Lesson 5 | Equations and Inequalities | 6th Grade Mathematics

Objective

Solve one-step equations with multiplication and division.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Foundational Standards

The foundational standards covered in this lesson

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.
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.

Criteria for Success

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

Understand that in the process of solving for a variable, whatever is done to one side of the equation must also be done to the other side in order to maintain the balance.
Use models and diagrams to solve for a variable.
Solve one-step multiplication and division equations algebraically.
Write and solve one-step multiplication and division problems for real-world contexts.
Interpret $$\frac{a}{b}$$ as $$a$$ divided by $$b$$, and where $$b$$ is a fraction, as $$a$$ multiplied by the reciprocal of $$b$$.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In this lesson, students encounter problems that draw on skills from the Number Sense domain, namely dividing by fractions and decimals. Students may need to review or recall these concepts in order to fully access this lesson.

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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

The following two diagrams represent the equation $${ 2x=8}$$.

a. Explain how you can use each diagram to find the value of $$x$$.

b. How can you solve the equation $${3.4m=13.6}$$ without using a diagram?

Guiding Questions

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Problem 2

a. Complete the tape diagram below to represent the equation $${{x\over5}=10}$$; then use it to find the value of $$x$$.

b. How can you solve the equation $${{p\over6}=9}$$ without using a diagram?

Guiding Questions

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Problem 3

Three equations are shown below. For each one, identify all of the ways that equation could be solved.

Equation A:

$$\frac{1}{5}x=15$$

$$x=15\times5$$
$$x=15\div5$$

Equation B:

$$\frac{3}{5}x=15$$

$$x=15\times\frac{3}{5}$$
$$x=15\times\frac{5}{3}$$
$$x=15\div\frac{3}{5}$$
$$x=15\div\frac{5}{3}$$

Equation C:

$$\frac{2}{5}x=\frac{4}{15}$$

$$x=\frac{4}{15}\times\frac{2}{5}$$
$$x=\frac{4}{15}\times\frac{5}{2}$$
$$x=\frac{4}{15}\div\frac{2}{5}$$
$$x=\frac{4}{15}\div\frac{5}{2}$$

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

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

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

Problem 1

Ron solved the equation $${ {{{2\over3}}}r=12}$$ by multiplying both sides by $${{2\over3}}$$ and he got $${r=8}$$. Explain the error that Ron made and then find the correct value for $$r$$.

Problem 2

Lee filled several jars with $${{1\over4}}$$ cup of water in each jar. He used a total of 8 cups of water. Let $$j$$ represent the number of jars that Lee filled.

Write and solve an equation to find out how many jars Lee filled.

Student Response

An example response to the Target Task at the level of detail expected of the students.

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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.

Include problems with procedural practice in solving multiplication and division equations, especially those involving decimals or fractions.
Include problems where students write an equation from a context (any operation, but focused on multiplication and division) and solve the equation.
Include error analysis problems of mistakes made in solving equations.
Include equations where students must divide a fraction by a fraction.

Open Up Resources Grade 6 Unit 6 Practice Problems — Lessons 4 and 5
Illustrative Mathematics Reciprocity
Open Middle Solving One-Step Equations — Challenge
EngageNY Mathematics Grade 6 Mathematics > Module 4 > Topic G > Lesson 27 — Exploratory Challenge, Exercises, and Problem Set

Lesson 4

Lesson 6

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

Expressions and Equations

Foundational Standards

Number and Operations—Fractions

Number and Operations—Fractions

The Number System

The Number System

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Problem Set

Target Task

Problem 1

Problem 2

Student Response

Additional Practice

Lesson Map

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