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Lesson Content

Lesson Plan Template:
Confirmatory or Structured Inquiry

Learning Objectives: What will students know and be able to do as a result of this lesson?
MAFS.4.NF.2.4
Given a situational word problem, students will create a model to find the correct product of a whole number and a fraction less than one.
MAFS.K12.MP.2.1
Given a situational word problem, students will identify quantities within the problem as a fraction and a whole number and recognize that a fractional amount is being repeated. Using various strategies students will have to manipulate the numbers to determine the total quantity needed and create a logical representation of the problem. Students will then relate that amount back to the context of the situation.
MAFS.K12.MP.3.1
Students, while in small groups, will: analyze the problem and construct an argument; justify conclusions with mathematical ideas; listen to the arguments of others and ask useful questions to determine if an argument makes sense; ask clarifying questions or suggest ideas to improve/revise the argument; compare two arguments and determine correct or flawed logic to determine the strategy to share with the whole group.

Prior Knowledge: What prior knowledge should students have for this lesson?
 Understanding of unit fractions
 Finding equivalent fractions, decomposing fractions, comparing fractions
 Modeling fractions with drawings, fraction bars/strips or circles, number lines, bar models
 Understanding multiplication is used when working with equalsized sets
 Multiplying whole numbers first by modeling with arrays, area model, partial product and then the standard algorithm
 Working within small groups, analyzing others' strategies, critiquing and receiving feedback from each other

Guiding Questions: What are the guiding questions for this lesson?
 How can multiplication of a whole number and a fraction be represented with a model?
 What do the numbers of the problem mean? How can I use what I have learned about multiplication to solve this problem?
 What questions can be asked to understand someone else's strategy? What should be done if I don't agree with that solution?

Introduction: How will the teacher introduce the lesson to the students?
The "Hook" and Activation of Prior Knowledge
A story set in the life of the students is used to engage students and provide a visualization of what is happening in the mathematics. Refer to the story in different parts of the lesson to help students visualize what is happening mathematically.
Present students with an animated story about the principal requesting some banners for a school project. This should be tied to an actual situation on campus.

Investigate: What question(s) will students be investigating? What process will students follow to collect information that can be used to answer the question(s)?
The problem can then be typed out on a mailing label and students can add it to their math notebooks to solve.
A sample is,
"Mrs. Russell wants us to have ribbons on each corner of our garden identifying the various crops. She determined 2/3 of a yard was the perfect length for each ribbon. Since ribbon is sold as a continuous strip as long as you need, how much should be purchased for all four corners? Create a representation to justify your thinking."
If no students use a multiplication model, give groups copies of the worked out solutions in the following attachment. Students are then asked to determine if the solutions and the representations on the paper are correct. I prefer to print out the papers and then cut them into cards. Looking at one solution at a time seemed to help focus the students.
Lesson1PossibleSolutions.docx

Analyze: How will students organize and interpret the data collected during the investigation?
Students should identify the information needed: the amount of ribbon for each corner and the number of corners.
They may use drawings, manipulatives, or a bar model to work a solution.
Please see the Feedback to Students for questions to ask students while they are working.
Allow students to work through the problem individually.
Once students have had time to record at least one strategy, have each person explain their strategy to their group. Other group members should ask questions for clarification.
Group members should then decide on a strategy to record on chart paper to share with the whole group.
Monitor the groups while they are recording on chart paper to determine the sequence of reporting out during the whole class discussion. Strategies should be shared with the whole class staring with the most basic, simplistic, or a misconception and ending with the most complex or abstract.
The following attachment is a repeat of the attachment in the Formative Assessment and shows some of my students' work on this problem.
Spies_Lesson1_StudentSample.docx

Closure: What will the teacher do to bring the lesson to a close? How will the students make sense of the investigation?
Facilitate whole group discussion of solution strategies.
Hang charts on board in order of the progression to be shared.
Have one member from each group explain the solution. During the explanations highlight examples of mathematical reasoning and have students critique any misconceptions and explain the error.
Prompts may include:
 You said what you did now tell us why you did that.
 What mathematical evidence would support your solution?
 How can we be sure that...?
 How could you prove that...?
 Will it still work if...?
 What were you considering when...?
 How did you decide to try that strategy?
 How did you test whether your approach worked?
 How did you decide what the problem was asking you to find? (What was unknown?)
 Did you try a method that did not work? Why didn't it work? Would it ever work? Why or why not?
 What is the same and what is different about...?
If no student solution shows (4x2)/(4x3), show work for the incorrect strategy and ask, A student last year did this. Does this work? Why or why not?
To summarize, ask what is the same about all of the strategies and about multiplying whole numbers?
Identify the key knowledge:
 Multiplication can be used to calculate repeated addition quantities with fractions, just like it can for whole numbers.
 Various models can prove this works.
Either at the end of the lesson or the next day, administer the Summative Assessment.

