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).
• 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.
• Above and Below Sea Level
The purpose of this task is to help students interpret signed numbers in a context as a magnitude and a direction and to make sense of the absolute value of a signed number as its magnitude.
• Absolute Value
This Math Shorts video uses a number line and a real-life example to explain the absolute value of a number. The classroom activity then has the student play a game where they move a penny in both positive and negative directions on a number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• 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.
This task is appropriate for assessing students understanding of repeated reasoning and generalizing that understanding to prepare them for deeper algebraic thinking needed in the expressions and equations domain.
• Adding Rational Numbers on the Number Line
In this interactive students must solve riddles about a wallaby jumping contest. But they must find equivalent fractions and common denominators to complete the riddle. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Addition and Subtraction of Integers
A card game in which positive and negative numbers are added together is the subject of this video teaching students how to add and subtract integers. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Bake Sale
This problem requires students to apply the concepts of factors and common factors in a context.
This task requires students to complete a series of steps in order to find a solution, and because they need to analyze constraints, it addresses some aspects of mathematical modeling. Students must first add fractions with familiar but unlike denominators, which is a skill developed in the 5th grade. Then students need to divide fractions by fractions.
• Batting Average
The goal of this task is to perform and analyze division with whole numbers in a sports context.
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.
• Chapter 3 - Student Workbook (UMSMP)
This is Chapter 3 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Extending the Number System.
• Chapter 3 - Mathematical Foundations (UMSMP)
This is Chapter 3 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Extending the Number System.
• Chapter 6 - Mathematical Foundations (UMSMP)
This is Chapter 6 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Expressions and Equations.
• Chapter 6 - Student Workbook (UMSMP)
This is Chapter 6 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Expressions and Equations.
• Comparing Temperatures
The purpose of the task is for students to compare signed numbers in a real-world context. It could be used for either assessment or instruction if the teacher were to use it to generate classroom discussion.
• Cup of Rice
Students are given a word problem "One serving of rice is 23 of a cup. I ate 1 cup of rice. How many servings of rice did I eat?" They must choose between 2 possible solutions and explain their reasoning.
• Dan's Division Strategy
The purpose of this task is to help students explore the meaning of fraction division and to connect it to what they know about whole-number division.
• Distances Between Points
The purpose of this task is for students to solve a mathematical problem using points in the coordinate plane.
• Distributive Property with Variables
Algebra tiles are used to generate equivalent expressions using the distributive property in this instructional video. The classroom activity asks student to further explore the distributive property. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Dividing by a Fraction is the Same as Multiplying by its Reciprocal
The purpose of this task is to help students understand that when we divide by a fraction, it is the same as multiplying by its reciprocal. By carrying out this work outside of a context, students can focus on numbers and operations. This task builds on the work of 5.NF.B, where students represented many different fractional multiplication and division contexts with diagrams.
• Drinking Juice, Variation 2
This task builds on a fifth grade fraction multiplication task, 5.NF Drinking Juice. This task uses the identical context, but asks the corresponding Number of Groups Unknown division problem. See Drinking Juice, Variation 3 for the Group Size Unknown version.
• Drinking Juice, Variation 3
This task builds on a fifth grade fraction multiplication task, 5.NF Drinking Juice. This task uses the identical context, but asks the corresponding Group Size Unknown division problem. See Drinking Juice, Variation 2 for the Number of Groups Unknown version.
• Equivalent Expressions with the Distributive Property
This animated Math Shorts video explains how the distributive property can help students model and create equivalent expressions. In the accompanying classroom activity, students play a quick game where they identify common factors within an expression and work on a series of problems that expand their understanding of how to apply the distributive property. While the problems begin with whole number expressions, students soon work toward algebraic notation and eventually develop the idea that ax + bx can be rewritten as x(a + b). NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Estimating Products of Decimals
The purpose of this task is to help students estimate products of decimal numbers. In order to promote sense-making, this task might work best before a teacher explains an algorithmic approach, so that students don't just carry out computations and look to see which choice is closest.
• Extending the Number Line
The purpose of this task is to understand that there are natural mathematical questions to ask for which there are no answers if we restrict ourselves to the positive numbers. The idea is to motivate the need for negative numbers and to see that there is a natural representation of them on the number line.
• Factors
This lesson is designed to help students understand factors of whole numbers.
• Factors and Common Factors
This task requires students to apply the concepts of factors and common factors in a context.
• Finding Factors
This lesson plan's activities give students practice in finding the factors of whole numbers.
• Fractions and Decimals from 0 to 1 on the Vertical Number Line
In this interactive students must solve riddles about jumping fleas by placing fractions and decimals on a 0 to 1 number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Fractions on the Number Line
In this task students are given a number line and they must label a number of fractions on the line. When given a selection of statements about inequality they must state which are true.
• Fractions, Mixed Numbers, and Decimals on the Number Line
Using the device of a frog-jumping contest, students learn about values on a number line by placing the frogs' jump distances on the line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• 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 Math Module 3: Rational Numbers (EngageNY)
Students are familiar with the number line and determining the location of positive fractions, decimals, and whole numbers from previous grades. Students extend the number line (both horizontally and vertically) in Module 3 to include the opposites of whole numbers. The number line serves as a model to relate integers and other rational numbers to statements of order in real-world contexts. In this module's final topic, the number line model is extended to two-dimensions, as students use the coordinate plane to model and solve real-world problems involving rational numbers.
In order to assist educators with the implementation of the Common Core, the New York State Education Department provides curricular modules in Pre-K-Grade 12 English Language Arts and Mathematics that schools and districts can adopt or adapt for local purposes.
• 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.
• Grade 6 Unit 3: Expressions (Georgia Standards)
In this unit students will represent repeated multiplication with exponents. Evaluate expressions containing exponents to solve mathematical and real world problems. Translate verbal phrases and situations into algebraic expressions. Identify the parts of a given expression. Use the properties to identify equivalent expressions. Use the properties and mathematical models to generate equivalent expressions.
• Grade 6 Unit 7: Rational Explorations: Numbers and their Opposites (Georgia Standards)
In this unit students will understand that positive and negative numbers are used together to describe quantities having opposite directions or values, understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. Recognize that the opposite of the opposite of a number is the number itself.
• Greatest Common Factor
This video from Math Shorts shows students how to find the greatest common factor of 2 numbers. The classroom activity shows them how Venn diagrams, multiplication and prime factors can also be used to find the greatest common factor. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Harry Makes a Big Splash with Positive and Negative Numbers
A video about a swim relay team opens this lesson on combining positive and negative numbers. The classroom activity involves using a number line and a game board and die. Students will write equations that represent the addition of positive and negative numbers. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Horizontal and Vertical Distances on the Cartesian Graph
In this activity students place marine animals on a Cartesian graph and then determine the horizontal and vertical distance between them. The classroom activity builds on the student's understanding of distances between points on a Cartesian graph. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• How many _______ are in. . . ?
This task provides a list of problems. They require that the students model each problem with some type of fractions manipulatives or drawings. The problems are meant to be a progression which require more sophisticated understandings of the meaning of fractions as students progress through them. If the task is used to help students see the connections to the invert-and-multiply rule for fraction division (as described in the solution) then they should already be familiar with and comfortable solving Number of Groups Unknown (a.k.a. "How many groups?") division problems with visual models.
• How Many Batches/What Fraction of a Batch?
The purpose of this task is to help students extend their understanding of multiplication and division of whole numbers to multiplication and division of fractions. The task does not ask students to find the product or quotient since the task is more about learning how to represent the situation, but teachers might choose to ask students to find or estimate the answers, if desired.
• How Many Containers in One Cup / Cups in One Container?
These two fraction division tasks use the same context and ask "How much in one group?" but require students to divide the fractions in the opposite order.
• How Many Staples?
The goal of this task is to perform long division with remainder in a context. Students are shown a box of staples and asked to find inconsistencies in the information on it.
• How Much in One Batch?
The purpose of this task is to help students extend their understanding of multiplication and division of whole numbers to multiplication and division of fractions.
• Integers on the Number Line 1
Given a number line, students are asked to find and label two numbers. Then given several inequalities, they must decide whether the inequality is true or false.
• Integers on the Number Line 2
The goal of this task is to study, with a number line, why it makes sense for a whole number a that -(-a)=a.
• Interpreting a Division Computation
In this task, students are shown a division problem and then asked to find the products of a group of numbers related to that problem.
• It's Warmer in Miami
The purpose of this task is for students to apply their knowledge of integers in a real-world context.
• IXL Game: Coordinate graphing
This game will help sixth graders learn to graph points on a coordinate plane. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use.
• 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.
• Jumping Flea
This purpose of this task is to help students understand the absolute value of a number as its distance from 0 on the number line. The context is not realistic, nor is meant to be; it is a thought experiment to help students focus on the relative position of numbers on the number line.
• Keep, Change, Flip
Students are taught the "Keep, Change, Flip" rule for dividing fractions by viewing this clever Flocabulary rap song. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Locating Points on the Cartesian Graph
In this activity students use logic and clues to plot the location of marine animals on a Cartesian graph. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Locations in the Coordinate Plane
The goal of this task is to introduce students to the relationships between the locations and coordinates of points graphed in all four quadrants of the coordinate plane. When describing the things they notice about the point locations and coordinates, the teacher should encourage students to use terms such as quadrant, distance, origin, sign, axis, and coordinate.
• Making Hot Cocoa, Variation 1
This is the first of two fraction division tasks that use similar contexts to highlight the difference between the "Number of Groups Unknown" a.k.a. "How many groups?" when the quotient is a fraction (or mixed number) greater than 1 (Variation 1) and when the quotient is a fraction that is less than 1 (Variation 2).
• Making Hot Cocoa, Variation 2
This is the second of two fraction division tasks that use similar contexts to highlight the difference between the "Number of Groups Unknown" a.k.a. "How many groups?" when the quotient is a fraction (or mixed number) greater than 1 (Variation 1) and when the quotient is a fraction that is less than 1 (Variation 2).
• Manipulating Graphs
This video demonstrates how to use the slope-intercept of a line to the graph of that line. The classroom activity has them demonstrate their understanding by finding equations for a set of lines through the origin. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Mile High
The first two parts of this task ask students to interpret the meaning of signed numbers and reason based on that meaning in a context where the meaning of zero is already given by convention.
• Modeling Fraction and Mixed Number Division Using Arrays
Students will learn how to solve word problems that involve dividing fractions and mixed numbers by using a visual model. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Modeling Fraction Division Using Comparison, Group Number Unknown
In this lesson students will learn how to solve a word problem involving the division of fractions by viewing an animation about a hedgehog's hibernation. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Modeling Fraction Division, Equal Groups, Group Size Unknown
The skill of dividing two fractions by groups of unknown size is the focus of this video. Students will learn how to solve a word problem using this process. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Modeling Fraction Division, Equal Groups, Number of Groups Unknown
This animated video shows students a model they can use to solve word problems involving the division of fractions. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• 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.
• Multiples and Common Multiples
This task requires students to apply the concepts of multiples and common multiples in a context.
The purpose of this task is for students to solve a real-world problem by interpreting and comparing points in the coordinate plane. This task focuses students' attention on the y-values of the points, asking for the greatest y-value and the least y-value, as well as the greatest difference between y-values when the x-values are the same.
• Patterns of Factors
This interactive activity asks students to sort numbers based on the number of factors or prime factors. They are asked to also identify one real-life example of the usefulness of divisibility. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Pennies to Heaven
The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task.
• Plotting Points in the Coordinate Plane
The goal of this task is to provide experience labeling coordinate axes appropriately to plot a given set of points, which will mean choosing an appropriate scale.
• Rational Numbers on the Number Line
Riddles about a wallaby jumping contest must be solved in this lesson. Students must place fractions, decimals or mixed numbers representing the lengths of jumps, on a number line from -5 to +5. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• 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.
• Reciprocity
The purpose of this task is to help students understand why dividing by a fraction gives the same result as multiplying by its reciprocal. This is accomplished by writing the division equation along with related multiplication equations and diagrams showing the situation for several different contexts.
• Reflecting Points Over Coordinate Axes
The goal of this task is to give students practice plotting points and their reflections.
• Running to School, Variation 2
This task builds on a fifth grade fraction multiplication task, "5.NF Running to School, Variation 1." This task uses the identical context, but asks the corresponding "Number of Groups Unknown" division problem. See "6.NS Running to School, Variation 3" for the "Group Size Unknown" version.
• Running to School, Variation 3
This task builds on a fifth grade fraction multiplication task, "5.NF Running to School, Variation 1." "6.NS Running to School, Variation 3" uses the identical context, but asks the corresponding "Group Size Unknown" division problem. See "6.NS Running to School, Variation 2" for the "Number of Groups Unknown" version.
• Sample Assessment Task: Cake Weighing
This four-part sample assessment task provides a novel situation in an authentic context that all students can understand, even though it is not likely to be in their everyday experience. Use the navigation at the upper right of this page to access the task.
• Sets and the Venn Diagram
This lesson is designed to help students understand the ideas surrounding sets and Venn diagrams.
• 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.
• Standing in Line
The purpose of this task is for students to solve a problem in context that can be solved in different ways, but in particular by dividing a whole number by a unit fraction.
• Tenths of (and So On)
The purpose of this task is to extend students' understanding of products of decimals by first focusing on products of unit fractions of the form 1/10n, where n is a positive integer. Students should have strategies from grade 5 for finding all of the products to the hundredths (5.NBT.B.7) that should easily extend to finding all of the products in this 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 Florist Shop
Students are given the scenario of a florist ordering roses and asked to find the smallest number of bunches she could order and explain their reasoning.
• 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.
• Traffic Jam
This task posits this word problem to students: "You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is 1 1/2 miles away. You are timing your progress and find that you can travel 2/3 of a mile in one hour. If you continue to make progress at this rate, how long will it be until you reach the exit? Solve the problem with a diagram and explain your answer."
• Using the Pythagorean Theorem on the Cartesian Graph
Students place animals on a Cartesian graph in this interactive activity. They then use the Pythagorean Theorem to determine the distance between the animals. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
• Video Game Credits
"It requires 1/4 of a credit to play a video game for one minute." Given this information, students are asked to answer questions about how long a student can play given a specific number of credits.
• 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.