Mathematics 7 (Grade 6/7)
Unit 1: The Number System (7.NS)
During this unit, students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers.
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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 (For students in Grade 6 only).
6.NS.C.7
Understand ordering and absolute value of rational numbers (For students in Grade 6 only).
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 (For students in Grade 6 only).
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 (For students in Grade 6 only).
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 (For students in Grade 6 only).
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 (For students in Grade 6 only).
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 (For students in Grade 6 only).
Apply and extend previous understandings of operations with fractions.
7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.1.A
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
7.NS.A.1.B
Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.
7.NS.A.1.C
Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.
7.NS.A.1.D
Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
7.NS.A.2.A
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.
7.NS.A.2.B
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.
7.NS.A.2.C
Apply properties of operations as strategies to multiply and divide rational numbers.
7.NS.A.2.D
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
7.NS.A.3
Solve realworld and mathematical problems involving the four operations with rational numbers.
Solve reallife and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3
Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.1
Write and evaluate numerical expressions involving wholenumber exponents (For students in Grade 6 only).
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 (For students in Grade 6 only).
 Check for Understanding: Interpreting Negative Numbers
 Review/Rewind: Intro to Negative Numbers
 Enrichment Tasks: It's Warmer in Miami  Mile High
6.NS.C.7
Understand ordering and absolute value of rational numbers (For students in Grade 6 only).
 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 (For students in Grade 6 only).
 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 (For students in Grade 6 only).
 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 (For students in Grade 6 only).
 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 (For students in Grade 6 only).
 Background Info. & Guided Practice
 Review/Rewind: Interpreting Absolute Values
 Check for Understanding: Interpreting 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 (For students in Grade 6 only).
 Check for Understanding: Coordinate Plane Problems in All 4 Quadrants
 Review/Rewind: Quadrants of the Coordinate Plane
 Enrichment Tasks: Distance Between Points
Apply and extend previous understandings of operations with fractions.
7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
 Background Info.
 Check for Understanding: Ordering Expressions  Adding Negative Numbers  Subtracting Negative Numbers  Negative Numbers Addition & Subtraction Word Problems
 Review/Rewind: Adding & Subtracting with Negatives on the Number Line
 Enrichment Tasks: Operations on the number line*  Differences and Distances
7.NS.A.1.A
Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.
 Background Info. & Guided Practice
 Review/Rewind: Number Opposites
 Enrichment Tasks: Bookstore Account  Distances on the Number Line 2*
7.NS.A.1.B
Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts.
 Background Info. & Guided Practice
 Check for Understanding: Absolute Value to Find Distance Challenge  Ordering Expressions
 Review/Rewind: Adding Negative Numbers  Adding Numbers with Different Signs
7.NS.A.1.C
Understand subtraction of rational numbers as adding the additive inverse, p  q = p + (q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts.
 Background Info. & Guided Practice
 Check for Understanding: Subtracting Negative Numbers
 Review/Rewind: Subtracting Integers by Additive Inverse
 Enrichment Tasks: Distances Between Houses*  Comparing Freezing Points*
7.NS.A.1.D
Apply properties of operations as strategies to add and subtract rational numbers.
 Background Info. & Guided Practice
 Check for Understanding: Adding and Subtracting Negative Fractions  Adding and Subtracting Rational Numbers
 Review/Rewind: Adding & Subtracting Positive/Negative Fractions
7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
 Background Info.
 Check for Understanding: Multiplying and Dividing Negative Numbers  Multiplying Positive and Negative Fractions
 Review/Rewind: Multiplying Fractions
7.NS.A.2.A
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (1)(1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts.
 Background Info. & Guided Practice
 Check for Understanding: Multiplying and Dividing Negative Numbers  Multiplying Positive and Negative Fractions
 Review/Rewind: Multiplying Positive & Negative Numbers  Dividing Positive & Negative Numbers  Multiply positive and negative integers using the distributive property
7.NS.A.2.B
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then (p/q) = (p)/q = p/(q). Interpret quotients of rational numbers by describing realworld contexts.
 Background Info. & Guided Practice
 Check for Understanding: Dividing Positive and Negative Fractions
 Review/Rewind: Understanding Fractions as Division  Negative Signs in Fractions
7.NS.A.2.C
Apply properties of operations as strategies to multiply and divide rational numbers.
 Background Info. & Guided Practice
 Check for Understanding: Multiplying Positive and Negative Fractions  Multiplying and Dividing Negative Numbers
 Review/Rewind: Dividing Negative Fractions  Multiplying Negative and Positive Fractions
7.NS.A.2.D
Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
 Background Info.
 Check for Understanding: Converting Fractions to Decimals  Writing Fractions and Repeating Decimals
 Review/Rewind: Converting Fractions to Decimals
 Enrichment Tasks: Equivalent fractions approach to nonrepeating decimals*  Repeating decimal as approximation*
7.NS.A.3
Solve realworld and mathematical problems involving the four operations with rational numbers.
 Background Info. & Guided Practice
 Check for Understanding: Adding & subtracting decimals word problems
 Review/Rewind: Comparing Negative Numbers
 Enrichment Tasks: Sharing Prize Money*  Anne's Family Trip  School Supplies
Solve reallife and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3
Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
 Background Info. & Guided Practice
 Check for Understanding: Discount, Tax, and Tip Word Problems  Markup and Commission Word Problems  Rational Number Word Problems
 Review/Rewind: Percent Word Problems
 Enrichment Tasks: Who is the better batter?  Spicy Veggies  Stained Glass*
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A.1
Write and evaluate numerical expressions involving wholenumber exponents (For students in Grade 6 only).
 Check for Understanding: Positive and Zero Exponents  Powers of Ten  Intro to Exponents  Exponents with Integer Bases  Evaluating Exponent Expression Word Problems
 Review/Rewind: Intro to Exponents
 Enrichment Tasks: Sierpinski's Carpet  Seven to the What?
What are some signs of student mastery?

Tools & Technology
Integer Football (Math Goodies): Students model integer addition and subtraction using a football field as a realworld number line. Digit Drop: Students are given a number sentence and they are required to drop the correct integer into the sentence to make a true statement. Zip, Zero, Zilch Game: Directions and rules for card game to help students practice applying properties of operations as a strategy. Multiplying Fractions Millionaire Game: Student can play as single player or play against his parents/friends. Game play is based on the Who Wants to be a Millionaire format and has 50/50 and phone a friend option. Game starts with multiplication of simple fractions and ends with multiplication of improper fractions. (Requires students simplify fractions as well.) Dividing Fractions Millionaire Game: Student can play as single player or play against his parents/friends. Game play is based on the Who Wants to be a Millionaire format and has 50/50 and phone a friend option. Game starts with division of simple fractions and ends with division of improper fractions. (Requires students simplify fractions as well.) These videos show how the number line and counters can support student understanding of: 
More 4 U
This video shows how problems can be solved using number line diagrams when dividing fractions. 