oriane lilley 5 grade fraction concepts, addition, and

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Oriane Lilley

5th Grade Fraction Concepts, Addition, and Subtraction

Winter 2016

Coit Creative Arts Academy

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Elements of the Lesson

Evidence that Documents the Elements

Table of Contents Assessment……………………………………………………….........4-16 Bibliography……………………………………………………..………….5 L1 Fraction Concepts and Fair Share Stories…………………………17 L2 Remainders……………………………………………………………24 L3 Addition and Subtraction of Fractions………………………………32 L4 Addition and Subtraction with Unlike Denominators………………38 Unit Reflection ……………………………………………………………45

Introduction This unit was taught with the fifth graders from a split fourth/ fifth grade classroom at Coit Creative Arts Academy. There were only seven students and most lessons were taught in the library with some lessons being taught in the classroom or conference room. The majority of students (five) were below grade level and the beginning of the unit was focused on review of fourth grade fraction concepts with the latter part of unit focusing on fifth grade fraction concepts. The included five lessons are part of a four week unit on Fraction Concepts, Addition, and Subtraction following The University of Chicago School Mathematics Project’s Everyday Mathematics Unit 3 that is mandated by the district.

Standards CCSS.MATH.CONTENT.4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. CCSS.MATH.CONTENT.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 1/2. 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. CCSS.MATH.CONTENT.5.NF.A.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. CCSS.MATH.CONTENT.5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. CCSS.MATH.CONTENT.5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed

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numbers, e.g., by using visual fraction models or equations to represent the problem.

Goals The learner will discover how to solve story problems that result in fractions. The learner will develop an understanding of fraction addition and subtraction. The learner will appreciate how fractions apply to their lives.

Lesson Objectives/ Targets Lesson One: I can explain why one fraction is larger or smaller than another. I can represent a fraction as a drawing. I can represent a number story by drawing a picture. Lesson Two: I can represent a number story by drawing a picture. I can create a number sentence out of a story problem. I can explain when it is appropriate to use a remainder. Lesson Three: I can add and subtract fractions. I can represent fraction addition and subtraction as a picture and a number sentence. Lesson Four: I can add and subtract fractions with unlike denominators. I can create equivalent fractions. I can represent fraction addition and subtraction as a picture and a number sentence.

Calendar Pre-Assessment: 2/1 Lesson One Fraction Concepts and Fair Share Stories: 2/ 8 Lesson Two Remainders: 2/17 (morning) Lesson Three Addition and Subtraction of Fractions: 2/17 (afternoon) Lesson Four Addition and Subtraction with Unlike Denominators: 2/22 Formative Assessment: 3/1 Summative Assessment: 3/3 The unit began on 2/1 and lasted until 3/4. Lessons were taught between three and five times a week depending on the school’s schedule.

Bloom’s Taxonomy/ Higher Level Thinking

Remembering: Lesson One, Lesson Three, Lesson Four Understanding: Lesson One, Lesson Two, Lesson Three Applying: Lesson One, Lesson Two, Lesson Three, Lesson Four Analyzing: Lesson One, Lesson Two, Lesson Four Evaluating: Lesson One, Lesson Two Creating: Lesson One, Lesson Four

Accommodations and Differentiations

Remediation/ intervention: Struggling students can ask a classmate for help or can raise their hand for help from the teacher. Manipulatives, drawings, and white boards will be used to help struggling students. These may be used in all lessons if necessary. Extension/ enrichment: Students that are excelling can help other students or complete any of the math boxes. These students may also be given additional, more challenging problems to solve. Learning styles: Math problems are worked through orally and visually on a whiteboard as a whole group or a small group. Most lessons have visual representations and illustrations. Most lessons have

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individual work that can also be completed with a partner.

Hands-on Learning Students will have the opportunity to use paper fraction circle pieces for support in several lessons. In all lessons students will have the opportunity to complete problems on the white board. Students will also be using individual white boards in one lesson. In all lessons students will be creating pictures that illustrate the math problems.

Subject Integration The visual arts integrated throughout the unit as students illustrate their thinking. Language Arts is also incorporated through the story problems that are used in many lessons.

Global/ Multicultural Fraction concepts will be tied into everyday life heavily during lessons one and two. Both lessons will use story problems that include fractions as an answer. Story problems will also be used during lessons three, four, and five. These story problems relate to everyday life when things need to be shared between multiple people or objects. Lessons three, four, and five deal with real life stories in situations where fractions are being added.

Technology Elmo projector, plastic/ paper manipulatives, white board, individual white boards.

Affective Domain The learner will appreciate how fractions apply to their lives. Through story problems in many lessons, students will be able to recognize and explain how fractions are used in every day life.

Classroom Setup The majority of lessons were taught in the library. There are about eight round tables with four chairs at each table. Students are allowed to sit at any of the tables, but will be separated if it is not conducive to learning. There is a projector, elmo, screen, and white board at the front of the room.

