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SDUSDFourthGradeUnit4Overview

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FOURTH GRADE Unit 4

Fractions 30 days

enVision 2.0 Topics 8-10

Overarching Understandings: The set of real numbers is infinite. Like whole numbers, fractions are real numbers. The fundamental principles of numbers (comparison, ordering, and equivalence) and operations of whole numbers (addition, subtraction, multiplication and division) also apply to fractions. A fraction describes the division of a whole (region, set, length) into equal parts. Essential Questions: • What is a fraction and how can it be represented? • How do I apply our understanding of fractions to real life? • How can I write a fraction to represent a part of a group? • How can I find equivalent fractions? • In what ways can I model equivalent fractions? • How are equivalent fractions related? • How can I compare and order fractions? • How can benchmark fractions be used to compare fractions? • How do I represent a fraction that is greater than one? • What is the relationship between a mixed number and an improper fraction? • How can I add factions with the same denominator? • Why does the denominator remain the same when I add or subtract fractions with like denominators? • How can I model multiplication of a whole number by a fraction?

Common Core State Standards: 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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 the principle to recognize and generate equivalent fractions. 4.NF.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 common denominators are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, <, and justify the conclusions, e.g., by using a visual fraction model. 4.NF.3 Understand a fractions a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of factions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify the decompositions by using a visual fraction model. c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and /or by using properties of operations and the relationship between addition and subtraction. D. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators. 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

SDUSDFourthGradeUnit4Overview

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a. Understand a fraction a/b as a multiple of 1/b. Represent 5/4 as the product of 5 x ¼. b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. c. Solve word problems involving multiplication of a fraction by a whole number using visual fraction models and equations to represent the problem. Key Vocabulary: fraction numerator denominator equivalent benchmark fraction mixed number improper fraction decompose compose

Sentence Frames: (fraction) is (numerator) pieces when the whole is equally divided into (denominator) pieces.

(fraction) is (numerator) divided by/shared by (denominator) . (1/2 is 1 divided by or shared by 2) is an equivalent fraction to because…

Suggested Materials: Fractions strips Pattern blocks Colored tiles Centimeter graph paper Centimeter dot paper Six, 10 ft. strings Paper clips Fraction cards (attached) Fraction bucket cards (attached) Fraction Strips (enVision Teaching Tool 13) Centimeter Graph Paper (enVision Teaching Tool 9) Centimeter Dot Paper (enVision Teaching Tool 11) Number Talks: Number Talks are used to build number sense, develop fluency, and make sense of problems. Problem Solving Number Strings Number Line

SDUSDFourthGradeUnit4 3

FOURTH GRADE

Unit 4 Fractions 30 days

28 lessons

1 Assessment Day 1 Re-teaching/Enrichment Day

Suggested Order of Lessons

Objective 1: Students will represent equivalent fractions and compare fractions by using benchmark fractions and equivalence. (4.NF.1, 4.NF.2) Lesson Source Title Page Number Notes 1 enVision 2.0 8-1 Equivalent Fractions Area

Models ENV TE p.411

2 SDUSD Color Tile Equivalencies

Unit p. 11

3 SDUSD Classroom Bake Sale

Unit p. 13

4 SDUSD Classroom Bake Sale

Unit p. 16

5 SDUSD Pattern Block Relationships

Unit p. 18

6 SDUSD Dot Paper Equivalencies

Unit p. 20

7 enVision 2.0 8-2 Equivalent Fractions: Number Lines

ENV TE p. 417

8 enVision 2.0 8-3 Generate Equivalent Fractions: Multiplication

ENV TE p. 423

9 SDUSD Fractions on a Number Line

Unit p. 24

10 enVision 2.0 8-5 Use Benchmarks to Compare Fractions

ENV TE p. 435

11 SDUSD Fraction Buckets

Unit p. 29

12 enVision 2.0 8-7 Construct Arguments ENV TE p. 447

Objective 2: Students will add and subtract fractions with the same denominator by using models and drawings. Students will add and subtract mixed numbers by using model and number lines. (4.NF.3.a, 4.NF.3.b, 4.NF.3.c, 4.NF.3.d)


13 enVision 2.0 9-1 Model Addition of Fractions ENV TE p. 465

14 enVision 2.0 9-2 Decompose Fractions ENV TE p. 471

15 enVision 2.0 9-3 Add Fractions with Like Denominators

ENV TE p. 477 Suggested “after” question: Why don’t you add the denominators when adding 2/5 and 1/5?

16 enVision 2.0 9-4 Model Subtraction of Fractions

ENV TE p. 483 Suggested “after” question: How does this work match the problem? (i.e. Where do you see the whole piece of plywood and where do you see the parts?)

17 enVision 2.0 9-5 Subtract Fractions With Like Denominators

ENV TE p. 489 Suggested “after” question: Why don’t you subtract the denominators when subtraction 5/10 from 8/10?

18 enVision 2.0 9-6 Add and Subtract Fractions With Like Denominators

ENV TE p. 495 Suggested “after” question: How does this work match the problem? (i.e. where do you see the charge of Sam’s phone when he started? Where do you see the part of the charge he used?)

