Grade 5 Mathematics, Quarter 2, Unit 2.4

Division With Whole Numbers and Fractions Overview Number of instruction days:

8–10

(1 day = 90 minutes)

Content to Be Learned

Mathematical Practices to Be Integrated



Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).

2 Reason abstractly and quantitatively.



Divide whole numbers (in problem situations) where the quotient can be expressed as a fraction or mixed number.

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Recognize that multiplication and division are inverse operations.

4 Model with mathematics.

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Represent problems using fraction models and equations.



Use visual fraction models to represent division involving a unit fraction.



Divide a unit fraction by a whole number, using a visual fraction model to show the quotient.

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Use a model to understand the algorithm for dividing fractions.



Divide a whole number by a unit fraction, using a visual model to show the quotient.



Use the relationship between division and multiplication in division situations involving unit fractions and whole numbers.



Create a story context when dividing unit fractions and whole numbers.



Solve real-world problems involving division of unit fractions and whole numbers.

6 Attend to precision. 

Attend carefully to the underlying unit quantities when solving problems.

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Specify the whole when dividing fractions.

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Clearly communicate the solution path and strategy used.

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What strategies can you use to solve division situations involving a unit fraction and a whole number?

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When solving division situations involving a unit fraction and a whole number, how can you apply and model the relationship between division and multiplication?

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How can you create and solve real-world problems when dividing unit fractions and whole numbers?

Essential Questions 

How can a fraction be interpreted as a division expression?

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How can you use visual fraction models to show your solution path when dividing a unit fraction by a whole number?



How can you use visual fraction models to show your solution path when dividing a whole number by a unit fraction?

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Grade 5 Mathematics, Quarter 2, Unit 2.4 Version 4

Division With Whole Numbers and Fractions (8–10 days)

Standards Common Core State Standards for Mathematical Content Number and Operations—Fractions

5.NF

Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.3

Interpret a fraction as division of the numerator by the denominator (a/b = a  b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

5.NF.7

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1 1

Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.

a.

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b.

Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c.

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Common Core State Standards for Mathematical Practice 2

Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents— and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of

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Division With Whole Numbers and Fractions (8–10 days)

Grade 5 Mathematics, Quarter 2, Unit 2.4 Version 4

quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

4

Model with mathematics.

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

6

Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

Clarifying the Standards Prior Learning In Grade 4, students used visual fraction models to generate equivalent fractions of the same whole. They applied this understanding when they compared fractions with different numerators and denominators. Students added and subtracted fractions and mixed numbers with like denominators. They also multiplied a fraction by a whole number.

Current Learning In Grade 5, students begin using unlike denominators when adding and subtracting fractions. They interpret a fraction as division of the numerator by the denominator. They solve word problems involving whole numbers with the results being fractions or mixed numbers. Students extend their understanding of the concept of division to divide unit fractions and whole numbers. They use visual fraction models to show the quotient when interpreting these situations. Students use their understanding of the relationship between division and multiplication in division situations involving unit fractions and whole numbers. This is a major cluster at this grade level.

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Grade 5 Mathematics, Quarter 2, Unit 2.4 Version 4

Division With Whole Numbers and Fractions (8–10 days)

Future Learning In Grade 6, students will use the meaning of fractions, the meaning of multiplying and dividing, and their relationship to each other to explain why the procedures for dividing fractions makes sense. They extend this knowledge to the full system of positive rational numbers. They will also extend previous understanding to divide fractions by fractions. They will work in ratio and proportional reasoning. In Grade 7, students will apply and extend their knowledge of fractions to include all operations of rational numbers.

Additional Findings Using the relationship between division and multiplication, students start working with simple fraction division problems. Having seen that division of a whole number by a whole number (for example, 5 ÷ 3) is the same as multiplying the number by a unit fraction (1/3  5), they now extend the same reasoning to division of a unit fraction by a whole number. (Progressions for the Common Core State Standards in Mathematics: 3–5, Number and Operation—Fractions, p. 12) Students use the meanings of fractions, multiplication, and division and the inverse relationship between multiplication and division to make sense of the procedures for multiplying and dividing fractions and explain why they work. (Curriculum Focal Points, p. 18) Representing numbers with various physical materials should be a major part of math instruction in the elementary grades. (Principals and Standards for School Mathematics, p. 33) Part of being able to compute fluently means making smart choices about which tools to use and when. (Principals and Standards for School Mathematics, p. 35)

Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students’ attainment of the mathematics within the unit. 

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).



Represent problems using fraction models and equations.



Divide whole numbers (in problem situations) where the quotient can be expressed as a fraction or mixed number.