Assessment Plan Pre-Assessment: Pre-test from The University of Chicago School Mathematics Project’s Everyday Mathematics Student Journal page 70. This assessment was given on 2/1. Students were instructed to only answer or attempt to answer the questions that they knew or thought they knew how to do, and that they should not guess the answers. This eliminated the confusion between guessing and misconceptions. Formative Assessment: Students’ progress and understanding was assessed daily during lessons using white boards, independent work, group work, and individual conferences. An exit slip was given at the end of Lesson Four on 2/22. All seven students completed this. Students were allowed to ask questions and receive help. A review paper was given 3/1 and 3/2. This review was given on 3/1 when there was a substitute. Only four students received the paper (the other three were given fourth grade work or were absent). Most students did not complete the assignment and it was reviewed whole group on 3/2. Summative Assessment: 3/4 written test from The University of Chicago School Mathematics Project’s Everyday Mathematics. This assessment was given on 3/4. Students were allowed to ask many questions and receive assistance from myself. Student Self-Assessment/ Reflection: Students’ attitudes and opinions of the entire unit were collected on 4/18. A self-created questionnaire (see attached) was used. This data was used to determine

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effectiveness of the unit and to inform further instruction.

Bibliography The University of Chicago School Mathematics Project. Everyday Mathematics Assessment Handbook. 4th ed. Columbus, OH: McGraw Hill Education, 2015. Print. The University of Chicago School Mathematics Project. Everyday Mathematics Student Math Journal. 4th ed. Vol. 1. Columbus, OH: McGraw Hill Education, 2015. Print. The University of Chicago School Mathematics Project. Everyday Mathematics Teacher’s Lesson Guide. 4th ed. Vol. 1. Columbus, OH: McGraw Hill Education, 2015. Print.

Pre-Assessment Data. Collected from seven students. Students were instructed to only answer the questions that they knew or thought they knew and that they were not supposed to guess.

Question Number

Number Attempted

Number Correct

Percent Correct of Attempted

Percent Correct of all (7) Students

Overall Grade (used only for comparison)

1 5 3 60% 43% S1 12.5%

2 3 0 0% 0% S2 20%

3 6 5 83% 71% S3 37.5%

4a 2 1 50% 14% S4 50%

4b 2 1 50% 14% S5 25%

5a 6 4 67% 57% S6 62.5%

5b 5 5 100% 71% S7 0%

5c 3 2 67% 29%

5d 3 3 100% 43%

5e 4 4 100% 57%

6 1 0 0% 0%

Formative Assessment Data 1. Collected from all seven students. Assessment was given on 2/22. See lesson four.

Student Story Problem Number Sentence Completed Total

1 1 0 0 1

2 0 0 0 0

3 1 1 1 3

4 1 1 0 2

5 1 1 0 2

6 1 1 1 3

7 1 1 1 3

Formative Assessment Data 2. Collected from four students. Remaining three students were either absent or given the wrong paper by the substitute teacher. Assessment was given on 3/1. See attachment.

Question Number Number Attempted Number Correct Percent Correct of Attempted

Percent Correct of all (4) Students

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1 4 1 25% 25%

2 3 0 0% 0%

3 2 0 0% 0%

4a 2 0 0% 0%

4b 2 0 0% 0%

5a 2 1 50% 25%

5b 2 1 50% 25%

5c 2 1 50% 25%

6 4 0 0% 0%

7 4 0 0% 0%

8 3 0 0% 0%

9a 3 0 0% 0%

9b 2 0 0% 0%

Summative Assessment Data. Collected from seven students. Assessment was given on 3/4. See attachment.

Question Number Number Correct Percent Correct Final Overall Grades

1 3 43% S1 10%

2a 4 57% S2 30%

2b 4 57% S3 55%

3 5 71% S4 65%

4 3 43% S5 70%

5a 4 57% S6 77.5%

5b 3 43% S7 87.5%

6 3 43%

7 6 86%

8a 5 71%

8b 3 43%

8c 5 71%

8d 3 43%

9 4 57%

10a 3 43%

10b 2 26%

11a 3 43%

11b 1 14%

12a 5 71%

12b 5 71%

12c 5 71%

12d 4 57%

13 3 43%

14a 4 57%

14b 5 71%

Student Reflection Assessment Data. Collected from all seven students on 4/18. See Attachment.

List some of the things that we learned about in our unit on fractions:

“fractions” “rounding” (remaining did not answer)

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Positive Response

Neutral Response

Negative Response

How did you feel about the unit on fractions? 1 5 1

How hard did you try? 3 4 0

How much did you learn? 3 4 0

How much do you remember? 1 4 2

Are you excited to learn more about fractions? 2 4 1

“dividing” “adding” “adding” “decimals” “decimals” “rounding”

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5th Grade Math Unit 3 Review

Name:______________________

1. I have 20 cookies and I want to share them evenly with 3 people. How many cookies will each

person get?

Solution:______________

Number Model:_____________________________

What did you do with the remainder? Why?

________________________________________________________________________________

________________________________________________________________________________

__________________________________________________

2. Write a number story with an answer of 1/4.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

_______________________________

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3. Divide the number line so that it is in thirds. Label the thirds on the number line.

