Draft Unit Plan: Grade 7 – Understand Ratio Concepts and...

DRAFT UNIT PLAN 8.EE.C.7-8: Analyze and Solve Linear Equations and Pairs of Simultaneous Linear Equations

Overview: The overview statement is intended to provide a summary of major themes in this unit.

This unit builds on prior knowledge of numerical expressions and equations and of algebraic expressions, using rational numbers to understand that: linear equations in one variable can have one, infinitely many or no solutions; solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs; and the points of intersection satisfy both equations at the same time.

Teacher Notes: The information in this component provides additional insights which will help educators in the planning process for this unit.

Students should:

add, subtract, multiply, and divide accurately within the system of rational numbers, particularly to solve real-life and mathematical problems.

be well-grounded in their ability to simplify algebraic expressions and solve algebraic equations, using rational numbers. be able to graph linear equations on a coordinate plane.

Enduring Understandings: Enduring understandings go beyond discrete facts or skills. They focus on larger concepts, principles, or processes. They are transferable and apply to new situations within or beyond the subject.

At the completion of this unit on analyzing and solving linear equations and pairs of simultaneous linear equations, the student will understand that:

systems of linear equations are a means of modeling some types of real-life and mathematical problems. systems of linear equations can be solved using more than one method, for example by elimination of a variable, graphing, or

substitution. information can be extracted from the written words in a mathematical problem and converted into an algebraic problem to be solved; the

result or solution can be used to answer the question in the original word problem.

Essential Question(s): A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourages re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

Draft Maryland Common Core State Curriculum Unit for Grade 8 Mathematics December 2012 of 19


What does the number of solutions (none, one, or infinite) of a system of linear equations represent? What are the advantages and disadvantages of solving a system of linear equations graphically versus algebraically? How can systems of equations be used to represent authentic scenarios and solve problems in life?

Content Emphases by Clusters in Grade 8:According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), some clusters require greater emphasis than others. The table below shows PARCC’s relative emphasis for each cluster. Prioritization does not imply neglect or exclusion of material. Clear priorities are intended to ensure that the relative importance of content is properly attended to. Note that the prioritization is in terms of cluster headings.

Key: ■ Major Clusters Supporting Clusters Additional Clusters

The Number SystemKnow that there are numbers that are not rational, and approximate them by rational numbers.

Expressions and Equations■ Work with radicals and integer exponents.

■ Understand the connections between proportional relationships, lines and linear equations.■ Analyze and solve linear equations and pairs of simultaneous linear equations.

Functions■ Define, evaluate and compare functions.Use functions to model relationships between quantities.

Geometry■ Understand congruence and similarity using physical models, transparencies or geometry software.

■ Understand and apply the Pythagorean Theorem Solve real-life and mathematical problems involving volume of cylinders, cones and spheres.



Statistics and Probability Investigate patterns of association in bivariate data.

Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):According to the Partnership for the Assessment of Readiness for College and Careers (PARCC), this component highlights some individual standards that play an important role in the content of this unit. Educators should give the indicated mathematics an especially in-depth treatment, as measured for example by the number of days; the quality of classroom activities for exploration and reasoning; the amount of student practice; and the rigor of expectations for depth of understanding or mastery of skills.

8.EE.C.7 This is a culminating standard for solving one-variable linear equations.

8.EE.C.8 When students work toward meeting this standard, they build on what they know about two-variable linear equations, and theyenlarge the varieties of real-world and mathematical problems they can solve.

Possible Student Outcomes: The following list provides outcomes that describe the knowledge and skills that students should understand and be able to do when the unit is completed. The outcomes are often components of more broadly-worded standards and sometimes address knowledge and skills necessarily related to the standards. The lists of outcomes are not exhaustive, and the outcomes should not supplant the standards themselves. Rather, they are designed to help teachers delve deeply into the standards and augment as necessary, providing added focus and clarity for lesson planning purposes. This list is not intended to imply any particular scope or sequence.

The student will be able to:

solve linear equations which have one solution, many solutions, or no solution. use the distributive property and collect like terms and solve equations. solve systems of equations in two variables algebraically and graphically. write an equation, given two points. write equations from context. interpret the solution to a system of equations in context.



