Algebra I Unit Plan 1
Running Head: ALGEBRA I UNIT PLAN
Applying Backwards Design to a Traditional Textbook:
An Algebra I Unit Plan
Meredith J. Patterson
Vanderbilt Patterson
Capstone
15 June 2011
Algebra I Unit Plan 2
Abstract
This Capstone project consists of a backwards design format unit plan based on
corresponding units from a specific traditional Algebra I textbook. The student goals that I have
identified are closely aligned with standards outlined by the National Council of Teachers of
Mathematics, as well as relevant Common Core Standards. The unit plan is designed to allow
for both conceptual and procedural fluency; since the textbook that serves as my basis stresses
skills, I have added to its activities and modified the sequence to emphasize the important role
that conceptual understanding plays in the ability to think algebraically. The rationales for my
individual and overarching modifications and design decisions are based on my coursework in
Mathematics Education and the accompanying literature. I have structured the unit plan using
the Understanding by Design framework developed by authors and educators Grant Wiggins and
Jay McTighe (2005). By integrating some of the valuable activities, exercises, and questions
provided in the teacher’s manual of the textbook with self-designed learning experiences, I have
created a unit on linear equations that provides more socially relevant opportunities for students
to develop a deeper conceptual understanding of linearity.
Algebra I Unit Plan 3
Introduction
Algebra I, which students traditionally complete in the eighth or ninth grade, is often said
to be the most important upper level math class, as it provides the basis upon which students will
build their understanding of geometry, trigonometry, calculus, and other higher level math
topics. In order for students to be able to succeed in high school mathematics, they must have a
solid mastery of the concepts and skill sets that are introduced in Algebra I. In addition, the
ability to think algebraically, regardless of the formal mathematics trajectory that will be
followed, is what Civil Rights activist and Algebra Project pioneer Robert Moses calls the
“gatekeeper for citizenship” (2001, p. 14). The topics addressed in the Algebra I curriculum
should be approached with the sense of urgency that is often reserved for the elementary school
reading classroom; children need to learn to think algebraically to be able to successfully and
meaningfully participate in life both inside and outside of the high school setting.
Glencoe Algebra I: Integrations, Applications, and Connections (2001) is a commonly
used textbook in Algebra I classrooms across the country. A perusal of its pages will reveal the
progress that has already been made in the attempt to make mathematics more relevant and
accessible to high school students. Unlike the Algebra I textbooks that many students used even
through the beginning of the twenty-first century, this text is overflowing with photographs of
situations that students could potentially encounter in their lives outside of the classroom. There
are numerous “real world application” problems labeled throughout the chapters, “internet
connection” suggestions that accompany many assignments, “what you’ll learn/why it’s
important” lists introducing each lesson, and, in the teacher’s manual, ideas about ways to
remodel concepts to reach students of all different learning styles. The strides that have been
made towards creating relevant and accessible math curricula are certainly encouraging.
Algebra I Unit Plan 4
However, there is still ample room for improvement in the reformation of math textbooks
and curricula. Despite the positive changes that have been implemented, there remain significant
holes in many commonly used math textbooks. Glencoe Algebra I, for example, strongly
emphasizes procedural fluency and deemphasizes the role of conceptual understanding. Students
are often deprived of valuable opportunities to construct their own learning and are instead
spoon-fed formulas, definitions, and properties in symbol-heavy language. The timelines that
are provided to students and teachers at the beginning of each unit clearly outline the goals of the
individual lessons as well as the National Council of Teachers of Mathematics (NCTM, 2000)
standards addressed; however, the order in which the lessons are generally presented is not
conducive to the development of conceptual understanding of the central ideas.
Traditionally, “curriculum” has been identified as the content of a course and perhaps, in
addition, the suggested timeline for addressing the topics within. Authors and educators Grant
Wiggins and Jay McTighe (2005) argue that “a curriculum is more than a traditional program
guide, therefore; beyond mapping out the topics and materials, it specifies the most appropriate
experiences, assignments, and assessments that might be used for achieving goals” (p. 6). They
propose the “backwards design” framework, in which educators should identify the desired
understandings and skills, determine how they can accurately assess students’ progress towards
these desired results, and then plan learning experiences accordingly.
Since Glencoe Algebra I is a widely-used text, and schools (or school districts or states)
often require teachers to use the provided textbook, it is important to look at how the curriculum
can be adapted rather than followed religiously or completely ignored. The following unit plan
was created by fusing elements of the backwards design framework with the suggested timelines
and activities provided in the Teacher’s Edition of Glencoe Algebra I. The desired results,
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assessments, and learning activities are generally aligned with the overall goals identified by the
authors of the unit and are carefully aligned with corresponding Common Core Standards that
will be addressed on cumulative standardized tests at the end of the course. The changes made to
the timeline and learning activities are intended to allow for deeper conceptual understanding by
focusing on the “big ideas” of the particular topics.
Unit Plan
Background:
Class: Algebra I
Grade: 9
Text: Glencoe Algebra I, 2001 edition
Unit: Linear Equations
Timeline: 11 days, 45 minute classes
Stage 1 – Desired Results
Established Goals:
Common Core Standards Addressed in Unit:
A-CED: Create equations that describe numbers or relationships.
2. Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or nonviable options in a modeling context. For
example, represent inequalities describing nutritional and cost constraints on combinations of
different foods.
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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.
A-REI: Reasoning with equations and inequalities:
10. Understand that the graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve (which could be a line).
Essential Questions:

What is the purpose of a linear equation? (topical)

What is the value of a mathematical equation? (overarching)

What does it mean to be linear? (topical)

What does it mean for quantities to relate to each other? (overarching)

What are different ways to express linear relationships? (topical)

What is slope? (topical)

How can change be represented symbolically? (overarching)
Wiggins and McTighe (2005) state that the role of the essential question is “to stimulate
thought, to provoke inquiry, and to spark more questions” (p. 106). Furthermore, they identify
the teacher’s intent as an integral aspect of these unit-driving questions. If a teacher were to pose
the question “what is slope?” and require that students answer “slope is equal to the rise divided
by the run,” this would not promote deeper inquiry. However, if the question “what is slope”
elicits from a student the response “what does it mean for values to change?” or “why is it that
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vertical lines are said to have no slope?” the teacher has managed to provoke further inquiry and
thus has successfully developed an essential question.
The above questions, therefore, do not have specifically worded answers that I intend to
have students reproduce on an assessment. During or after the unit’s implementation, I would
consider it a very positive result if students were to respond to the essential question “what is
slope?” with one of the two examples listed above. It would be even better (and it is probably
more likely) if a student were to raise a question that I cannot foresee.
It is ineffective to introduce essential questions and assume that students will be aware of
how these questions should be used to guide their learning throughout the unit. Wiggins and
McTighe point out that “students need to see how penetrating questions and arguments produce
knowledge and understanding” (2005, p. 122). It is important for the teacher to voice essential
questions in a way that truly elicits and encourages creative thinking. The 1991 Standards
released by the National Council of Teachers of Mathematics claim that “well-posed questions
can simultaneously elicit and extend students’ thinking. The teacher’s skill at formulating
questions to orchestrate the oral and written discourse in the direction of mathematical reasoning
is crucial” (NCTM, 1991.) Essential questions, even well-developed ones, can be meaningless if
the teacher does not constantly help students return to them; I will therefore continue to make
references to how they should be addressed throughout the unit in my unit calendar.
Understandings:

Linear equations can be used to model everyday phenomena.

A linear equation represents all the points on the line and can be used to determine
unknown independent or dependent values.
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
There are a variety of ways that linear relationships can be expressed, and the “best”
representation changes as the context changes.

Slope is another way of saying “rate of change” and describes the ratio of the change in
the dependent values to the change in the independent values.
The understandings have been articulated based on a combination of potential “answers”
to the essential questions and desired end-of-unit student “know-hows.” Counter-intuitively, it
would be ideal if, when students were asked at the end of the unit what they were now able to
understand, there was not a single response that directly mirrored any of the above statements.
Furthermore, “evidence of understanding does not require that students state the understanding in
words primarily” (Wiggins & McTighe, 2005, p. 142). The above statements summarize the
relevant highlights from the unit that I hope students will be able to transfer into other units,
classes, and outside contexts. I expect that students would come away with variations on these
main points that I have stated; while their “big ideas” would be similar at the core to those listed,
it is likely that they would be worded differently and more personally.
These understandings, at the core, should all share the common root of “linear
relationships and their expressions.” This is the major theme that connects all of the desired
understandings of the unit. Bransford, Brown, and Cocking (2000, p. 139) stress the importance
of giving students an “overall picture” when addressing multiple objectives. The above
understandings are important and meaningful by themselves, but it is the common theme among
them that must be highlighted in order to help students “learn their way around” linearity.
Students will know:
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Definitions:

relation

function

domain

range

quadrant

slope

linear

dependent variable

independent variable

y-intercept

x-intercept
It is essential that students be able to define the words listed above not only in the
language that the textbook provides and that they might see on a standardized test, but also in
language that they can use to communicate with people who are not members of the Algebra I
class or who have never taken Algebra I. In other words, by the end of this unit, a student should
be able to recognize that slope is equivalent to the “change in y over the change in x” AND be
able to explain what that means in less formal language that an outsider could understand (e.g.
“slope describes the way that a something changes as another element changes, like distance
over time). The definitions listed above are the necessary pieces of knowledge that students will
draw upon as they construct deeper conceptual understanding of linearity.
Algebra I Unit Plan 10
Students will be able to…

Given any form of linear equation, produce the corresponding graph.

Given a graph of a line, write the standard, point-slope, and slope-intercept forms of the
equation represented.

Given information about a relationship between two values, model the information with a
linear equation.

Determine the slope of a linear equation (given any form: graphical or algebraic).

Transition between standard, point-slope, and slope-intercept form as well as other linear
relationship expressions (e.g. tables, charts, etc.).

Algebraically manipulate equations in order to change them from one form to another.
These skills are the other pieces that students will need in order to develop conceptual
understanding of linearity. While being able to determine the slope of a linear equation is not an
understanding in and of itself, it is impossible for students to be able to demonstrate
understanding of linearity without the ability to identify the slope of a line. The value of these
skills is overemphasized in the Glencoe Algebra I curriculum; however, it is important not to
underemphasize their role in conceptual understanding. These skills are frequently addressed
throughout the unit and are therefore assessed continually as students acquire them. In addition,
they provide the basis for a large portion of the culminating assessment. Due to their role in
students’ conceptual understanding as well as their prevalence on standardized tests, it would be
ineffective to ignore them.
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Stage 2 – Assessment Evidence
Wiggins and McTighe (2005) define the following types of assessments in the Understanding by
Design framework:
Performance Tasks (PT): Complex challenges that mirror the issues and problems faced by
adults. Ranging in length from short-term tasks to long-term, multistaged projects, they yield
one or more tangible products and performances…
Academic Prompts (AP): Open-ended questions or problems that require the student to think
critically, not just recall knowledge, and to prepare a specific academic response, product, or
performance…
Quiz and Test Items (QT): Familiar assessment formats consisting of simple, content-focused
items…
Informal Checks for Understanding (IC): Ongoing assessments used as part of the
instructional process. Examples include teacher questioning, observations, examining student
work, and think-alouds. These assessments provide feedback to the teacher and the student.
They are not typically scored or graded.
The following types of assessment appear in this unit:
(PT/AP) Mid-way through unit:
Create two linear equations/functions that would result in your getting $134 from the walkathon.
Identify the slopes and y-intercepts of these equations. What are the situations that result in
these two different lines? Be creative!
(PT/AP) Before Culminating Assessment:
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AUNT BEE’S PHONE BILL
Your aunt has the following phone bills. She is unsure of exactly how she is being charged.
Help her determine the breakdown of the texting plan that is being billed below:
February: $9.80 for 156 text messages
March: $11.65 for 193 text messages
April: $7.85 for 117 text messages
Represent this in the way that you think best communicates it to her. Use tables, graphs, charts,
equations, sentence explanations, or a combination of these. Help her to understand how you
arrived at the answer.
(QT) Culminating Assessment I – Unit Test
(Adapted from exercises in Glencoe Algebra I)
1. Graph the line 3x + 2y = -6
2. Write the equation for the line that passes through the point (-3, 1) and has a slope of ½
3. Express the following equation in standard form: y – 3 = -4(x – 1)
4. Identify the slope and y-intercept of the line y = -x + 5
5. Write an equation that represents the following relationship:
x
0
1
2
y
-1
3
7
6. Write the equation of the line that passes through the point (1, 5) and (4, -4)
7. Identify the y-intercept in the line -4x + 2y = -8
8. What is the slope of the line that is perpendicular to the line 2x + 3y = -7
9. Which of the following lines have positive slopes? (circle all that apply)
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a. 7x + y = -4
b. y = 2x + 1
c. y – 1 = 2(x + 3)
d. -4x – 4y = 8
e. y = 3
10. Write the equation of a line that goes through the point (1, 7) and is parallel to the line
