4.NF.4 - About the Math, Learning Targets, and Rigor

Grade 4 Fractions

4.NF.4

Full Standard

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

  1. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
  2. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
  3. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

Note: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

 

Measurement Topic

This standard is reported on the report card in these quarters as follows:

4th Grade Students Learning 4th Grade Standards
Quarter 1 Quarter 2 Quarter 3 Quarter 4

 

 

Report Card Measurement Topic: Demonstrates understanding of multiplication. (A&B parts of the standard)

Report Card Measurement Topic: Demonstrates understanding of patterns, expressions, equations, and algebraic thinking. (C part of the standard)
3rd Grade Students Learning 4th Grade Standards
Quarter 1 Quarter 2 Quarter 3 Quarter 4

 

 

Report Card Measurement Topic: Demonstrates understanding of multiplication. (A&B parts of the standard)

Report Card Measurement Topic: Solves one and two-step word problems with any operation. (C part of the standard)

 

Learning Targets

  • Explain a fraction as a multiple of a unit fraction. For example, 3/4 = 1/4 + 1/4 + 1/4 or 3 x 1/4.
  • Multiply a whole number by a fraction using visual models.
  • Use the associative property to explain that n x (a/b) = (n × a)/b). For example, 5 x 7/8 = 5 x (7 x 1/8) = (5 x 7) x 1/8 = 35 x 1/8 -or- 35/8.
  • Solve word problems by multiplying a whole number by a fraction.

 

About the Math

  • Explain a fraction as a multiple of a unit fraction. For example, 3/4 = 1/4 + 1/4 + 1/4 or 3 x 1/4.

picture of 3 x 2/5Unit fractions always have a numerator of one. Students need to see that fraction parts can be counted, just like we count whole numbers. So if we count 1 orange, 2 oranges, 3 oranges, etc., we can also count 1 fourth, 2 fourths and 3 fourths. So if 3 oranges can be thought of as 3 groups of one orange, then ¾ can be represented as 3 groups of ¼ or 3 x ¼.

  • Multiply a whole number by a fraction using visual models.

Prior to teaching the procedure for multiplying a whole number by a fraction, students need to understand conceptually why the answer is reasonable. If I have 5 groups of 1/6, how many 1/6s are there? You can add 1/6 + 1/6 + 1/6 + 1/6 + 1/6 to equal 5/6. If you want to multiply 4 x 2/3, students need to think of this as 4 groups of 2/3 or 8 groups of 1/3. When added to show four groups of two-thirds, 2/3 + 2/3 + 2/3 + 2/3 = 8/3 or 2 2/3. Or this can be shown as 4 X 2/3 or 4/1 x 2/3= 8/3 or 2 2/3.

  • Use the associative property to explain that n x (a/b) = (n × a)/b). For example, 5 x 7/8 = 5 x (7 x 1/8) = (5 x 7) x 1/8 = 35 x 1/8 -or- 35/8.

The Associative Property can be used to explain what is happening when we mutliply a fraction by a whole number. As noted above, 5 x 7/8 can be thought of in this way. 7/8 is the same as 7 x 1/8. Therefore, we can think of 5 x 7/8 as 5 x (7 x 1/8) or (5 x 7) x 1/8.

  • Solve word problems by multiplying a whole number and a fraction

Students need to understand what a reasonable answer looks like when multiplying fractions.  Between what two whole numbers does your answer lie? If students have been led down the incorrect idea that when you multiply whole numbers the product is always greater. Giving students “rules” such as that sets them up for confusion as they continue to learn about rational numbers. This is not true when dealing with fractions. Students should focus on what an answer will look like prior to actually calculating the answer. Questions like: Does this answer make sense? How do you know it is going to be less than a certain number? Using word problems in a context helps students make sense.
Examples of Word Problems:

  • Kim runs 2/3 mile every day. How far does she run in one week?
  • Ms Howard is making punch. The punch uses ¾ cup of orange juice for one serving. If she makes 8 servings, how many cups of orange juice does she need?

Math In Minutes:  Multiplying  Whole Numbers with  Fractions


The Illustrative Mathematics tasks below demonstrate expectation for this standard

 

 

 

 

Progression of Standard within Grade 4

This progression informs how to develop the standard within the grade level. This progression is provided by HCPSS Elementary Mathematics.

Progression Throughout Year
Quarter 1 Quarter 2 Quarter 3 Quarter 4
 

 

 
  • Explain a fraction as a multiple of a unit fraction. For example, 3/4 = 1/4 + 1/4 + 1/4 or 3 x 1/4.
  • Multiply a fraction by a whole number using visual models.
  • Use the associative property to explain that n x (a/b) = (n × a)/b). For example, 5 x 7/8 = 5 x (7 x 1/8) = (5 x 7) x 1/8 = 35 x 1/8 -or- 35/8.
  • Solve word problems that multiplying a whole number and a fraction.
 

