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Module Focus: Grade 4 – Module 5 Sequence of Sessions

Overarching Objectives of this February 2014 Network Team Institute Module Focus sessions for K-5 will follow the sequence of the Concept Development component of the specified modules, using this narrative as a tool

for achieving deep understanding of mathematical concepts. Relevant examples of Fluency, Application, and Student Debrief will be highlighted in order to examine the ways in which these elements contribute to and enhance conceptual understanding.

High-Level Purpose of this Session Focus. Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for

teaching these modules. Coherence: P-5. Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that

develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same.

Standards alignment. Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.

Implementation. Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.

Related Learning Experiences● This session is part of a sequence of Module Focus sessions examining the Grade 4 curriculum, A Story of Units.

Key Points The learning of fractions follows the same instructional sequence as the learning of whole numbers. Fractional units behave just as all other units do, and can be manipulated like whole numbers. Decomposing fractions strengthens the part-whole relationship. Fraction equivalence and comparison are supported by visual models.

Session Outcomes

What do we want participants to be able to do as a result of this session?

How will we know that they are able to do this?

Focus. Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for teaching these modules.

Coherence: P-5. Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same . (Specific progression document to be determined as appropriate for each grade level and module being presented.)

Standards alignment. Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.

Implementation. Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.

Participants will be able to articulate the key points listed above.

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction to the Module

20 minsEstablish the instructional focus of Grade 4 Module 5.

Grade 4 Module 5 PPT Facilitator Guide

Review Grade 4 Module 5.

Topic A Lessons 40 mins Examine the lessons of Topic A. Grade 4 Module 5 PPT Grade 4 Module 5 Lesson

Excepts

Review Topic A.

Topic B Lessons 20 mins Examine the lessons of Topic B. Grade 4 Module 5 PPT Grade 4 Module 5 Lesson

ExceptsReview Topic B.

Topic C Lessons 40 mins Examine the lessons of Topic C. Grade 4 Module 5 PPT Grade 4 Module 5 Lesson

ExceptsReview Topic C.

Topic D Lessons 30 mins Examine the lessons of Topic D. Grade 4 Module 5 PPT Grade 4 Module 5 Lesson

ExceptsReview Topic D.

Topic E Lessons 40 mins Examine the lessons of Topic E. Grade 4 Module 5 PPT Grade 4 Module 5 Lesson

ExceptsReview Topic E.

Topic F Lessons 25 mins Examine the lessons of Topic F. Grade 4 Module 5 PPT Grade 4 Module 5 Lesson

ExceptsReview Topic F.

Topic G Lessons 20 mins Examine the lessons of Topic C. Grade 4 Module 5 PPT Grade 4 Module 5 Lesson

ExceptsReview Topic D.

Summary / Closing 20

Reiterate the key points of the lessons and faciliatate a discussion of the final exploratory lesson and end-of-module assessment.

Grade 4 Module 5 PPT Review Topic H and End-of-Module Assessment.

Session Roadmap

Section: Grade 4 Module 5 Time: 255 minutes

[255 minutes] In this section, you will… Materials used include:

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

1 min 1. Welcome! In this Module Focus Session, we will examine Grade 4 – Module 5.

1 min 2. Our objectives for this session are the following:•Examination of the development of mathematical understanding across the module with a focus on the Concept Development within the lessons•Introduction to mathematical models and instructional strategies to support implementation of A Story of Units

1 min 3. We will begin by exploring the Module Overview to understand the purpose of this module, and then we will dig into the math of the module. We’ll lead you through the teaching sequence, paying close attention to the mathematics that is taught within the Concept Developments and to the progression of the mathematics as students move through each of the lessons. We’ll examine the other lesson components as well and how they function in collaboration with the Concept Development. Finally, we’ll take a look back at the module, reflecting on all the parts as one cohesive whole.

1 min 4. The fifth module in Grade 4 is Fraction Equivalence, Ordering, and Operations. This module includes 41 lessons and is allotted 45 instructional days.

This module builds on understandings established in Module 5 of Grade 3, Fractions as Numbers on the Number Line. It also incorporates understandings of multiplication and division from Grade 3 and from Module 3 of Grade 4. This module prepares students for work in Module 6 where they will extend their learning to express decimal fractions as decimals and to add and subtract decimal fractions. Students will also build the foundational knowledge necessary for success with fractions in Grade 5.

10 min

5. Allow 8 minutes for participants to read through the Module Overview. Encourage participants to underline familiar components. Highlight what is new or unfamiliar for teachers and/or students.

This session will strive to cover many of the unfamiliar components in the module including the models and methods used to build a deeper conceptual understanding of fractions. Further information about fractions can be found in the Progressions. In order to allot more time to the models of this module, we will not be reading the Progressions.

