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NSU | Spring 2014

Internship Outcome 1Teachers show commitment to students and their learning.

The lessons that I have chosen for this outcome are included in our

fractions unit. This is an addition to the curriculum this year for our fifth

grade students. Throughout the unit students will work with percents,

fractions, compare fractions, order fractions, add and subtract fractions,

and multiply and divide fractions.

This particular investigation focuses on understanding, comparing,

and solving problems with fractions and percents. Throughout the course

of this investigation students develop ideas about representing the

meaning of fractions, decimals, and percents and the relationships among

them; comparisons and equivalents of fractions, decimals, and percents;

and the development of strategies for adding and subtracting fractions

and decimals.

The final investigation focuses multiplying and dividing fractions.

The only experience students have to this point is multiplying and dividing

whole numbers.

Students took a learning style test at the beginning of the school

year which helped me identify the best way to design instruction to meet

their needs. The results of that survey are charted on the next page.

Internship Outcome 1 Page 1

Learning Style Inventory Results

Student Auditory Score Visual Score Kinesthetic ScoreJA 7 4 7AB 4 10 4CB 7 2 9RB 3 3 12TC 6 7 5MD 5 10 3AF 4 10 4KF 8 4 6JG 3 6 9KG 5 1 12MG 14 0 4AH 5 1 12EH 5 3 10MH 6 7 5NH 2 6 10MH 8 7 3DH 3 12 3NJ 10 4 4BM 6 5 7RM 6 6 6AM 8 5 5HM 5 3 10DO 7 4 7SP 4 6 8AR 4 4 10ER 4 5 9JS 6 4 8ES 5 9 4GT 5 3 10CT 7 5 6

Total Number

of Students

9 students 8 students 17 students


The chart below lists the strengths, areas of concern, and

accommodations that each child needs in order to be successful in various

aspects of the classroom.

Student Strengths Areas for

Improvement Accommodations

Jade Strong social skills, auditory and

kinesthetic learner

Organization, reading comprehension skills,

attitude toward academics

Daily assignment book, binder to keep

assignments organized, positive

reinforcement, matching books

Ashlyn Good study skills, well-behaved, well-

mannered, visual learner

Very shy – needs to work on class participation

Positive reinforcement and be sure she

knows the answer when calling on her.

CamdenKinesthetic learner

with auditory learning being very strong.

Well-mannered, fun to be around

OrganizationBinder organization on a weekly basis, daily

assignment book

Reece Strong academic skills, kinesthetic

learnerBehavior Partner with a variety

of other students.

Trey Well-mannered and persistent.

Visual and auditory learner

IEP for reading, math, written expression,

and speech/language

Daily assignment book, binder to keep materials organized, educational assistant in classroom during language arts and

mathematics classroom instruction,

small group instruction, and extra

one-on-one help.Morgan Visual learner,

participates well in class

OrganizationDaily assignment

book, binder to keep materials organized

AbbyVisual learner Staying on task when

working with partners

Working with a variety of partners with

different strengths




Kylie Auditory learner, organized, polite,

behavior

Gaining confidence in self

Provide positive feedback when

possible

Jaydon Kinesthetic learner, strong academic

student in all areas

Talkative, maintaining focus

Provide frequent breaks or opportunities

to share ideas with peers

Kaden Academically gifted student, kinesthetic

learner

IEP for behavior – following instructions, on task, homework

completion, and accepting criticism

Boys Town/Girls Town Model – daily point

sheet, provide learning opportunities that allow him to move

about and work with others

MadysonAuditory learner Staying on task during

independent work time

Circulate throughout the room during all

work time

AshtonKinesthetic learner

Being an advocate for himself in the

classroom, homework and assignment

completion

Daily assignment book, hands on

activities to actively engage him

EstelleKinesthetic learner Off task

Verbal and visual reminders to get back

on task

Margare

t Auditory and visual learner.

