Essential Questions:
Lesson Plan C.7a: Determining the Number of Solutions to Linear Equations
Lesson Plan C.7b: Solving Linear Equations
Lesson Seed C.7a: Equations with Different Solutions
Lesson Seed C.7b: Linear Equations – Place Mats
Lesson Seed C.7b: Equations with Distributive Property
Lesson Seed C.7b: Linear Equations – Rally
Lesson Seed C.8a: Systems of Equations on Coordinate Plane
Lesson Seed C.8c: Two-Variable Real World Problems
Unit Overview
Content Emphasis By Clusters in Grade 8
Progressions from Common Core State Standards in Mathematics
Send Feedback to MSDE’s Mathematics Team
Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.
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.
Students should:
At the completion of this unit on analyzing and solving linear equations and pairs of simultaneous linear equations, the student will understand that:
Students will be able to:
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.
Students may
Interdisciplinary Connections:
Interdisciplinary connections fall into a number of related categories:
Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:
Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be 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 Cluster Analyze 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 solutions: 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) = 3x^{2}+12x
collecting like terms: This process is also known as “combining” like terms. Example: 2x + 3 + 5x = 7x +3
Solve Systems of Equations:
Using elimination by addition or subtraction, the y variables will cancel out, and x = 5
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.
Solve for x or y in either equation. In this example it is easier to solve for y because it has a coefficient of 1.
Since x = 5, substitute 5 for x in either equation and then solve for y.
The solution to the system of equations is: (x, y) = (5, -1)
Graph each equation. Where they intersect on the coordinate plane is the solution.
Part II – Instructional Connections outside the Focus Cluster
Additional Resources: