Mathematics 6 (Grade 6)
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 should my child know and be able to do?
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 real-world 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 real-world 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 real-world 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 real-world 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 real-world 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 real-world 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 3/4-cup 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 multi-digit numbers and find common factors and multiples.
6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.
6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
6.EE.A.1
Write and evaluate numerical expressions involving whole-number 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 1-100 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 real-world 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 real-world 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 real-world 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 real-world 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 real-world 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 real-world 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 3/4-cup 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 multi-digit numbers and find common factors and multiples.
6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.
- Check for Understanding: Multi-Digit 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 multi-digit 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 whole-number 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 1-100 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?
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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) Pick-a-Path (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. |