Grade 5 Place Value and Decimals

5.NBT.6

About the Math

Full Standard

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.

Measurement Topic

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

**5th Grade Students Learning 5th Grade Standards**
Quarter 1	Quarter 2	Quarter 3	Quarter 4
Report Card Measurement Topic: Demonstrates understanding of multiplication and division of different number types.

**4th Grade Students Learning 5th Grade Standards**
Quarter 1	Quarter 2	Quarter 3	Quarter 4
Report Card Measurement Topic: Demonstrates understanding of division.

Learning Targets (I can)

Explain the difference between partitive (how large are the groups?) and measurement (how many groups?) division.
Represent division of multi-digit whole numbers with base ten models (equal groups) and arrays/area models.
Relate division of multi-digit numbers to multiplication of those numbers.
Connect representations to a partial quotient algorithm.
Use partial quotients to divide multi-digit whole numbers.
Explain partial quotients.
Solve problems using multi-digit division (including remainders).

About the Math

5th-grade students do not work with the traditional, long-division algorithm. That "standard" algorithm is taught in 6th grade.

Explain the difference between partitive (how many groups?) and measurement (how large are the groups?) division.

There are two common types of division problems. The first type is called repeated subtraction. You know the total amount and the amount in each group and you are looking for the number of groups. How many packages of 6 ping pong balls can be made from 42 ping pong balls? There are 7 groups of 6 in 42 so you 7 packages can be made.

The second type is called sharing division. You know the total amount and the number of groups and you need to find out how much are in each group. Jody has 42 apples. She puts them into 6 equally sized groups. How many are in each group? There are 7 in each group.

Represent division of multi-digit whole numbers with base ten models (equal groups) and arrays/area models.

Students should use models of larger numbers to model division of those numbers. Students must connect partial quotients and/or the partial quotients algorithm to these models from the very beginning.

Relate division of multi-digit numbers to the multiplication of those numbers.

As students use partial quotients, they should connect the process to multiplication. In other words, when a student gives out 25 coins to 3 boxes (the act of dividing), they should connect that to 3 groups of 25 (75) coins that have been given out.

Connect representations to a partial quotient algorithm.

As noted above, the partial quotients algorithm must be connected to representations used for early instruction of division with multi-digit dividends.

Use partial quotients to divide multi-digit whole numbers.

Understanding of the partial quotients algorithm has many layers. One layer is that partial quotients connect to a model or drawing (as mentioned above). Another layer is that students are able to take out tens and one. A third layer is that students are able to take out groups of tens and ones. And yet another layer is that students are able to take out larger, convenient groups of numbers.

Explain partial quotients.

Students should explain partial quotients through the algorithm, models, or drawings. Additionally, they should use multiplication to show how groups of a number are being removed from the dividend.

Essential vocabulary for this standard includes dividend, divisor, array , partial quotients, and quotient (online dictionary, visual math dictionary).

Math in Minutes: This video highlights partial quotients.

Math In Minutes: This Video highlights partitive and measurement division

Division can be completed through partials. This supports our ability to develop mental mathematics. Greg Tang's Division Partials (pictured below) explains how these ideas work.

Progression of Standard within Grade 5

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 the difference between partitive (how many groups?) and measurement (how large are the groups?) division. Represent division of multi-digit whole numbers with base ten models (equal groups) and arrays/area models. Relate division of multi-digit numbers to multiplication of those numbers. Connect representations to a partial quotient algorithm. Use partial quotients to divide 2-digit, 3-digit, and 4-digit quotients by 1-digit divisors. Explain partial quotients.	Continue practice through routines, independent practice and homework.	Continue practice through routines, independent practice and homework.	Continue as needed.

Progression of this Standard Across Grades

This progression is informed by the Achieve the Core Coherence Map. Information is not the complete standard.