3 min 6. Studying this document prior to teaching the first half of the module allows a teacher to see where he/she needs to take the students. Sometimes teachers can get stuck on a lesson where students may not be reaching mastery. Remember to think what lies ahead and how reaching success in each lesson, not necessarily 100% mastery, will allow teachers to stay on pace, and allow students to continue learning new concepts as they strengthen the ones from the days before. This module moves slowly and comprehensively across the fraction standards.

Take a quick glance at the mid-module assessment to see where we will be going as we study the mathematics in the 1st half of the module.

1 min 7. We will now take a look at the lessons, focusing on each Concept Development. The goal is for you to get a general sense of the module and of the progression of concepts throughout.

1 min 8. Topic A builds on student learning of unit fractions from Grade 3. Students work with non-unit fractions, decomposing them to find the unit fraction equivalence. Students express fractions using addition and multiplication. Paper folding activities work, at the concrete level, to bridge Grade 3 knowledge of work with unit fractions. Tape diagrams and the area model are used to support learning and conceptual understanding at the pictorial level. Students are able to ‘see’ the equivalence rather than simply learning the ‘tricks’.

6 min 9. Complete a paper folding activity to begin the slide:1.Decompose 1 strip into thirds.a) Draw a number bond to represent the whole decomposed into thirds.b) Ask for a number sentence. (1 = 1/3 + 1/3 + 1/3)c) Place your finger between the 2nd and 3rd units. Ask for a new number sentence. (1 = 2/3 + 1/3)2.Decompose 1 strip into sixths, shade 5 sixths. (Repeat with steps a-c.)3.Decompose 2 strips into fourths, shade 7 fourths. (Repeat with steps a-c.)

(Click to advance the slide.) Each bullet appears with a corresponding image.We begin the module with a direct reference to models and understanding from Grade 3 - Module 5 where students used paper strips and number bonds to identify and compose fractions. Now we are introducing them to the additive nature of fractions.

1 min 10. As we saw with our fraction strip representing 1 ¾, we can also decompose fractions greater than 1 whole. Moving past the concrete stage of the paper folding strips, we represent fractions pictorially as tape diagrams, which was also done in Grade 3. Here we introduce students to modeling fractions greater than 1 with tape diagrams. We also discover that there can be multiple ways to decompose a fraction. We move from the pictorial number bond to a more abstract representation, horizontal addition number sentences.

3 min 11. Lesson 2 is a continuation of Lesson 1. Here students are called upon to find the multiple decompositions of a given fraction.

(Click to advance the slide.)Debrief: Think to yourself, how is decomposing a fraction similar to the work done in Grades K and 1 with whole numbers?Turn and talk. Will anyone share?(GK-1 decompose whole numbers, first using counting pieces, then using 5 or 10 frames, and then using number bonds as they prepare to write addition and subtraction number sentences.)

2 min 12. Students can express non-unit fractions as the sum of unit fractions, which they have been doing for two lessons, but now they use multiplication. This is familiar territory for them, as:

3 bananas = 1 banana + 1 banana + 1 banana = 3 x 1 banana, or

3 tens = 1 ten + 1 ten + 1 ten = 3 x 1 ten.

Easily, students make the connection to these prior experiences and the visual of a tape diagram to see 3 fourths equals 3 times 1 fourth.

Students also use the distributive property to decompose 5 times 1 third into 2 multiplication equations, thus “pulling out one”. This concept of “pulling out one” is prevalent in this module, specifically the 2nd half of the lessons, and is connected to pulling out a ten or hundred to decompose when subtracting.

2 min 13. Just as we can decompose 1 ten into 10 ones or 1 hundred into 10 tens, we can decompose a unit fraction, such as 1 fifth, into smaller units. Again, working with the tape diagram and number bonds keeps this work visually supportive and accessible for students. Using what we learned in the first 3 lessons, the unit or non-unit fractions can be decomposed using addition or multiplication, as shown in the 2nd graphic.

4 min 14. (Click to advance the slide.) AP Problem appears.Solve this problem using the models and learning from the first 4 lessons of this module. Compare your answers with your table partners.

(Click to advance the slide.) AP sample answer appears.Here, a student drew a tape diagram to represent the loaf of bread cut into 6 slices. Each slice is decomposed into 2 equal pieces creating twelfths. The equivalence of 2/6 to 4/12 can be shown using multiplication or division or addition.

5 min 15. (Click 2 TIMES to advance the slide.)Lesson 5 is the first lesson in the module and the curriculum where the area model is introduced to decompose fractions into smaller units. The area represents 1 whole and can be decomposed vertically, as shown here, into fifths. A horizontal line further decomposes the fifths in half, creating tenths. This is a consistent model used in G4 to find equivalent fractions and is used as well as in G5-M3 to add fractions with related denominators.

(Click to advance the slide.) (Switch to document camera.)Model for participants how to show ½ = 5/10.

12 min

16. Lesson 6 is a continuation of Lesson 5. Lesson 6 focuses on decomposing non-unit fractions to find equivalent fractions.

