ALEX Classroom Resource

  

Solving Linear Equations with Negative Numbers

  Classroom Resource Information  

Title:

 Solving Linear Equations with Negative Numbers

URL:

 https://aptv.pbslearningmedia.org/resource/mgbh.math.ee.equation/solving-linear-equations-with-negative-numbers/

Content Source:

 PBS
Type: Audio/Video

Overview:

Solve a linear equation that has negative numbers and a variable. This video focuses on using inverse operations to solve for a variable.

This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers and is part of the Math at the Core: Middle School collection.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 6
15. Write, read, and evaluate expressions in which letters represent numbers in real-world contexts.

a. Interpret a variable as an unknown value for any number in a specified set, depending on the context.

b. Write expressions to represent verbal statements and real-world scenarios.

c. Identify parts of an expression using mathematical terms such as sum, term, product, factor, quotient, and coefficient.

d. Evaluate expressions (which may include absolute value and whole number exponents) with respect to order of operations.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given contextual or mathematical problems both when known models exist (for example formulas) or algebraic models are unknown,
  • Interpret the parts of the model in the original context.
  • Create the algebraic model of the situation when appropriate.
  • Use appropriate mathematical terminology to communicate the meaning of the expression.
  • Evaluate the expressions for values of the variable including finding values following conventions of parentheses and order of operations.
Teacher Vocabulary:
  • Expressions
  • Term
  • Coefficient
  • Sum
  • Product
  • Factor
  • Quotient
  • Variable
  • Constant
  • Difference
  • Evaluate
  • Order of Operations
  • Exponent
  • Absolute Value
Knowledge:
Students know:
  • Correct usage of mathematical symbolism to model the terms sum, term, product, factor, quotient, variable, difference, constant, and coefficient when they appear in verbally stated contexts.
  • Conventions for order of operations.
  • Convention of using juxtaposition (5A or xy) to indicate multiplication.
Skills:
Students are able to:
  • Translate fluently between verbally stated situations and algebraic models of the situation.
  • Use operations (addition, subtraction, multiplication, division, and exponentiation) fluently with the conventions of parentheses and order of operations to evaluate expressions for specific values of variables in expressions.
  • Use terminology related to algebraic expressions such as sum, term, product, factor, quotient, or coefficient, to communicate the meanings of the expression and the parts of the expression.
Understanding:
Students understand that:
  • The structure of mathematics allows for terminology and techniques used with numerical expressions to be used in an analogous way with algebraic expressions, (the sum of 3 and 4 is written as 3 + 4, so the sum of 3 and y is written as 3 + y).
  • When language is ambiguous about the meaning of a mathematical expression grouping, symbols and order of operations conventions are used to communicate the meaning clearly.
  • Moving fluently among representations of mathematical situations (words, numbers, symbols, etc.), as needed for a given situation, allows a user of mathematics to make sense of the situation and choose appropriate and efficient paths to solutions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.15.1: Define algebraic expression and variable.
M.6.15.2: Convert mathematical terms to mathematical symbols and numbers.
M.6.15.3: Translate verbal and numerical expression using all operations.
M.6.15.4: Define coefficient, constant and term.
M.6.15.5: Match mathematical terms with correct mathematical symbols.
M.6.15.6: Convert mathematical terms to mathematical symbols and numbers.
M.6.15.7: Calculate an expression in the correct order. with or without a calculator (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
M.6.15.8: Choose the correct value to replace each variable in the algebraic expression (Substitution).
M.6.15.9: Calculate a numerical expression, with or without a calculator (Ex. V=4x4x4).
M.6.15.10: Recognize the correct order to solve expressions with more than one operation.