Divide a unit fraction by a whole number.



Divide a whole number by a unit fraction.

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Providence Public Schools

Division With Whole Numbers and Fractions (8–10 days)

Grade 5 Mathematics, Quarter 2, Unit 2.4 Version 4



Use the relationship between division and multiplication in division situations involving unit fractions and whole numbers.



Create a story context when dividing unit fractions and whole numbers.



Solve real world problems involving division of unit fractions and whole numbers.

Instruction Learning Objectives Students will be able to: 

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).



Divide whole numbers (in problem situations) where the quotient can be expressed as a fraction or mixed number.



Represent problems using fraction models and equations.



Divide a unit fraction by a whole number.



Divide a whole number by a unit fraction.



Use the relationship between division and multiplication in division situations involving unit fractions and whole numbers.



Create a story context when dividing unit fractions and whole numbers.



Solve real world problems involving division of unit fractions and whole numbers.



Demonstrate understanding of the concepts and skills learned in this unit.

Resources enVision Math Grade 5, Pearson Education, Inc., 2009 

Topic 11, Multiplying Fractions and Mixed Numbers, Teacher Edition

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Teacher Resource Masters

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Student Edition

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Also see Supplemental Insert: Lesson 11-5A Dividing Unit Fractions by Non-Zero Whole Numbers

Investigations in Numbers, Data, and Space – Grade 5, Pearson Education, Inc., 2008 

Implementing Investigations at Grade 5- Implementation Guide



Unit 4, What’s That Portion? Teacher Edition Investigation 4A, Multiplying and Dividing Fractions

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Grade 5 Mathematics, Quarter 2, Unit 2.4 Version 4



Division With Whole Numbers and Fractions (8–10 days)

See Supplemental Insert: Session 4A.8 Dividing a Whole Number by a Fraction Session 4A.9 Dividing a Fraction by a Whole Number Session 4A.10 Assessment: Dividing with Fractions

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Teacher Resource Binder

Pearson Online Success Net: www.pearsonsuccessnet.com/snpapp/login/login.jsp Implementing Investigations Site: http://investigations.terc.edu Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations.

Materials Fraction bars/strips/ tiles, centimeter grid paper

Instructional Considerations Key Vocabulary benchmark fractions

resizing

compatible numbers

scaling

Planning for Effective Instructional Design and Delivery The focus of this unit is on using strategies, models, and the relationship between division and multiplication to solve problems involving the division of a whole number by a unit fraction (e.g. 8 ÷ ½ ) and with problems involving the division of a unit fraction by a whole number (e.g. 1/6 ÷ 3). Students solve real world problems by using visual models and equations to represent the problem. They use visual representations to divide whole numbers by fractions (and vice versa), and connect the representation to its corresponding equation. To support conceptual understanding, model and represent division involving fractions in various ways using fraction strips, a number line and an array model (find the area of a rectangle with fractional side lengths by drawing on grid paper or tiling with appropriate unit fraction lengths). Always begin by identifying the whole unit and name what that unit represents in the context of the situation. It is important to note that grade 5 students do not need to divide a fraction by a fraction. This is indicated in the CCSS document (see footnote for standard 5.NF.7). Reasoning and estimation is important when making calculations with fractions. Throughout the unit, students can practice their estimation skills by engaging in the Ten-Minute Math Activity titled Estimation and Number Sense: Closest Estimate. (See Investigations Unit 4, Investigation 4A Multiplying and Dividing Fractions, Supplemental Insert, for examples. Encourage students to explain how they chose an estimate, including how they thought about each of the numbers. Encourage students to share their thinking with their peers. When solving word problems, ensure student representations and written equations correctly represent the story.

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Division With Whole Numbers and Fractions (8–10 days)

Grade 5 Mathematics, Quarter 2, Unit 2.4 Version 4

Incorporate Ten Minute Math Activities, the Problem of the Day, the Daily Spiral Review and Quick Checks that are aligned to The Common Core State Standards for Mathematics. EnVision Math Center Activities and Investigations Activities offer additional practice for student learning and support small group differentiated instruction. Use teacher created common tasks as formative assessments to monitor student progress and understanding of critical content and essential questions. Use data from formal and informal assessments to guide your instruction and planning. For planning considerations, read through the teacher editions for suggestions about scaffolding techniques, using additional examples, and differentiated instruction as suggested by the envision and Investigations resources.

Notes

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Grade 5 Mathematics, Quarter 2, Unit 2.4 Version 4

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Division With Whole Numbers and Fractions (8–10 days)

Providence Public Schools