Call attention to the importance of labeling the whole. The first two area models don’t show the whole, but, with the fractions below it, one could determine that the first image is ¾. Without the fraction labeling the area model, the first image could also represent 3/2.

Instruct participants to complete 1-7 of the Problem Set. Allow 10 minutes to do so.Encourage participants to share their work with table partners as opposed to providing an ‘answer key’. This process would then mirror what occurs in a classroom during a Debrief – students sharing work and critiquing peers (MP3 and MP6).

4 min 17. Allow 3 minutes for participants to discuss at tables.Allow 1 minute for share out.

Sample answers:•Decomposing fractions leads to better understanding of the representation of non-unit fractions and to fraction equivalence because they are able to use reasoning to explain why two different fractions can represent the same part of a whole.

1 min 18. Topic B builds on student learning from Topic A. In Topic B, students begin to generalize and to see that multiplication and division can be used to create equivalent fractions. The terms numerator and denominator are introduced to the students in Lesson 7. The learning is supported throughout Topic B by use of the area model and tape diagrams and, finally, with the number line.

1 min 19. The example shown here of student work for this Application Problem is supported by student understanding from Topic A. It allows students to see that there is more than one way to decompose a fraction. Here, each seventh is decomposed in half and in thirds. Similarly, one could decompose sevenths into fourths, fifths, tenths, or hundredths. That is not, however, realistic to show in an area model. Let’s use Topic B to discover a more efficient method using multiplication and division.

3 min 20. Lessons 7 and 8 develop the concept of finding equivalent fractions, those with smaller units, using multiplication. First, an area model is decomposed and evaluated. Then students are introduced to the concept that each unit was doubled or tripled and, thus, the units selected were doubled or tripled. When we double or triple whole numbers, we multiply. So here we multiply the numerator by 2 or 3, and we multiply the denominator by 2 or 3.The last image shows how students must reason using an area model or multiplication when a number sentence may be untrue, such as ¾ being equivalent to 6/12.

4 min 21. Although much of the module’s work is with abstract numbers, all Application Problems put the learning into a context for students. It is important to stress the real-world connections we can make with fractions. That connection is also brought out in particular Debrief questions.

Solve this Application Problem and discuss your solutions with those at your table.

(Click to advance the slide.)Solution 1 shows finding equivalent fractions using multiplication as done here in Topic B. Solution 2 refers back to both Grade 3 and Topic A of Grade 4 using number bonds as a model.Accept all reasonable answers and explanations, and probe students to explain their thinking. This encourages development of many mathematical practices.

1 min 22. Starting with non-unit fractions, students compose larger fractional units using an area model. Relating this work to decomposing fractions, they find division can be used as an abstract method to find an equivalent fraction.

1 min 23. Students consider the numerator and the denominator as a single number when composing. “What must be done to the top number, must be done to the bottom number.” Using what they learned about factors in Module 3, they can quickly determine equivalent fractions. Finding the greatest common factor of both the numerator and the denominator helps us to simplify fractions. Remember, the CCSS do not require students to simplify fractions unless it aids in finding a solution or in making sense of a problem. Here 8/12 can also be written as 2/3 or 4/6. Just as we learned in younger grades that addition and subtraction are related operations and are taught together, so must the composition and decomposition of numbers be taught in union, so as to build greater fluency in fractions.

7 min 24. Another pictorial model used to show the equivalence of fractions is the number line. The number line is introduced by drawing it directly below a tape diagram so that students may see the decomposition, or composition, is the same.

Allow participants 5 minutes to complete #s 8-10 of the Problem Set.Encourage participants to share their work with table partners as opposed to providing an ‘answer key’. This process would then mirror what occurs in a classroom during a Debrief – students sharing work and critiquing peers (MP3 and MP6).


Sample Answer:•The models allow students to clearly SEE that although equivalent fractions may “look” very different they represent the same portion of the whole.

1 min 26. The focus of Topic C is comparison of fractions. The topic begins with comparing fractions using benchmark numbers, such as 0, ½ and 1. Students then reason about the comparison of fractions using common units. Number lines and tape diagrams can be decomposed to show or plot the related denominators for comparison. Finally, students use the area model to show equivalence for denominators that are not related.

3 min 27. Present the following script from Problem 1 of Lesson 12 to engage participants.

2 min 28. Practice with the benchmarks between 0 and 1 allows students to compare larger fractions.

The first number line asks for students to recognize fractions greater than one using whole numbers as benchmarks. The second number line compares fractions with the same whole number but different fractional units.

Students can pull out the whole number from mixed numbers or improper fractions using a number bond. If the whole numbers are equal to each other, the fractional parts can be compared, referring to their Lesson 12 learning.

Here both fractions have a whole number of 1. It is easiest, therefore, to simply compare the fractional units of 3/8 and 4/6 which they learned in the previous lesson.

3 min 29. Subtraction with fractions can be complicated. Always converting to an improper fraction is not always a viable method, especially as we begin to work with larger numbers. We show a variety of methods that students can employ and apply to any numbers.