Shyness, mathematics, quality

of work produced, increase intrinsic

motivation

Track students called on during class to

make sure she participates, one-on-one help for math,

peer tutor

NickKinesthetic learner

Struggles academically, homework and

assignment completion

Daily assignment book, binder to keep material organized, teacher and parent signing of planner

every night, homework folder

Maddie Auditory and visual learner, eager to share

thoughts during discussions

Staying on-task when working with peers

and during independent work time

Daily assignment book, teacher

circulation in the room to raise level of

concernDylan

Visual learnerVery pessimistic attitude, intrinsic

motivation

Model appropriate behavior daily, weekly

counseling visits




NathanAuditory learner

Reading fluency, reading

comprehension, mathematics, rushes through assignments

Daily assignment book, one-on-one

help, Reading Club

Brady Kinesthetic learner, very likable kid, loyal

IEP for reading, math, written expression,

and speech/language, homework and


Daily assignment book educational assistant in classroom during

classroom instruction, small group

instruction, social stories, social group, after school tutoring

Rachel Auditory, visual, and kinesthetic learner

Homework and assignment completion

Daily assignment book, binder to

organize materials, teacher and parent sign planner every

nightAlecia

Auditory learner Mathematics After school tutoring.

Harmon

y Kinesthetic learner

Off task behavior, homework and


Daily assignment book, binder to keep materials organized, teacher and parent

signature required in planner every night, after school tutoring,

verbal and visual reminders, preferential

seating, homework folder

Dawson Auditory and kinesthetic learner

IEP for reading and written expression,

homework and assignment completion

Daily assignment book educational assistant in classroom during

classroom instruction, small group

instruction, teacher and parent signature

required every night in planner, after school

tutoring




Skylar Kinesthetic learner, very positive outlook

on school

Working with students who she would not choose to work with

Weekly seating arrangements that allow for various

shoulder partners to share thinking with

Ava Kinesthetic learner, eager to share thoughts during

discussions

OrganizationDaily assignment

book, binder to keep material in

Eian Kinesthetic learner, excellent reader

Very sensitive, low self-esteem, math skills, gives up too

easily

Provide a great deal of positive feedback and confidence boosters,

one-on-one math help

Jaden Kinesthetic learner, willing to try new

strategies

Working with students outside of his social

circle

Provide enrichment activities for when

homework is completed

EmilyVisual learner Social skills

Provide a variety of social settings for her

to interact in

Grace Kinesthetic learner, loves mathematics, outgoing, fun to be

around

Staying on task during independent work time

and group work

Provide several opportunities for

students to share their thinking with a partner throughout instruction

Claire Auditory and kinesthetic

Social skills, emotional well-being,

organization skills

Daily assignment book, binder to keep materials in, social skills group, after school tutoring


Unit 4, Session 1.1 Lesson Plan

Activity Name: Everyday Uses of Fractions, Decimals, and Percents

Grade Level: Fifth Grade

Major Concepts: Identifying daily usage of fractions, decimals, and percents.

Materials, Resources and Technology needed for the lesson: Copy of Investigations – Unit 4 “What’s That Portion?” Chart: “Everyday Uses” Digit Cards Student Activity Book p.1-4 Student Math Handbook Copy of Family Letter SMART Board Projector SMART Notebook Software

Rationale: According to Lev Vygotsky, cooperative learning is beneficial to

students because “peers are usually operating within each other’s zones of proximal development, [and] they often provide models for each other of slightly more advanced thinking” (Slavin 2012, pg. 42). This lesson will implement cooperative learning by having students work together in pairs, groups of 4, and as a class.

Objectives: 1. Interpreting everyday uses of fractions, decimals, and percents2. Finding fractional parts of a whole or of a group (of objects, people,

and so on)3. Finding a percentage of a group (of objects, people, and so on)


Standards:

CCSS.Math.Practice.MP2 – Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

5.NF – Use equivalent fractions as a strategy to add and subtract fractions.

1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

Behavioral or Observable Objectives:As a result of this lesson, students will be able to:

- Find fractional parts of a whole or of a group- Find a percentage of a group

Assessment: Observe students as they are working on problems that relate

fractions and percents. Collect Student Activity Book page 1 and do a quick tally of the

number of correct and incorrect answers to see whether there are any questions that many students found difficult.


http://www.corestandards.org/Math/Practice/MP2

Student Activity Book p.1-2

Procedure: 1. Ten-Minute Math: Estimation and Number Sense

Use digit cards to make two 3-digit plus 1-digit addition problems. Give students 30 seconds to mentally estimate a sum as close as possible to the exact answer. Student may write down partial sums if they wish. Some students may be able to determine the exact answer. Have two or three students explain their work, and record these strategies on the board.

2. Let students know that in this unit they are going to be studying fractions, decimals, and percents and remind them that they already know a great deal about these numbers.

3. Ask students for some examples of how they have seen percents in everyday situations. Record the student suggestions on chart paper under the heading “Everyday Uses.” If the student only gives a number, ask them to give a real-world context for its use.

4. When the class has suggested something for each list, ask them to compare the uses of the different notations.

5. Write one half on the board using all three notations. (½ 0.5 50%) Discuss ways in which each form is used. For example, we may talk about 0.5 inch, or ½ inch, but probably not 50% of an inch.

6. Students will complete Student Activity Book p.1-2 and may discuss their work with a partner. The purpose of this activity is to remind students about what they already know about fractions and percents.

7. Observe students while they work to learn how they are thinking about the problems and to see whether any students will need to review basic ideas about fractions and their notation. Choose one or two problems for the discussion at the end of class.


8. When most students have completed the activity page, bring students together to go over two or three problems, sharing what they know. Emphasize how students understood and solved the problems and how they chose their answers.

9. Assign Student Activity Book page 3-4

Input: Make sure students understand that fractions, decimals, and percents are all ways of referring to parts of a whole or a group, or different ways to show amounts less than one or amounts between two whole numbers.

As students discuss the various notations it will be important to be alert to misconceptions students have and plan to address these as the unit progresses.

Check for Understanding: Do students provide a percent, a fraction, or a whole number as the

problem specifies? Can students represent a portion of a group of things (2 out of 5) as

a fraction (2/5) (Problem 1)? Can students find the missing portion of a whole (Problems 3 and

6b)? Do students understand that fractions are equal pieces of a whole?

Can they find a fractional part of a group of things? Do students know that 100% of a quantity is the whole amount and

50% is half the amount (Problem 4)? Can students compare related fractions (Problems 7a and 7b)? Can they add familiar, related fractions (Problem 7c)?

Closure: Students will need to turn to a shoulder partner and share two things that they learned about fractions, decimals, and percents.


Modifications for Special Needs or Cultural Differences:Differentiation of Instruction

Intervention Students who have difficulty understanding these problems may benefit from working together. Encourage them to make sketches to help clarify unfamiliar fractions on the Student Activity Book page. Ask them whether they remember ways that they have represented fractional parts in the past. Help them plan what they might share in the discussion at the end.

Extension Ask students who finish early to go back through the problems and write the numbers in the problems in other ways that make sense—as equivalent fraction or percents.


Proof of Student Learning

Students completed Student Activity Book p.1-2. The maximum number of points a student could earn on this assessment was 25 points.

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Lesson Reflection

Overall, the lesson went pretty well. Most students were able to

come up with a place where they had seen a decimal, fraction, or percent

used. The students that struggled are the ones that come family

backgrounds in which limited exposure to print materials is available. The

next time I teach this lesson I think I will introduce the concept a few days

ahead of time and have students look for print materials at school or at

home that contain decimals, fractions, or a percent. This may add to the

real world context of the lesson.

There were nine students that scored below a 19 (76%) on the

assessment. Several of these students are struggling to identify what the

whole is when calculating the percent of a number. Additional one-on-one

or small group help may need to be provided to these students.


Unit 4, Session 4A.1 Lesson Plan

Activity Name: Multiplying a Whole Number by a Fraction


Major Concepts: Multiplying a fraction and a whole number.

Materials, Resources and Technology needed for the lesson: Copy of Investigations – Unit 4 Common Core Guide Student Activity Book p.71-73 Fraction Bars (several per child) SMART Board Projector SMART Notebook Software SMART Document Camera

Rationale: According to Lev Vygotsky, cooperative learning is beneficial to

students because “peers are usually operating within each other’s zones of proximal development, [and] they often provide models for each other of slightly more advanced thinking” (Slavin 2012, pg. 42). This lesson will implement cooperative learning by having students work together in pairs, groups of 4, and as a class.

Objectives: 1. Using a representation to multiply a fraction and a whole number. 2. Extending understanding of the operation of multiplication to

include fractions. 3. Writing multiplication equations for multiplying a fraction and a

whole number.


Standards:

CCSS.Math.Practice.MP2 – Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

CCSS.Math.Practice.MP4 – Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5.NF.4 - 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.)

5.NF.5 - Interpret multiplication as scaling (resizing), by:a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.




5.NF.6 - Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.


- Using a representation to multiply a fraction and a whole number.

- Extend understanding of the operation of multiplication to include fractions.

- Write multiplication equations for multiplying a fraction and a whole number.