 

 

Progression of this Standard Across Grades

This progression is informed by the Achieve the Core Coherence Map Links to an external site.. Information is not the complete standard.

Progression Across Grades
Grade 3 Grade 5
  • Understand unit fractions (3.NF.1)
  • Interpret products of whole numbers (3.OA.1) 
  • Solve word problems with x/÷ (3.OA.3)
  • Multiply a fraction or whole number by a fraction (5.NF.4)  
  • Divide unit fractions by whole numbers and whole numbers by unit fractions (5.NF.7 )

 

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TASKS

These tasks can be used with small group or whole group instruction. 

4.NF.4a

  • Begin by reviewing the meaning behind the expression 4 x 6. One way students should be able to describe this expression is 4 groups of 6. Another way students should be able to describe this expression is 6 + 6 + 6 + 6. Ask students to make a connection to the meaning of the expression 4 x 1/6. Encourage students to describe the expression in words and with a connection to repeated addition. Finally, students should use a visual representation or a fraction manipulative to represent the same expression. Continue with additional expressions that involve multiplying a whole number and a fraction.
  • Use an open number line to represent the fraction 5 x 1/3. Students should be encouraged to identify the equivalent fractions on the number line. One possible differentiation would be to begin by representing 5 x 1/3 using a fraction manipulative such as the fraction tiles, Cuisenaire rods, or fraction towers. Once the expression is represented with these manipulatives, place the manipulative on top of the open number line and draw a jump to represent repeated addition. Continue with other expressions that involve multiplying a whole number and a fraction.
  • Josh noticed that ⅓ + ⅓ + ⅓ +⅓ was the same as 4 x ⅓. Do you agree or disagree with Josh’s observation? Explain your thinking.
  • What pattern do you notice when multiplying a whole number by a fraction? Why do you think this pattern occurs?
  • When multiplying a whole number by a fraction, what happens to your product? Why is your product less than your original whole number? (also works for 4.NF.4b)
  • What two factors can be multiplied to equal a product of 6/8?

4.NF.4b

  • What is the relationship between decomposing fractions and multiplying fractions by a whole a number? (halving and doubling)
  • Use fraction tiles or Cuisenaire rods to prove how the halving and doubling strategy results in the same product.  
  • Why is 3 x 2/6 the same as 6 x ⅙?
  • What equation could equal the same product as 2 x 8/10?

 

SLIDE-BASED TASKS 

 These links are HCPSS created instructional tasks. These tasks are provided in Google slides. When selected, a copy of the file is added to your drive for classroom use. These tasks should be used for inspiration and resources, but instruction should start with students having the opportunity to engage with the math first (often involving physical and/or visual models) followed by discussion and explicit instruction to ensure student understanding.

 

 

Additional Tasks 

These links provide instructional ideas connected to this standard. 

 

 

Tasks From Print Resources

These publications have been provided for each school. They are typically stored in team closets or the media center. Check with your team leader if you cannot find them. 

Print Resources
Book Thumbnail Book Title Grade Pages
Teaching Student-Centered Mathematics 3-5 How Close Can You Get?, Activity 3.10, Page 93
Hands-On Standards, Common Core 4 80-92 (Multiply Fractions Lessons 1-3)
Hands-On Standards 5-6 46 (Multiply with Fractions)

20 Thinking Questions for Fraction Circles

 

 3-6 70-81

Math In Practice

Teaching Fourth-Grade Math

 4 Module 9

 

 

 

 

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Centers

These print resources can be used during independent or center time. These resources could also be used as lesson seeds.

 

 

INDEPENDENT PRACTICE/HOMEWORK/ASSESSMENT

 These resource sheets can be used for independent practice, homework, or assessment. They are intended to reinforce procedures and concepts. They should not be used as a source of direct instruction or whole-group practice.

 

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Learning Targets

rubric

  • Explain a fraction as a multiple of a unit fraction. For example, 3/4 = 1/4 + 1/4 + 1/4 or 3 x 1/4.
  • Multiply a fraction by a whole number using visual models.
  • Use the associative property to explain that n x (a/b) = (n × a)/b). For example, 5 x 7/8 = 5 x (7 x 1/8) = (5 x 7) x 1/8 = 35 x 1/8 -or- 35/8.
  • Solve word problems that multiplying a whole number and a fraction.

 

Learning targets identify what students should be able to do. This rubric can be applied to tasks and observations for assessment and/or grading. 

 

 

Visit the SBIR (Standards Based Instruction and Reporting) tab in Course Essentials for more information and clarification. There you will find the measurement topic crosswalk, report card comments, links to professional learning/resources and guidance.

 

 

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 This course content is offered under a CC Attribution Non-Commercial Share Alike Links to an external site. license. Content in this course can be considered under this license unless otherwise noted.