Summative Assessment
1.At the conclusion of this lesson, students will be given an incorrect solution and asked to determine whether or not it is correct and to justify their thinking. See attached Lesson 1 Summary document.
SpiesLesson1Summary.docx

Formative Assessment
1.Teacher will circulate during independent work time noting correct or incorrect strategies and solutions. The teacher should make a note of strategies that may need to be or should be discussed whole group and identify those who may be struggling with understanding the task itself. Be sure those who have struggled are listening carefully to the whole group discussion. You may pull them to a small group the next day and use fraction strips to explain the problem.
2. During small group discussion teacher will listen for explanations with solid mathematical reasoning or those with misconceptions or errors. Those will be highlighted and discussed during the whole group review. During whole group discussion hand signals of thumbs up or down will be used to check students level of understanding.
The attached document shows various student solutions to the task in this lesson.
Spies_Lesson1_StudentSample.docx
SpiesLesson1Summary.docx

Feedback to Students
1. Monitor students during individual work time by circulating and seeing what is being recorded as they work the problem. Encourage students working out a correct solution strategy and challenge them to find another way to confirm their solution. Prompt students who are struggling or not addressing the task with the following possible questions:
What do we know about the situation?
Can a drawing help you see what is happening?
Is there another way to prove your answer is correct?
What does that amount tell you?
Does this look like any problems we have done before? Why or why not?
How did you determine that amount?
What do the numbers used in the problem represent?
What does the 4 tell us? The 2/3?
How is the number of ribbons related to 2/3 yards?
What operations might we use to find a solution?
How did you decide in this task that you needed to use...?
Could we have used another operation or property to solve this task? Why or why not?
2. During the group sharing and discussion encourage students to share and listen to others. Acknowledge those asking questions of each other that promote deeper understanding. (This addresses the Math Practice Standard: Construct viable arguments and critique the reasoning of others.)
3. In preparation for the instruction after this lesson, sort the summative assessments according to understanding. Form small groups from this work. Once small groups are assembled according to understanding, point out misconceptions and use fraction bars, the area model, or number line to illustrate why the solution was incorrect.
Assessment
 Feedback to Students:
1. Monitor students during individual work time by circulating and seeing what is being recorded as they work the problem. Encourage students working out a correct solution strategy and challenge them to find another way to confirm their solution. Prompt students who are struggling or not addressing the task with the following possible questions:
What do we know about the situation?
Can a drawing help you see what is happening?
Is there another way to prove your answer is correct?
What does that amount tell you?
Does this look like any problems we have done before? Why or why not?
How did you determine that amount?
What do the numbers used in the problem represent?
What does the 4 tell us? The 2/3?
How is the number of ribbons related to 2/3 yards?
What operations might we use to find a solution?
How did you decide in this task that you needed to use...?
Could we have used another operation or property to solve this task? Why or why not?
2. During the group sharing and discussion encourage students to share and listen to others. Acknowledge those asking questions of each other that promote deeper understanding. (This addresses the Math Practice Standard: Construct viable arguments and critique the reasoning of others.)
3. In preparation for the instruction after this lesson, sort the summative assessments according to understanding. Form small groups from this work. Once small groups are assembled according to understanding, point out misconceptions and use fraction bars, the area model, or number line to illustrate why the solution was incorrect.
 Summative Assessment:
1.At the conclusion of this lesson, students will be given an incorrect solution and asked to determine whether or not it is correct and to justify their thinking. See attached Lesson 1 Summary document.
Lesson1Summary.docx
Accommodations & Recommendations
Accommodations:
Model a similar situation with manipulatives such as fraction strips.
Support English Language Learners with translations, definitions, and examples of unfamiliar vocabulary.
Extensions:
It is expected that following the whole group discussion students will be able to correctly find the product of 3 x 2/5 as 6/5 or 1 1/5 using a representation or model of choice. They should be able to correctly explain and justify their strategy. Their level of understanding will be rated 04 and what is recorded will be used to guide further instruction. Scores will be determined as described below. 0  incorrect product, no strategy shown or explained 1  incorrect product, partial strategy? 2  correct product, no strategy or only partial strategy? 3  correct product, partial strategy or explanation. 4  correct product, complete and thorough explanation.

Suggested Technology: Document Camera
Special Materials Needed:
 Typed problem on mailing labels for math journals or posted for all students to read
 Math journals, notebooks, or recording paper
 Fraction manipulatives (fraction bars, strips, pieces are possibilities)
 Chart paper/markers for each group
 Printed sample solution cards
Further Recommendations:
 Have "Possible solution strategies for students to analyze" cards colored, printed and laminated for multiple use.
Additional Information/Instructions
By Author/Submitter
This lesson also addresses MAFS.K12.MP.2.1 Reason abstractly and quantitatively and MAFS.K12.MP.3.1 Construct viable arguments and critique the reasoning of others. This is the first of 2 lessons addressing the multiplication of fractions. The second lesson is Multiple Bake Sale Cookie Recipes with Fractional Ingredients, Resource 45577.
Source and Access Information
Contributed by:
Amy Spies
Name of Author/Source: Amy Spies
District/Organization of Contributor(s): Volusia
Is this Resource freely Available? Yes
Access Privileges: Public
* Please note that examples of resources are not intended as complete curriculum.