19 enVision 2.0 9-7 Estimate Fraction Sums and Differences

ENV TE p. 501

20 enVision 2.0 9-8 Addition and Subtraction of Mixed Numbers

ENV TE p. 507

21 enVision 2.0 9-10 Subtract Mixed Numbers ENV TE p. 519 In the “after” focus on Mark’s work. Suggested question: How does Mark’s work match the problem? (i.e. Where do you see the 2 1/8 miles


Mark has to walk to get to his Aunt’s house? Where do you see the 6/8 of a mile he has already walked? Where do you see the distance he has left to walk?)

22 enVision 2.0 9-11 Model With Math

ENV TE p. 525

23 SDUSD Class Logo

Unit p. 34

24 SDUSD Class Logo Unit p. 40

Objective 3: Students will solve problems involving multiplication of a fraction and a whole number by using models and drawings (4.NF.4)

25 enVision 2.0 10-1 Fractions as Multiples of Unit Fractions: Use Models

ENV TE p. 543

26 enVision 2.0 10-2 Multiply a Fraction by a Whole Number: Use Models

ENV TE p. 549

27 enVision 2.0 10-3 Multiply a Fraction by a Whole Number: Use Symbols

ENV TE p. 555 Suggested “after” question: In the model, where do you see the number of cups needed for 8 gallons of fruit punch?

28 enVision 2.0 10-6 Model With Math

ENV TE p. 573

29 Assessment 30 Re-Teach/Enrichment Note: enVision 2.0 Lessons 8-4,8-6, 9-9, and 10-4 were not included in the suggested order of lessons due their focus on procedure versus strategy development. enVision 2.0 Lesson 10-4 was not included in the suggested order of lessons because the concept it addresses is not a focus of this unit.


SDUSD Math Lesson Map

The structure of math lessons should follow the Launch, Explore, Summarize format. This structure allows students to explore mathematical concepts with rigor (fluency, concept development, and application) to develop understanding in ways that make sense. Some rich tasks may take multiple days for students to explore. In these cases, each day should still follow the Launch, Explore, Summarize format.

EXPLORE (15–20 minutes)

The teacher provides opportunities and support for students to develop conceptual understanding by providing meaningful explorations and tasks that promote active student engagement. The teacher monitors the development of student understanding by conferring with students and asking students questions in order to understand and stimulate their thinking. The teacher uses this information to plan for the Summarize and, if needed, to call the students together for a mid-Explore scaffold to focus or propel student thinking. The students are actively engaged in constructing meaning of the mathematical concept being taught. Students engage in private reasoning time before working with partners or groups. Students use multiple representations to solve rich tasks and communicate their mathematical understanding.

INDIVIDUAL, PAIRS, OR SMALL GROUP

SUMMARIZE (15–20 minutes)

The teacher provides opportunities to make public the learning that was accomplished by the students by sharing evidence of what was learned, and providing opportunities for students to analyze, compare, discuss, extend, connect, consolidate, and record thinking strategies. A summary of the learning is articulated and connected to the purpose of the lesson. The students are actively engaged as a community of learners, discussing, justifying, and challenging various solutions to the Explore task. The students are able to articulate the learning/understanding of the mathematical concept being taught either orally or in writing. Students can engage in this discussion whether or not they have completed the task.

WHOLE GROUP

PRACTICE, REFLECT, and APPLY (10–15 minutes) This time is saved for after the Summarize so students can use what they have learned to access additional tasks. The opportunities that teachers provide are responsive to student needs. The students may have the opportunity to: revise their work, reflect on their learning, show what they know with an exit slip, extend their learning with a similar or extension problem, or practice with centers or games. The teacher confers with individual students or small groups.

INDIVIDUAL, PAIRS, OR SMALL GROUP

Number Talks 15 minutes

Number Talks are a chance

for students to come together to practice fluency and share their mathematical thinking by

engaging in conversations and discussions around

problem solving and number sense activities.

LAUNCH (5–10 minutes) The teacher sets the stage for learning by ensuring the purpose and the rationale of the lesson are clear by connecting the purpose to prior learning, posing the problem(s), and introducing the Explore task for students. During this time the teacher is identifying the tools and materials available, reviewing academic vocabulary, and setting the expectations for the lesson. The students are actively engaged in a short task or discussion to activate prior knowledge in preparation of the Explore task. Students may be using tools and/or manipulatives to make sense of the mathematical concept.

WHOLE GROUP

FO

RM

ATIVE A

SSESSMEN

T

The teacher determines w

hat students are learning and are struggling with by conferring w

ith students and by examining student w

ork throughout the lesson. This form

ative assessment inform

s ongoing adjustments in the lesson and next steps for the class and each student.

The students are actively engaged in showing their learning accom

plishments related to the m

athematical concept of the lesson.


SDUSD Mathematics Units We understand that for deep and sustainable change in mathematics to take place, teachers, students, and leaders must grapple with what the rich mathematics asked for by Common Core State Standards-Mathematics looks like in the classroom, in pedagogical practice, in student work, in curriculum, and in assessments. It is our goal that teachers and site leaders work collaboratively toward a shared vision of math instruction that develops mathematically proficient students as defined by the CCSS-Mathematics. It is our hope that these units provide a common instructional foundation for this collaboration. The SDUSD Mathematics Units are designed to support teachers and students as we shift from a more directive style of teaching mathematics toward a more inquiry-based style. In problem-based learning, students develop the habits of mind and interaction of mathematicians through engaging in mathematical discourse, connecting representations, asking genuine questions, and justifying and generalizing ideas. These mathematical habits reflect the shifts in pedagogy required to support the Common Core Standards for Mathematical Practice.