(Click to advance.)Lesson 17 begins with subtracting from 1. Easily the whole number can be renamed as a fraction, such as 5 fifths, and the students can subtract.

4 min 30. Use the following script, adapted from Problem 1 of Lesson 14, to introduce the next lesson.

T: Which is greater, 1 apple or 3 apples?S: 3 apples.T: Which is greater, 1 fourth or 3 fourths?S: 3 fourths.T: What do you notice about these

statements? (Hold up white board with comparisons.)S: The units are the same. It is easy to

compare because the units are the same.T: Which is greater, 1 fourth or 1 fifth?S: I can’t compare them. The units aren’t the

same. 1 fourth because fifths are smaller fractional units than fourths.

T: Which is greater, 2 fourths or 2 sixths?S: 2 fourths.T: What do you notice about these

statements? (Hold up white board with comparisons.)

S: The numerators are the same.

(CLICK TO ADVANCE THE SLIDE.)Here, we see a student comparing two fractions with unlike denominators, but related numerators. 1 seventh is greater than 1 twelfth, sevenths having a larger fractional size than twelfths. 5/7 is, therefore, greater than 5/12.

(CLICK TO ADVANCE THE SLIDE.)On occasion, it is possible for students to compare fractions with related numerators. Students can make related numerators into like numerators by decomposing or composing a fraction, as seen in the second image. Although this method is valuable and valid, it is not stressed and is not a major component of this Topic.

5 min 31. Comparing fractions with related denominators is the subject of the remainder of Lesson 14. A student can compare 3/5 to 7/10 easily when finding that fifths and tenths are related. Decomposing fifths into tenths can be done using a tape diagram or a number line. Once both fractions have like units, the comparison is simple. Students are armed with several methods for comparing any two fractions and practice their reasoning during the Concept Development and Debrief of the lesson.

(Switch to document camera.)With participants following along, model how to draw tape diagrams and a number line to show the comparison of 5/6 and 9/12.

After drawing ask:1.How could I reason about these fractions without drawing a model?2.Could changing 9/12 to ¾ simplify this problem?3.How could I move to the abstract level to solve without drawing a model?4.How could I back up and support students at a concrete level?

16 min

32. (Click to advance the slide.)Compare ¾ to 4/5.Can I use benchmarks? (Yes, but fourths and fifths are tricky to see.)Can I use common numerators? (Sure, but they aren’t related so I would have to decompose both fractions.)Can I use common denominators? (Sure, but they aren’t related so I would have to decompose both fractions.)A tape diagram or number line is not the place to be comparing fractions such as these with unrelated denominators or unrelated numerators. To make like units, it is easiest to use the area model. Developed further in Grade 5 to add fractions with unrelated denominators, decomposing area models to make like units for two fractions is introduced in Lesson 15.

(Click to advance the slide.)First, an area model is drawn for each fraction with vertical lines to decompose one area and horizontal lines to decompose the other. Next, students overlap each model with the other model’s lines. Fifths are drawn on the ¾ model. Fourths are drawn on the 4/5 model. Each model now is composed of twentieths.

(Click to advance the slide.)When numbers start to increase their value and we

must compare fractions greater than 1, number bonds are used to pull out the whole and parts of improper fractions, thus allowing students to see if the wholes are the same, then the parts can be compared.

Note this lesson is heavily weighted with pictorial representations to present the math. By the end of Lesson 15, students are encouraged to begin comparing fractions with unrelated denominators abstractly by using multiplication to create like units for both fractions in order to compare. A concrete representation could use paper area models that can be cut and rearranged for students to count the number of units.(Switch to document camera.)Model for participants how to compare 2/3 and ¾.

Allow participants 9 minutes to complete Problem Set #11- 17.


Sample answers:•Comparison of fractions with like numerators and denominators, and the use of a tape diagram are familiar concepts.

•Fraction comparison using benchmarks, area models, and number lines are new.

•Each method shines a different light on the skill of comparing fractions .Exposure to and practice with all of the methods will enable to students to choose the one most appropriate to use when they need to compare

fractions.

1 min 34. The focus of Topic D is addition and subtraction of fractions. Students work with fractions less than one and fractions greater than one up to two. Addition and subtraction of fractions begins with common units and then extends to related units. Students apply what they have learned to solve word problems involving the addition and subtraction of fractions.

3 min 35. (Click to advance the slide.)The language of units is imperative to stress in the younger grades so its importance can once again be strengthened and applied when working with fractions.

T: 1 banana plus 2 bananas equals?S: 3 bananas.T: 1 banana plus 2 apples equals?S: I can’t solve. The units are not like.T: How can I make like units?S: Both are fruits. 1 piece of fruit plus 2 pieces of fruit equals 3 pieces of fruit.

