enVision Mathematics Topic 4
6th Grade; November – December (5 weeks); 2nd Quarter
Embedded Files
Topic Title(s):
Represent and Solve Equations and Inequalities
Prepared Graduates:
MP2. Reason abstractly and quantitatively.
MP4. Model with mathematics.
MP6. Attend to precision.
Standard(s):
2. Algebra and Functions
The highlighted evidence outcomes are the priority for all students, serving as the essential concepts and skills. It is recommended that the remaining evidence outcomes listed be addressed as time allows, representing the full breadth of the curriculum.
Students Can (Evidence Outcomes):
6.EE.B. Expressions & Equations: Reason about and solve one-variable equations and inequalities.
Describe solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. (CCSS: 6.EE.B.5)
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; recognize that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. (CCSS: 6.EE.B.6)
Solve real-world and mathematical problems by writing and solving equations of the form x ± p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. (CCSS: 6.EE.B.7)
Write an inequality of the form x > c, x ≥ c, x < c, or x ≤ c to represent a constraint or condition in a real-world or mathematical problem. Show that inequalities of the form x > c, x ≥ c, x < c, or x ≤ c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. (CCSS: 6.EE.B.8)
6.EE.C. Expressions & Equations: Represent and analyze quantitative relationships between dependent and independent variables.
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. (CCSS: 6.EE.C.9)
Investigate unknown values to form hypotheses, make observations, and draw conclusions. (Entrepreneurial Skills: Inquiry/Analysis)
Reason about the values and operations of an equation both within a real-world context and abstracted from it. (MP2)
State precisely the meaning of variables used when setting up equations. (MP6)
Analyze relationships between dependent and independent variables.
Reason about the operations that relate constant and variable quantities in equations with dependent and independent variables. (MP2)
Model with mathematics by describing real-world situations with equations and inequalities. (MP4)
Inquiry Questions
What are the different ways a variable can be used in an algebraic equation or inequality? For example, how are these uses of the variable x different from each other? (a) x + 5 = 8; (b) x = 1/2; (c) x > 5.
How is the solution to an inequality different than a solution to an equation?
How can you determine if a variable is the independent variable or the dependent variable?
What are the advantages of showing the relationship between an independent and dependent variable in multiple representations (table, graph, equation)?
Coherence Connections
This expectation represents major work of the grade.
In previous grades, students write simple expressions that record calculations with numbers, interpret numerical expressions without evaluating them, and generate ordered pairs from two numerical rules.
In Grade 5, students analyze numerical patterns and relationships, including generating and graphing ordered pairs in the first quadrant.
This expectation connects to several others in Grade 6: (a) understanding ratio concepts and use ratio reasoning to solve problems, (b) applying and extending previous understandings of multiplication and division to divide fractions by fractions, (c) applying and extending previous understandings of numbers to the system of rational numbers, (d) applying and extending previous understandings of arithmetic to algebraic expressions, and (e) representing and analyzing quantitative relationships between dependent and independent variables.
In Grade 7, students solve real-life and mathematical problems involving two-step equations and inequalities. In Grade 8, students work with radicals and integer exponents and solve linear equations and pairs of simultaneous linear equations.
In Grade 7, students decide if two quantities are in a proportional relationship and identify the unit rate in tables, graphs, equations, diagrams, and verbal descriptions.
Academic Vocabulary & Language Expectations:
Equation, solution of an equation, Addition Property of Equality, Subtraction Property of Equality, Multiplication Property of Equality, Division Property of Equality, inverse relationship, inequality, dependent variable, independent variable
Assessments:
Instructional Resources & Notes:
enVision Mathematics Topic 4
Let's Investigate! Tasks
Let's Investigate! Operation Equation (TE) (relates to Lesson 4-2)
Let's Investigate! The Write Way (TE) (relates to Lesson 4-5)
5th-to-6th Grade Warm-Ups for Activating Prior Knowledge (Geoff Krall)
Tier 1 Intervention & Supports (i-Ready Tools for Instruction):
Tier 1 Intervention: Test Solutions of Equations, Solve Equations, Write Inequalities, Represent Inequalities on Number Lines, Relate Independent and Dependent Variables
Coherence Map/Concept Progressions: 6.EE.B.6, 6.EE.B.7, 6.EE.B.8, 6.EE.C.9
enVision Mathematics 6-8 & Number Worlds Connections (for SVVSD Special Education teachers only)
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