Prior Knowledge Skills:
  • Recognize key terms to solve word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Recognize key terms to solve word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Define simple expression.
  • Recall simple equations.
  • Recognize properties of addition and multiplication.
  • Recall addition, subtraction, multiplication, division symbols.
  • Define parentheses, braces, and brackets.
  • Define numerical expression.
  • Recognize expressions.
  • Apply properties of operations as strategies to add and subtract.
  • Recall properties of operations as strategies to add and subtract.
  • Represent addition and subtraction with objects, mental images, drawings, expressions, or equations.
  • Use addition, subtraction, multiplication and division to solve one- and two-step word problems.
  • Recognize key terms to solve word problems.
  • Apply properties of operations as strategies to multiply and divide.
  • Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.
  • Recall the formula for area (L × W).
  • Recognize that unit squares are equal.
  • Recall the formula for perimeter (P= L+L+W+W or P=2L + 2W).
  • Recall basic addition and multiplication facts.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.15 Evaluate algebraic expressions when given specific values for the variables (e.g. x + 2, where x = 4).
Mathematics
MA2019 (2019)
Grade: 7
9. Use variables to represent quantities in real-world or mathematical problems and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality, and interpret it in the context of the problem.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Write and solve mathematical equations (or inequalities) to model real-world problems.
  • Interpret the solution to an equation in the context of a problem
  • Interpret the solution set of an inequality in the context of a problem.
  • Graph the solution to an inequality on a number line.
Teacher Vocabulary:
  • Algebraic expressions
  • Equations
  • Inequalities
  • Greater than
  • Greater than or equal to
  • less than
  • less than or equal to
Knowledge:
Students know:
  • p(x + q) = px + pq, where p and q are specific rational numbers.
  • When multiplying or dividing both sides of an inequality by a negative number, every term must change signs and the inequality symbol reversed.
  • In the graph of an inequality, the endpoint will be a closed circle indicating the number is included in the solution set (≤ or ≥) or an open circle indicating the number is not included in the solution set ( < or >).
Skills:
Students are able to:
  • use variables to represent quantities in a real-world or mathematical problem.
  • Construct equations (px + q = r and p(x + q) = r) to solve problems by reasoning about the quantities.
  • Construct simple inequalities (px + q > r or px + q < r) to solve problems by reasoning about the quantities.
  • Graph the solution set of an inequality.
Understanding:
Students understand that:
  • Real-world problems can be represented through algebraic expressions, equations, and inequalities.
  • Why the inequality symbol reverses when multiplying or dividing both sides of an inequality by a negative number.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.7.9.1: Define equation, inequality, and variable.
M.7.9.2: Set up equations and inequalities to represent the given situation, using correct mathematical operations and variables.
M.7.9.3: Calculate a solution or solution set by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.4: Test the found number or number set for accuracy by substitution.
M.7.9.5: Recall solving one step equations and inequalities.
M.7.9.6: Recognize properties of numbers (Distributive, Associative, Commutative).
M.7.9.7: Define equation and variable.
M.7.9.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.
M.7.9.9: Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.10: Test the found number for accuracy by substitution.
Example: Is 5 an accurate solution of 2(x + 5)=12?.
M.7.9.11: Identify the unknown, in a given situation, as the variable.
M.7.9.12: List given information from the problem.
M.7.9.13: Recalling one-step equations.
M.7.9.14: Explain the distributive property.
M.7.9.15: Define inequality and variable.
M.7.9.16: Set up an inequality to represent the given situation, using correct mathematical operations and variables.
M.7.9.17: Calculate a solution set to an inequality by combining like terms, isolating the variable, and/or using inverse operations.
M.7.9.18: Test the solution set for accuracy.
M.7.9.19: Identify the unknown, of a given situation, as the variable.
M.7.9.20: List information from the problem.
M.7.9.21: Recall how to graph inequalities on a number line.
M.7.9.22: Recall how to solve one step inequalities.

Prior Knowledge Skills:
  • Define inequality.
  • Define equivalent, simplify, term, distributive property, associative property of addition and multiplication, and the commutative property of addition and multiplication.
  • Define equation, solution of an equation, solution of an inequality, and inequality.
  • Compare and contrast equations and inequalities.
  • Determine if an inequality is by replacing the variable with a given number.
  • Determine if an equation is true by replacing the variable with a given number.
  • Simplify a numerical sentence to determine equivalence.
  • Recognize the symbols for =, >, <, ?, and ?.
  • Define equation and variable.
  • Set up an equation to represent the given situation, using correct mathematical operations and variables.
  • Identify the unknown, in a given situation, as the variable.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.7.9 Use the properties of operations to solve one-step equations and inequalities from real-world and mathematical problems.
Mathematics
MA2019 (2019)
Grade: 7
Accelerated
18. Use variables to represent quantities in a real-world or mathematical problem and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
[Grade 7, 9, and linear portion of Algebra I with Probability, 11]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Write and solve mathematical equations (or inequalities) to model real-world problems.
  • Interpret the solution to an equation in the context of a problem.
  • Interpret the solution set of an inequality in the context of a problem.
Teacher Vocabulary:
  • Algebraic expressions
  • Equations
  • Inequalities
  • Greater than
  • Greater than or equal to
  • less than
  • less than or equal to
Knowledge:
Students know:
  • p(x + q) = px + pq, where p and q are specific rational numbers.
  • When multiplying or dividing both sides of an inequality by a negative number, every term must change signs and the inequality symbol reversed.
  • In the graph of an inequality, the endpoint will be a closed circle indicating the number is included in the solution set (≤ or ≥) or an open circle indicating the number is not included in the solution set ( < or >).
Skills:
Students are able to:
  • Use variables to represent quantities in a real-world or mathematical problem.
  • Construct equations (px + q = r and p(x + q) = r) to solve problems by reasoning about the quantities.
  • Construct simple inequalities (px + q > r or px + q < r) to solve problems by reasoning about the quantities.
  • Graph the solution set of an inequality.
Understanding:
Students understand that:
  • Real-world problems can be represented through algebraic expressions, equations, and inequalities.
  • The inequality symbol reverses when multiplying or dividing both sides of an inequality by a negative number, and why.
Diverse Learning Needs:
Tags: inverse operations, linear, opposite, variable
License Type: Public Domain
For full descriptions of license types and a guide to usage, visit :
https://creativecommons.org/licenses
AccessibilityVideo resources: includes closed captioning or subtitles
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Author: Kristy Lacks
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