Draft Unit Plan

5.NF.3-7: Apply and Extend Previous Understanding of Multiplication and Division to Multiply and Divide Fractions

Overview:The overview statement is intended to provide a summary of major themes in this unit.

This unit extends knowledge of multiplication and division from whole numbers and decimals into fractions. Modeling plays a significant role in developing students’ understanding of algorithms for multiplication and division of fractions. Students develop proficiency with number concepts and operations through working with authentic situations.

Teacher Notes: The information in this component provides additional insights which will help the educator in the planning process for the unit.

  • Students should be able to show division of fractions in more than one way.
  • Teachers should be using models for multiplication and division of fractions.
  • Make sure that students understand that a unit fraction is written in the form of one over a non-zero number(ex. , , …).
  • Students should be able to explain the result ofmultiplying a fraction by one,a number greater than one and a number less than one.
  • Teacher and students should know that division of fractions in grade five is limited to a whole number divided by a unit fraction and a unit fraction divided by a whole number.

Enduring Understandings: Enduring understandings go beyond discrete facts or skills. They focus on larger concepts, principles, or processes. They are transferable and apply to new situations within or beyond the subject.

At the completion of the unit on multiplication and division of fractions, the student will understand that:

  • multiplication does not always make the product larger than the factors and division does not always make the quotient smaller than the divisor.
  • multiplication and division can increase, decrease, or keep a number the same size.
  • a fraction is relative to the size of the whole or unit.

Essential Question(s): A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

  • How do operations with fractions compare/relate to operations with whole numbers and decimals?
  • How is multiplying or dividing whole numbers similar to multiplying or dividing fractions?
  • How can multiplying fractions be modeled using area, a number line, or measurement models?
  • How can dividing fractions be modeled using area, sets, or a number line?
  • How do fractions impact multiplication and division?

Content Emphases by Cluster in Grade 5:

According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), some clusters require greater emphasis than others. The table below shows PARCC’s relative emphasis for each cluster.Prioritization does not imply neglect or exclusion of material. Clear priorities are intended to ensure that the relative importance of content is properly attended to. Note that the prioritization is in terms of cluster headings.

Key: ■ Major Clusters  Supporting Clusters  Additional Clusters

Operations and Algebraic Thinking

Write and interpret numerical expressions.

Analyze patterns and relationships.

Number and Operation in Base Ten

■ Understand the place value system.

■ Perform operations with multi-digit whole numbers and with decimals to hundredths.

Number and Operation - Fractions

■ Use equivalent fractions as a strategy to add and subtract fractions.

■ Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Measurement and Data

Convert like measurement units within a given measurement system.

Represent and interpret data.

■ Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Geometry

Graph points on the coordinate plane to solve real-world and mathematical problems.

Classify two-dimensional figures into categories based on their properties.

Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):

According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), this component highlights some individual standards that play an important role in the content of this unit. Educators should give the indicated mathematics an especially in-depth treatment, as measured for example by the number of days; the quality of classroom activities for exploration and reasoning; the amount of student practice; and the rigor of expectations for depth of understanding or mastery of skills.

  • 5.NBT.6 The extension from one-digit divisors to two-digit divisors requires care. This is a major milestone along the way to reaching fluency with the standard algorithm in grade 6 (6.NS.2).
  • 5.NF.4 When students meet this standard, they fully extend multiplication to fractions, making division of fractions in grade 6 (6.NS.1) a near target.

Possible Student Outcomes:

The following list provides outcomes that describe the knowledge and skills that students should understand and be able to do when the unit is completed. The outcomes are often components of more broadly-worded standards and sometimes address knowledge and skills necessarily related to the standards. The lists of outcomes are not exhaustive, and the outcomes should not supplant the standards themselves. Rather, they are designed to help teachers delve deeplyinto the standards and augment as necessary, providing added focus and clarity for lesson planning purposes. This list is not intended to imply any particular scope or sequence.

The student will be able to:

  • Compute multiplication and division of fractions.
  • Interpret a fraction as division of the numerator by the denominator.
  • Extend previous understanding of multiplication and division of whole numbers and decimals to fractions.
  • Solve authentic problems involving multiplication and division of fractions using visual models and/or equations.

Progressions from Common Core State Standards in Mathematics: For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:

The Common Core Standards Writing Team (10 September 2011). Progressions for the Common Core State Standards in Mathematics(draft), accessed at:

Vertical Alignment: Vertical curriculum alignment provides two pieces of information:

  • A description of prior learning that should support the learning of the concepts in this unit
  • A description of how the concepts studied in this unit will support the learning of additional mathematics
  • Key Advances from Previous Grades:
  • Between grade 4 and grade 5, students grow in their ability to analyze multiplication and division of fractions.
  • apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction .
  • In grade 4 students also learn the relationship between fractions and division.
  • In grade 4 students work in finding whole-number quotients with a one-digit divisors.
  • Additional Mathematics:

Students will:

  • extend their work with multiplication and division to fractions in grade 6.
  • apply their work to the four basic operations and culminatetheir work with rational numbersin grade 7.
  • work to expand the operations to real numbers in grade 8.

