Math / 6th Grade / Unit 6: 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.
Math
Unit 6
6th Grade
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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)
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The following assessments accompany Unit 6.
Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment after lesson 6.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment
Post-Unit Assessment Answer Key
Post-Unit Student Self-Assessment
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
Unit Launch
Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.
The central mathematical concepts that students will come to understand in this unit
Terms and notation that students learn or use in the unit
dependent variable
equation
inequality
independent variable
percent equation
solution
substitution
To see all the vocabulary for Unit 6, view our 6th Grade Vocabulary Glossary.
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 6th Grade Course Material Overview.
Topic A: Reasoning About and Solving Equations
Represent equations in the form $${ x+p=q }$$ and $${px=q}$$ using tape diagrams and balances.
6.EE.B.6 6.EE.B.7
Define and identify solutions to equations.
6.EE.B.5
Write equations for real-world situations.
Solve one-step equations with addition and subtraction.
Solve one-step equations with multiplication and division.
Solve percent problems using equations.
6.EE.B.7 6.RP.A.3.C
Solve multi-part equations leading to the form $${x+p=q }$$ and $${px=q}$$.
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Topic B: Reasoning About and Solving Inequalities
Define and identify solutions to inequalities.
6.EE.B.5 6.EE.B.8
Write and graph inequalities for real-world conditions. (Part 1)
6.EE.B.8
Write and graph inequalities for real-world conditions. (Part 2)
Solve one-step inequalities.
6.EE.B.6 6.EE.B.8
Topic C: Representing and Analyzing Quantitative Relationships
Write equations for and graph ratio situations. Define independent and dependent variables.
6.EE.C.9 6.RP.A.3.A
Represent the relationship between two quantities in graphs, equations, and tables. (Part 1)
Represent the relationship between two quantities in graphs, equations, and tables. (Part 2)
Key
Major Cluster
Supporting Cluster
Additional Cluster
The content standards covered in this unit
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.
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.
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.
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.
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.
Standards covered in previous units or grades that are important background for the current unit
6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.2 — Write, read, and evaluate expressions in which letters stand for numbers.
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.
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 — 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 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
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.
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 — 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.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 — Fluently divide multi-digit numbers using the standard algorithm.
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 — 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 — Understand ordering and absolute value of rational numbers.
Standards in future grades or units that connect to the content in this unit
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 — 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 — 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 — Solve linear equations in one variable.
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.
7.RP.A.2 — Recognize and represent proportional relationships between quantities.
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.
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.
Unit 5
Numerical and Algebraic Expressions
Unit 7
Geometry