The SDUSD math units are compiled with multiple sources to ensure students have a variety of mathematical experiences aligned to the CCSS. All lessons should follow the structure of Launch, Explore, and Summarize. The following document will guide teachers in planning for daily lessons, by helping them understand the structures of each of the sources.

Structure for enVision 2.0 Lessons

Use Step 1 Develop: Problem-Based Learning is the Launch, Explore, and Summarize for every enVision 2.0 Lesson.

Launch: (Before) Start with the Solve-and-Share problem. Pose the problem to the students making sure the problem is understood. This does not mean you explain how to do the problem, rather you ensure that students understand what the problem is about. Establish clear expectations as to whether students will work individually, in pairs, or in small groups. This includes making sure students know which representations and tools they might be using or if they will have a choice of materials. Explore: (During) Students engage in solving the problem using a variety of strategies and tools. Use the suggested guiding questions to check in briefly with students as needed, in order to understand and push student thinking. You may want to use the “Extension for Early Finishers” as needed. Summarize: (After) Select student work for the class to analyze and discuss. If needed, use the Sample Student Work provided for each lesson in enVision 2.0. Practice, Reflect, Apply: (Select Problems from Workbook Pages, Reteach, Games, Intervention Activity) During this time, students may revise their work from the Explore time or you may use pieces of Step 2 Develop: Visual Learning and Step 3 Assess and Differentiate. Note: The Quick-Check component is now a few select problems that are highlighted with a pink checkmark in the Teacher’s Edition. This time provides an excellent opportunity to pull small groups of students that may need additional support.


Structure for Engage NY Lessons

Launch/Explore: (Concept Development) The Concept Development constitutes the major portion of instructional time when new learning is introduced. During this time, the lessons move through a deliberate progression on material, from concrete to pictorial to abstract. Your word choice may be slightly different from that in the vignettes, and you should use what works from the suggested talking points to meet your students’ needs. Summarize: (Student Debrief) The student debrief piece helps develop students’ metacognition by helping them make connections between parts of the lesson, concepts, strategies, and tools on their own. The goal is for students to see and hear multiple perspectives from their classmates and mentally construct a multifaceted image of the concepts being learned. Through questions that help make these connections explicit, and dialogue that directly engages students in the Standards for Mathematical Practice, they articulate those observations so the lesson’s objective becomes eminently clear to them. Practice, Reflect, Apply: (Problem Set/Exit Ticket) The Problem Set often includes fluency pertaining to the Concept Development, as well as conceptual and application word problems. The primary goal of the Problem Set is for students to apply the conceptual understandings learned during the lesson. Exit Tickets are quick assessments that contain specific questions to provide a quick glimpse of the day’s major learning. The purpose of the Exit Ticket is twofold: to teach students to grow accustomed to being individually accountable for the work they have done, and to provide you with valuable evidence of the efficacy of that day’s work which is indispensible for planning purposes. This time provides an excellent opportunity to pull small groups of students that may need additional support.


Structure for Georgia Standards Lessons The Georgia Standards tasks have been included in the units to provide students opportunities for rich, engaging, real-world mathematical experiences. These tasks allow students to develop conceptual understanding over time and may take more than one math lesson to complete. The extra time for these lessons has been allotted for in the units. When planning for a Georgia Task, it is suggested that you start by doing the mathematics the students will be engaging in before presenting it to the students. Launch: You may need to activate prior knowledge for some of the tasks that will be presented by showing images, letting students engage in partner talk about real-life situations, or using the suggested activity from the background knowledge component. Pose the task to the students making sure the task is understood. This does not mean that you explain how to do the problem, rather you ensure that students understand what the problem is about. You establish clear expectations as to whether students will work individually, in pairs, or in small groups. This includes making sure students know which representations and tools they might be using or if they will have a choice of materials. Explore: Students will engage in working on the task using a variety of strategies and tools. You may use the Essential Questions or Formative Assessment questions provided in the lesson as needed in order to understand and prompt student thinking. Summarize: Select student work for the class to analyze and discuss. Use partnerships and whole-class collaborative conversations to help students make sense of each others’ work. The Formative Assessment questions may also be used during this time to facilitate the conversation. Practice, Reflect, Apply: At this time, provide students time to reflect and revise their work from the Explore after they have engaged in the conversation in the Summarize portion of the lesson. This time provides an excellent opportunity to pull small groups of students that may need additional support.


