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Algebra 1 Unit F: Systems of Equations
Unit Overview
Common Core State Standards
Content Standards
There are several ways to solve systems of equations and inequalities, including graphing and using equivalent forms
of equations and inequalities within the system. You can represent many real-world mathematical problems with
systems and interpret their solutions as viable or non-viable.
Focus Standards:
MAFS.912.A-CED.1.3
MAFS.912.A-REI.3.5
MAFS.912.A-REI.3.6
MAFS.912.A-REI.4.11
MAFS.912.A-REI.4.12
Standards for
Mathematical Practice
MAFS.K12.MP.4.1
MAFS.K12.MP.5.1
Supporting Standards:
MAFS.912.N-Q.1.2
MAFS.912.N-Q.1.3
Textbook Resources
Pearson Prentice Hall Algebra 1 copyright 2011
Pearson SuccessNet
Sections: 6.1 – 6.6, 6.6 Concept Byte
Mathematics Formative Assessment System Tasks
The system includes tasks or problems that teachers can
implement with their students, and rubrics that help the
teacher interpret students' responses. Teachers using MFAS
ask students to perform mathematical tasks, explain their
reasoning, and justify their solutions. Rubrics for interpreting
and evaluating student responses are included so that
teachers can differentiate instruction based on students'
strategies instead of relying solely on correct or incorrect
answers. The objective is to understand student thinking so
that teaching can be adapted to improve student
achievement of mathematical goals related to the standards.
Like all formative assessment, MFAS is a process rather than
a test. Research suggests that well-designed and
implemented formative assessment is an effective strategy
for enhancing student learning.
Mathematics Formative Assessment System Tasks
This a working document that will continue to be revised and improved taking your feedback into consideration.
Other Resources
Algebra Nation
Online Graphing Calculator
Canvas Mathematics Website
Exploring Systems of Equations using Graphing
Calculators
Systems of the Linear Round Table
When Two Lines Meet
Graphing vs. Substitution. Which would you
Choose?
Solving Systems of Equations by Substitution
Defining Regions using Inequalities
Pairs of Whole Numbers
Pasco County Schools, 2014-2015
Algebra 1 Unit F: Systems of Equations
Unit Scale (Multidimensional) (MDS)
The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for
measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1,
“What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps
students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the
academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and
declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards.
4.0
In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these
standards. Examples:

3.0
2.0
1.0
Create a real-world context that can be modeled by a system of linear equations or linear inequalities. Students will write the system, graph, and
interpret the solution.
The Student will:
 Define appropriate quantities for the purpose of descriptive modeling. (MAFS.912.N-Q.1.2)
 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (MAFS.912.N-Q.1.3)
 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable
options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
(MAFS.912.A-CED.1.3)
 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a
system with the same solutions. (MAFS.912.A-REI.3.5)
 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (MAFS.912.AREI.3.6)
 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x);
find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases
where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (MAFS.912.A-REI.4.11)
 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution
set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (MAFS.912.A-REI.4.12)
The student will recognize or recall specific vocabulary, such as:
 System, substitution, elimination, strict inequality, half-plane, linear inequality
The student will perform basic processes, such as:
 Solve linear equations and inequalities
 Graph a linear equation
 Evaluate expressions
With help, partial success at 2.0 content but not at score 3.0 content
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Algebra 1 Unit F: Systems of Equations
Unpacking the Standard: What do we want students to Know, Understand and Do (KUD):
The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question
#1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what
student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional
scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are
essential for mastery.
Domain: Algebra: Reasoning with Equations & Inequalities
Cluster: Represent and solve equations and inequalities graphically (Major Cluster)
Standard: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to
a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (MAFS.912.A-REI.4.12)
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Students will understand what the shaded region of a graph represents and how dashed and solid lines are incorporated into the solution.
Know
Declarative knowledge: Facts, vocab., information



Key Vocabulary: half-plane and boundary line
Which region to shade based on the
inequality sign
How to distinguish between strict (< 𝑜𝑟 >)
and not strict (≤ 𝑜𝑟 ≥) boundary lines
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Analysis


Graph the solution to a linear inequality in two variables
Graph the solution set to a system of linear inequalities in two variables
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Graph linear equations, substitute an ordered pair into an inequality, and solve for y in an inequality
Moving Beyond:
Use constraints in practical applications.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Algebra 1 Unit F: Systems of Equations
Uni-Dimensional, Lesson Scale:
The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of
knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown
below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to
further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1,
“Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the
actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level,
not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard.
Common Core Standard
MAFS.912.A-REI.4.12: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set
to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
Score
Learning Progression
You can work at most 8 hours tomorrow. You need to earn at least $20 to buy your new
shirt. Your dog-walking job pays $5 per hour and your job as a car wash attendant pays
$6.25 per hour.
I can…
4.0
3.5



Sample Tasks
Create a system of inequalities from a given context
Graph the solution to the system
Identify possible solutions
a)
Write and graph a system of linear inequalities to model the situation.
b) Find two solutions to the system and explain their meaning.
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
Graph and identify one solution
3.0
2.5
I can…
 Graph a linear inequality with two variables
 Graph a system of linear inequalities
o Identify one solution for the system
2𝑥 + 3𝑦 < 6
1
𝑦 ≥ 𝑥−4
3
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Algebra 1 Unit F: Systems of Equations
Which graph represents the given system of inequalities?
−2𝑥 + 𝑦 < 3
𝑥+𝑦≥4
I can…
2.0


Identify the graph that corresponds to a
given linear inequality with two variables
Identify the graph that corresponds to a
given system of linear inequalities
A.
1.0
B.
C.
D.
I need prompting and/or support to complete 2.0 tasks.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Algebra 1 Unit F: Systems of Equations
Sample High Cognitive Demand Tasks:
These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply
demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them
as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and
common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?”
and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New
Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors).
MAFS Mathematical Content Standard(s)
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two
variables. (MAFS.912.A-REI.3.6)
Design Question 1; Element 1
MAFS Mathematical Practice(s)
Make sense of problems and persevere in solving them. (MAFS.K12.MP.1.1)
Design Question 1; Element 1
Marzano’s Taxonomy
Teacher Notes
Questions to develop mathematical thinking,
possible misconceptions/misunderstandings,
how to differentiate/scaffold instruction,
anticipate student problem solving strategies
Task
*These tasks can either be teacher created or
modified from a resource to promote higher
order thinking skills. Please cite the source for
any tasks.
Knowledge Utilization
This task addresses A-REI.C.6, solving systems of linear equations exactly, and provides a simple example of a system with
three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the
given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find
the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given
information. This equation is then combined with the given information about the sums of pairs of numbers to deduce
what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of
equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It
provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
Licensed Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015