4. Rename these fractions:

a. 10/4

b. 12/3

5. Write a fraction to make each number story true.

a. ___________ + 1/4 > 1

b. _____________+1/2 > 2/3

c. 2-___________> 1

6. 1/3 + 1/6 =____________

7. 10/12 – 1/4 = ___________

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8. I bought 30 cookies and gave 1/3 of them to my mom. How many cookies did my mom get?

Answer:_____ cookies

9. What is:

a. 1/4 of 24? ___________

b. 1/2 of 18? ___________

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Name:________________

Fractions Unit Student Reflection

List some of the things that we learned about in our unit on fractions:

How did you feel about the unit on fractions?

I really liked it! It was okay. I did not like it.

How hard did you try during this unit?

I tried really hard. I tried some. I did not try very hard.

How much did you learn during this unit?

I learned a lot. I learned some. I did not learn a lot.

This is how much I remember from the unit:

A lot. Some. A little bit.

I am excited to learn more about fractions.

Yes! Maybe. Not really.

What is your favorite thing that you learned during the fractions unit?

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Lesson One

2/8

Fraction Concepts and Fair Share Stories



Standards CCSS.MATH.CONTENT.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 1/2. 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. CCSS.MATH.CONTENT.5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. CCSS.MATH.CONTENT.5.NF.B.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.

Objectives/Targets

I can explain why one fraction is larger or smaller than other. I can represent a fraction as a drawing. I can represent a number story by drawing a picture. Using number stories, the learner will be able to draw pictures to represent the elements of the story. These pictures will help the learner to visualize the story and see how elements are broken apart and fractions become the answer. Each student will be formatively assessed during the lesson either by demonstrating their understanding to the class by leading an example or will work through a problem individually with the teacher.

Lesson Management: Focus and Organization

“Thank you ___ for doing your work” “I like the way ___ is working quietly” “I’ll wait until you’re quiet” Also calling on students who get distracted easily and checking in with every student individually and often. If students are continuously off topic then they will be asked to move to another table. Free time and recess will also be taken away for inappropriate behavior.

Introduction: Creating Excitement and Focus

Warm-up problems. Three warm-up problems will be displayed using the elmo projector. See attachment. This is a daily routine and students are expected to work individually and quietly. This activity will give students practice visualizing fractions. It will also show if students are able to determine the size of a fraction relative to another fraction and if they are able to draw a picture to go along with it.

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The warm-up activity will be discussed and reviewed whole group. To engage students and create excitement, discuss how these fractions relate to their lives by using real world examples. (Would you rather have 1/2 or 1/4 of a candy bar?) During the discussion frequently call on students and have students illustrate their thinking.

Input: Setting up the Lesson Task analysis: 1. (warm-up) Students will sit at tables at the library. Have the

warm-up displayed using the elmo projector. Students should begin working independently as soon as they are seated. If they do not, remind students that they are working independently and quietly on their warmup. Allow about 5 minutes for students to complete the warm-up.

2. (warm-up review) Review the warm-up as a whole group. TT: now that most of you are done I need all attention on the front board so that we can review the warm-up together. Call on a student to complete the first problem. Call on a second student to complete the second problem. Call on another student to complete the third problem. Before calling on the student, be sure that they have the correct answer on their warm-up paper. If they do not have the correct answer then talk through the problem with them as they complete it.

3. (transition-expectations) TT: now that we’ve completed our warm-up we’re going to continue working on number stories that deal with fractions. First we’re going to review a couple problems from yesterday’s work and then you will do some independent work.

4. (modeling) Call on someone to read problem 4 from page 71. Work through this problem on the white board. Explain all steps clearly and explain your thinking. Draw a picture to explain your thinking.

5. (together) Have a student read problem 3 from page 71. Again, work on the problem on the white board, but call on multiple students to answer all of the questions. Have students draw the pictures on the white board to explain their thinking.

6. (transition) TT: are there any questions? We’re going to do some independent work. If you have any questions please raise your hand. I would like you to work on page 75 in your workbook. Before you leave, you need to have at least two problems finished and check by myself.

7. (independent work) As students work on page 75 in their workbook walk around the room and check in with all students. Students may work in pairs or small groups as long as they are being productive and helping one another. If there are multiple students struggling with the same problem then work it out together on the white board.

8. (transition) TT: Since most of you have completed the page, I would like to review problem 2 together. I’ll give you one more minute to finish up before we begin.

9. (closure) Call on a student to read problem 2 aloud. Work through the problem on the white board calling on students to

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answer all questions. Either illustrate their thinking, or have them draw their own picture. TT: What did you learn about fractions and number stories today? What are the different ways that you represent fractions and fractions stories?

10. (dismissal) TT: once you gather all of your things you may go back to the classroom. Be sure to be quiet because Mrs. Potgeter’s group may still be working.

Thinking levels: questions to engage students’ thinking

Remembering: Restate what the fraction is named. Understanding: Identify the larger/ smaller fraction. Applying: Illustrate your thinking. Analyzing: Compare the two drawings. Compare the two strategies that we used. Evaluating: Support your answer with writing or a drawing. Creating: Construct your own strategy. Devise a solution.

