6th Grade Mathematics | Equations and Inequalities

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 to 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 2021-2022 school year, see our 6th Grade Scope and Sequence Recommended Adjustments.

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Assessment

This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Unit Launch

To learn more about how to prepare a unit, internalize a lesson, and understand the different components of a Fishtank ELA lesson, visit our Preparing to Teach Fishtank ELA Teacher Tool.

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

The central mathematical concepts that students will come to understand in this unit

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

The materials, representations, and tools teachers and students will need for this unit

Optional: Balance scale (Teacher set)
Optional: Graph Paper (2-3 sheets per student)

To see all the materials needed for this course, view our 6th Grade Course Material Overview.

Vocabulary

Terms and notation that students learn or use in the unit

inequality

equation

solution

substitution

percent equation

independent variable

dependent variable

To see all the vocabulary for Unit 6, view our 6th Grade Vocabulary Glossary.

Lesson Map

Topic A: Reasoning About and Solving Equations

1

Represent equations in the form $${ x+p=q }$$ and $${px=q}$$ using tape diagrams and balances.

6.EE.B.6 6.EE.B.7

2

Define and identify solutions to equations.

6.EE.B.5

3

Write equations for real-world situations.

6.EE.B.6 6.EE.B.7

4

Solve one-step equations with addition and subtraction.

6.EE.B.6 6.EE.B.7

5

Solve one-step equations with multiplication and division.

6.EE.B.6 6.EE.B.7

6

Solve percent problems using equations.

6.EE.B.7 6.RP.A.3.C

7

Solve multi-part equations leading to the form $${x+p=q }$$ and $${px=q}$$.

6.EE.B.6 6.EE.B.7

Topic B: Reasoning About and Solving Inequalities

8

Define and identify solutions to inequalities.

6.EE.B.5 6.EE.B.8

9

Write and graph inequalities for real-world conditions. (Part 1)

6.EE.B.8

10

Write and graph inequalities for real-world conditions. (Part 2)

6.EE.B.8

11

Solve one-step inequalities.

6.EE.B.6 6.EE.B.8

Topic C: Representing and Analyzing Quantitative Relationships

12

Write equations for and graph ratio situations. Define independent and dependent variables.

6.EE.C.9 6.RP.A.3.A

13

Represent the relationship between two quantities in graphs, equations, and tables. (Part 1)

6.EE.C.9 6.RP.A.3.A

14

Represent the relationship between two quantities in graphs, equations, and tables. (Part 2)

6.EE.C.9 6.RP.A.3.A

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Expressions and Equations

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.

Expressions and Equations

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.
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.

Expressions and Equations

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.
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.

Expressions and Equations

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.
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.

Expressions and Equations

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.
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.

Expressions and Equations

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

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.

Ratios and Proportional Relationships

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.
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.

Ratios and Proportional Relationships

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.
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.

Ratios and Proportional Relationships

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

Standards covered in previous units or grades that are important background for the current unit

Expressions and Equations

6.EE.A.1

Expressions and Equations

6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2

Expressions and Equations

6.EE.A.2 — Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.A.3

Expressions and Equations

6.EE.A.3 — Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Number and Operations—Fractions

5.NF.A.1

Number and Operations—Fractions

5.NF.A.1 — Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
5.NF.B.3

Number and Operations—Fractions

5.NF.B.3 — Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.B.4

Number and Operations—Fractions

5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Operations and Algebraic Thinking

5.OA.B.3

Operations and Algebraic Thinking

5.OA.B.3 — Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

Ratios and Proportional Relationships

6.RP.A.1

Ratios and Proportional Relationships

6.RP.A.1 — Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
6.RP.A.2

Ratios and Proportional Relationships

6.RP.A.2 — Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."
6.RP.A.3

Ratios and Proportional Relationships

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.

The Number System

6.NS.A.1

The Number System

6.NS.A.1 — Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
6.NS.B.2

The Number System

6.NS.B.2 — Fluently divide multi-digit numbers using the standard algorithm.
6.NS.B.3

The Number System

6.NS.B.3 — Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
6.NS.C.6.C

The Number System

6.NS.C.6.C — Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.C.7

The Number System

6.NS.C.7 — Understand ordering and absolute value of rational numbers.

Future Standards

Standards in future grades or units that connect to the content in this unit

Expressions and Equations

7.EE.B.4

Expressions and Equations

7.EE.B.4 — Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
8.EE.B.5

Expressions and Equations

8.EE.B.5 — Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6

Expressions and Equations

8.EE.B.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
8.EE.C.7

Expressions and Equations

8.EE.C.7 — Solve linear equations in one variable.

Functions

8.F.A.1

Functions

8.F.A.1 — Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8.

Ratios and Proportional Relationships

7.RP.A.2

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.
7.RP.A.3

Ratios and Proportional Relationships

7.RP.A.3 — Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.

Equations and Inequalities

Unit Summary

Assessment

Unit Prep

Intellectual Prep

Internalization of Standards via the Post-Unit Assessment

Internalization of Trajectory of Unit

Unit-Specific Intellectual Prep

Essential Understandings

Materials

Vocabulary

Lesson Map

Common Core Standards

Core Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Ratios and Proportional Relationships

Ratios and Proportional Relationships

Ratios and Proportional Relationships

Ratios and Proportional Relationships

Foundational Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Number and Operations—Fractions

Number and Operations—Fractions

Number and Operations—Fractions

Number and Operations—Fractions

Operations and Algebraic Thinking

Operations and Algebraic Thinking

Ratios and Proportional Relationships

Ratios and Proportional Relationships

Ratios and Proportional Relationships

Ratios and Proportional Relationships

The Number System

The Number System

The Number System

The Number System

The Number System

The Number System

Future Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Functions

Functions

Ratios and Proportional Relationships

Ratios and Proportional Relationships

Ratios and Proportional Relationships

Standards for Mathematical Practice

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