4.NBT.6 - About the Math, Learning Targets, and Rigor

Grade 4 Multiplication and Division

4.NBT.6

Full Standard

Find whole-number quotients and remainders with up to four-digit dividends and one-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.

 

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 division.

 

Report Card Measurement Topic: Demonstrates understanding of division.
3rd Grade Students Learning 4th Grade Standards
Quarter 1 Quarter 2 Quarter 3 Quarter 4

 

Report Card Measurement Topic: Demonstrates understanding of division.

 

Report Card Measurement Topic: Demonstrates understanding of division.

 

Learning Target

  • Use division to determine the size of each group when the number of groups is known (partitive, i.e. 12 apples in 3 bags. How many are in each bag?)
  • Use division to determine the number of groups when the size of each group is known. (quotative or measurement, 12 apples with 4 in each bag. How many bags?)
  • Represent multi-digit division (up to four-digit dividends by one-digit divisors) with models and drawings that include remainders.
  • Write an equation for division situations.
  • Use partial quotients to divide multi-digit dividends by one-digit divisors.

 

About the Math

Two types of division problems are sharing (also call partitioning division) and repeated subtraction (also called measurement division.) In partition problems the whole is shared or distributed among a known number of sets to determine the size of each. When the number of sets is unknown but the size of the equal sets is known, the problems are measurement division.

  • Use division to determine the size of each group when the number of groups is known (partitive, i.e. 12 apples in 3 bags. How many are in each bag?)

Partition Division:
Jake has 15 oranges. He wants to share them equally among his 3 friends. How many oranges does each friend get?

  • Use division to determine the number of groups when the size of each group is known. (quotative or measurement, 12 apples with 4 in each bag. How many bags?)

Measurement Division:
Jake has 15 oranges. He puts them into baskets containing 3 oranges each. How many baskets did he use?

  • Represent multi-digit division (up to four-digit dividends by one-digit divisors) with models and drawings that include remainders.

Not all division situations come out evenly. Students need to understand the meaning of remainders. Initial instruction should include using counters and arrays to show how sometimes counters are "left over." Remainders can have different effects on answers depending on the context of the problem. Students need to interpret the meaning of the remainder. The remainder may be discarded because it has no effect on the answer. The remainder can force the answer to go to the next whole number.

 

Interpreting the Remainder (Reasoning Abstractly and Quantitatively):
When solving a word problem, there are three different ways to interpret the remainder. Interpreting is developed through many, many experiences. There are different ways to interpret a remainder. This should not be presented as a procedure. Students must be able to justify how they interpret a remainder.

  •     The answer is only the quotient.   

Movie tickets cost $5. How many can be purchased with $39?
39 ÷ 5 = 7 Drop the remainder 4 and the answer is 7 tickets because you cannot buy another ticket with the remainder. 

  •     You have to add one to the quotient.

There were 39 people on a bus when it broke down. People were driven home, 5 in each car. How many cars were used?
39 ÷ 5 = 7 R 4. You need to use the next whole number, adding one to the quotient because another car is needed for the leftover people.

  •     The solution is only the remainder.

A total of 39 trophies were purchased for the basketball league. Each box holds 5 trophies. How many trophies will be in the partially filled box? 4 trophies are in a box that isn't filled.

 

Write an equation for division situations.

As with all other operations and situations, students must be able to write and connect problems and situations to an equation.

 

Use partial quotients to divide multi-digit dividends by one-digit divisors.

Partial quotients is a division strategy in which the dividend is broken apart into convenient parts. Those parts are divided separately, then these partial quotients are added together. Partial quotients can be represented using an area model or symbolically.

See the video below for more about partial quotients.

 Math In Minutes:  The video below highlights partial quotients.

Math In Minutes: The video below highlight partitive and measurement division.

 

 

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
  • Use division to determine the size of each group when the number of groups is known (partitive, i.e. 12 apples in 3 bags. How many are in each bag?)
  • Use division to determine the number of groups when the size of each group is known. (quotative or measurement, 12 apples with 4 in each bag. How many bags?)
  • Represent two-digit by one-digit division with models and drawings.
  • Write an equation for division situations.
  • Use partial quotients to divide two-digit dividends by one-digit divisors.

 

Divide 2-, 3-, and 4-digit dividends by 1-digit divisors
  • Use division to determine the size of each group when the number of groups is known (partitive, i.e. 12 apples in 3 bags. How many are in each bag?)
  • Use division to determine the number of groups when the size of each group is known. (quotative or measurement, 12 apples with 4 in each bag. How many bags?)
  • Represent multi-digit division (up to four-digit dividends by one-digit divisors) with models and drawings.
  • Write an equation for division situations.
  • Use partial quotients to divide multi-digit dividends by one-digit divisors.