Accommodations: implementing differentiation principles

Remediation/ intervention: Struggling students can ask a classmate for help or can raise their hand for help from the teacher. Manipulatives, drawings, and white boards will be used to help struggling students. Extension/ enrichment: Excelling students can help other students in class. They may also work on any of the math boxes pages in their workbook. Learning styles: Information will be given and discussed with students orally and will be presented visually. Students visually represent their work. Students will be allowed to work individually or in pairs or small groups.

Methods, Materials and Integrated Technology

Instructional methods: Modeling, direct instruction, individual work, small group/ partners, investigation, group discussion Engagement strategies: Gradual release, making connections to everyday life, group work. Materials needed/ prepared: Individual math journals, warm up, warm up paper, white board, markers, plastic/paper manipulatives (if necessary) Integrated technology: Elmo projector, plastic/ paper manipulatives (if necessary), white board.

Modeling: “I DO”

Review problem 4 from page 71 in the student workbook. See attachment. Show that the number story can be solved multiple ways. Draw both ways on the whiteboard.

Checking for Understanding

Throughout the lesson, call on all students multiple times. During independent work be sure that all students are engaged and are understanding the content. Students will be formatively assessed either by working through a problem with the class or by working through a problem with the teacher during independent work time.

Guided Practice: “WE DO”

Review problem 3 from page 71 in the student workbook. See attachment. Show that the number story can be solved multiple ways. Draw both ways on the whiteboard. Call on students to answer all questions. Have students illustrate their solutions on the whiteboard.

Collaborative (“YOU DO TOGETHER”) and/or Independent Practice (“YOU

Page 75 of student workbook. See attachment. Students will work individually to complete the remaining problems. This activity will give students practice drawing and solving number

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DO”)

stories that have a fraction for an answer.

Closure After all students have completed page 75 of the workbook, review either number 2 or 3 whole group. Frequently call on individual students to answer the questions. Draw pictures on the white board to illustrate student’s thinking. If possible, show multiple ways that the problem can be solved. TT: What did you learn about fractions and number stories today?

Assessment

Page 75 of the student workbook will be used as a formative assessment. The students’ answers during the review portion will also be used as an assessment. All workbook pages will be looked at and checked for both completion and for correct answers and illustrations.

Reflection The majority of students were engaged during the lesson. There were some students off task or disengaged, but were able to be redirected and focus their efforts on the task. Many students struggled initially. Some students were given one or two simpler problems to practice before beginning the other problems. By guiding students through the questions and representing them visually, students were better able to understand the process of how to solve a story problem. One student used a method that I had not thought of. It was very useful and we shared it with the rest of class and used it to solve multiple problems. Using multiple methods and visually representing the story problems helped students to see and understand the process.

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Lesson Two

2/17

Lesson 3-3 pg. 77-78

Remainders



Standards CCSS.MATH.CONTENT.5.NF.B.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Objectives/Targets I can represent a number story by drawing a picture. I can create a number sentence out of a story problem. I can explain when it is appropriate to use a remainder. Students will use number stories to write equations and draw pictures that match the stories. Students will be able to determine when it is necessary to use a remainder, when it is necessary to round up or down, and when it is necessary to use a fraction. Drawing pictures will help students to visualize the story problem. Writing number sentences will help students to see fractions as they are normally written and will apply to later operation of fractions in lessons three and four. Each student will be formatively assessed during the lesson either by demonstrating their understanding to the class by leading an example or will work through a problem individually with the teacher.




Warm-up problems. These warm-up problems will be displayed using the elmo projector. See attachment. This is a daily routine and students are expected to work individually and quietly. The warm up problem is something that has been worked on for several prior lessons. In previous problems students have always divided up the remainder, but in this problem they are asked not to. This will help students to think about why we sometimes divide the remained and use fractions. The warm-up activity will be discussed and reviewed whole group. To engage students and create excitement, connect this story problem to past problems and to real life experiences. (How is this similar to/ different from other problems? Have you ever encountered a similar problem in your life?) Having a new aspect

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introduced to an old problem helps students to think critically at the beginning of the lesson. During the discussion frequently call on students and have students illustrate their thinking.

Input: Setting up the Lesson Task analysis: 1. (warm-up) Students will sit at tables at the library. Have the

warm-up displayed using the elmo projector. Students should begin working independently as soon as they are seated. If they do not, remind students that they are working independently and quietly on their warmup. Allow about 5 minutes for students to complete the warm-up.

2. (warm-up review) Review the warm-up as a whole group. TT: now that most of you are done I need all attention on the front board so that we can review the warm-up together. Work through the first problem calling on several students to share and explain their answers. What do you do with the extra pizza? Cut it up? Throw it away?

3. (transition-expectations) TT: now that we’ve completed our warm-up we’re going to continue working on number stories that deal with fractions and remainders. The number stories that we are going to do today are going to be very similar to the ones that we have been working on, but they are a little bit different. We’re going to focus on remainders and when we should use them.

4. (modeling) TT: please open your books to page 77. _____ please read problem 1. (Draw a picture on the white board showing 16 pancakes and five people) Since we have 16 pancakes and 5 people, each person will get 3 pancakes and we will have 1 left over, or one that is remaining. What should we do with the pancake? Throw it away? Cut it up? Give it to one person? Explain that in this situation it makes sense to cut it up and share it with everyone.