that goes through (3, -2) and (0, 4).
The unit test serves to assess essential skills that students need to have developed by the
end of this unit. Each question has only one correct answer. Similar questions will guide
students’ nightly procedural practice homework assignments. The skills assessed in this test
come directly from the procedural goals (“students will know how…”) indicated in Part I.
The goals listed in Part I and assessed here in the unit test are intended to be closely
aligned with the corresponding standards. Algebraic manipulation of equations is an absolutely
essential skill at all levels of high school and college mathematics and in important contexts
outside of the formal math classroom. Moses (2001) describes the continually increasing
importance of algebraic manipulation skills in the twenty-first century. Computer technology, he
states, “places abstract symbolic representations front and center. These representations are the
tools to control the technology, and in order to use this technology to organize work you have to
understand these symbolic representations and the place that society has assigned for young
people to learn this symbolism – this is algebra” (p. 13). To design a unit on linear equations
without heavily emphasizing the value of symbolic representation and manipulation would be
irresponsible.
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It is important to point out that though this test focuses completely on procedures, it is not
representative of the entire unit; that is why it is only one part of the culminating assessment.
Procedural fluency is not a flaw in and of itself. In fact, it is a vital component of mathematical
understanding. However, where my proposed portion of the culminating assessment differs from
the traditional culminating assessment and from the unit test proposed in the prescribed Glencoe
Algebra I curriculum is in its close relationship to the more conceptually oriented assessments.
Thompson, Philipp, Thompson, and Boyd (1994) contrast calculationally oriented teaching and
more conceptually oriented instruction. The latter often aims to “focus students’ attention away
from the thoughtless application of procedures toward a rich conception of situations, ideas, and
relationships among ideas” (p. 86). Since this portion of the assessment is given in conjunction
with a task that brings the calculations, vocabulary words, and procedures to life, it does not
deprive the unit of its overall conceptual orientation; it merely ensures that concepts can be
mastered through the necessary skills that accompany the deeper-level creative thinking that
students must demonstrate.
(PT) Culminating Assessment II
Design-a-Plan
You have been hired by Phony Phone Wireless to design three cell phone plans. The
company draws customers of all ages; some are students who text often, some are business
people who need many minutes, and some are very young children who only have cell phones
for safety purposes. Generally, they would like a plan for each of these three ‘categories’ of
costumers.
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The company tries to pull in between $45 and $95 per month per customer, so your plans
should be designed accordingly. Obviously, some customers will hardly use their phones, but
your goal is to come up with reasonably and competitively priced plans that will end up being in
this price range for the average customer. The company does not have many set guidelines, but
they do ask that you have some sort of monthly base fee.
After you design the plans, you need to present them to the CEO of Phony Phones
Wireless and the executive team. You may do so in any way that you like, but keep in mind that
multiple representations of a mathematical situation are often helpful to people. Remember also
that the executives might not be familiar with all of our ‘textbook terms’ (slope, y-intercept, etc.)
It is your job to communicate with the executives in a way that will convince them that your
plans are mathematically sound.
Be creative! You are not just demonstrating your understanding to me; you are making a
sales pitch to a team of executives with high expectations. How will you share your newly
developed expertise?
RUBRIC FOR CULMINATING ASSESSMENT:
Points:
Presentation
How will you
pull together all
of your findings
to impress the
executives
during your sales
pitch?
Communication
How will you
interpret your
mathematical
findings to an
audience outside
0
Presentation does
not include all
necessary
information.
1
Presentation
includes
necessary
information but
lacks
organization and
clarity.
2
Presentation
includes all
necessary
information and
is well-crafted,
neat, and clear.
3
Meets all criteria
for 2 points
PLUS the student
includes multiple
forms of media
in a creative,
meaningful way.
Ideas are not
articulated
through words
OR
student uses only
‘textbook
Some graphs and
equations left
without
explanations
OR
‘textbook
Student explains
all ‘textbook
language’ in
detail.
Explanations
could be
Meets all criteria
for 2 points
PLUS student
includes written
dialogue of sales
pitch to
Algebra I Unit Plan 16
of the classroom? language.’
Mathematics
How well do you
demonstrate an
understanding of
linearity, slope,
and y-intercept in
a real-world
setting and in
symbolic form?