(Click to advance the slide.)Lesson 16 begins with relating units used in previous modules, like ones and meters, to the addition and

subtraction of fractions.A number line is used to “slide” left or right to model the addition and subtraction.The new term ‘mixed units’ is introduced in this lesson, allowing students to communicate precisely when renaming fractions is necessary or possible. For example, 9 fifths minus 3 fifths equals 6 fifths. A number bond can show us that 6 fifths can be renamed as 1 and 1 fifth by pulling out one.

1 min 36. Again, the number bond can be used to rename the final sum or difference. It is not, however, required that students do so. When applying these sums and differences to word problems, a context will be clearer if a student reports they drank 1 and ¾ cups of milk rather than 7/4 cups of milk which is an unconventional way to report the sum based on a context.

2 min 37. In this lesson, students work together to show multiple ways of adding and subtracting more than two fractions. Collaboration and discussion as well as critical analysis are all important parts of this lesson. Particular attention to making one or pulling out one is a focus as it will strengthen their fraction fluency for the 2nd half of lessons which include larger numbers.

3 min 38. In this lesson, students work together to show multiple ways of adding and subtracting more than two fractions. Collaboration and discussion as well as critical analysis are all important parts of this lesson. Particular attention to making one or pulling out one is a focus as it will strengthen their fraction fluency for the 2nd half of lessons which include larger numbers.

4 min 39. Lesson 19 is a Word Problem Lesson wherein students use the Problem Set of word problems to solve in class. As students gain fluency with fractions, various solution strategies will arise. Share different strategies, as one may be more efficient than another strategy, or a student may learn a new strategy they had yet to consider.

(Click to advance the slide.)Use Read, Draw, Write to solve. If you finish early, try finding an alternative solution strategy.

(Click to advance the slide.)Read the problem with participants.

Encourage them to first draw what they know. Hint that a bar model would be acceptable.

Allow 2 minutes to work.

(Click to advance the slide.)Show Solution 1. Discuss it is straightforward adding and renaming the sum.

(Click to advance the slide.)Shows Solution 2. Discuss it is pulling out one to make a whole and a part before finding the sum. This solution also renames to a larger unit, 3/5.

(Click to advance the slide.)Students should always write a response so as to put their answer back into context to check for a reasonable response.

3 min 40. A special note: the Grade 4 standards limit students to adding fractions with like units. However, 4.NF.5 asks students to add tenths and hundredths by converting tenths to hundredths. This work is limited mostly to work with decimal fractions and will be heavily covered in Module 6. To prepare students for this work, and to use the work of converting fractions to different units learned in previous Topics, Lessons 20 and 21 extend beyond Grade 4 expectations to add fractions with related units. It is possible to skip these lessons, not effecting the remainder of the module’s learning, if your students are struggling with Grade level concepts. However, challenge others with this work as you remediate others. This concept is not assessed. It will be revisited in Grade 5 Module 3.

(Click to advance the slide.)Again, related units would be ones such as thirds, sixths, and twelfths. Or halves, fourths, and eighths. Or halves, fourths, and twelfths.

(Click to advance the slide.)Beginning pictorially, students rename thirds as sixths, which is previous grade level work, to add like units using tape diagrams.

(Click to advance the slide.)Pictorially, this time on a number line, students decompose and rename sixths as twelfths, sliding along the number line and recording numerically.

(Click to advance the slide.)Abstractly, students can use multiplication to rename fractions, such as fifths as tenths and, again, add numerically.

12 min

41. In Lesson 21, tape diagrams, number lines, and number bonds are used to solve addition problems involving fractions with related units whose sum is greater than 1.

The image on the left shows a tape diagram model for adding fractions. The number bond is used to rename the sum as a mixed number.

The image on the right shows the number line model and, again, the number bond to assist in the renaming of the sum.

Note that subtraction with related units is not taught because subtraction of decimal fractions is not a Grade 4 standard.

Allow participants 10 minutes to complete Problem Set #18 - 24.


Sample Answers:•The use of unit language in this module enables students to relate subtraction of fractions to their work with subtracting whole numbers. Students still see a part, part, whole relationship, as with whole numbers, modeled with the tape diagram.

•Because the number bond is such a familiar model for students by Grade 4, using it with fractions is comfortable transition for them. They easily see the appropriateness of using this model to express these new “part-part-whole” relationships. It shows more connectedness of fractions and whole numbers.

1 min 43. Now we have reached the 2nd half of the module. The 1st half of the module kept all fractions within 2 - All sums, all fractions decomposed, all wholes were never greater than 2. It’s purposeful, just as in Kindergarten where students aren’t expected to decompose 86 or 147 but rather numbers 0 to 19. Now that students have spent 21 lessons building upon their Grade 3 experience with fractions, they are prepared to decompose, find equivalence, order, add, and subtract fractions with greater values. This will strengthen their understanding of fractions.

The focus of Topic E is to extend the learning from Topics A-D and to relate it to fractions greater than 1. At the end of the Topic, students apply what they have learned about fractions to solve word problems involving the addition and subtraction of fractions and

representation of the data on a line plot.