Common Core Approach to Assessment Assessments provide ongoing opportunities for students to show their learning accomplishments in addition to offering students a pathway to monitor their progress, celebrate successes, examine mistakes, uncover misconceptions, and engage in self-reflection and analysis. A central goal of assessments is to make students aware of their strengths and weaknesses and to give them opportunities to try again, do better and, in doing so, enjoy the experience of seeing their hard work pay off as their skill and understanding increases. Furthermore, the data collected as a result of assessments represent invaluable tools in the hands of teachers and provides specific data about student understanding that can inform instructional decisions. For each Topic in enVision 2.0 the following assessments are available: In the Student Workbook: -Topic Assessment -Performance Assessment Online Teacher’s Edition: -Additional topic assessment Black-line Master -Additional performance assessment Black-line Master Online Student Assessment -Teacher can modify the number of items on an assessment -Teacher can rearrange order of problems All of the assessment items for enVision 2.0 are aligned to the types of problems students may encounter on state testing. We have found enVision 2.0 has an excessive amount of items suggested for each topic. To avoid over-assessing, we recommend that school sites work collaboratively in grade-level teams to determine how to best use all the assessment resources available to evaluate student understanding and reduce the amount of items assessed. The SDUSD math units have grouped related topics together within a unit. Sites may choose to only give an assessment at the end of each unit, consisting of items from multiple topics, rather than using multiple days to assess each topic individually.

SDUSDFourthGradeUnit4Lesson2 11

Fourth Grade Unit 4 Lesson 2

LESSON FOCUS Students will be able to explain why two fractions are equivalent even though they use different numbers.

MATERIALS Color Tiles, 1 inch graph paper LAUNCH Make color tiles in all 4 colors available to students, who will work in pairs,

small groups, or individually. Students who do not use color tiles can draw their designs on grid paper. Say: “Using the tiles, create a design that is one half blue.” Private Reasoning Time: Allow students a minute or two to create their design. As they do, circulate around the room looking for simple and creative examples to share with the class. After students complete their designs, discuss some of the differences in the class. Ask: • Did everyone use the same colors? • Does everybody’s design look the same? Why not? How can that be since half of the design had to be blue? • Did everyone use the same amount of tiles? Why or Why not? • How did you decide what you were going to do to create this pattern? • If we created another design, would you do it differently? How? You may need to repeat this activity a few times before starting the Explore section of this lesson. Before moving on, students should see that there are many different options for each design. Just because the problem calls for a fraction in fourths, doesn’t mean they need to use four tiles. They also need to understand that they may only receive part of the information needed to solve the problems; they will need to fill in the rest.

EXPLORE Students work in pairs or threes, to build designs with one-inch tiles, based on the description given on a task card. (See task cards attached.) Each student individually builds their representation for the card. Once all students in the group have finished, they discuss their designs and decide on which one they will use for their representation for the class. Once the students agree upon the design, each student will copy it onto a sheet of 1- inch graph paper. Below the picture they are to write a description and an equation of all the colors used in their design. For example: “Our design for card C has !

! yellow, !

! green and !

! red. !

! + !

! + !

! = !

! or 1

whole.”

SDUSDFourthGradeUnit4Lesson2 12

Start with Card A and work towards Card H. Most groups will not be able to finish all 8 cards in the time allotted for the lesson. Teacher note: As students are building the designs circulate around the room checking for misunderstandings. Are students using only the minimum number of tiles? Challenge them to use more. How did they make the decision to use the number of tiles they did, and why did they choose these colors?

SUMMARIZE Refocus the class by sitting in a large circle for a whole class discussion.

Share one of the group’s designs. Ask: " How could you prove that this group's designs fits the directions." or Compare two different designs. Ask, "How are these two designs similar and different?" Have students turn and talk to a partner, then facilitate a whole class discussion.

REFLECT, APPLY and EXTEND

Give the students 8 tiles and tell them that as a class you need to make a design that is !

! red, !

! green, !

! yellow and !

! blue. Ask students to describe

how they know how many tiles of the region match up to a specific fraction. Have students create their own task cards. Students should use 24 total tiles and use the denominators 2, 3, 4, 6, 8 and 12. Students need to make sure that the fractions add up to !"

!" or 1 whole. As students work, check to

make sure that they have completed the puzzle and have written fractions in simplest form.

SDUSDFourthGradeUnit4Lesson3

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LESSON FOCUS Naming different fractional pieces of the same whole. Equivalence does not necessarily mean congruence.

MATERIALS BLM of square cakes (see master at end of lesson) and 1-centimeter paper (enVision Teaching Tool 9)

LAUNCH

Begin a class discussion of scenario for this Task: Bake Sale to Raise Money to Fund the Classroom Makeover.

Be sure to clarify that a whole is the same as one whole, and one-half is the same as a half. Also, the plural of a half is halves; and one and one-half (1!! ) is equal to one whole plus one-half. Many students will see !

! and

mistakenly call it one and one-half. Be sure to connect the symbols for fractions to the written words. Ex: !

! = one-third.

EXPLORE Working in small groups, students need to determine and explain what fraction of the whole cake each piece represents. They need to write equations for at least 3 different cakes that correctly show the fractional pieces equal to 1.

SUMMARIZE Refocus the class by sitting in a large circle for a whole class discussion. Select student work for one cake in which there are disagreements or misconceptions. Post examples and ask: “I notice we have some different ideas about what fraction of the whole cake this piece represents.” “What do you notice about the thinking in these representations?” Possible confusions:

• When naming the pieces in the second cake, some students may think the large piece is 1/3 of the whole while others may think it’s ½ of the whole.

