers and/or symbols.

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Attachment #2

Activity 1, C

Elementary Recycling Survey Results

Wye Oak Elementary

Strategy 1

(later label as Addition Equation to show total number of bins filled)

Strategy 2

(later label as Multiplication Equation to show total number of bins filled)

Oriole Elementary Strategy 1

(later label as Addition Equation to show total number of bins filled)

Strategy 2

(later label as Multiplication Equation to show total number of bins filled)

Number of Classes Surveyed

8 10

How much of the paper bin is recycled in each class?

56

78

How much of the plastic recycling bin is filled in each class?

15

26

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Attachment #3

Activity 1, D

Grade Level Recycling Contest

Grade Level Number of Bins Fraction of bins filled

Total amount of bins filled

3 5 23

4 5 12

5 5 35

6 4 67

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Attachment # 4

Name: ___________________

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.

________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________

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Attachment # 5Activity 2, B

Eco Park Planning Chart Eco Park Section Fraction of the

ParkTotal Square Yards

of SectionVegetable Garden 3

8

Butterfly Garden 16

Water Ecosystem 14

Animal Habitat 324

Reflection Bench 112

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Attachment #6

Activity 2, B

Eco Park Section PlannerEco Park Section Fraction of the Park Total Square Yards of

Section

Vegetable Garden 38

= 924 9

Butterfly Garden 16

=

Water Ecosystem 14

=

Animal Habitat 324

=

Reflection Bench 112 =

Total 2424 24 square yards

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Attachment #7

“Suppose the principal now has given you 36 square yards for the Eco Park. How many square yards would the Butterfly

Garden ( 16 )Water Ecosystem ( 1

4 ), and Reflection Bench ( 112 ) be?”

Expanded Eco Park Section PlannerEco Park Section Fraction of the

ParkTotal Square Yards

of Section

Butterfly Garden 16 =

Water Ecosystem 14 =

Reflection Bench 112 =

Total 3636 36 square yards

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Attachment #8

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Attachment #9

Activity 3

When a fraction is multiplied by another fraction, will the product increase or decrease? Why? Prove your answer using words, numbers, models and/or symbols in your explanation.

________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________

________________________________________________________________________________________________________________________

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