Lesson 13 | Equations and Inequalities | 6th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

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.

Foundational Standards

The foundational standards covered in this lesson

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.
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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Identify the independent and dependent variables in a relationship between two quantities.
Create a table of values for a relationship.
Plot values from a table in a coordinate plane.
Write equations to represent the relationship between two associated quantities.
Interpret points in a coordinate plane in context of the situation.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Lessons 13 and 14, similar to Lesson 12, bring together several concepts from standards such as 6.RP.3, 6.EE.7, and 6.NS.8 to represent relationships between quantities in different ways. In this lesson, students look at proportional relationships in the form $${px=q}$$.

Lesson Materials

Graph Paper (2-3 sheets per student)

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Sarah is purchasing pencils to share. Each package has 12 pencils. The equation $${n=12p}$$, where $$n$$ is the total number of pencils and $$p$$ is the number of packages, can be used to determine the total number of pencils Sarah purchased.

a. Determine which variable is dependent and which variable is independent.

b. Make a table showing the number of pencils purchased for 3 to 7 packages.

c. Plot the points in a coordinate plane.

d. If Sarah has 168 pencils, how many packages did she purchase?

Guiding Questions

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References

EngageNY Mathematics Grade 6 Mathematics > Module 4 > Topic H > Lesson 31 — Exercise 1

Grade 6 Mathematics > Module 4 > Topic H > Lesson 31 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

The graph below shows the relationship between the number of miles, $$m$$, and number of hours, $$h$$, spent traveling on Veronica’s road trip.

a. Which variable is the independent variable? The dependent variable?

b. What does the point (4, 200) mean in context of the situation?

c. Describe the relationship between the two quantities.

d. Write an equation to represent the relationship.

Guiding Questions

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References

EngageNY Mathematics Grade 6 Mathematics > Module 4 > Topic H > Lesson 32 — Example 2

Grade 6 Mathematics > Module 4 > Topic H > Lesson 32 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 3

At a store closing, everything in the store is 50% its original price.

a. Write an equation to represent the discounted price, $$d$$, of any item in the store with an original price $$p$$.

b. Create a table of values with at least three or four different original prices.

c. Plot the values in the coordinate plane to show the relationship between the original price and the discounted price of items in the store.

d. Explain how you could use your graph to find the discounted price of an item that originally cost $15.

Guiding Questions

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Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Fishtank Plus Content

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Arian wants to save 20% of his paychecks in a savings account.

a. Write an equation to represent the amount Arian should save, $$s$$, from a paycheck in the amount of $$p$$ dollars.

b. Create a table of values with at least three or four different paycheck amounts.

c. Plot the values in a coordinate plane to show the relationship between the amount Arian saves and the amount Arian earns.

d. Explain how you could use your graph to find how much of a $60 paycheck Arian should put into his savings account.

Student Response

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Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Illustrative Mathematics Chocolate Bar Sales
Open Up Resources Grade 6 Unit 6 Practice Problems — Lesson 16 #2–3 and Lesson 17 #2
EngageNY Mathematics Grade 6 Mathematics > Module 4 > Topic H > Lesson 31 — Problem Set; focus on problems in the form px=q

Lesson 12

Lesson 14

Lesson Map

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.

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

Solve one-step equations with addition and subtraction.

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

Solve one-step equations with multiplication and division.

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

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 $${px=q}$$.

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

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)

6.EE.B.8

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)

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

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

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

Equations and Inequalities

Objective

Common Core Standards

Core Standards

Expressions and Equations

Ratios and Proportional Relationships

Foundational Standards

Operations and Algebraic Thinking

The Number System

Criteria for Success

Tips for Teachers

Lesson Materials

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem 3

Guiding Questions

Problem Set

Target Task

Student Response

Additional Practice

Lesson Map

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