3 min 44. The opening lesson of the 2nd half of the module has students apply many concepts they are already familiar working with, but now with larger numbers. Again, the same models can be used, but as students strengthen their fluency of fraction concepts, the work becomes more and more abstract.

(Click to advance.)Students add fractions to whole numbers and subtract fractions from whole numbers using tape diagrams as models, accompanied by a number sentence.

(Click to advance.)Students are reminded, by writing fact family number sentences, of how fractions and whole numbers behave the same when adding and subtracting.

(Click to advance.)A number line is an alternative model to the tape diagram to show subtraction from a whole number.A number bond also models for students that subtracting a fraction less than one from a whole number can simply be modeled as subtraction from one and adding the remaining whole. The last image has a student thinking about subtracting 5/12 from 12/12. 9 can be decomposed as 8 and 12/12. The remaining whole is 8. The difference is 7/12. The answer is, therefore, 8 7/12.

4 min 45. Students must recall that multiplication is repeated addition. Just as 2+2+2+2+2+2=12, so does 6x2=12. Instead of adding 6 halves, we multiple 6 times 1 half. The number line allows us to quickly see how we can rename 6 halves as 3.

The associative property is then brought out to show that 6 halves can actually be renamed as 3 x (2 halves). This really helps to see that 6 times 1 half is 3.

(Click to advance.)From here, students are ready to express a whole number times a unit fraction as a mixed number.

Practice expressing 13 x 1/5 with participants.

Ask participants to brainstorm a context for this type of problem.(Takes 1/5 minute to run around his house, ran 13 times, how long?)(Making 1/5 pound burgers, 13 guests, how many pounds to buy?)

2 min 46. First, students use what they know about fraction decomposition to rename improper fractions as wholes and parts, shown with this number bond. 6 thirds is renamed as 2 ones. The number line models the addition of the wholes and parts to make 7/3, and students find that 7/3 is equal to 2 1/3 on the number line.

(Click TWICE to advance.)Next, students simplify that method in conjunction with what they learned in the previous lesson. Moving to the abstract, students convert improper fractions to mixed numbers using multiplication. This is the most efficient method when working with larger values.

5 min 47. Application Problems allow students to place this fraction work within a context. Use RDW to solve this problem.

Allow participants about 2 minutes to solve.

Share out solutions at tables or with the whole group.

Share student sample work by clicking to animate.

(Click to advance the slide.)This student drew a number line showing the 13- 1 sixth mile laps. Decomposing 13 sixths as 12 sixths and 1 sixth, the students finds 2 1/6 is equivalent to 13/6.

(Click to advance the slide.)Here a student shows how multiplication can be used to find the same solution.

(Click to advance the slide.)

Finally, a statement is written to contextualize the answer.

3 min 48. Now students work in the opposite direction. Instead of finding equivalent mixed numbers, the mixed number is given and the fraction greater than one is solved for. Students convert a mixed number to a fraction. First, they use models. Next, using mental math instead, they calculate without a model.

(Click to advance the slide.)Working with the numbers from the Application Problem, the Concept Development opens with decomposing 2 1/6 into 12/6 and 1/6. Of course, instead of the addition, one could associate the numbers and use multiplication. A number line models this work, counting by sixths to see 2 1/6 is equal to 13/6.

(Click to advance the slide.)Soon, the work becomes more abstract, associating the mixed number to find a fraction greater than one.

5 min 49. Using their knowledge of benchmark fractions and comparison and ordering of fractions from the 1st half of lessons, students apply that work with larger-valued fractions.

Work with a partner for 1 minute to solve this problem. Don’t forget to work pictorially. Models not only support the abstract work but provide reasoning for an answer.

(Click to advance the slide.) Shows the word problem.

(After working…)What are some strategies you and your partner used to solve this problem?

(Renamed all fractions as fractions greater than one.)

(Drew a number line and plotted points.)(Renamed all fractions as eighths so I could work

with like units.)(Thought about benchmarks of 0, ½, and 1.

Renamed all fractions as mixed numbers.)(Renamed 3 6/8 as 3 ¾.)

(Click to advance the slide.)Here is a student sample number line.

(Discuss the results and possible incomplete student answer.)

6 min 50. Comparing using tape diagrams for related denominators or using the area model and creating common denominators for unrelated denominators is not new to students. But when faced with larger valued fractions, mixed number or improper, students must reason about the whole and parts of each number. If the wholes are the same, the parts are compared using any method they already learned. Students may also create like denominators or like numerators using multiplication.

(Click to advance the slide.)Work with your partner to compare 7 3/5 to 7 4/6. Try to find 2 solutions.

(Allow several minutes to work. Share the below images by modeling the multiplication. Draw area models for like denominators if helpful.)

7 min 51. Students are introduced formally to line plots in Lesson 28. Here, students use the Problem Set as part of the Concept Development. Students use measurement with a ruler to precisely decompose a number line into equal-sized units. A table with data is provided and students plot points and answer questions related to the line plot within a context.



