Equations and Inequalities
Students discover how to use equations and inequalities to model relationships between quantities, and investigate the meaning of having a solution to an equation or an inequality.
Unit Summary
In Unit 6, sixth graders move from expressions to equations and inequalities. They revisit familiar diagrams such as tape diagrams to model equations, and they discover new models such as balances and hanging mobiles. Students investigate what it means to be a solution to an equation or an inequality and how use equations and inequalities to model relationships between quantities. When using an equation or inequality to represent real-world situations, students must decontextualize the situation to represent it using variables and symbols and then recontextualize in order to interpret what their answer means in regard to the situation at hand (MP.2). In this unit, students bring concepts from three domains together: Ratios and Proportions, Number Sense, and Expressions and Equations. They re-visit percentages from Unit 2 and solve percent problems using equations. They study relationships between different quantities and draw on their ratio reasoning where relevant. A note on fluency: solving equations provides a good opportunity for students to continue development of and to demonstrate fluency with decimal operations and fraction division. Several problems involve computing with decimals and dividing by fractions; include additional problems in practice for students as needed.
Several prior skills support students in this unit. In fifth grade, students analyzed patterns and relationships when they studied standard 5.OA.3. They also observed what happened when these relationships were plotted on the coordinate plane. In previous sixth-grade units, students studied algebraic and numerical expressions and collections of equivalent ratios. Students draw on all of these concepts and skills in this unit.
There are many future connections to the standards in this unit. In seventh grade, students will deeply investigate proportional relationships in the form $$y=rx$$, understanding the value of $$r$$ as the constant of proportionality. They’ll further investigate the graphs of these equations, and in eighth grade, students will compare across multiple representations of proportional relationships. Students will also become exposed to increasingly more complex equations and inequalities to solve.
Pacing: 17 instructional days (14 lessons, 2 flex days, 1 assessment day)
For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 6th Grade Scope and Sequence Recommended Adjustments.
Assessment
This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.
Unit Prep
Intellectual Prep
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Internalization of Standards via the Post-Unit Assessment
- Take the Post-Unit Assessment. Annotate for:
- Standards that each question aligns to
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that Assessment points to
Internalization of Trajectory of Unit
- Read and annotate the Unit Summary.
- Notice the progression of concepts through the unit using the Lesson Map.
- Do all Target Tasks. Annotate the Target Tasks for:
- Essential Understandings
- Connection to Post-Unit Assessment questions
- Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.
Unit-Specific Intellectual Prep
- Read Progressions for the Common Core State Standards in Mathematics, 6-7, Ratios and Proportional Relationships and Progressions for the Common Core State Standards in Mathematics, 6-8, Expressions and Equations for the standards relevant to this unit.
- Read the following table that includes models used throughout the unit.
| Model | Example |
| Tape diagram |  |
| Balance/mobile |  |
Essential Understandings
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- A solution to an equation or inequality represents the value(s) for the variable that, when substituted in, make the equation or inequality a true statement. The solution set to an inequality can be represented using a ray on a number line.
- Equations are statements of balance between two expressions. In order to solve for a variable in an equation, whatever actions are taken on one side of the equation must also be taken on the other side in order to maintain the balance.
- An equation can be used to model the association between two quantities where one quantity is considered the independent variable and the other quantity is the dependent variable. These relationships can be represented in a table of values and graphed in the coordinate plane.
Materials
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- Tape diagram
- Balance/mobile (optional)
- Number line
- Coordinate plane
- graph paper (optional)
Vocabulary
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inequality
equation
solution
substitution
percent equation
independent variable
dependent variable
Related Teacher Tools:
Lesson Map
Topic A: Reasoning About and Solving Equations
1
6.EE.B.6
6.EE.B.7
Represent equations in the form $${ x+p=q }$$ and $${px=q}$$ using tape diagrams and balances.
2
6.EE.B.5
Define and identify solutions to equations.
3
6.EE.B.6
6.EE.B.7
Write equations for real-world situations.
4
6.EE.B.6
6.EE.B.7
Solve one-step equations with addition and subtraction.
5
6.EE.B.6
6.EE.B.7
Solve one-step equations with multiplication and division.
6
6.RP.A.3.C
6.EE.B.7
Solve percent problems using equations.
7
6.EE.B.6
6.EE.B.7
Solve multi-part equations leading to the form $${x+p=q }$$ and $${px=q}$$.
Topic B: Reasoning About and Solving Inequalities
8
6.EE.B.5
6.EE.B.8
Define and identify solutions to inequalities.
9
6.EE.B.8
Write and graph inequalities for real-world conditions. (Part 1)
10
6.EE.B.8
Write and graph inequalities for real-world conditions. (Part 2)
11
6.EE.B.6
6.EE.B.8
Solve one-step inequalities.
Topic C: Representing and Analyzing Quantitative Relationships
12
6.RP.A.3.A
6.EE.C.9
Write equations for and graph ratio situations. Define independent and dependent variables.
13
6.RP.A.3.A
6.EE.C.9
Represent the relationship between two quantities in graphs, equations, and tables. (Part 1)
14
6.RP.A.3.A
6.EE.C.9
Represent the relationship between two quantities in graphs, equations, and tables. (Part 2)
Common Core Standards
Key: Major Cluster Supporting Cluster Additional Cluster
Core Standards
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Expressions and Equations
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6.EE.B.5 — Understand 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.
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6.EE.B.6 — Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
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6.EE.B.7 — 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.
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6.EE.B.8 — Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
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6.EE.C.9 — 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.
Ratios and Proportional Relationships
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6.RP.A.3 — Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
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6.RP.A.3.A — Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
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6.RP.A.3.C — Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
Foundational Standards
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Expressions and Equations
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6.EE.A.1
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6.EE.A.2
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6.EE.A.3
Number and Operations—Fractions
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5.NF.A.1
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5.NF.B.3
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5.NF.B.4
Operations and Algebraic Thinking
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5.OA.B.3
Ratios and Proportional Relationships
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6.RP.A.1
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6.RP.A.2
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6.RP.A.3
The Number System
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6.NS.A.1
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6.NS.B.2
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6.NS.B.3
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6.NS.C.6.C
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6.NS.C.7
Future Standards
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Expressions and Equations
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7.EE.B.4
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8.EE.B.5
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8.EE.B.6
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8.EE.C.7
Functions
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8.F.A.1
Ratios and Proportional Relationships
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7.RP.A.2
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7.RP.A.3
Standards for Mathematical Practice
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CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
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CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
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CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
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CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
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CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
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CCSS.MATH.PRACTICE.MP6 — Attend to precision.
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CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
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CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.