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:
Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 |
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Report Card Measurement Topic: Demonstrates understanding of division. |
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Report Card Measurement Topic: Demonstrates understanding of division. |
Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 |
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Report Card Measurement Topic: Demonstrates understanding of division. |
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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.
Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 |
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Divide 2-, 3-, and 4-digit dividends by 1-digit divisors |
Divide 2-, 3-, and 4-digit dividends by 1-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.
Grade 3 | Grade 5 |
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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:
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- 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.
Module 6 • Multiplication & Division (2-Digit by 1-Digit) and Connect Basic Facts Review (8s) to Factors and Multiples
Divide 2-digit numbers by 1-digit using representations and partial quotients:
Module 7 • Multiplication & Division (3- and 4-Digit by 1-Digit)
Divide 3-digit and 4-digit numbers by 1-digit numbers using representations and partial quotients:
Module 12 • Multiplication (2-Digit by 2-Digit) & Division and Solve Multiplication & Division Word Problems
Divide 4-digit dividends by 1-digit divisors, 3-digit dividends by 1-digit divisors, 2-digit dividends by 1-digit divisors:
Module 15 • Measurement and Problem Solving
Divide 4-digit dividends by 1-digit divisors, 3-digit dividends by 1-digit divisors, 2-digit dividends by 1-digit divisors:
Additional Tasks
These links provide instructional ideas connected to this standard.
- Mental Division Strategy Links to an external site. (Illustrative Math)
- What is 2500 Divided by 300? Links to an external site. (Georgia Department of Education, pg. 90-95)
- Partial Quotients Strategy Guide Download Partial Quotients Strategy Guide (ppt)
- Compatible Numbers to Estimate Links to an external site. (Georgia Department. of Education, pg. 81-84)
- Brain Only! Links to an external site.(Georgia Department of Education, pg. 85-89)
- Dividing with Skittles Download Dividing with Skittles (Utah Core Academy Lesson)
- Going Camping Download Going Camping (Utah Core Academy Lesson)
- Healthy Snacks Download Healthy Snacks (Utah Core Academy Lesson)
- Birthday Division Download Birthday Division (Utah Core Academy Lesson)
- Calves on the Farm Download Calves on the Farm (Utah Core Academy Lesson)
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.
Book Thumbnail | Book Title | Grade | Pages |
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Teaching Student-Centered Mathematics | 3-5 |
The Broken Division Key, Activity 2.26, Page 65
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Hands-On Standards | 3-4 | 38-39 (Dividing with one-digit divisors) | |
Super Source - Color Tiles
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3-4 | 58-61 | |
Developing Mathematics with Base Ten |
2-6 | 71-73
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Math Intervention: Building Number Power | 3-5 | 121-123 | |
NumberSense |
3-4 |
92-96 |
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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.
- Greatest Quotient Download Greatest Quotient (HCPSS-adapted print resource)
- Break Apart the Dividend Download Break Apart the Dividend (HCPSS-adapted print resource)
- Divide and Conquer Download Divide and Conquer (HCPSS-adapted print resource)
- Least Quotient Download Least Quotient (HCPSS-adapted print resource)
- Remainder Count Download Remainder Count (print resource)
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.
- Divide Two-Digit by One-Digit a Links to an external site.
- Divide Two-Digit by One-Digit b Links to an external site.
- 2-digit by 1-digit: Represent and Solve Links to an external site.
- 2-digit by 1-digit: Represent and Solve_2 Links to an external site.
- 2-digit by 1-digit: Represent and Solve_3 Links to an external site.
- Divide Three-Digit by One-Digit a Links to an external site.
- Divide Three-Digit by One-Digit b Links to an external site.
- Divide Three-Digit by One-Digit c Links to an external site.
- 3-digit by 1-digit: Represent and Solve Links to an external site.
- 3-digit by 1-digit: Reason About Division Links to an external site.
- Divide Four-Digit by One-Digit a Links to an external site.
- Divide Four-Digit by One-Digit b Links to an external site.
- 4-digit by 1-digit: Represent and Solve Links to an external site.
- 4-digit by 1-digit: Reason About Quotient Links to an external site.
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Learning Targets
- 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.
Rubric for Tasks Links to an external site.
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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Links to an external site. license. Content in this course can be considered under this license unless otherwise noted.