Sample Answer:•The content of this topic builds on the learning of Topics A-D by giving students the opportunities to apply the same models and strategies they used with fractions less than one to fractions greater than one. Students are successful because this opportunity to link new learning to old.

1 min 53. Topic F provides students with the opportunity to use their understanding of fraction addition and subtraction as they explore mixed number addition and subtraction by decomposition. Students first estimate the sums and differences and then work to find the exact sums and differences using a variety of different strategies. This Topic F closely mirrors Topic D from the 1st half of lessons. Again, a variety of models are used to bridge the work of the 1st half to this more complex work with larger numbers.

2 min 54. Students use number lines to roughly plot a rounded point and add on. They round fractions to benchmark numbers. They find that rounding fractions and estimating the sums is usually quite accurate.More care must, however, be taken when estimating a difference.In subtracting 4 8/9 minus 3 1/5, each fraction can be rounded to the nearest whole because each is simply one unit away from the whole and those units are relatively small.

(Click to advance.)3 ¾ minus 3 1/7, however, does not fall into the same category. Here we show on the number line the actual difference.

(Click to advance.)Since each fraction is just 1 unit from the nearest whole, if we round to the nearest whole, our estimated difference actually increases, much beyond the actual difference. By rounding one number up and another number down, the estimate is not reasonable.

(Click to advance.)Here we find that in rounding both numbers, still to benchmarks but in the same direction of the number line, the estimated difference will likely be much more accurate and reasonable compared to the actual difference. Students practice this reasoning with several carefully chosen subtraction problems to illustrate this fact.

How is this similar to whole number estimation of the difference?(Explain using this image.)

2 min 55. Just as with addition of whole numbers, there are many solution strategies in the addition of fractions. In Kindergarten and Grade 1, we work on making 5 and making 10 and then work to count on or make 10 by decomposing and addend.(Show example of how many more make 10. 7? (3.) Next, 7 + 5. Count on past 7- 8, 9, 10, 11, 12. Or decompose 5 into 3 and 2. 7 + 3 =10 and 10 +2 = 12.)

Counting in unit form, making 10, and counting on are all whole number strategies to be applied to fraction addition.

(Click to advance the slide.)Unit Counting - This is similar to G4-M1 and the addition algorithm. We add like units. The number line models and supports this.

(Click to advance the slide.)Making 10 - In fractions, we make a whole or the next whole. For example, from 3 1/8, how many more eighths make 4? (7 eighths.)

2 min 56. Continuation from previous slide’s discussion…

(Click to advance the slide.)Renaming units - 7 ones plus 5 ones is? (12 ones.) We rename as 1 ten 2 ones. Here 5 fourths is renamed as 4/4 and ¼. 4/4 is 1 so the sum is 6 ¼.

(Click to advance the slide.)Making 10 - We make 10 before adding by decomposing an addend. 7 + 5 is rather 10 + 2. So by decomposing ¾ to make a whole, we simplified the addition.

(Click to advance the slide.)Lastly, we can show counting on the “arrow way” which is a strategy used in the lower grades and one used in Module 2 with the addition of metric units to make the next unit. Here we add 2/4 to make the next whole, 6.

1 min 57. Here is where teachers may continue to see a difference in the solution strategies that students use to solve. When adding a mixed number to a mixed number, add like units: ones with ones, fourths with fourths. The length to which that is recorded is up to the mental math a student may sustain. Some may need to record a long chain of their work. Others may compute mostly in their heads. Continue to probe students for reasoning and understanding. Strategies for making one still apply here, but we encourage students to add the ones first. If needed, students may still choose to model with number lines.

1 min 58. Write on your white boards the following problem 34 – 13. Solve. (21.) Why was that so easy? (There were enough of each unit to subtract.)

(Click to advance the slide.)Are there enough fifths to subtract 3/5 from 3 4/5? (Yes.) So the subtraction here is simple. A number line can model this for us.

3 min 59. On your personal boards solve 33-14. (19.) What was a strategy used here to solve? (Rename 1 ten as 10 ones.)

(Click to advance the slide.)Are there enough fourths to subtract? (No.) Just as we did before, let’s decompose the subtrahend. 3/5 minus 3/5 is zero. Now the problem is 3-1/5 which we solved before. We also show the counting up strategy to keep in mind the relationship that subtraction and addition have.

(Click to advance the slide.)A different solution strategy used for when there are not enough units is that we can decompose the minuend, or the total, to pull out one. 3 1/5 becomes 2 1/5 plus 1. We learned how to subtract fractions from 1 in Topic D, so we have simplified this subtraction problem. Again, we can add up as well.

There are many strategies. Remember that we don’t

need each student to be fluent with them all. Encourage students to find a strategy that works. It is important that they are exposed to many strategies. This helps increase their fluency and conceptual understanding of fractions. It also helps them to see the correlation fractions and whole number addition and subtraction have.

