Mathematics 7 (Grade 6/7)
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.
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 (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 real-world 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 real-world 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 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 (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 real-world 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 real-world 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 real-world 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 non-zero 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 real-world 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 real-world and mathematical problems involving the four operations with rational numbers.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3
Solve multi-step real-life 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 whole-number 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 real-world 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 real-world 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 real-world 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).
- 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 (For students in Grade 6 only).
- Review/Rewind: Interpreting Absolute Values
- Check for Understanding: Interpreting 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 (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.
- 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.
- 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 real-world contexts.
- 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 real-world contexts.
- 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.
- 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.
- 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 real-world contexts.
- 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 non-zero 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 real-world contexts.
- 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.
- 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.
- Check for Understanding: Converting Fractions to Decimals | Writing Fractions and Repeating Decimals
- Review/Rewind: Converting Fractions to Decimals
- Enrichment Tasks: Equivalent fractions approach to non-repeating decimals* | Repeating decimal as approximation*
7.NS.A.3
Solve real-world and mathematical problems involving the four operations with rational numbers.
- 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 real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3
Solve multi-step real-life 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.
- Check for Understanding: Discount, Tax, and Tip Word Problems | Mark-up 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 whole-number 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?
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Tools & Technology
Integer Football (Math Goodies): Students model integer addition and subtraction using a football field as a real-world 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. |