Lesson 6 | Linear Expressions & Single-Variable Equations/Inequalities | 9th Grade Mathematics

Objective

Write equations using defined variables to represent a contextual situation.

Common Core Standards

Core Standards

The core standards covered in this lesson

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Building Functions

F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Creating Equations

A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

High School — Number and Quantity

N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Interpreting Functions

F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

The foundational standards covered in this lesson

8.F.B.4

Functions

8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Criteria for Success

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

Define units associated with variables.
Write simple expressions and equations to model contextual situations.
Use known quantities to try the algebraic model developed and determine reasonability of the model.
Assign domain restrictions based on context and algebraic relationships.
Rearrange the equation used to model a relationship to highlight a quantity of interest.
Use nested expressions to describe a larger model.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson only contains one anchor problem so that a performance task from Mathematics Vision Project (listed under Problem Set Guidance, first bullet) can be addressed fully.
This lesson is the first in a series of five lessons on defining variables, modeling a contextual situation, and applying the model to specific situations.

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Anchor Problems

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

Below is a formula that describes a quantity related to getting to a destination via walking and riding the bus.

$${S_{b}}$$ represents the average speed of the bus.
$${D_{b}}$$ represents the distance you travel on the bus.
$${D_{w}}$$ represents the distance you walk to your final destination.
$${S_{w}}$$ represents the average speed you walk
$$h$$ represents the amount of time you wait for the bus.

What does the variable $$R$$ mean in the formula below?

$${D_{b}}={S_{b}}\left ( R-\frac{{D_{w}}}{{S_{w}}}-h \right )$$

Guiding Questions

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Problem Set

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

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

Here is a parallelogram.

Which of these are correct expressions for the area of this parallelogram?

$${{ab}}$$ $$\frac{1}{2}{{ab}}$$ $${{ah}}$$ $$\frac{1}{2}{{ah}}$$ $${{2a+2b}}$$ $${{2(a+b)}}$$ $${{ab}}h$$

Which of these are correct expressions for the perimeter of the parallelogram?

$${{ab}}$$ $$\frac{1}{2}{{ab}}$$ $${{ah}}$$ $$\frac{1}{2}{{ah}}$$ $${{2a+2b}}$$ $${{2(a+b)}}$$ $${{ab}}h$$

Here is a trapezoid. It is made up of two triangles, each with height $$h$$.

Write a formula, using the area of each triangle, that describes the area of the trapezoid.

References

Inside Mathematics Performance Assessment Tasks Grades 3-High School Expressions

Expressions of the Performance Assessment Tasks created by the by the Mathematics Assessment Resource Service (MARS) of the Shell Centre for Mathematical Education, University of Nottingham, England are made available by Inside Mathematics under a license from Shell Centre Publications. Accessed Sept. 12, 2017, 4:46 p.m..

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.

Mathematics Vision Project: Secondary Mathematics One Module 4: Equations & Inequalities — Lesson 4.2, Elvira's Equations, Questions #1-10 (This section should be done in its entirety and in order.)

Lesson 5

Lesson 7

Lesson Map

Topic A: Properties and Solutions of Single-Variable Linear Expressions and Equations

Identify properties of operations that result in equivalent linear expressions.

A.REI.A.1 A.SSE.A.1 A.SSE.A.2

Use properties of equations to analyze and write equivalent equations.

A.REI.A.1

Solve single-variable linear equations using properties of equality.

A.REI.A.1 A.REI.B.3

Solve equations with a variable in the denominator.

A.REI.B.3 F.IF.A.1

Solve for a variable in an equation or formula.

A.CED.A.4

Topic B: Modeling with Single-Variable Linear Equations

Write equations using defined variables to represent a contextual situation.

A.CED.A.2 A.CED.A.4 F.BF.A.1 F.IF.B.5 N.Q.A.1

Define variables; write and solve equations to represent a contextual situation.

A.CED.A.1 A.CED.A.2 A.CED.A.4 F.BF.A.1 F.IF.B.5 N.Q.A.1

Write and solve equations to represent contextual situations where estimations and unit conversions are required.

A.CED.A.1 A.CED.A.2 A.CED.A.4 F.BF.A.1 F.IF.B.5 N.Q.A.1

2 days

Model a contextual situation and make an informed decision based on the model.

A.CED.A.1 A.CED.A.2 A.CED.A.4 F.BF.A.1 F.IF.B.5 N.Q.A.1

Topic C: Properties and Solutions of Single-Variable Linear Inequalities

Solve unbounded single-variable inequalities in contextual and non-contextual situations.

A.CED.A.3 A.REI.A.1 A.REI.B.3 A.SSE.B.3

Write and graph compound single-variable inequalities to describe the solution to contextual and non-contextual situations.

A.CED.A.3 A.REI.B.3

Solve and graph compound inequalities where algebraic manipulation is necessary in contextual and non-contextual situations.

A.CED.A.3 A.REI.A.1 A.REI.B.3

Linear Expressions & Single-Variable Equations/Inequalities

Objective

Common Core Standards

Core Standards

Building Functions

Creating Equations

Creating Equations

High School — Number and Quantity

Interpreting Functions

Foundational Standards

Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Guiding Questions

Problem Set

Target Task

References

Additional Practice

Lesson Map

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