5. (together) Have another student read problem 2. Call on another student to draw the picture of 32 water bottles and 6-piece carriers. Give students a few minutes to think. Call on a student to show how they answered the problem. What did they do with the extra 2 bottles? Use a whole other carrier? Divide the bottle between the carriers? Can you divide a bottle? Leave them on the bus? What makes sense for this situation?

6. (transition) TT: are there any questions or clarifications? Do we need to do one more together? (if yes, do 19 jelly beans between 6 people as a group) I would like you to independently complete the two problems on page 78 of your math journal. Please raise your hand or ask another classmate if you have any questions.

7. (independent work) As students work on page 78 in their workbook walk around the room and check in with all students. Students may work in pairs or small groups as long as they are being productive and helping one another. If there are multiple students struggling with the same problem then

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work it out together on the white board. 8. (transition) TT: Since most of you have completed the page, I

would like to review problem 4 together. I’ll give you one more minute to finish up before we begin.

9. (closure) Call on a student to read problem 4 aloud. Work through the problem on the white board calling on students to answer all questions. Either illustrate their thinking, or have them draw their own picture. TT: So, 2 books a week is only 72 books, but 3 books a week is 108? Could he just read a couple extra books? Or could he read a portion (fraction) of a book each week? What would you do? Either answer is correct. How were the problems today similar to and different from the problems that we have previously worked on? Have you ever encountered a similar problem in your life? How does this compare to the other problems that we did today?

10. (dismissal) TT: once you gather all of your things you may go back to the classroom. Be sure to be quiet because Mrs. Potgeter’s group may still be working.


Understanding: Recognize when it is appropriate to use a remainder. Applying: Illustrate your thinking. Analyzing: Compare different word problems. Evaluating: Support why a remainder was reported.


Remediation/ intervention: Struggling students can ask a classmate for help or can raise their hand for help from the teacher. Manipulatives, drawings, and white boards will be used to help struggling students. Extension/ enrichment: Excelling students can help other students in class. They may also work on any of the math boxes pages in their workbook. Learning styles: Information will be given and discussed with students orally and will be presented visually. Students will be allowed to work individually or in pairs or small groups.


Instructional methods: Direct instruction, individual work, small group/ partners, investigation, group discussion Engagement strategies: Gradual release, making connections to everyday life, group work. Materials needed/ prepared: Individual math journals, warm up, warm up paper, white board, markers, plastic manipulatives (if necessary) Integrated technology: Elmo projector, plastic/ paper manipulatives (if necessary), white board.

Modeling: “I DO” Page 77 problem 1 of student workbook. See attachment.. Illustrate this problem on the whiteboard. Show that there is one pancake left over and explain that there are several different things that you could do with it. You could throw it away, give it all to one person, or split it up between the five people. Explain that in previous lessons we have always divided it up and although that makes sense in this example, it doesn’t always make sense for all examples.

Checking for Understanding Throughout the lesson, call on all students multiple times. During

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independent work be sure that all students are engaged and are understanding the content. Students will be formatiely assessed either by working through a problem with the class or by working through a problem with the teacher during independent work time.

Guided Practice: “WE DO”

Page 77 problem 2 of student workbook. See attachment.. Have one student read the problem aloud. After giving students a couple minutes to think about the problem, call on a student to illustrate the problem. Call on multiple other students to solve the problem together as a group. What did they do with the extra 2 bottles? Use a whole other carrier? Divide the bottle between the carriers? Can you divide a bottle? Leave them on the bus? What makes sense for this situation? Guide students towards thinking that a fraction does not make sense in this situation because you cannot split up a water bottle. You either have to use another carrier with just two bottles, or you can choose to leave those two bottle out. Either answer is acceptable.

Collaborative (“YOU DO TOGETHER”) and/or Independent Practice (“YOU DO”)

Page 78 of student workbook. See attachment. Students will work on these two problems independently or in small groups. These two problems will give students practice solving story problems that have remainders. Students will need to decide what to do with the remainder in each situation. The first problem will have a remainder left over. The remainder in the second problem will cause the quotient to be rounded up or can be reported as a fraction.

Closure Page 78 problem 4 of the student workbook. See attachment. Call on multiple students to explain and illustrate their thinking. So, 2 books a week is only 72 books, but 3 books a week is 108? Could he just read a couple extra books? Or could he read a portion (fraction) of a book each week? What would you do? Either answer is correct.

Assessment Page 78 of the student workbook will be used as a formative assessment. The students’ answers during the review portion will also be used as an assessment. All workbook pages will be looked at and checked for both completion and for correct answers and illustrations. Students need to be able to show what they did with the remainder and explain why that makes sense and why other options do not.

Reflection Students were very engaged and were able to understand the concept of remainders well. Students were able to explain when and why we treat remainders differently. The majority of students struggled with problem 4. They were unsure of rounding up because it was a new concept for them. For many problems, there were several different solutions that were acceptable and some students had a difficult time accepting that. Some students did get off task, but with redirection were able to refocus their attention. I think that this lesson would have worked better as the first lesson rather than the second. This lesson introduces students as to why we sometimes need to split the remainder up into fractions. Some students had difficulty with the first couple problems because up until this point they had always represented the remainder as a fraction and they were confused because they did not know there

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was another way.

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Lesson Three

2/17

Addition and subtraction of fractions



Standards CCSS.MATH.CONTENT.5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Objectives/Targets I can add and subtract fractions. I can represent fraction addition and subtraction as a picture and a number sentence. Students will add and subtract fractions with like denominators by drawing pictures and using paper fraction circle pieces. Students will also represent the addition and subtraction of fraction with a number sentence. Using multiple methods of expression allows students to gain a deeper understanding of the process of how and why fractions are added or subtracted from one another. Students will be formatively assessed by the answers on their white boards and each student will work through at least one problem with the teacher or demonstrate their understanding to the class by leading an example.




Warm-up problems. These warm-up problems will be displayed using the elmo projector. See attachment. This is a daily routine and students are expected to work individually and quietly. The warm-up activity will be discussed and reviewed whole group. To engage students and create excitement, ask students if they have seen or heard of these types of problems before. Ask students to create story problems to go along with the fraction drawings. During the discussion frequently call on students and have students illustrate their thinking.

Input: Setting up the Lesson Task analysis: 1. (warm up) Students will be sitting in their individual desks in

the classroom. (fourth graders will be gone on a field trip). Display the attached warm-up paper on the white board using the elmo projector. Students will be working on this individually at their seats. Remind students that they also

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need to write a number sentence for each problem. 2. (transition) TT: Since most of you are done with your warm-

up we will be reviewing it in one minute. 3. (warm- up review) review the warm- up whole group. Call on

multiple students to answer each part of the questions and to illustrate their thinking.

4. (transition-expectations) TT: today we will be working on addition and subtraction of fractions. Just like in your warm-up we will be drawing a picture and writing a number sentence for each problem that we work on today. You will be using individual white boards and I expect that those white boards will be used only for math problems and not for anything else.

5. (modeling) write one illustrated addition and one illustrated subtraction problem from the attached paper on the white board.

6. Complete these problems on the board. Show how the parts come together to make a larger part. Emphasize that the top part of the fraction gets added together, but the part always stays the same, the size of the piece does not change.

7. (together) write one illustrated addition and one illustrated subtraction problem on the board. Do these problems together and write the number sentences. Frequently call on multiple students to answer each part of the questions. Each student will be completing each problem on their individual white boards.

8. (transition) TT: we are going to start doing some independent work with adding and subtracting fractions. Each of you will complete these on your white board. When have you have completed all of the problems on the board raise your hand so that I can check your work. Do not erase your board if I have not looked at it.

9. (independent work) write four problems at a time on the board from the attached paper. Mix addition and subtraction and illustrations and number sentences. If students finish them quickly give individual students more problem on their individual board. If students are struggling pair them with someone else or work through several problems with them. Once all students have completed the problems erase them and write four new problems. If many students are struggling with the same problem review that one whole group on the large white board.

10. (transition) TT: after everyone has finished this set of problems, we will review all of them together before we leave for recess.

11. (closure) complete these problems together as a whole group. Call on multiple students to answer different parts of the questions. Do you add the top or bottom numbers? Why don’t you add the bottom numbers together? Do the pieces ever change size? Ensure that these questions are clear with

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every student. TT: Today we learned a lot about adding and subtracting fractions. How does drawing a picture help you? How does writing the number sentence help you? How are some ways that you might use this outside of school? When will you be able to add or subtract fractions?

12. (dismissal) TT: once you have cleaned up your area and have put your white board and marker away you may line up in the back of the room for lunch recess. When everyone is lined up then we will leave.


Remembering: Explain why the denominators do not get added together. Understanding: Express a drawing as a number sentence. Applying: Illustrate a number sentence with circle drawings. Apply fraction addition concepts to real-world problems and situations.




Instructional methods: Direct instruction, individual work, small group/ partners, investigation, group discussion Engagement strategies: Materials needed/ prepared: warm up, warm up paper, individual white boards, markers, plastic manipulatives (if necessary) Integrated technology: Elmo projector, plastic/ paper manipulatives (if necessary), white boards.

Modeling: “I DO” Complete one illustrated addition problem and one illustrated subtraction problem from the attached sheet on the white board. Show that the pieces can only be added or subtracted because they are the same size. (both fourths, both thirds, etc.) Emphasize that only the top numbers get added or subtracted and that the bottom number always stays the same.

Checking for Understanding Throughout the lesson, call on all students multiple times. During independent work be sure that all students are engaged and are understanding the content. Students will be formatively assessed either by working through a problem with the class or by working through a problem with the teacher during independent work time.

Guided Practice: “WE DO” Complete one illustrated addition problem and one illustrated subtraction problem from the attached sheet on the white board. Have students complete this together on their individual white boards. Write number sentences for both problems.

Collaborative (“YOU DO TOGETHER”) and/or Independent Practice (“YOU

Write three to five problems from the attached paper on the board at a time. Mix addition, subtraction, illustrations, and number sentences. Students will complete each set of problems on their

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DO”) individual white boards. Check each student’s white board for effort, correct answers, and misconceptions. If multiple students are struggling on the same problem then complete it together on the white board.