Assessment: Observe students as they are working on problems that require

them to solve multiplication problems involving fractions and whole numbers.

Student Activity Book p.73


Use digit cards to make two 5-digit plus 5-digit addition problems. Give students 30 seconds to mentally estimate a sum as close as possible to the exact answer. Student may write down partial sums if they wish. Some students may be able to determine the exact answer. Have two or three students explain their work, and record these strategies on the board.

2. Share the following information with students about The Big Bicycle Race: The race is 480 miles long.At the end of Day 1: Nora has completed 1/6 of the race.


Stuart has completed 1/10 of the race. Margaret has completed 1/8 of the race.

3. Explain to students that we’ve been adding and subtracting fractions, and now we’re going to multiply fractions. Share a story context that uses the information shared with the class.

4. Ask students to think about what they already know about comparing fractions to think about who is winning the race at the end of Day 1. Who’s in first place and who’s in last place? How do you know? Ask students to turn and talk to their shoulder partner about their responses. Share answers and reasoning with the large group.

5. Continue the discussion by asking students what would need to be done to know exactly how many miles each person had cycled. Again, have students turn and talk with their shoulder partner. Ask students to think about how this problem could be represented.

6. Distribute a copy of Fraction Bars to each student and display one on the SMART Board. Students should use the length of the fraction bar to stand for the entire length of the race or 480 miles. Again, ask students to work with their shoulder partner to show how far Norah has gone at the end of Day 1. After a minute or two ask students to share their solutions and explain their thinking. Record responses and save for the discussion later.

7. Students should work with a partner to complete Student Activity Book p.71-72.

8. Discussion: Write 1/6 of 480 miles is 80 miles on the board and gather students’ ideas on how you would write this as an equation. Record suggestions. Allow students a few minutes to explain their ideas.

9. Draw an additional fraction bar on the board and pretend that this time the race is 100 miles long. Ask the students how they would represent a student that biked two of these races. How would you write that as an equation?


10. Ask students how they would write the equation if the child’s bike broke down and the child could only bike half of the 100-mile race.

11. Tell students that when using fractions, a fraction of something indicates multiplication. Just like 2 groups of something is multiplication, ½ group of something is also multiplication.

12. Assign Student Activity Book p.73.

Input: It is important in the first sessions of this Investigation to establish and use the fraction bar as a representation. Some students may want to use a number line to solve these problems. A number line is a representation that works to find the solutions to these problems, but challenge students to also use the fraction bar. Later on in this Investigation students will be asked to represent multiplying a fraction by a fraction using the fraction bar. It is important that they have experience with the fraction bar before those sessions.

Check for Understanding: How are students solving the problems?

Are they using fractions bars to help them solve the problems?Are they dividing 480 by the denominator to determine the total miles completed?

Are students noticing the relationship between the problems? How are students dividing the fraction bar for Day 4?

Do students understand that fraction bars should be divided into equal parts? Are students using the relationship between sixths and thirds; between fourths, eighths, and sixteenths?

Closure: Students will be allowed to check their answers to their independent

work with a partner. Hand out an index card to each child a few minutes before the end of class. Ask students to write down an example of an equation with a representation that explains the equation.



Intervention Some students may not know how to approach solving these problems. Ask these students to only work on Day 1 and to use the fraction bars to help them solve the problem.

Extension For students who quickly solve all the problems, ask them to estimate the fraction of the race each participant had cycled on Day 3.

ELL Some student may still have difficulty with fraction names. Write the words sixth, tenth, and eighth on the board. Have students copy the words and then write the fraction and draw a picture for each. As students explain their reasoning, check to see that relate sixths to dividing by 6, tenths to dividing by 10, and so on.



Students completed Student Activity Book p.73. The maximum number of points a student could earn on this assessment was 10 points. Students were able to use fraction bars to illustrate the fraction of the trail that had been hiked at various points in time during the day.

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Lesson Reflection

The use of the fraction bar proved to be a very effective strategy for the vast majority of the class. Students at this age are still concrete thinkers, so this supports their stage of cognitive development. It is difficult for students at this age to think of dividing something up into parts abstractly. The use of a fraction bar gives them the visual representation they need to make sense of the mathematics. The most difficult part for students seemed to be figuring out how to get the fraction bar divided into equal sections. Once students mastered that concept the writing of the equations came pretty quickly.