Possible Organization of Unit Standards:This table identifies additional grade-level standards within a given cluster that support the over-arching unit standards from within the same cluster. The table also provides instructional connections tograde-level standards from outside the cluster.

OverarchingUnit Standards / Related Standards
within the Cluster / Instructional Connections
outside the Cluster
5.NF.3:Interpret a fraction as division of the numerator by the denominator (= ). 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. / 5.NBT.6:
  • Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

5.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. / 5.NF.4a:
  • Interpret the product () × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.
5.NF.4b:
  • Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths.
  • Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
/ 5.NBT.2:
  • Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponentsto denote powers of 10.

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

5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5.NF.7:Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions / 5.NF.7a:
  • Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.
5.NF.7b:
  • Interpret division of a whole number by a unit fraction, and compute such quotients.
5.NF.7c:
  • 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.

Connections to the Standards for Mathematical Practice: This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

In this unit, educators should consider implementing learning experiences which provide opportunities for students to:

  1. Make sense of problems and persevere in solving them.
  • Analyze a problem and decide if it is multiplication of fractions or division of fractions problem.
  • Consider the best way to solve a problem.
  • Can interpret the meaning of their answer to a given problem.
  1. Reason abstractly and quantitatively
  • Consider the ideas that multiplication is repeated addition and division is repeated subtraction.
  1. Construct Viable Arguments and critique the reasoning of others.
  • Justify the process of working a multiplication or division problem involving fractions.
  • Justify an argument with a model for multiplication or division of fractions.
  1. Model with Mathematics
  • Draw a diagram that represents multiplication or division of fractions.
  • Analyze an authentic problem and use a nonverbal representation of the problem.
  1. Use appropriate tools strategically
  • Use virtual media and visual models to explore fractional multiplication or division problems.
  • Use appropriate manipulatives.
  1. Attend to precision
  • Demonstrate an understanding of the mathematical processes required to solve a problems by communicating all of the steps in solving the problem.
  • Label appropriately.
  • Use the correct mathematics vocabulary when discussing problems.
  1. Look for and make use of structure.
  • Look at a multiplication table and recognize the relationship between multiplication and division.
  1. Look for and express regularity in reasoning
  • Pay special attention to details and continually evaluate the reasonableness of answers.

Content Standards with Essential Skills and Knowledge Statements and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.

Standard / Essential Skills and Knowledge / Clarification
5.NF.3: Interpret a fraction as division of the numerator by the denominator (= ). 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 as the result of dividing 3 by 4, noting that multiplied by 4 equals 3 and that when 3 wholes are shared equally among 4 people each person has a share of size . 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.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
4a: Interpret the product () × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show () × 4 = , and create a story context for this equation. Do the same with × () =. (In general,
() × () =)
4b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.5: Interpret multiplication as scaling (resizing) by:
5a: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
5b: Explaining why multiplying a given number by a fraction greater than one results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence to the effect of multiplyingby 1.
5.NF.6: Solve real world problemsinvolving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5.NF.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (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.)
7a: Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for () ÷ 4 and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that () ÷ 4 = because () × 4 = .
7b: Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ () and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ () = 20 because 20 × () = 4.
7c: 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 lb of chocolate equally? How many -cup servings are in 2 cups of raisins? / •Ability to recognize that a fraction is a representation of division.
• See the skills and knowledge that are stated in the Standard.
• Knowledge of unit fractions to multiply all fractions. (4.NF.3)
• Knowledge of using rectangular arrays to find area using rational numbers. (4.NBT.5)
• See the skills and knowledge that are stated in the Standard.
• See the skills and knowledge that are stated in the Standard.
• See the skills and knowledge that are stated in the Standard.
• See the skills and knowledge that are stated in the Standard.
• See the skills and knowledge that are stated in the Standard.
• Knowledge of the relationship between multiplication and division.(4.NBT.6), (5.NF.7a), (5.NF.7b) / 5.NF.4b: unit fractions to multiply all fractions: Refer to 4.NF.3 for background knowledge.

Connect these to form an array.

5.NF.5: scaling (resizing): Refer to 4.OA.1 for background knowledge. Scaling is a multiplicative comparison. It compares the size of a product to the size of one factor on the basis of the size of the other factor. Example: 12 = 3 4 so 12 is 3 times as many as 4 and 4 times as many as 3.
5.NF.6: real world problems: A problem that has context not just symbols. It is a practical world problem as opposed to an academic world problem. They are drawn from actual events or situations.

Evidence of Student Learning: The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.