Progressions from Common Core State Standards in Mathematics: For an in-depth discussion of the overarching, “big picture” perspective on student learning of content related to this unit, see:

The Common Core Standards Writing Team (10 September 2011). Progressions for the Common Core State Standards in Mathematics (draft), accessed at: http://commoncoretools.files.wordpress.com/2011/04/ccss_progression_ee_2011_04_25.pdf

Vertical Alignment: Vertical curriculum alignment provides two pieces of information: A description of prior learning that should support the learning of the concepts in this unit A description of how the concepts studied in this unit will support the learning of additional mathematics

Key Advances from Previous Grades: Students build on previous work with proportional relationships, unit rates, and graphing to connect these ideas as to understand that the points (x, y) on a non-vertical line are the solutions of the equation y = mx + b, where me is the slope of the line as well as the unit rate of a proportional relationship (in the case b = 0). Students also formalize their previous work with linear relationships by working with functions – rules that assign to each input exactly one output.

Additional Mathematics: Students analyze and solve linear equations and pairs of simultaneous linear equations:o in algebra when modeling a situation, when comparing linear equations to quadratic equations, and when studying various types of

functionso in algebra when graphing the line of an equation on the coordinate planeo in algebra and geometry when expanding into real numbers and beyond.

Possible Organization of Unit Standards: This table identifies additional grade-level standards within a given cluster that support the overarching unit standards from within the same cluster. The table also provides instructional connections to grade-level standards from outside the cluster.

Overarching Unit Standards. Supporting Standards within the Cluster

Instructional Connections outside the Cluster

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

8.EE.C.7a:Give examples of 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



Overarching Unit Standards. Supporting Standards within the Cluster

Instructional Connections outside the Cluster

with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form of x = a, a = a, or a = b results (a and b are different numbers).

8.EE.C.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

pairs consisting of an input and the corresponding output (function notation is not required in Grade 8).

8.EE.C.8: Analyze and solve pairs of simultaneous linear equations

8.EE.C.8a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.C.8b: Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

8.EE.C.8c:Solve real-world and mathematical problems leading to two linear equations in two variables.



Connections to the Standards for Mathematical Practice: This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

In this unit, educators should consider implementing learning experiences which provide opportunities for students to:

1. Make sense of problems and persevere in solving them. Analyze a problem and depict a good way to solve the problem. Consider the best (most straightforward) way to solve a problem. Interpret the meaning of their answer to a given problem.

2. Reason abstractly and quantitatively Consider examples of linear equations in one variable which could have one solution, no solution, or infinitely many solutions. decide what the number of solutions represents in the context of a problem or task. Determine whether or not their solution connects to the question.

3. Construct Viable Arguments and critique the reasoning of others. Justify the process used to determine the solution to a pair of simultaneous linear equations. Justify an argument for one solution, no solution, or infinitely many solutions.

4. Model with Mathematics Graph a pair of simultaneous linear equations on the coordinate plane to show their intersection/solution. Analyze an authentic problem and create a nonverbal representation for it.

5. Use appropriate tools strategically Use virtual media and visual models to explore the process of solving word problems involving linear equations with rational

numbers. Use manipulatives to solve a problem.

6. Attend to precision Demonstrate their understanding of the mathematical processes required to solve a problems by communicating all of the steps in

solving the problem.



Label appropriately. Use the correct mathematics vocabulary when discussing problems.

7. Look for and make use of structure. Compare, reflect upon, and discuss multiple solution methods.

8. Look for and express regularity in reasoning Pay special attention to details and continually self-evaluate the reasonableness of their answers. Use mathematical principles to help them solve the problem.

Content Standards with Essential Skills and Knowledge Statements and Clarifications: The Content Standards and Essential Skills and Knowledge statements shown in this section come directly from the Maryland State Common Core Curriculum Frameworks. Clarifications were added as needed. Educators should be cautioned against perceiving this as a checklist. All information added is intended to help the reader gain a better understanding of the standards.

Standard Essential Skills and Knowledge Clarification

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

8.EE.C.7a: Give examples of linear

equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler

Ability to build on prior knowledge of solving linear equations

one solution: An example of a linear equation in one variable with one solution is x = 5.

infinitely many solutions: An example of a linear equation in one variable with infinitely many solutions is x = x.

no solution: An example of a linear equation in one variable with no solutions is x + 1 = x + 2.




forms, until an equivalent equation of the form of x = a, a = a, or a = b results (a and b are different numbers).

8.EE.C.7b: Solve linear equations

with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

See the skills and knowledge that are stated in the Standard.

equivalent equation: Equations with the same value are equivalent, for example: 2 + 3 = 11 − 6 is equivalent to 5 = 5 OR y +3 = 5 is equivalent to y = 2.

distributive property: The distributive property states that for all numbers a, b, and c, a(b + c) = ab + ac

Examples are a(b + c) = ab + ac and 3x(x+4) = 3x(x) + 3x(4) = 3x2+12x

collecting like terms: This process is also known as “combining” like terms. Example: 2x + 3 + 5x = 7x +3

8.EE.C.8: Analyze and solve pairs of simultaneous linear equations8.EE.C.8a: Understand that solutions

to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Ability to solve systems of equations algebraically or by graphing




8.EE.C.8b: Solve systems of two

linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

8.EE.C.8c:Solve real-world and mathematical problems leading to two linear equations in two variables.