There were four students that performed poorly on the assessment. Part of the problem for one of the children was that he refused to represent his work on the fraction bar. He was insistent that he could “just figure it out in his head.” Unfortunately, he thinks he is an abstract thinker, but he’s not quite there. For one of the other children, she rushed through her paper and didn’t bother to follow instructions. I will have this child go back and redo her paper over again. The other two children are having a difficult time getting the whole divided up into equal parts. I will continue to work with these two children one-on-one to help them develop the concept of using the denominator to tell you how many parts the whole is being divided up into.



Activity Name: Multiplying Whole Numbers by Fractions and Mixed Numbers


Major Concepts: Writing multiplication equations involving a fraction and a whole number.

Materials, Resources and Technology needed for the lesson: Copy of Investigations Common Core Guide Copies of Fraction Bars (several copies per student) Student Activity Book p.74-76 SMART Board Projector SMART Notebook Software SMART Document Camera

Rationale:Children at this age are in Piaget’s concrete operational stage

Student in this stage of development can form concepts, see relationships, and solve problems, but only as long as they involve objects and situations that are familiar. To assist students in their learning Fraction Bars will be provided as a concrete model for students to use in their learning.

Objectives: 1. Writing and interpreting multiplication equations involving a fraction

and a whole number.2. Using a representation and reasoning to multiply a whole number by

a fraction or mixed number.


Standards: CCSS.Math.Practice.MP1 – Make sense of problems and persevere in

solving them.Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.




5.NF.5 - Interpret multiplication as scaling (resizing), by:a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.b. Explaining why multiplying a given number by a fraction greater than 1 results




in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.



- Write multiplication equations involving a fraction and a whole number.

- Use a representation to multiply a whole number by a fraction or mixed number.

Assessment: Observe students at work and collect data on the following:

How do students solve the problems? Do they use the fraction bars? Do they use relationships to other fractions to help them? For the mixed numbers, do they multiply first by the whole number and then by the fraction and add the products together?

Observe students at work and collect data on the following: Do students correctly represent the problem in a multiplication equation?

Student Activity Book p.76 – Another Bicycle Race


Use digit cards to make two 5-digit minus 4-digit subtraction problems. Give students 30 seconds to mentally estimate differences as close as possible to the exact answer. Student may write down partial differences if they wish. Some students may be


able to determine the exact answer. Have two or three students explain their work, and record these strategies on the board.

2. Write 2/3 x 90 on the board. Discuss what this equation means. Ask for a volunteer to tell a story the equation could represent in the race context.

3. Ask students to represent the equation with a fraction bar. Students should what their representation means in the race context and find the answer. Ask one or two students to share their answer with the class.

4. Continue with another example. 1 ½ x 16 =5. Give students copies of Fraction Bars and have use them to

complete Student Activity Book p.74-75.6. Discuss students’ solutions to Problem 1 on Student Activity Book

p.74. Ask 1 or 2 students to explain their solution, including the answer on the fraction bar.

7. Assign Student Activity Book p.76 to assess student understanding of this lesson.

Input: Encourage students to draw fraction bars to help them find the

answers. Students should also be reminded that they can use what they know about multiplication and fractions as well.

Check for Understanding: Do students use the fraction bars to solve problems? Do they use relationships to other fractions to help them? For the mixed numbers, do they multiply first by the whole number

and then by the fraction and add the products together? Do students correctly represent the problem in a multiplication

equation?


Closure: As the class comes to a close, students will be asked to share one

way they used the fraction bar to help them represent and solve an equation in today’s lesson.


Intervention For students who are still working on making sense of multiplying with fractions, have them only solve the problems that involve fractions and not those with mixed numbers. If there are students who find it challenging to multiply by a non-unit fraction, ask them to use the fraction bars to solve the problem. Have them first solve a related problem with a unit fraction and then solve the actual problem.

Extension Challenge students to write their own problems that require multiplying a fraction or a mixed number by a whole number. Students can then exchange their problems with a classmate’s problems to solve and check.

ELL Some English Language Learners may have difficulty making sense of the idea of 1 ½ times the length. Talk through similar situations that use objects you have available in the room or by drawing pictures. First talk through a whole number problem, then a mixed number problem. For example, draw a tree that is 3 times as tall as a person or show a pencil that is 1 ½ times as long as another pencil.



Students completed Student Activity Book p.76. The maximum number of points a student could earn on this assessment was 10 points. Students were able to use fraction bars to illustrate the fraction of the bicycle race that had been completed.