2 min 60. Parallel to mixed number addition where we add like units, in subtraction we subtract like units. We encourage students to subtract the wholes first so as to simplify the problem. Now they can use any of the strategies learned in the previous lessons. Here are a few. Look them over and discuss them with your partner.

7 min 61. This lesson is a continuation of Lesson 33 and, again, shows another solution method. This time the focus is on the decomposition of the whole to make more units available to subtract. This aligns nicely to the subtraction algorithm as we are renaming the whole and making more of the smaller units.



Sample Answer:•After being introduced to and working with multiple strategies, students will have the freedom to use the strategy that is most comfortable, or makes the most sense, to them.

1 min 63. Topic G builds on the concept of representing a fraction as a whole number times a mixed number. Students use both the associative and distributive properties to express a representation of the multiplication. At the end of the Topic, students solve word problems and, once again, interpret a line plot and answer questions regarding the points that are plotted.

2 min 64. Students continue using the associative property to represent fractions as multiplication, building upon their Topic E experience.To show how this number sentence can associate, first students are given the example of 4 times 3 centimeters.

Write 4 x 3 centimeters = 4 x (3 centimeters) = (4 x 3) centimeters = 12 centimeters.

By showing this in unit form, students can make the connection to the numerical form, supported by the area model visual.

4 min 65. Lesson 36 further develops the concepts of Lesson 35 and places the problems into a context.

Let’s use RDW together to solve.

Solve problem with participants.

(Click to advance to show the solution.)

4 min 66. Let’s try another. This time, I’ll give you 2 minutes to get started. Then we will work together to finish the problem. If you finish early, compare your solution strategy with your partner’s.

Solve problem together, showing multiple solutions.

(Click to advance the slides to show the solution strategies.)

3 min 67. Lesson 38 advances in complexity as students are multiplying mixed numbers times whole numbers. To do so, the distributive property is used. As shown here, it is introduced by rearranging a tape diagram. The top tape shows 2 copies of 3 1/5. The bottom tape groups the whole numbers and the fractions.

(Click to advance the slide.)Next, students write out how to solve the problem. First they see 2 copies of 3. We multiply 2 times 3. There are also 2 copies of 1/5. So we multiply 2 times 1/5. The products are added together to find the final answer.

(Click to advance the slide.)As a scaffold and support of the pictorial drawings of tape diagrams, students may write the expression as repeated addition, decomposing each mixed number.

Providing a context to these abstract problems allows students to contextualize the meaning of the numerical work. They decontextualize to solve and then place their answer back into context to check for the reasonableness of their answer. As students progress through fractions and deepen their experiences, they further develop mathematical practices if allowed such opportunities. Lessons 37 and 38 offer various word problems to do just this.

4 min 68. As we have discussed today, the learning of fractions follows a similar path as the learning of whole numbers. In the previous lessons, we thought of the multiplication as copies. Remember, Patti needed 6 pieces of the same length of yarn. Now we bring it further and apply the multiplication of a whole number times a fraction as comparison using the familiar phrase from Grade 4-Module 3 “times as many as”.

Let’s solve this problem together.

Use RDW to solve showing 2 solutions.

(Click to advance the slides when finished to show the solutions.)

1 min 69. Topic G culminates with data for students to create a line plot and then answer several questions related to the line plot. Students have a chance to practice addition, subtraction, and multiplication of fractions.



Sample Answer:•Students are successful with this topic because the strategies taught allow them to apply what they’ve learned earlier. They are able to decompose the mixed number and then apply the distributive property, a concept they are quite familiar with by this point.

2 min 71. In this final lesson of the module, students explore fractions even further. First they are given cards, each card labeled with a fraction for each unit between 0 and 1. For example, 0 sixths, 1 sixth, 2 sixths…all the way to 6 sixths. They are asked to arrange the cards so that they can find the sum of all the fractions in an efficient manner. Students work together, critique each other, and solve various other similar problems.

5 min 72. Allow participants a few minutes to look at the End-of-Module Assessment. If time allows, participants may work to complete the assessment. If time does not allow, encourage a brief discussion regarding the assessment and how the work within the module prepares students for success.

1 min 73.

3 min 74. Take two minutes to turn and talk with others at your table. During this session, what information was particularly helpful and/or insightful? What new questions do you have?

Allow 2 minutes for participants to turn and talk. Bring the group to order and advance to the next slide.

2 min 75. Let’s review some key points of this session.

Use the following icons in the script to indicate different learning modes.

Video Reflect on a prompt Active learning Turn and talk

Turnkey Materials Provided

Grade 4 Module 5 PPT Grade 4 Module 5 Faciliator Guide Grade 4 Module 5 Lesson Excerpts

Additional Suggested Resources

● How to Implement A Story of Units● A Story of Units Year Long Curriculum Overview● A Story of Units CCLS Checklist

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