Progression Across Grades
Grade 4	Grade 6
Fluently add and subtract multi-digit whole numbers using the standard algorithm (4.NBT.4) Division with multi-digit dividends (4.NBT.6)	Fluently divide multi-digit numbers using the standard algorithm (6.NS.2) Fluently add, subtract, multiply, and divide decimals using the standard algorithm (6.NS.3)

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Instructional Tasks

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

Ask students to explain the meaning of the expression 375 ÷ 3. Students might share the following statements: How many groups of 3 are in 375? or 375 is split evenly into 3 groups, how many are in each group? If students do not share both examples of division, facilitate a discussion of the two different types of division (partitive and quotative).
Using the same expression (375 ÷ 3), ask students to share how they could break apart 345 in different ways so that it would be easier to divide by 3. Examples might include (300 + 75, 300 + 60 + 15, 300 + 30 + 30 +15, 300 + 60 + 12 + 3, etc). Ask students to use the breaks they created to solve using the partial quotients strategy. Allow students to share and facilitate a discussion about which decomposition was the most efficient and why.
If the quotient is 34, what could a possible dividend and divisor be? (use either a three- or four-digit dividend)
How do you know that 34 is not the quotient of 1,216 ÷ 4?
Find a number that when divided by either 2, 3, or 5 has a remainder of 1.
List three numbers that when divided by 5 each have a remainder of 1.
Using the digits 4, 9, 7, 1 and 5, create a division sentence with a two-digit divisor and the greatest possible quotient.
Write a division problem that has a four-digit dividend and a one-digit divisor with a quotient that is even.
What is the relationship between multiplication and division? Provide examples to show your thinking.
Use two different division strategies to solve 9,754 ÷ 5.
How does understanding place value help when solving 439 ÷14 with the partial quotient strategy? Explain by using examples.

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 3 • Division

Divide Multi-Digit Dividends by 1-Digit Divisors Using Partial Quotients:

Divide Multi-Digit Dividends by 2-Digit Divisors Using Partial Quotients:

Additional Tasks

These links provide instructional ideas connected to this standard.

Minutes and Days (Links to an external site.)(Illustrative Mathematics)
Tomato Tomato (3 Act Task, G.Fletcher)
Measurement Word Problems
The Grass is Always Greener (Georgia Department of Education, pgs. 73-76)
Division Four in a Row (Georgia Department of Education, pgs. 77-81)
Are These All 364 ÷ 15? (pages 81-86) (Georgia Department of Education)
Start of the Year Celebration (pages 87-91) (Georgia Department of Education)
Library Books (Utah Core Academy Lesson)
Classroom Makeover (Utah Core Academy Lesson)
Katy Perry (Utah Core Academy Lesson)
Katy Perry Student Sheet (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.

Print Resources
Book Title	Grade	Pages
Teaching Student-Centered Mathematics	3-5	The Broken Division Key, Activity 2.26, Page 65
Nimble With Numbers	5-6	62-63 "Divisible Products" 70 "Finding Quotients Practice"
Nimble With Numbers	4-5	91-92 "Remainders Race" 93-94 "Remainders Score" 95-97 "Creating Quotients"
Fundamentals	5-6	8-11 "Division Dash" 24-27 "Does It Divide?" 28-31 "Remainder Race"
Brain-Compatible Activities for Mathematics	4-5	19-23 'Blocks of Division" 29-31 'Divisibility Dash" 32-35 "Scuba Division" 25-28 "Words of Division"
Math Intervention: Building Number Power	3-5	121-122 "Partial Quotients" 123 "Prove It"
Mental Math in the Middle Grades	3-5	99-104 Lesson 30, 31, 32
Math In Practice Teaching Fifth-Grade Math	5	Module 4

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Independent Work

Centers

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

Least Quotient (HCPSS-adapted print resource)
Greatest Quotient (HCPSS-adapted print resource)
Write and Solve a Division Problem (HCPSS-adapted print resource)
The Range Game (print resource)
Division Partials (GregTang 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.

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Assessment

Learning Targets

rubric

Explain the difference between partitive (how many groups?) and measurement (how large are the groups?) division.
Represent division of multi-digit whole numbers with base ten models (equal groups) and arrays/area models.
Relate division of multi-digit numbers to multiplication of those numbers.
Connect representations to a partial quotient algorithm.
Use partial quotients to divide multi-digit whole numbers.
Explain partial quotients.

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

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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