• When naming the pieces in the third cake some students may name each piece1/2. They may not be convinced that each ¼ of the same whole can be a different configuration as long as it has the same area.

Record student responses on a chart to be referred to tomorrow. An actual consensus won’t be needed until tomorrow, so you can leave students thinking.


14

Classroom Bake Sale

Students determine the fractional size of cake pieces and how much each piece costs.

Your fourth grade class is holding a bake sale to raise money for your classroom makeover. You will be selling cakes baked in a square pan. The cakes below can be cut into pieces in the following ways. Each cake is different: Chocolate, Angel Food, Red Velvet, Vanilla, Tres Leches, and Cheesecake.

Part A: Your job is to determine what fraction of the whole pan each piece represents. Explain your thinking. Write an equation to show that the sum of the fractional pieces of each pan is equal to 1. Ex: !

! + !

! + !

! + !

! + !

! = 1. You should use words, numbers, and/or pictures.


15

Classroom Bake Sale

The fourth grade is holding a bake sale to raise money for your classroom makeover. You will be selling cakes baked in a square pan. The cakes can be cut into fractional pieces as seen below. You need to determine the cost for each fractional piece of cake to sell. Each whole cake costs $10.

Part A. Label the fraction of the whole pan each part represents. Explain how you labeled the fractional pieces.

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

__________________________________________________________


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LESSON FOCUS Naming Different Fractional Pieces of the Same Whole. MATERIALS BLM of square cakes (see master at end of lesson) and 1-centimeter paper

(enVision Teaching Tool 9) LAUNCH Refer to the thinking from the previous day about what fractional part of the

whole each piece is. Come to consensus on the fractional names for each piece. Review vocabulary. Tell students that each cake sold at the bake sale costs $10. Today their challenge is to determine and explain the cost of each piece. They need to write equations for at least 3 different cakes that correctly show how the cost of the pieces of each cake equal $10.00 when added.

EXPLORE Students work in partnerships or triads to determine the cost of the pieces. Confer with students as they work. Question them to understand their thinking as well as push their thinking when necessary.


Select student work for one cake Ask students to decide if they agree or disagree with the pricing. Have them turn and talk to a partner to explain and justify their thinking. Listen in on students’ conversations. Facilitate a class discussion about the students reasoning for their pricing. Or Select student work for one cake in which there are disagreements or misconceptions about the pricing. Post examples and ask: “I notice we have some different ideas about the price of this piece.” “What do you notice about the thinking in these representations?”

Record student responses on a chart. Come to consensus on the cost for each piece of each cake.

REFLECT APPLY and EXTEND

Students create a cake selection with a value of $10 and at least 3 different kinds of cake. Students will explore the idea of whether or not their selection needs to be a square. They should explain their reasoning and create 2 correct equations: one with fractions, the other with money.


17

Classroom Bake Sale

Part B. Label the price for each fractional piece. Explain how you know the price for each fractional piece if each whole cake costs $10.

___________________________________________________________________________________________________

___________________________________________________________________________________________________

___________________________________________________________________________________________________

___________________________________________________________________________________________________

___________________________________________________________________________________________________


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LESSON FOCUS Naming Fractional Relationships in Pattern Blocks MATERIALS Pattern Blocks and recording sheet (see master at end of lesson) LAUNCH

Focus the class by sitting in a large circle for a whole class discussion. Show a green triangle pattern block and a blue rhombus pattern block on the doc cam or on a flip chart. Give students pattern blocks if you wish. Ask: What fractions could you use to describe the relationship between these two pattern blocks? Give students private reasoning time. After about a minute, have students turn and talk. Listen in to decide which ideas to bring forth. Facilitate a class discussion: Record students’ Ideas about the fractional relationships. Introduce the following frames: 1. The shape is fraction of a shape . 2. If the shape is a whole then a shape is worth fraction .

Ask students to try on each frame with a partner. Record student ideas.

EXPLORE Give students sets of pattern blocks and the recording sheet. Students work to explore and name the fractional relationships of pattern blocks when the whole changes. Confer with partnerships and groups to find out how students are making sense of the relationships. Ask questions that will support and push their thinking. Listen for ideas you may want to bring forward to the group for discussion.

SUMMARIZE Refocus the class by sitting in a large circle for a whole class discussion. Students bring their recording sheets to the group. Have students discuss their reasoning about the fractional relationships they found with a partner. Aks: What are you thinking about the fractional relationships? Explain your reasoning to your partner.” Listen in to select ideas you want to bring forward to the group. Select students to put forth an idea. Ask: “What are you thinking about this idea/relationship? Does it make sense to you? Why or why not?

Create class chart that matches the task.


19

Pattern Block Relationships

Complete the table below. Explore the relationships among the pattern blocks when the unit fractional value of the yellow hexagon is 1; the red trapezoid is 1; the blue rhombus is 1; and the green triangle is 1. Describe some patterns you noticed from the table below, and be prepared to share them with the class.

When the whole is ?, What is the value of…? = 1 = 1 = 1 = 1

Hexagon

Trapezoid

Rhombus

Triangle


20


LESSON FOCUS Develop a conceptual understanding of equivalent fractions. Consider patterns in equivalent fractions.

MATERIALS Centimeter dot paper (enVision 2.0 Teaching Tool 11), Fraction Names worksheet (see master at end of lesson)

LAUNCH Focus the class by sitting in a large circle for a whole class discussion .

On dot paper outline (and project) a 3 by 3 rectangle that has 2/3 of it shaded.

Tell this story: Two students looked at this picture. Each saw a different fraction. Nancy saw !

! , but Elijah saw !

! . Ask: How can they see the same picture but see

different fractions? Why?

Give students a minute of private think time and have them turn and talk to a partner about to explain their thinking. .

Provide dot paper to partners. Partners need to agree and have an explanation ready. They can use dot paper drawings to support their explanation.

Facilitate a class discussion and explain and justify equivalent ways to name the space that is shaded.

EXPLORE Students work individually, in partnerships or in groups.

Students find as many fraction names as possible for each shaded region on the worksheet. (BLM L-5) Provide extra dot paper for students.

For students who are having trouble, draw a fractional part for them. For example, for 1 draw a two-square rectangle. Tell them this story: This is a candy bar. How many parts does it have? What fractional name can we give for each part? Try not to give more assistance than is absolutely necessary to get students on track.

Challenge fast finishers to find more fractional names. For example, in number 2, a small triangle can be used as a unit to produce 12/24.


Create a class chart of all the fraction names that students have found for the shaded region. Record these without comment even if some are incorrect. Then have a class discussion to come to consensus around the explanations. Students may have used different shapes to construct a unit fraction. For example, in the first drawing, five squares make !

! and can be used to name the

shaded region as 4/6. Some students may have used a column of five squares and others a different arrangement of 5 squares.

• The first region can be named, !!, !!, !!"

, !"!"

, or !"!" …

• The second region can be named !! , !! , !! , !! , and !

!"…


21

• The last region also has lots of names from !! to !

!"…

• Focus attention on all the names for one region and discuss any patterns that students observe. This is a very good conceptual connection to the algorithm for finding equivalent fractions.


22

3x3 square master


23

Fraction Names

Find fraction names for each shaded region. Explain how you saw each name you found. 1.

2.

3.


24


LESSON FOCUS Place fraction cards on an open a number line. Focus on comparing, ordering and equivalence.

MATERIALS 6-10 foot pieces of string (one per group), Fraction Cards for each group (see master at end of lesson), paper clips, math notebooks, graph paper

LAUNCH Prior to beginning this lesson, set up strings around the room where students are going to work in groups of four. Have one string available for a whole group discussion. Focus the class by sitting in a large circle or semi-circle. Discuss with the class about using a number line as another way to represent a fraction.

Ask: Where do we see fractions on number lines in the real world? Examples include measuring distances, thermometers, graphs, coordinate grids. Holding up a ruler to give students the context of !

!, !!, and !

! inches

would be helpful. Distribute the whole number cards (0-4) to each group of students. Discuss how we could arrange them on our number lines. (We don’t want a discussion about leaving space between the numbers at this time; we are just looking for the order.) Have one student share how they would arrange the cards on the whole group’s string. Once students are in agreement, dismiss them to set the whole number cards on their own string, and return to the large group. Once the students have returned, discuss any differences between the groups’ number lines. Hand out the halves number cards next. Explain to the class that they are going to add these cards to their number line. Students will not have to return this time to the large group.

EXPLORE Students place the halves number cards on their number line where they belong. They can adjust the whole number cards if they need to. If a half number card has the same value (is equivalent to) as a whole number card, the students should reason about where to hang it (below the whole number card with a paper clip.) Mid Explore Check in: Once the majority of the class has set up their number line, stop everyone focus their attention by calling them to a circle or semi-circle and discuss the work they have done up to this point. Ask: • What changes to the number line did you need to make? Why? • What did you do when you had !

! as a card? Why did you place it under

the 2? • Are 2 and !

! the same number? Why?

• Is there anything else that you notice about our number line at this time? • Do you see any patterns beginning to form? Hand out the remaining number cards (fourths, eights, and mixed numbers). Each group will need to place the cards on the number line. They


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are allowed to change or modify their number line at any time to make the task easier. They cannot remove any card from the number line, only move it. As the students are completing the number line, the teacher is moving from group to group discussing patterns that students see, any problems they may be having, confirming or questioning students’ conjectures, and redirecting students that need help. Suggested questions as students work: Could you add any of the half and fourth cards as well? Where would they go?

SUMMARIZE Refocus the class by sitting in a large circle for a whole class discussion. Facilitate a class discussion about where to place given fractions on the group number line. Ask students, “Where do you think this number will go? Explain why.” Have students turn and talk to a partner about their thinking. As a group, come to consensus as a class and place the numbers in the agreed upon places. Call out sets of equivalent fractions and ask, “Why are these two numbers located in the same place on our number line. Call out a mixed number an and an improper fraction (1 ¾ and 7/4) and ask, “Why are these two numbers located in the same place on our number line?”

REFLECT, APPLY, EXTEND

Challenge students to recreate the number line in their math notebooks. They can also add any notes or findings they have made during the activity. Repeat this activity but use fraction cards that have thirds, sixths, and twelfths as denominators.


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27


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LESSON FOCUS Compare two fractions with different numerators and different denominators by comparing them to benchmarks of 𝟏

𝟐 and 1.

MATERIALS Fraction Bucket Cards (see master at end of lesson), Fraction Cards (see master at end of lesson)

LAUNCH Focus the class by sitting in a large circle for a whole class discussion. Using the “Fraction Bucket” cards, have classroom discussion about how to place the fraction bucket cards on a continuum form least to greatest. Discuss any misunderstandings about the buckets.

Model the game with 5 cards, one for each bucket, to demonstrate how to place them correctly. Using one of the cards, the teacher places it on the correct bucket card while thinking out loud to the class. “I have the card !

!, so I am

thinking that if I had four parts out of 4 total parts, I would have all the parts. So I would have a whole. I am going to place this card in the one whole bucket.” Repeat with another card.

As a class, determine where the next three cards would be placed. Discuss possible reasons why the card belongs where it is placed. Look for multiple reasons. Would it be a better fit in a different bucket? Can students support each other’s ideas? Explain that in a few moments students will be working with a partner to place many different fractions in the correct bucket.

EXPLORE Each pair/group of students receives a copy of the “Fraction Bucket” cards and a set of fraction cards. Students lay out the fraction bucket cards in the correct order. Shuffle the fraction cards and place them face down in front of themselves. Take turns flipping over a fraction card and placing it on the correct bucket. As the card is being placed on the bucket, the student must explain why they are choosing that particular bucket. If the partner agrees with the explanation, another card is flipped and the students continue. If the partner does not agree with the explanation, they get a turn to explain where they think it goes. Both students must agree on which bucket each card will be placed in. If the pair cannot agree, they can place the fraction card to the side for later. Repeat until all the cards have been placed. Circulate around the room to observe the students at work. Listen to students reasoning as they place a card. Ask student to re-explain why a card is place in a certain bucket. If there is a card that is not agreed upon, listen to both arguments, and help students find other cards that may help them make a final decision. Notice any fractions that are tricky for most students to place. These may be good to bring to the whole group in the summarize discussion. After all the fraction cards have been placed, students can check with fraction bars and ask other groups. Students can keep notes of the cards they used in


30

their notebook as a reference. Shuffle the fraction cards and repeat


Ask: Were there any cards that were difficult to decide where to place? When students bring forth a fraction that was difficult for them to place, ask the whole group: Ask: What might make this card difficult? Give students a minute to think, then have them talk to a partner. Facilitate a whole class discussion about the cards that are tricky and why they are tricky.


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Bucket Cards

Between One Half and One Whole

Less than One Half

More than One Whole

One Half

One Whole


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33


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LESSON FOCUS Students will add unit fractions with unlike denominators resulting in mixed numbers.

MATERIALS pattern blocks, class chart from Lesson 5 LAUNCH Refer to the thinking from Lesson 5 about what fractional part of the whole each

pattern block represents when the Hexagon is the whole. Discuss the scenario for this task: Class Logo. Given the value of the yellow hexagon as 1, students will complete the worksheet. Students can refer to their math journals and the chart from Lesson 5. Recording Sheet for Launch Each yellow hexagon pattern block represents one whole. The other colored pattern blocks represent a fraction of the whole yellow hexagon. You may refer to your math journal. Write the fraction of the yellow hexagon each colored shape represents. 1 yellow hexagon represents 1 yellow hexagon. 1 red trapezoid represents ____ yellow hexagon. 1 blue rhombus represents ____ yellow hexagon. 1 green triangle represents ____ yellow hexagon. Share out of the answers to the worksheet. Let students discuss disagreements about values and come to consensus. Present the Logo Task: You will create your own design for the logo and calculate its value given that the yellow hexagon is 1 whole. Include a clear concise explanation of your thinking and calculations on the recording sheet. (An additional page can be used if necessary.)

EXPLORE Students may work in partnerships, triads or small groups to design and calculate the value of their Class Logo. Recording Sheet for Explore Use the pattern block shapes to create your own logo for the class. Count all the pattern blocks you used in your design. Write the total number of each pattern block you used in your design. We used ____ yellow hexagon(s). We used ____ blue rhombus (rhombi). We used ____ red trapezoid(s). We used ____ green triangle(s).


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When one yellow hexagon is the whole, calculate the following: We used ____ red trapezoid(s). This represents ____ yellow hexagon(s). We used ____ blue rhombus (rhombi). This represents ____ yellow hexagon(s). We used____ green triangle(s). This represents ____ yellow hexagon(s). Confer with students as they work. Question them to understand their thinking as well as push their thinking when necessary.


Select one or two students Class Logos to present to the whole class. Ask students to consider the logo design and the thinking used to determine it’s value. Ask students to decide if they agree or disagree with the thinking. Give students private reasoning time. Have them turn and talk to a partner to explain and justify what they notice about the logo. Listen in on students’ conversations. Facilitate a class discussion about the students reasoning for the value of the Logo. Or Select student work for in which there are disagreements or misconceptions about the value. Facilitate a discussion about the disagreements. As time permits students do a gallery walk to see all the logos.