 

Divide 2-, 3-, and 4-digit dividends by 1-digit divisors

  • Use division to determine the size of each group when the number of groups is known (partitive, i.e. 12 apples in 3 bags. How many are in each bag?)
  • Use division to determine the number of groups when the size of each group is known. (quotative or measurement, 12 apples with 4 in each bag. How many bags?)
  • Represent multi-digit division (up to four-digit dividends by one-digit divisors) with models and drawings.
  • Write an equation for division situations.
  • Use partial quotients to divide multi-digit dividends by one-digit divisors.


Divide 2-, 3-, and 4-digit dividends by 1-digit divisors

 

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
  • Commutative, Associative, Distributive Properties (3.OA.5) 
  • Division as unknown factor (3.OA.6) 
  • Fluently x/÷ facts (3.OA.7) 
  • Add/subtract within 1000  (3.NBT.2)
  • Divide multi-digit dividends  (5.NBT.6)
  • Add, subtract, multiply, and divide decimals (5.NBT.7)

 

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TASKS

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

Instructional Materials for the various computation strategies can be found in Figuring Out Fluency Multiplication and Division with Whole Numbers (this book has been provided to schools) as follows:

Screenshot 2023-10-10 at 11.47.13 AM.png
Book Cover for Figuring Out Fluency Multiplication and Division with Whole Numbers

 

    • Think Multiplication (to divide)-- pages 102-107
    • Partial Quotients-- pages 122-127

 

 

 

 

 

 

 

 

 

 

  • Use the GeoGebra web app Links to an external site. to create a visual representation of the two different types of division for the expression 21÷3. Ask students if this modeling would be efficient when solving a problem such as 97÷3. How else could we use manipulatives to help us divide larger numbers? Provide students with base 10 blocks and place value disks to help them discover the importance of using partial quotients to divide.
  • Pose the following prompt to students. The 4th grade at Howard Elementary School is going on a field trip. There are 96 students and 3 buses. If each bus needs to hold an equal number of students, how many students will fit on each bus? Allow students to work independently or in a small group to complete the task using place value manipulatives (base 10 blocks or place value disks) and writing tools. While students are working, ask prompting questions to enhance their thinking (Examples: What expression can be used to represent this word problem? How did you create the number 96 with your base 10 manipulatives? What do the base 10 blocks/place value disks represent? What did you use to represent 3 buses? Were you able to share the tens and ones evenly? Will you always be able to share tens and ones evenly?) Pose several additional division problems (including some that require students to regroup the dividend) for students to solve with partners.
  • What is the relationship between multiplication and division? Provide examples to show your thinking.
  • How does knowing 5 x 5 help you to solve 75 ÷ 5? Explain.
  • How many different ways can you solve 84 ÷ 6?
  • If the quotient is 15, what could your possible dividend and divisor be?
  • How does changing the value of your divisor affect the quotient? (e.g., 350 ÷ 5 vs. 350 ÷ 50?)
  • Using the digits 4, 9, 7, and 5, create a division sentence with the greatest possible quotient.
  • Which division strategy (partial quotients, rectangular array, area model) do you think is best? Justify your answer.

 

SLIDE-BASED TASKS 

 These links are HCPSS created instructional tasks. These tasks are provided in Google slides. 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

The Broken Division Key, Activity 2.26, Page 65
How Close Can You Get?, Activity 3.10, Page 93


 
Hands-On Standards 3-4 38-39 (Dividing with one-digit divisors)

Super Source - Color Tiles

 

 3-4 58-61

Developing Mathematics with Base Ten

 2-6 71-73


 
Math Intervention: Building Number Power 3-5 121-123
NumberSense

3-4

92-96
  Mental Math in the Middle Grades   99-104 (Lesson 30, 31 & 32)

Math In Practice

Teaching Fourth-Grade Math

 4 Module 6

 

 

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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

  • Use division to determine the size of each group when the number of groups is known (partitive, i.e. 12 apples in 3 bags. How many are in each bag?)
  • Use division to determine the number of groups when the size of each group is known. (quotative or measurement, 12 apples with 4 in each bag. How many bags?)
  • Represent multi-digit division (up to four-digit dividends by one-digit divisors) with models and drawings that include remainders.
  • Write an equation for division situations.
  • Use partial quotients to divide multi-digit dividends by one-digit divisors.

 

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.