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Lesson Reflection

The students are adapting very well to using the Fraction Bars to

represent their equations. A continued struggle for some students is using

the denominator of the fraction as the number of parts that the whole will

be divided into. An added struggle for some is that the value of the whole

needs to remain the same from one problem to the next.

There were eight students that really struggled to show their

representation appropriately on the Fraction Bar. Students seemed to be

able to write the equation, but were not able to correctly represent the

equation. These students will be pulled to work in a small group during

class in order to provide some remedial instruction.



Activity Name: Multiplying Fractions or Mixed Numbers


Major Concepts: Multiplying Fractions or Mixed Numbers with Whole Number

Materials, Resources and Technology needed for the lesson: Copy of Investigations Common Core Guide Copies of Fraction Bars (several copies per student) Student Activity Book p.74-78 SMART Board Projector SMART Notebook Software SMART Document Camera

Rationale: As Jean Piaget believed, sometimes non-formal settings help

students learn material better. To support this theory, many of the activities in this lesson will take place in a less-structured classroom math workshop setting. Also, students will be able to get a better understanding of the material because it will be presented to them in several different contexts.

Objectives: 1. Multiplying a fraction or mixed number and a whole number.2. Using a representation and reasoning to multiply a whole number by

a fraction or mixed number.


Standards: CCSS.Math.Practice.MP1 – Make sense of problems and persevere in solving.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.


5.NF.5 - Interpret multiplication as scaling (resizing), by:b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.


Behavioral or Observable Objectives:



As a result of this lesson, students will be able to:- Multiply a fraction or mixed number and a whole number.- Use a representation and reasoning to multiply a whole

number by a fraction or mixed number.

Assessment: Observe students as they are working on problems that relate to a

bicycle race or cycling and running. Student Activity Book p.77 – Cycling and Running

Procedure: 1. Ten-Minute Math: Estimation and Number Sense: Closest Estimate

Write each of the problems on the board, one at a time:1. 5 x 7/8 ~ 1 5 102. 36 x 2/5 ~ 7 15 45Give students approximately 30 seconds to look at the three possible estimates and determine which is the closest to the actual answer. Ask several students to explain how they chose an estimate, including how they thought about each of the numbers.

2. Students will be completing two different activities in a math workshop setting today. Students will continue to work on Bicycle Race Training using fraction bars or reasoning about fractions and multiplication.

3. Students will continue by completing cycling and running. In this lesson, students will continue solving multiplication problems involving fractions and mixed numbers using Student Activity Book p.77.

4. Bring the students back together to discuss problem number 5 on Student Activity Book p.77. Solicit responses from students as to how they solved the problem. How many used fraction bars? How many used another way to solve the problem?


5. Ask students to turn and talk to their shoulder partner about what they are thinking about whether or not we are really multiplying. Allow students time to share their responses with the large group.

6. Assign Student Activity Book p.78.

Input: We’ve been multiplying whole numbers by fractions and mixed numbers. When we started, some of you weren’t sure these were really multiplication problems. I’m curious what people are thinking now. What makes these multiplication problems?

Check for Understanding: Students will complete Student Activity Book p.77-78. How do students solve the problems?

Closure: This lesson will close with a discussion about the problems the

students had been solving individually. Students will share a variety of responses for solving the problems.


Intervention For students who are still working on making sense of multiplying with fractions, have them solve only the problems that involve fractions and not those with mixed numbers.

Extension Challenge students to write their own problems involving multiplying fractions, mixed numbers, and whole numbers. Students can exchange problems with a classmate to solve and check.



Students completed Student Activity Book p.77. The maximum number of points a student could earn on this assessment was 20 points. Students were able to use fraction bars to illustrate the fraction of the race cyclists had completed. Students also wrote equations for each problem.

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Lesson Reflection

This lesson proved to be somewhat of a challenge for several of the

students. This was the first time the students did not have a fraction bar

model printed on the page for them. The students that used the Fraction

Bars handout had a great deal more success than those that did not.

The children that earned the lowest scores were really having a bad

day and struggled to stay on task during math workshop time. A future

modification of this lesson would be to pull some students for small group

while the rest of the class works independently.