Closure Choose three problems from the attached paper. Complete these together. Call on multiple students to answer each part of the problem and to illustrate their thinking. Do you add the top or bottom numbers? Why don’t you add the bottom numbers together? Do the pieces ever change size?

Assessment Students will be formatively assessed on their white board answers? Each on answer on each student’s white board will be looked at for not only accuracy, but also for effort. Students will not only have the answer, but will also have a correct number sentence written for each problem.

Reflection Students were able to grasp this concept quickly, especially addition. The drawings were a good visual representation and students were able to see the correlation quickly. The drawings allowed students to see the pieces and to see how the pieces came together. Subtraction was a little more difficult, but students were able to understand this concept completely after practicing. Addition and subtraction with only the number sentences and not the pictures was more difficult, but drawing the pictures helped students to complete the number story portion of the practice problems. Some students were off task and drawing on the white boards. With redirection, students were able to focus on the problems and understand and complete them. Although there was some drawing, I think that using the white boards was a good idea, especially since there were so many practice problems. The boards allowed students to change their drawings and to show me their answers quickly and clearly.

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Lesson Four

2/22

Addition and Subtraction with Unlike Denominators



Standards CCSS.MATH.CONTENT.4.NF.A.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. CCSS.MATH.CONTENT.5.NF.A.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. CCSS.MATH.CONTENT.5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Objectives/Targets I can add and subtract fractions with unlike denominators. I can create equivalent fractions. I can represent fraction addition and subtraction as a picture and a number sentence. Students will be able to represent the addition and subtraction of fraction in both pictures and number sentences. The pictures will help students to visualize the fractions and how the denominator changes the fraction and the size of the pieces. The images students create will help them to see that fractions cannot be added or subtracted without first making them the same size piece (equivalent). In order to make the fraction pieces the same size, students will also learn the process of creating equivalent fractions. Each student will be formally assessed during the lesson either by demonstrating their understanding to the class by leading an example or will work through a problem individually with the teacher. Students will also be formally assessed on their exit slip completed at the end of the class period.


“Thank you ___ for doing your work” “I like the way ___ is working quietly” “I’ll wait until you’re quiet” Also calling on students who get distracted easily and checking in with every student individually and often. If students are continuously off topic then they will be asked to move to another table. Free time and recess will also be taken away for inappropriate

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behavior.


Warm-up problems. These warm-up problems will be displayed using the elmo projector. See attachment. This is a daily routine and students are expected to work individually and quietly. The warm-up activity will be discussed and reviewed whole group. During the discussion frequently call on students and have students illustrate their thinking. To engage students and create excitement, relate the size of the pieces to everyday life. Would you rather have 1/2 or 2/4 of a pizza? Why? Does it make a difference? Can you add the pieces together right away? What if they’re not the same size? What can you do to make them the same size? How did you make the pieces of the circle the same size?

Input: Setting up the Lesson Task analysis: 1. (warmup) Students will sit at tables in the library. Have the

warmup displayed using the elmo projector. Students should begin working independently as soon as they are seated. If they do not, remind students that they are working independently and quietly on their warmup. Allow about 5 minutes for students to complete the warm-up.

2. (warmup review) Review the warm-up as a whole group. TT: now that most of you are done I need all attention on the front board so that we can review the warm-up together. Have each student complete a problem. Ask the questions: Can you add them right away? What if the pieces aren’t the same size? How do you make them the same size? Can you draw it as a picture? Would you rather have 1/2 or 2/4 of a pizza? Why? Does it make a difference?

3. (transition- expectations) TT: You might have noticed that the warmup problems today were different than the problems that we have been working on. Some of the fraction pieces were not the same size. When we wrote the number sentences we could see that the denominators were different. When the denominators are different, we cannot just add up the top pieces like we do when fractions have the same denominator. Today we are going to learn how to do this with pictures and with number sentences.

4. (modeling) TT: before we try some problems together, I’m going to show you how I add fractions when the pieces aren’t the same size. (Draw the second problem (1/2 +1/4) from the warmup on the board.) Before I begin I can see that each circle has one piece shaded. But I can also see that these pieces are not the same size. The first piece is a lot bigger. So the first step is making the pieces the same size. To do this, I am going to make the first one have the same number of pieces as the second one. Right now it just has 2 pieces and I need it to have 4, so I’m just going to draw one more line to cut them in half. Now, I can see that there are four pieces and now two of them are shaded in. Now that the pieces are the same size, I can add them just like we usually

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do. I can see that I have three total pieces shaded and the pieces are part of a whole that has been cut into 4s, so my answer is 3/4. Now, we’re not done yet. Now we need to write the number sentence. Writing the number sentence helps us to see the numbers involved, not just the pictures. So, we started out with 1/2 + 1/4 = _____. The first thing we did was change the one half into a different fraction. I left a space next to it so that we could write our new fraction next to it. We know by looking at our drawing that it changes from 1/2 into 2/4. So I am going to write the new fraction of 2/4 right next to the old fraction. (Should look like: 1/2 (2/4) + 1/4 =___) But how did I get that? Without looking at the drawing how do I know that it is 2/4? I can use multiplication to get from 1/2 to 2/4. The most important part when making new fractions is that you have to always do the same thing to both the top and the bottom parts of the fractions. When you make these new fractions that are equal to each other we call them equivalent.