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Class Logo

Students are asked to design a class logo using pattern blocks.

First they need to calculate the unit fraction value of each pattern block, when the value of the yellow hexagon is 1.

Students create their own design and calculate the total value of the pattern blocks in their design using the values from the previous step.

Students are then given a challenge of creating a new design using values equal to 5!! yellow

hexagons. There must be one of each shape (pattern block: at least one yellow, one red, one blue, and one green) in the mural. Scenario: Your class has been asked to create a class logo. Your teacher has only one rule. Each design created must be made entirely of pattern blocks. Part B You will create your own design for the logo and calculate its value given that the yellow hexagon is 1 whole. You will transfer your design to a recording sheet that your teacher gives you. Include a clear concise explanation of your calculations. You may use an additional page if necessary.


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Name_________________________________Date_______________RecordingSheet-ClassLogoPartA(Launch)Eachyellowhexagonpatternblockrepresentsonewhole.Theothercoloredpatternblocksrepresentafractionofthewholeyellowhexagon.Youmayrefertoyourmathjournal.Writethefractionoftheyellowhexagoneachcoloredshaperepresents.1yellowhexagonrepresents1yellowhexagon.1redtrapezoidrepresents____yellowhexagon.1bluerhombusrepresents____yellowhexagon.1greentrianglerepresents____yellowhexagon.


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Name_____________________________________________Date____________RecordingSheet-ClassLogoPartB(Explore)Usethepatternblockshapestocreateyourownlogofortheclass.Countallthepatternblocksyouusedinyourdesign.Writethetotalnumberofeachpatternblockyouusedinyourdesign.Weused____yellowhexagon(s). Weused____bluerhombus(rhombi).Weused____redtrapezoid(s). Weused____greentriangle(s).Whenoneyellowhexagonisthewhole,calculatethefollowing:Weused____redtrapezoid(s).Thisrepresents____yellowhexagon(s).Weused____bluerhombus(rhombi).Thisrepresents____yellowhexagon(s).Weused____greentriangle(s).Thisrepresents____yellowhexagon(s).Draw/traceyourdesignhere.


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SampleStudentLogoForPartBStudentdesignswillvary.

2yellowhexagon(s) 5bluerhombus(rhombi)4redtrapezoid(s) 7greentriangle(s)---------------------------------------------------------------------------------------------------------4redtrapezoid(s)represent 𝟒

𝟐(or2)yellowhexagons.

5bluerhombus(rhombi)represent𝟓

𝟑(or1𝟐

𝟑)yellowhexagons.

7greentriangle(s)represent𝟕

𝟔(or1𝟏

𝟔)yellowhexagons.


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LESSON FOCUS Students will add unit fractions with unlike denominators resulting in mixed numbers.

MATERIALS math journals, open number line, index cards or post-it notes for mixed numbers, pattern blocks

LAUNCH Focus class by gathering in a large circle. Create a class open number line. Explain the class number line will contain the values of all the class logos that were designed yesterday. Ask: What beginning and ending values do you think will be needed on the number line. Why? Give students some private time to think. Have students partner talk about the beginning and ending values and the placement of their design values on the line.

Give students some time to begin place the values from each of the designs from yesterday on the number line. Share out students ideas. Facilitate a class discussion about any disagreements about values and come to consensus. Present today’s task: Students will create a pattern block design with a value equal to exactly 5 !

!

(given that the yellow hexagon is one whole). Students must have at least one of each pattern block in their design.

EXPLORE Students may work individually, in partnerships or in small groups. Students will create a pattern block design with a value equal to exactly 5 !

!

(given that the yellow hexagon is one whole). Students must have at least one of each pattern block in their design. Confer with students as they work. Ask what they are thinking. Note their struggles to bring forth in the summarize discussion. Then pose a question that advances their thinking. Possible extensions include: Consider the whole as 2 yellow hexagons instead of 1. Calculate the value of the design you made today. When you calculate the value of the design you made in will everyone get the same answer? Show your work and explain your thinking.


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Now consider the whole as 2 blue rhombi. What is the value of the design you made today ? Will everyone get that same answer?


Discuss difficulties students had with the constraint of having a value of exactly 5 !

! .

As time permits, do a gallery walk so students can see many different possibilities.


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SampleResponseForPartD Studentcreatesanewlogo,usingatleastoneofeachpatternblock,thatisequivalentto5!

!yellow

hexagons.Oneexampleisshownbelow;however,therearemanywaystocorrectlycompletethispart.

Studentexplainshowthepicturehasavaluethatisequalto5!

!yellowhexagons.

Example:“Iknowmypictureequals5!

!becauseittakes2trapezoidstomake1hexagon,3rhombito

make1hexagon,and6trianglestomake1hexagon.Istartedwith2hexagons.ThenIused4trapezoids,whichequals2morehexagons.Ittakes2trianglestomake1rhombus,so2triangles+2rhombi=1hexagon.Thatmakes5hexagons.Theremainingtriangleis!

!ofahexagon,soIhavethe

sameas5!!hexagonsinmypicture.

envision 2.0 fourth grade unit 4 2.0 fourth...sdusd fourth grade unit 4 overview 2 a. understand a...

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