ALEX Classroom Resource

  

Grade 7 Mathematics Module 2, Topic C: Applying Operations With Rational Numbers to Expressions and Equations

  Classroom Resource Information  

Title:

 Grade 7 Mathematics Module 2, Topic C: Applying Operations With Rational Numbers to Expressions and Equations

URL:

 https://www.engageny.org/resource/grade-7-mathematics-module-2-topic-c-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Topic C, students problem-solve with rational numbers and draw upon their work from Grade 6 with expressions and equations. They perform operations with rational numbers, incorporating them into algebraic expressions and equations.  They represent and evaluate expressions in multiple forms, demonstrating how quantities are related. The Integer Game is revisited as students discover “if-then” statements, relating changes in players’ hands (who have the same card-value totals) to changes in both sides of a number sentence. Students translate word problems into algebraic equations and become proficient at solving equations of the form px + q = r and p(x + q) = r, where p, q, and r, are specific rational numbers. As they become fluent in generating algebraic solutions, students identify the operations, inverse operations, and order of steps, comparing these to an arithmetic solution. The use of algebra to represent contextual problems continues in Module 3. 

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 7
4. Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.

a. Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.

b. Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts.

c. Explain subtraction of rational numbers as addition of additive inverses.

d. Use a number line to demonstrate 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.

e. Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.

f. Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.

g. Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Explain situations where opposite quantities combine to make zero, known as additive inverses.
  • Apply their knowledge of addition and subtraction of rational numbers to describe real-world contexts.
  • Add and subtract rational numbers using number lines to show connection to distance
  • Explain the connection between subtraction and addition of additive inverses.
  • Model multiplication and division of rational numbers (number horizontal and vertical number lines, integer chips, bar models).
  • Use properties of operations to multiply signed numbers.
  • Convert rational numbers to a decimal using long division and determine if the result is terminating or repeating.
Teacher Vocabulary:
  • Integers
  • Rational numbers
  • Additive inverses
  • opposite quantities
  • Absolute value
  • Terminating decimals
  • Repeating decimals
Knowledge:
Students know:
  • a number and its opposite have a sum of 0.
  • A number and its opposite are called additive inverses.
  • Strategies for adding and subtracting two or more numbers.
  • Absolute value represents distance on a number line, therefore it is always non-negative.
  • Strategies for multiplying signed numbers.
  • 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).
  • The decimal form of a rational number terminates or eventually repeats.
Skills:
Students are able to:
  • add rational numbers.
  • Subtract rational numbers.
  • Represent addition and subtraction on a number line diagram.
  • Describe situations in which opposite quantities combine to make 0.
  • Find the opposite of a number.
  • Interpret sums of rational numbers by describing real-world contexts.
  • Show that the distance between two rational numbers on the number line is the absolute value of their difference.
  • Use absolute value in real-world contexts involving distances.
  • Multiply and divide rational numbers.
  • Convert a rational number to a decimal using long division.
Understanding:
Students understand that:
  • finding sums and differences of rational numbers (negative and positive) involves determining direction and distance on the number line.
  • Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
  • If a factor is multiplied by a number greater than one, the answer is larger than that factor.
  • If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
  • Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
  • Integers can be divided, provided that the divisor is not zero.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.7.4.1: Define rational numbers, horizontal, and vertical.
M.7.4.2: Recall how to extend a horizontal number line.
M.7.4.3: Recall how to extend a vertical number line.
M.7.4.4: Demonstrate addition and subtraction of whole numbers using a horizontal or vertical number line.
M.7.4.5: Give examples of rational numbers.
M.7.4.6: Define absolute value and additive inverse.
M.7.4.7: Explain that the sum of a number and its opposite is zero.
M.7.4.8: Locate positive, negative, and zero numbers on a number line.
M.7.4.9: Recall properties of addition and subtraction.
M.7.4.10: Model addition and subtraction using manipulatives.
M.7.4.11: Show addition and subtraction of 2 or more rational numbers using a number line within real-world context.
M.7.4.12: Define absolute value and additive inverse.
M.7.4.13: Show subtraction as the additive inverse.
M.7.4.14: Give examples of the opposite of a given number.
M.7.4.15: Show addition and subtraction using a number line.
M.7.4.16: Discuss various strategies for solving real-world and mathematical problems.
M.7.4.17: Identify properties of operations for addition and subtraction.
M.7.4.18: Recall the steps for solving addition and subtraction of rational numbers.
M.7.4.19: Identify the difference between two rational numbers on a number line.
M.7.4.20: Recall the steps of solving multiplication of rational numbers.
M.7.4.21: Identify the pattern for multiplying signed numbers.
M.7.4.22: Recall the steps of solving division of rational numbers.
M.7.4.23: Explain that dividing a rational number zero is undefined.
M.7.4.24: Recall that a fraction can be written as a division problem.
M.7.4.25: Recall the steps to divide two rational numbers.
M.7.4.26: Identify whether a decimal is terminating or repeating.

