Download Unit B: Equations, Inequalities, and Problem Solving

Math for College Readiness | Unit B: Equations, Inequalities, and Problem Solving
Unit Overview
Math Florida Standards
Content Standards
Major Focus: Students will master the techniques of solving linear equations and inequalities; students will build, solve,
and interpret equations and inequalities in order to model and solve contextual problems.
Tasks:
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Students will use the properties of equality to solve equations, including literal equations.
Students will use the properties of inequalities to solve inequalities.
Students will create equations and inequalities given a modeling situation and will interpret solutions as viable
or non-viable.
Students will use units to guide the solution of multi-step problems and will choose and interpret units
consistently in formulas.
Students will choose a level of accuracy when reporting units in a modeling situation.
Textbook Resources
Martin-Gay, E. (2013). Intermediate algebra: Math for
college readiness. Boston, MA: Pearson Learning
Solutions.
Sections: 1.4, 2.1, 2.2, 2.3, 2.4, 2.5
Tips for Success in Mathematics pp. 2-6
Study Skills Builder: pp. 663-671
Bigger Picture Study Guide Outline: pp. 672-674
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.
http://www.cpalms.org/resource/mfas.aspx
This a working document that will continue to be revised and improved taking your feedback into consideration.
MAFS.7.EE.2.4
MAFS.912.A-CED.1.1
MAFS.912.A-CED.1.3
MAFS.912.A-CED.1.4
MAFS.912.A-REI.1.1
MAFS.912.A-REI.2.3
MAFS.912.N-Q.1.1
MAFS.912.N-Q.1.2
MAFS.912.N-Q.1.3
Standards for
Mathematical Practice
MAFS.K12.MP.1.1
MAFS.K12.MP.4.1
MAFS.K12.MP.5.1
MAFS.K12.MP.6.1
Other Resources
Mathematics Assessment Resource Service
College Readiness Math Resources
PERT Resources
Algebra Nation
Online Graphing Calculator
National Library of Virtual Manipulatives
Geogebra
Virtual Nerd
YouTube
Khan Academy—Math
Engage NY
TI Nspired Resource Center for Educators
Pasco County Schools, 2014-2015
Math for College Readiness | Unit B: Equations, Inequalities, and Problem Solving
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:
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
3.0
Given a linear equation or inequality, design a problem that could be modeled by that equation, paying close attention to the use of appropriate units in
the context of the problem.
Create absolute value equations and inequalities in one variable, and use them to solve problems.
The Student will:
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Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning
about the quantities. (MAFS.7.EE.2.4)
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Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions. (MAFS.912.A-CED.1.1)
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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)
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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. (MAFS.912.A-CED.1.4)
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Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a solution method. (MAFS.912.A-REI.1.1)
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Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (MAFS.912.A-REI.2.3)
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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. (MAFS.912.N-Q.1.1)
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Define appropriate quantities for the purpose of descriptive modeling. (MAFS.912.N-Q.1.2)
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Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (MAFS.912.N-Q.1.3)
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Math for College Readiness | Unit B: Equations, Inequalities, and Problem Solving
2.0
The student will recognize or recall specific vocabulary, such as:
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Variable, equation, properties of equality, inequality, formula, units, descriptive modeling, level of accuracy, limitations on measurement
The student will perform basic processes, such as:
1.0
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Combine like terms in an expression or equation
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Use the distributive property to simplify an expression
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Choose appropriate units for a solution
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Given an equation with a correct solution, identify the steps taken to achieve the solution.
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Solve simple equations such as ax + b = c, where a, b, and c are constants.
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Identify and represent important information from a modeling context.
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
Math for College Readiness | Unit B: Equations, Inequalities, and Problem Solving
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: Solve equations and inequalities in one variable (Major)
Standard: MAFS.912.A-REI.2.3: (Solve) linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Students will understand that many processes may be involved in order to achieve a solution to and equation/inequality. They should also be able to comprehend the nature
of mathematical equality/inequality, namely that an equal sign does not mean “compute”, but rather means “is the same as”.
Know
Declarative knowledge: Facts, vocab., information
Vocabulary: Linear equation in one variable,
linear inequality in one variable, solve, solution,
coefficient
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Solve a linear equation in one variable (with and without variable coefficients).
Solve a linear inequality in one variable (with and without variable coefficients).
Properties of Equality
Properties of Inequality
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Distributive Property, Combining Like Terms, Basic understanding of a variable, Skill with a number line (in preparation for representing solutions to linear inequalities)
Learning Goals:
Students can solve a linear equation in one variable.
Students can solve a linear inequality in one variable.
Moving Beyond:
MPRCC2—Know how and when to apply standard algorithms or concepts, and perform them flexibly, accurately, and efficiently.
MPRCC22—Solve linear equations in one variable using manipulations guided by the rules of arithmetic and the properties of equality.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Math for College Readiness | Unit B: Equations, Inequalities, and Problem Solving
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.
MAFS.912.A-REI.2.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Score
4.0
3.5
3.0
Learning Progression
I can…
 Solve a complicated linear equation containing both numerical and
variable coefficients.
 Solve compound inequalities and represent their solution graphically
and in interval notation.
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
I can…
 Solve a linear equation in one variable (with and without variable
coefficients)
 Solve a linear inequality in one variable (with and without variable
coefficients)
Sample Tasks
1. Solve for x: 3(a + x) + 4ax = 2(5x + 4ax) – 7
2. Solve the following inequality, graph the solution on a number line, and
write the solution in interval notation. 18 < 5x – 2 < 33
1. Solve for q: 9q – 3(6 – 5q) = q – ⅓(6q + 9)
2. Solve for n: an – b = c
3. Solve for x, graph the solution on a number line, and write the solution in
interval notation.
2.5
2.0
1.0
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
I can…
 Solve a simple linear equation in one variable.
 Graph the solution of a linear inequality on a number line.
 Solve a simple linear inequality in one variable.
I need prompting and/or support to complete 2.0 tasks.
1 2 x
+ ³
2 3 6
1. Solve for x: 2(x – 4) = 10
2. Solve the following inequality and graph the solution on a number line.
3x – 5 < 10
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Math for College Readiness | Unit B: Equations, Inequalities, and Problem Solving
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)
Design Question 1; Element 1
MAFS Mathematical Practice(s)
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
MAFS.912.A-CED.1.1: Create equations and inequalities in one variable and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them.
MAFS.K12.MP.4.1: Model with mathematics.
MAFS.K12.MP.5.1: Use appropriate tools strategically.
MAFS.K12.MP.6.1: Attend to precision.
Level 4-Knowledge Utilization: “Experimenting”
Questions:
How much fence do you think you will need?
Does your answer make sense with the picture?
What mathematical symbol goes with the phrase “at most”?
Misconceptions:
Incorrect computations when solving the inequality.
The shape of the garden (it does not have to be square, but it does have to be a rectangle)
Differentiation:
Change the shape of the gardens for extending students that might finish early
Equations of perimeter and a step by step guide for struggling students
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Math for College Readiness | Unit B: Equations, Inequalities, and Problem Solving
In a community garden, you want to fence in a vegetable garden that is adjacent to your friend’s garden. You
have at most 42 ft. of fence. Create an inequality to represent your problem.
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What are the possible lengths of your garden?
What would happen if you use the fence from the adjacent garden?
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.
Design Question 4; Element 22
From Algebra I, Pearson.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015