Strand: THE NUMBER SYSTEM (6.NS)

Apply and extend previous understandings of multiplication and division of whole numbers to divide fractions by fractions (Standard 6.NS.1). Compute (add, subtract, multiply and divide) fluently with multi-digit numbers and decimals and find common factors and multiples (Standards 6.NS.2–4). Apply and extend previous understandings of numbers to the system of rational numbers (Standards 6.NS.5–8).

Standard 6.NS.3

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

• 12 Rectangular Units
The purpose of this task is for students to notice how the decimal point behaves when numbers in different place-value places (for example, 0.04 and 0.3) are multiplied.
• 2 Units Wide and 3 Units Long
The purpose of this task is for students to notice how the decimal point behaves when numbers in the same place (both in the hundreds, both in the thousandths, etc) are multiplied.
• Adding Base Ten Numbers, Part 1
The goal of this task is to demonstrate that since digits in the same place represent the same-sized units, we can always add digits in the same place. This is one of three tasks relating to this.
• Adding Base Ten Numbers, Part 2
This is the second in a set of three tasks generalizing an addition algorithm whole numbers to all base-ten numbers.
• Adding Base Ten Numbers, Part 3
This is the third in a set of three tasks generalizing an addition algorithm from whole numbers to all base-ten numbers.
There are two aspects to fluency with division of multi-digit numbers: knowing when it should be applied, and knowing how to compute it. While this task is very straightforward, it represents the kind of problem that sixth graders should be able to recognize and solve relatively quickly.
• Changing Currency
The purpose of this task is for students to notice that if the dividend and divisor both increase by a factor of 10, the quotient remains the same. This sets them up to understand the rules for moving decimal points when performing long division.
• Chapter 0 - Mathematical Foundations (UMSMP)
This is Chapter 0 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Fluency.
• Chapter 0 - Student Workbook (UMSMP)
This is Chapter 0 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Fluency.
• Gifts from Grandma, Variation 3
The purpose of this task is to show three problems that are set in the same kind of context, but the first is a straightforward multiplication problem while the other two are the corresponding "How many groups?" and "How many in each group?" division problems.
• Grade 6 Math Module 2: Arithmetic Operations Including Division of Fractions (EngageNY)
In Module 2, students complete their understanding of the four operations as they study division of whole numbers, division by a fraction and operations on multi-digit decimals. This expanded understanding serves to complete their study of the four operations with positive rational numbers, thereby preparing students for understanding, locating, and ordering negative rational numbers (Module 3) and algebraic expressions (Module 4).
• Grade 6 Unit 1: Number System Fluency (Georgia Standards)
In this unit students will find the greatest common factor of two whole numbers less than or equal to 100. Find the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. Interpret and compute quotients of fractions. Solve word problems involving division of fractions by fractions using visual fraction models and equations to represent the problem. Fluently divide multi-digit numbers using the standard algorithm. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
• Jayden's Snacks
Building on their fifth grade experiences with operations on decimal numbers, sixth grade students should find the task to be relatively easy. The emphasis in this task is on whether students are actually fluent with the computations, so teachers could use this as a formative assessment task if they monitor how students solve the problem.
• Movie Tickets
The purpose of this task is for students to solve problems involving decimals in a context involving a concept that supports financial literacy, namely inflation.
• Reasoning about Multiplication and Division and Place Value, Part 1
The three tasks in this set are not examples of tasks asking students to compute using the standard algorithms for multiplication and division because most people know what those kinds of problems look like. Instead, these tasks show what kinds of reasoning and estimation strategies students need to develop in order to support their algorithmic computations.
• Reasoning about Multiplication and Division and Place Value, Part 2
The three tasks (including part 1 and part 3) in this set are not examples of tasks asking students to compute using the standard algorithms for multiplication and division because most people know what those kinds of problems look like. Instead, these tasks show what kinds of reasoning and estimation strategies students need to develop in order to support their algorithmic computations.
• Setting Goals
The purpose of this task is for students to solve problems involving division of decimals in the real-world context of setting financial goals. The focus of the task is on modeling and understanding the concept of setting financial goals, so fluency with the computations will allow them to focus on other aspects of the task.
• Tenths of Tenths and Hundredths of Hundredths
The purpose of this task is to relate what students know about multiplication, area, and fractions to multiplying decimals for powers of ten that are less than 1. The questions are carefully sequenced so that students are lead to construct an argument for why, from a geometric perspective, 0.10.1 is 0.01 and 0.010.01 is 0.0001. Eventually, students should generalize their understanding and know that the number of decimal places in a product is the same as the total number of decimal places in the factors. This task gives a geometric basis for understanding why that is true.
• The Number System (6.NS) - 6th Grade Core Guide
The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - The Number System.
• What is the Best Way to Divide?
The purpose of this task is to have students think strategically about their method for solving a division problem. This task shows an example of focusing on the choice of strategy as opposed to applying an algorithm without first considering options.

http://www.uen.org - in partnership with Utah State Board of Education (USBE) and Utah System of Higher Education (USHE).  Send questions or comments to USBE Specialist - Lindsey Henderson and see the Mathematics - Secondary website. For general questions about Utah's Core Standards contact the Director - Jennifer Throndsen .

These materials have been produced by and for the teachers of the State of Utah. Copies of these materials may be freely reproduced for teacher and classroom use. When distributing these materials, credit should be given to Utah State Board of Education. These materials may not be published, in whole or part, or in any other format, without the written permission of the Utah State Board of Education, 250 East 500 South, PO Box 144200, Salt Lake City, Utah 84114-4200.