Prior Knowledge Skills:
  • Define parentheses, braces, and brackets.
  • Recall addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • Identify two fractions as equivalent (equal) if they are the same size or the same point on a number line.
  • Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
  • Generate equivalent fractions.
  • Show on a number line that numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.
  • Define rational number.
  • Arrange integers and /or rational numbers on a horizontal or vertical number line.
  • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
  • Identify a rational number as a point on the number line.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.7.4 Add and subtract integers up to 15.

Mathematics
MA2019 (2019)
Grade: 7
5. Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Apply their knowledge of addition, subtraction, multiplication and division of rational numbers to describe real-world contexts.
  • Solve multi-step problems using numerical expressions that involve addition, subtraction, multiplication, and/or division of rational numbers, including problems that involve complex fractions
Teacher Vocabulary:
  • Rational numbers
  • Complex fractions
  • properties of operations
Knowledge:
Students know:
  • how to model real-world problems to include situations involving elevation, temperature changes, debits and credits, and proportional relationships with negative rates of change.
  • how to evaluate numerical expressions with greater fluency, using the properties of operations when necessary.
Skills:
Students are able to:
  • Solve real-world and mathematical problems involving the four operations with rational numbers.
Understanding:
Students understand that:
  • rational numbers can represent values in real-world situations.
  • properties of operations learned with whole numbers in elementary apply to rational numbers.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.7.5.1: Discuss various strategies for solving real-world and mathematical problems.
M.7.5.2: Recall steps for solving fractional problems.
M.7.5.3: Identify properties of operations for addition and multiplication.
M.7.5.4: Recall the rules for multiplication and division of rational numbers.
M.7.5.5: Recall the rules for addition and subtraction of rational numbers.

Prior Knowledge Skills:
  • Recall addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • Define rational number.
  • Arrange integers and /or rational numbers on a horizontal or vertical number line.
  • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
  • Identify a rational number as a point on the number line.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.7.5 Solve multiplication problems up to fifteen with whole number factors.
Mathematics
MA2019 (2019)
Grade: 7
Accelerated
8. Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.

a. Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses. 

b. Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts.

c. Explain subtraction of rational numbers as addition of additive inverses.

d. Use a number line to demonstrate 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.

e. Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.

f. Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.

g. Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats. [Grade 7, 4]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Apply their knowledge of addition and subtraction of rational numbers to describe real-world contexts.
  • Use physical and visual models to add and subtract integers.
  • Add and subtract rational numbers.
  • Model multiplication and division of rational numbers.
  • Apply the distributive property to rational numbers.
  • Convert rational numbers to a decimal using long division to determine if the result is terminating or repeating.
Teacher Vocabulary:
  • Integers
  • Rational numbers
  • Additive inverses
  • opposite quantities
  • Absolute value
  • Terminating decimals
  • Repeating decimals
Knowledge:
Students know:
  • a number and its opposite have a sum of 0.
  • A number and its opposite are called additive inverses.
  • properties of operations.
  • Absolute value represents distance on a number line, therefore it is always non-negative.
  • 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).
  • The decimal form of a rational number terminates in 0s or eventually repeats.
Skills:
Students are able to:
  • add rational numbers.
  • Subtract rational numbers.
  • Represent addition and subtraction on a number line diagram.
  • Describe situations in which opposite quantities combine to make 0.
  • Find the opposite of a number.
  • Interpret sums of rational numbers by describing real-world contexts.
  • Show that the distance between two rational numbers on the number line is the absolute value of their difference.
  • Use absolute value in real-world contexts involving distances.
  • Multiply and divide rational numbers.
  • Convert a rational number to a decimal using long division.
Understanding:
Students understand that:
  • p + q is the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative.
  • Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
  • If a factor is multiplied by a number greater than one, the answer is larger than that factor.
  • If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
  • 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.
  • Integers can be divided, provided that the divisor is not zero.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 7
Accelerated
9. Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable. [Grade 7, 5]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Apply knowledge of addition and subtraction of rational numbers to describe real-world contexts.
  • Solve multi-step problems using numerical expressions that involve addition, subtraction, multiplication, and/or division of rational numbers, including problems that involve complex fractions.
Teacher Vocabulary:
  • Rational numbers
  • Complex fractions
  • properties of operations
Knowledge:
Students know:
  • Include situations involving elevation, temperature changes, debits and credits, and proportional relationships with negative rates of change.
  • Develop greater fluency with evaluating numerical expressions, using the properties of operations to increase their flexibility in approach.
Skills:
Students are able to:
  • Interpret products and quotients of rational numbers by describing real-world contexts.
  • Solve real-world and mathematical problems involving the four operations with rational numbers.
Understanding:
Students understand that:
  • it important to be able to write numeric expressions in multiple ways.
  • Rational numbers can represent values in real-world situations.
  • properties of operations learned with whole numbers in elementary apply to rational numbers
Diverse Learning Needs:
Tags: add, complex fraction, decimals, divide, fractions, integer, multiply, operations, properties, rational number, realworld, signed fraction, subtract, whole numbers
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There are seven lessons in this topic.

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Author: Hannah Bradley
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