Mathematics 6 (Grade 6)
Unit 1: The Number System (6.NS)
Throughout this unit, students use the meaning of fractions, the
meanings of multiplication and division, and the relationship between
multiplication and division to understand and explain why the
procedures for dividing fractions make sense. Students use these
operations to solve problems. Students extend their previous
understandings of number and the ordering of numbers to the full system
of rational numbers, which includes negative rational numbers, and in
particular negative integers. They reason about the order and absolute
value of rational numbers and about the location of points in all four
quadrants of the coordinate plane.
What will my child learn?
Students will:
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.5
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.
6.NS.C.6
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.
6.NS.C.6.A
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, e.g., (3) = 3, and that 0 is its own opposite.
6.NS.C.6.B
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.C.6.C
Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.7
Understand ordering and absolute value of rational numbers.
6.NS.C.7.A
Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret 3 > 7 as a statement that 3 is located to the right of 7 on a number line oriented from left to right.
6.NS.C.7.B
Write, interpret, and explain statements of order for rational numbers in realworld contexts. For example, write 3ºC > 7ºC to express the fact that 3ºC is warmer than 7ºC.
6.NS.C.7.C
Understand the absolute value of a rational numbers as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of 30 dollars, write 30 = 30 to describe the size of the debt in dollars.
6.NS.C.7.D
Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than 30 dollars represents a debt greater than 30 dollars.
6.NS.C.8
Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Solve realworld and mathematical problems involving area, surface area, and volume.
6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.
6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.
Compute fluently with multidigit numbers and find common factors and multiples.
6.NS.B.2
Fluently divide multidigit numbers using the standard algorithm.
6.NS.B.3
Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.
6.EE.A.1
Write and evaluate numerical expressions involving wholenumber exponents.
6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and 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 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Students will:
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.C.5
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.
 Check for Understanding: Interpreting Negative Numbers
 Review/Rewind: Intro to Negative Numbers
 Enrichment Tasks: It's Warmer in Miami  Mile High
6.NS.C.6
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.
 Review/Rewind: Missing Numbers on the Number Line Examples
 Enrichment Tasks: Logical Leaps  Fraction Game
6.NS.C.6.A
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, e.g., (3) = 3, and that 0 is its own opposite.
 Check for Understanding: Negative Numbers on the Number Line  Negative Numbers on the Number Line Without Reference to 0  Number Opposites
 Review/Rewind: Number Opposites
 Enrichment Tasks: Integers on the Number Line 2  Zip Zilch Zero
6.NS.C.6.B
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
 Check for Understanding: Graphing Points and Naming Quadrants  Points on the Coordinate Plane  Reflecting Points on the Coordinate Plane
 Review/Rewind: Points of the Coordinate Plane Examples
 Enrichment Tasks: Reflecting Points over Coordinate Axes
6.NS.C.6.C
Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
 Check for Understanding: Ordering Negative Numbers  Decimals on the Number Line
 Review/Rewind: Decimals and Fractions on a Number Line
 Enrichment Tasks: Cake Weighing  Line 'EM Up!
6.NS.C.7
Understand ordering and absolute value of rational numbers.
 Check for Understanding: Finding Absolute Values  Comparing Absolute Values
 Review/Rewind: Absolute Value of Integers
 Enrichment Tasks: Jumping Flea  Above and Below Sea Level
6.NS.C.7.A
Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret 3 > 7 as a statement that 3 is located to the right of 7 on a number line oriented from left to right.
 Check for Understanding: Comparing Positive and Negative Numbers on a the Number Line
 Review/Rewind: Ordering Negative Numbers
 Enrichment Tasks: Fractions on the Number Line  Integers on the Number Line
6.NS.C.7.B
Write, interpret, and explain statements of order for rational numbers in realworld contexts. For example, write 3ºC > 7ºC to express the fact that 3ºC is warmer than 7ºC.
 Check for Understanding: Writing Numerical Inequalities
 Review/Rewind: Negative Numbers, Variables, and Number Lines
 Enrichment Tasks: Comparing Temperatures
6.NS.C.7.C
Understand the absolute value of a rational numbers as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of 30 dollars, write 30 = 30 to describe the size of the debt in dollars.
 Review/Rewind: Absolute Value and Number Lines
 Enrichment Tasks: Zip, Zilch, Zero  How Much Did the Temperature Drop?
6.NS.C.7.D
Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than 30 dollars represents a debt greater than 30 dollars.
 Check for Understanding: Interpreting Absolute Values
 Review/Rewind: Comparing Absolute Values
6.NS.C.8
Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
 Check for Understanding: Coordinate Plane Problems in All 4 Quadrants
 Review/Rewind: Quadrants of the Coordinate Plane
 Enrichment Tasks: Distance Between Points
Solve realworld and mathematical problems involving area, surface area, and volume.
6.G.A.3
Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.
 Check for Understanding: Drawing Polygons  Drawing Polygons 2  Rectangles on the Coordinate Plane
 Review/Rewind: Drawing a Quadrilateral on the Coordinate Plane Example
 Enrichment Tasks: Polygons in a Coordinate Plane
6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?.
 Check for Understanding: Understanding Dividing Fractions by Fractions  Dividing Positive Fractions  Dividing Fractions by Fractions Word Problems
 Review/Rewind: Understanding Division of Fractions
 Enrichment Tasks: How Many Containers in One Cup /Cups in One Containers?  Traffic Jam
Compute fluently with multidigit numbers and find common factors and multiples.
6.NS.B.2
Fluently divide multidigit numbers using the standard algorithm.
 Check for Understanding: MultiDigit Division
 Review/Rewind: Intro to Long Division
 Enrichment Tasks: Interpreting a Division Computation  How Many Staples?
6.NS.B.3
Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.
 Check for Understanding: Adding Decimals  Subtracting Decimals  Adding and Subtracting Decimals Word Problems  Multiplying Decimals  Dividing Decimals
 Review/Rewind: Operations with Decimals I  Operations with Decimals 2  Operations with Decimals Subtract  Operations with Decimals Subtract 2
 Enrichment Tasks: Jayden's Snacks  Pennies to Heaven  Setting Goals
6.EE.A.1
Write and evaluate numerical expressions involving wholenumber exponents.
 Check for Understanding: Intro to Exponents  Positive and Zero Exponents  Writing Expressions with Variables  Exponents  Evaluating Expressions with Exponents
 Review/Rewind: Order of Operations Examples: Exponents
 Enrichment Tasks: Sierpinski's Carpet  Seven to the What?
6.NS.B.4
Find the greatest common factor of two whole numbers less than or equal to 100 and 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 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
 Check for Understanding: Greatest Common Factor (GCF)  Least Common Multiple (LCM)  GCF and LCM Word Problems  Distributive Property
 Review/Rewind: Greatest Common Factor Explained  Least Common Multiple
 Enrichment Tasks: Adding Multiples  Florist Shop
What are some signs of student mastery?

Tools & Technology Catch the Fly (Hotmath): Students name the coordinates the fly has to land on so the frog can catch the fly. (Adobe Flash required) Billy Bug (Oswego): Students move Billy Bug using arrows to the coordinates of the hidden grub. (Adobe Flash required) PickaPath (NCTM Illuminations): Students can use all operations as they navigate a maze involving integers fractions, decimals, and exponents. (Downloadable as an app) 
More 4 U
View a video on how to divide fractions with tape diagrams. 