Ability to solve systems of two linear equations in two variables algebraically using elimination by addition or subtraction; using substitution

Ability to discuss solution methods with a system of equations - algebraically and graphically Ability to recognize that systems equations may have one solution, infinitely many solutions, or no solution.

Solve Systems of Equations:

1. Elimination by addition or subtraction:

2x + y = 9 3 x – y = 16 5x = 25 x = 5 2(5) + y = 9 or 3(5) – y = 16 10 + y = 9 15 – y = 16 y = –1 y = –1 The solution to the system of equations is: (x, y) = (5, –1) 2. Elimination by substitution :

2x + y = 9 3x – y = 16Solve for x or y in either equation. In this example it is easier to solve for y because it has a coefficient of 1.


Once y has been eliminated to determine the value of x, 5 can be substituted into either of the equations to determine the value of y.



2x + y = 9 Therefore: 3x – (9 – 2x) = 16 (substituted 9-2x for y) -2 x = -2 x 5x – 9 = 16 y = 9 -2x + 9 = + 9

5x = 25 x = 5

Evidence of Student Learning: The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.

Fluency Expectations and Examples of Culminating Standards: This section highlights individual standards that set expectations for fluency, or that otherwise represent culminating masteries. These standards highlight the need to provide sufficient supports and opportunities for practice to help students meet these expectations. Fluency is not meant to come at the expense of understanding, but is an outcome of a progression of learning and sufficient thoughtful practice. It is important to provide the conceptual building blocks that develop understanding in tandem with skill along the way to fluency; the roots of this conceptual understanding often extend one or more grades earlier in the standards than the grade when fluency is finally expected. 8.EE.C.7: Students have been working informally with one-variable linear equations since as early as kindergarten. This important line

of development culminates in grade 8 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions, as well as cases require algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational number.

Common Misconceptions: This list includes general misunderstandings and issues that frequently hinder student mastery of concepts related to analyzing and solving linear equations and pairs of simultaneous linear equations.



Students may:

Have difficulties manipulating signed values and variables when solving simultaneous equations. Have misunderstandings about the use and meaning of variables, particularly as systems of equations require students to solve for multiple

variables. Confuse the sequence of steps inherent in determining the solution of linear equations (substitution versus elimination), and may combine

aspects of both methods. Stop after solving for only one variable. When solving for the second variable in pairs of simultaneous equations, may substitute into the same equation that was used to determine

the value of the first variable. Forget to multiply entire row (all terms) of the equation when using the elimination method. Not re-write the equations in standard form when using the elimination method. Not recognize situations in which the use of one method is more expedient than another. Not realize that they need both equations in slope-intercept form in order to solve graphically. Be unsure of what to do when all variables are eliminated, or what this situation represents.

Interdisciplinary Connections: Interdisciplinary connections fall into a number of related categories: Literacy standards within the Maryland Common Core State Curriculum Science, Technology, Engineering, and Mathematics standards Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level

coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.

Model Lesson Plan Chart:

Available Model Lesson Plan(s)



The lesson plan(s) have been written with specific standards in mind. Each model lesson plan is only a MODEL – one way the lesson could be developed. We have NOT included any references to the timing associated with delivering this model. Each teacher will need to make decisions related to the timing of the lesson plan based on the learning needs of students in the class. The model lesson plans are designed to generate evidence of student understanding.

This chart indicates one or more lesson plans which have been developed for this unit. Lesson plans are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

Standards Addressed Title Description/Suggested Use

8.EE.C.7a8.EE.C.7b

Equation SolutionsSolving Equations

Model Lesson Seed Chart:

Available Lesson Seed(s)



The lesson seed(s) have been written with specific standards in mind. These suggested activity/activities are not intended to be prescriptive, exhaustive, or sequential; they simply demonstrate how specific content can be used to help students learn the skills described in the standards. Seeds are designed to give teachers ideas for developing their own activities in order to generate evidence of student understanding.

This chart indicates one or more lesson seeds which have been developed for this unit. Lesson seeds are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

Standards Addressed Title Description/Suggested Use

8.EE.C.7a Equations with Different Solutions

8.EE.C.7b Linear Equations (Place Mat)

8.EE.C.7b Equations with Distributative Property

8.EE.C.7b Linear Equations (Rally)

8.EE.C.8a Systems of Equations on Coordinate Plane

8.EE.C.8b Two-Variable Real World Problems

Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include: Items purchased from vendors PARCC prototype items PARCC public release items Maryland Public release items



Formative Assessment

Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will included later, as available from the vendor(s) producing the modules.)

Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.

Part I – Focus ClusterAnalyze and Solve Linear Equations and Pairs of Simultaneous Linear Equations

one solution: An example of a linear equation in one variable with one solution is: x = 5.

infinitely many solutions: An example of a linear equation in one variable with infinitely many solutions is: x = x.

no solution: An example of a linear equation in one variable with no solutions is: x + 1 = x + 2.

families of graphs: The graph below shows a few family members for the linear equation: y = x.



equivalent equation: Equations with the same value are equivalent, for example: 2 + 3 = 11 − 6 is equivalent to 5 = 5 OR y +3 = 5 is equivalent to y = 2.

distributive property: The distributive property states that for all numbers a, b, and c, a(b+c) = ab + ac Examples are a(b + c) = ab + ac and 3x(x+4) = 3x(x) + 3x(4) = 3x2+12x

collecting like terms: This process is also known as “combining” like terms. Example: 2x + 3 + 5x = 7x +3

Solve Systems of Equations:1. elimination by addition or subtraction: 2x + y = 9 2x + y = 9 3x – y = 16 3 x – y = 16 5x = 25 x = 5


The red line, y = x, is considered the “parent.” All members of a family of graphs have the same slope, but different y-intercepts. Families of graphs can have infinitely many members.

Using elimination by addition or subtraction, the y variables will cancel out, and x = 5.


2(5) + y = 9 or 3(5) – y = 16 10 + y = 9 15 – y = 16 y = –1 –y = 1 y = –1

2. elimination by substitution:

2x + y = 93x – y = 16Solve for x or y in either equation. In this example it is easier to solve for y because it has a coefficient of 1. 2x + y = 9 Therefore: 3x – (9 – 2x) = 16 (substituted 9-2x for y) -2x = -2x 3x – 9 + 2x = 16 y = 9 – 2x 5x – 9 = 16 + 9 = + 9 5x = 25 x = 5Since x = 5, substitute 5 for x in either equation and then solve for y.2x + y = 9 becomes 2(5) + y = 9 10 + y = 9 y = -1The solution to the system of equations is: (x, y) = (5,-1)


Once y has been eliminated to determine the value of x, 5 can be substituted into either of the equations to determine the value of y.

The solution to the system of equations is: (x, y) = (5, –1)


3. graphically: Graph each equation. Where they intersect on the coordinate plane is the solution.2x + y = 93x – y = 16

Part II – Instructional Connections outside the Focus Cluster

proportional relationship: A comparison of two variable quantities having a fixed (constant) ratio is considered to be a proportional relationship. Example: 50 miles on 3 gallons is a proportional relationship to 100 miles on 6 gallons and 150 miles on 9 gallons.

constant rate of change/slope:


The line graphed on this coordinate plane has a slope of 32

. This means as one moves up or down the line, the ratio of the change in units on the x-axis, 2, is constant with the change in units on the y-axis, 3.

slope = vertical change

horizontal change = riserun

http://en.wikipedia.org/wiki/File:Slope_of_lines_illustrated.jpg


direct variation:

zero slope:


When two variables are related in such a way that the ratio of their values always remains the same, the two variables are said to be in direct variation.

In the diagram, the vertical change of the line, 3 units, is in direct variation with its horizontal change, 2 units. In other words, the vertical change is in direct variation to

the horizontal change in a ratio of 32 .

As shown in the diagram, when the slope of a line is undefined, the line is vertical and parallels the y-axis. As one moves up or down a line, the value for x does not change. The x-value for the line on the right is always 3, regardless of the y-value.

We know that division by zero is impossible because regardless of the value of the dividend, it can never be reproduced through multiplication by a divisor of zero because 0 x 0 = 0.

So, if we think of yx as

all values0 , then the slope of a vertical line is

incapable of being defined. For example, if you move on the line

from the point (3, 7) to the point (3, 2), then yx =

‾ 50 = undefined.

Conversely, if you move from (3, 2) to (3, 7), then yx =

50 =

http://en.wikipedia.org/wiki/File:Slope_of_lines_illustrated.jpg


Resources: This section contains links to materials that are intended to support content instruction in this unit.

http://commoncoretools.files.wordpress.com/2011/04/ccss_progression_ee_2011_04_25.pdf


http://commoncoretools.files.wordpress.com/2011/04/ccss_progression_ee_2011_04_25.pdf

Draft Unit Plan: Grade 7 – Understand Ratio Concepts and...

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