Activity Name: Multiplying Fractions by Fractions


Major Concepts: Multiplying a Fraction by a Fraction

Materials, Resources and Technology needed for the lesson: Copy of Investigations Common Core Guide Copies of Fraction Bars (several copies per student) Student Activity Book p.79-82 Chart Paper SMART Board Projector SMART Notebook Software SMART Document Camera

Rationale: As Erik Erikson describes in his fourth stage of psychosocial

development, children at this age are much more socially active than before. To allow for social interaction in this lesson, we will use “Think, Pair, Share” during our mathematics discussion. This gives students the opportunity to share their ideas with their classmates in an organized fashion and also creates a more student-centered classroom environment.

Objectives: 1. Multiplying a fraction by a fraction.2. Representing a fractional part of a fractional quantity.3. Understanding the relationship between the denominators of the

factors and the denominators of the product.


Standards:



CCSS.Math.Practice.MP7 - Look for and make use of structure.Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x- y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.


Behavioral or Observable Objectives:



As a result of this lesson, students will be able to:- Multiply a fraction by a fraction- Representing a fractional part of a fractional quantity

Assessment: Observe students as they are working and collect data on the

following: How do students find the fraction of ½ or 1/3 in the problem?How do students find the product?What do students notice about the denominators?

Student Activity Book p.81 – Fraction Match

Procedure: 1. Ten-Minute Math: Estimation and Number Sense: Closest Estimate

Write each of the problems on the board, one at a time:1. 5/8 x 425 ~ 50 100 2502. 230 x 7/15 ~ 115 230 345Give students approximately 30 seconds to look at the three possible estimates and determine which is the closest to the actual answer. Ask several students to explain how they chose an estimate, including how they thought about each of the numbers.

2. Draw a fraction bar divided into two halves with ½ shaded. Ask students to turn and talk to their partner and then tell you where to put a line that shows ½ of the half that is shaded. Draw a dotted vertical line to show ½ of the half and add stripes.


3. Talk through what each of the numbers refers to in the picture. Add examples on chart paper.

4. Ask students to think about why this is a multiplication problem

privately and then turn and talk to their shoulder partner. Ask students to share their responses with the class.

5. Complete a few more examples together in class using the same procedure. Have students work on Student Activity Book p.79-80. Students may choose to work independently or with a partner.

6. Bring the class back together as a large group and fill in the rest of the chart paper. Discuss various problems using the Think-Pair-Share strategy.

7. Ask students to formulate a conjecture about how to figure out what the denominator is when multiplying a fraction and a fraction. Record their conjectures on chart paper.

8. Assign Student Activity Book p.81 as an assessment for this lesson.

Input: Students need to think about what the representation of ½ of the half shows. Students should examine what part of the whole bar is striped.

Check for Understanding: How do students find the fraction of ½ or 1/3 in the problems?


How are they dividing the shaded part into the given fraction? How do students find the product? Do they divide the unshaded part into the given fractional parts and

then count all the pieces? (Problem3 on p.79) If they divided the shaded half into fifths, do they realize the

unshaded part has to be the same to have tenths? What do students notices about the denominators? Are they beginning to notice that multiplying the two denominators

of the factors gives the denominator of the product?

Closure: The closure for this lesson will be the list of conjectures that the class generates and posts on chart paper. These conjectures will remain posted for the next lesson, so students can refer to them as they encounter new problems.

Modifications for Special Needs or Cultural Differences:

Differentiation of InstructionIntervention If students have difficulty figuring out the size of the fractional

pieces that are the answers to the problems, tell them to first divide the ½ or 1/3 into the fractional pieces indicated and then continue to divide the rest of the bar into equal-size pieces. Remind students that fractions need to be equal-size pieces of a whole.

Extension If students easily solve the problems, ask them to write word problems with other contexts that involve multiplication of a fraction by a fraction to solve them.



Students completed Student Activity Book p.81. The maximum number of points a student could earn on this assessment was 15 points. Students were able to use fraction bars to illustrate a fraction times a fraction.

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Lesson Reflection

I felt as though the students were able to use the fraction bars very well in this lesson to solve a fraction times a fraction. I attribute some of this to the work we did earlier on in this unit with fractions, decimals, and percents. Interestingly enough students had an easier time dividing the fraction bars into fractional parts and then subdividing into fractional parts again. This proved to be a much easier task for the students than using the fraction bar as the whole (number of miles in the race) and dividing it into fractional parts.

Overall, student performance on the lesson assessment was very good. Part of the success can be attributed to the use of a concrete model for students to represent each part of the equation with.


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