5. (together) TT: We’re also going to do a problem together. There are a lot of steps and a lot of things that need to happen, so be sure that you are listening and also writing everything down on your own paper. Complete the problem 1/3 + 3/6=____ together on the board. Both illustrate the problem and write the number sentence. Have students complete each step of the problem. Illustrate this or have them illustrate this on the board. Emphasize that the fraction needs to multiplied, not added and that the top and bottom need to have the same thing happen to both of them.

6. (transition) TT: I am going to leave this example up on the board while you work on journal page 95. If you have any questions or need any help raise your hand. You do not have to estimate, only complete the problems.

7. (independent work) As students work on page 95 in their workbook walk around the room and check in with all students. Students may work in pairs or small groups as long as they are being productive and helping one another. If there are multiple students struggling with the same problem then work it out together on the white board.

8. (transition) TT: since most of you are finishing up with your journal page, I will give about 3 more minutes before we complete one together and you are given your exit ticket information.

9. (closure) Complete 1/2 + 1/5= ___ together on the board. Call on students to complete all parts of the problem. TT: what were some of the most important things that you learned today about adding fractions and about adding fractions that have different denominators? How were these similar to and different from the problems that we have done before?

10. (exit ticket- formative assessment) TT: In the next ten minutes on a blank piece of paper, you need to write your own story problem that includes adding fractions with

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different denominators. You need to have a story written and a number sentence written. You also need to solve your number sentence.

11. (dismissal) TT: once I have your exit ticket and you gather all of your things you may go back to the classroom. Be sure to be quiet because Mrs. Potgeter’s group may still be working.


Remembering: Explain why the denominators of fractions need to be the same before they added. Applying: Apply the previously learned rules of fraction addition to addition of fractions with unlike denominators. Analyzing: Compare fractions with unlike denominators. Creating: Design and solve your own story problem.




Instructional methods: Direct instruction, individual work, small group/ partners, investigation, group discussion Engagement strategies: Materials needed/ prepared: Individual math journals, warm up, warm up paper, white board, markers, plastic manipulatives (if necessary) Integrated technology: Elmo projector, plastic/ paper manipulatives (if necessary), white board.

Modeling: “I DO”

Model 1/2+1/4=___ on the white board. Draw the circle picture and show how the pieces can be cutup to make the pieces same size. Also write the number sentence and show how the equivalent fraction can be made with multiplication.

Checking for Understanding Throughout the lesson, call on all students multiple times. During independent work be sure that all students are engaged and are understanding the content. Students will be formatively assessed either by working through a problem with the class or by working through a problem with the teacher during independent work time. Students will also be formatively assessed through their exit slips.

Guided Practice: “WE DO” Complete 1/3 + 3/6=____ together on the white board. Call on students to complete all parts of the problem. Show the circle drawing and how it can be cutup. Show the number sentence and how the fraction gets multiplied to create an equivalent fraction.

Collaborative (“YOU DO TOGETHER”) and/or Independent Practice (“YOU DO”)

Students will complete journal page 95. See attachment. Students will work individually or in small groups. This page will give students practice solving addition problems with fractions of unlike denominators.

Closure After all or most students have completed the journal page complete

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1/2 + 1/5= ___ on the board together. Call on multiple students to answer questions. Students will also complete an exit slip before leaving.

Assessment Students will be formatively assessed through their journal page. Students will also be formatively assessed on their exit slip. Three components of the exit slip will be considering for accuracy. Students will need to write the number story, write the number sentence, and solve the number sentence.

Reflection Adding fractions with unlike denominators and creating equivalent fractions is a difficult concept to explain and even more difficult to write. My students were able to grasp it although not quite as quickly or easily as I would have hoped. By the end of the lesson, they were able to complete the journal pages efficiently. Many students asked questions. All students were able to explain why the denominators had to be the same before the fractions were added.

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Unit Reflection

Overall, I think that my unit went well. I think that as a whole my students understand the concept of

fractions, how fractions are made, what they mean, and how they are added together. Based on

assessment data, students are able to explain the reasoning behind their work, which is very

important because they are able to use that in many other situations. I also think that my students

have a good understanding of how fractions are used in their everyday lives.

However, I think that there is always room for improvement. I don’t think that the students had a good understanding of how the different concepts came together and how they were able to relate to one

another. Based on assessment data, students do not understand adding or subtracting fractions in

context.

I learned a lot through teaching this unit and I have already been applying this knowledge to the math

unit that I am teaching now. I am more conscious of relating the topics to each other and building one

concept on another. I am also more conscious of challenging my high achieving students. Finding

challenging pages for them before class to continue to build their skills is more effective than having

them help another student or coming up with additional problems on the spot for them. I am also

more conscious of how much students are involved. I was able to notice that students scored better

on concepts that were taught in a more involved lesson.