Company Sales
Goal
How close do
you come to
helping the
company collect
their desired $45$95 per month
per customer?
Representation
How efficiently
did you portray
the mathematical
concepts
involved through
symbolic
equations,
graphs, tables,
and words?
Equations and
graphs
demonstrate no
understanding of
linear
relationships.
language’ is not
explained in
enough detail
Graphs/equations
written correctly
but demonstrate
misunderstanding
about the
concepts of slope
and/or yintercept.
Models do not
resemble attempt
to help company
reach sales goals.
Only one or two
models are
within range of
sales goals.
No graphs
OR
no equations
OR
no tables
Some
mathematical
concepts are
modeled using
multiple
representations.
understood by a
diverse audience.
executives.
Equations,
graphs, tables,
and language
demonstrate a
clear
understanding of
linearity and its
various
components.
All models are
within range of
sales goals.
Student extends
project to
determine when
certain plans are
cheaper/more
expensive based
on minutes/texts
used.
All mathematical
concepts are
modeled using
multiple,
appropriate
representations.
Meets criteria for
2 points PLUS
representations
are justified (e.g.
“this would be a
good way of
explaining the
plan to
customers…”
All models are
within range of
sales goals and
detailed
explanation/
rationale is
provided.
The culminating assessment, which gives students the opportunity to create cell phone
plans using their newly developed understanding of linear relationships, was designed with
hopes of integrating elements of both backwards design and culturally responsive pedagogy.
The task is purposefully open-ended and “messy.” Since there are so many open-ended
components (e.g. how the mathematics is represented symbolically, how the ideas are
communicated, how the plans are mathematically structured, etc.), I would not expect any two
student groups to have the same or even similar projects. I recognize that the results will
Algebra I Unit Plan 17
probably be “messy.” For example, students might use multiple equations for each plan to
include both texting and minute allotments, or they might invent a plan that will probably end up
costing most customers a little less than $45 or a little more than $95. As discussed by Wiggins
and McTighe (2005), in order for a performance task to be “authentic,” it must “involve a real or
simulated setting and the kind of constraints, background ‘noise,’ incentives, and opportunities
an adult would find in a similar situation” (p. 153). I want the mathematics to be realistic, and as
noted by the authors of the Understanding by Design framework, mathematics in the world
outside of the classroom is often “messy.”
I chose the context of a cell phone company because I am aware of how huge a role cell
phones play in the social lives of high school students. The world of cell phones therefore
creates a context in which students can see the relevance of mathematics in their lives outside of
school. Daniel Chazan (2000) defines a real-world problem as one that “someone, perhaps even
the student, might encounter outside of school” (p. 40). Many of the real-world problems that
are found throughout the Glencoe Algebra I text are indeed applications to life outside of the
classroom; however, they lack close connections to the lives of most high school students. For
instance, the context that frames a real-world application problem on the text’s proposed unit test
involves the growth rate of bluegill fish. By framing the final performance task in the context of
cell phone plans, I have created an application more likely to apply to students’ lives while
maintaining the core mathematical themes of the chapter.
In developing and fine-tuning this culminating assessment, I tried to analyze the task
through the five criteria developed by Leah, Hoover, and Kelly (1992) and adapted by
Zawojewski (1996). As a result, I can answer in the affirmative to the five questions that
Algebra I Unit Plan 18
Zawojewski (1996) poses that can help determine whether an assessment is worthwhile and
effective:
1. Does the problem elicit important mathematics?
The task requires that students can correlate a “monthly base fee” to a y-intercept. It also
requires students to be able to think about the slope of a line in terms of the real-world context of
cents per minute or text message. Students must create tables, equations, and graphs to portray
this information, which address important mathematical goals for the unit. They must also
communicate what they find in ways that could be understood outside the classroom. All of
these mathematical understandings are closely aligned with the goals indicated by the Common
Core Standards listed in Part I.
2. Do all students have access to the task?
Since cell phones are such a significant part of high school life in general, the context
should make sense to all students. After ensuring that all students in the class understand the
contextual terms and meanings in the problem, I would be confident that students can access the
task. The project falls at the end of the unit and all students should by now have an
understanding of rate of change and y-intercept. Any student who has “kept up” in the class (that
is, participated in group work, completed homework and quizzes, and engaged in class
discussion) should have access to the task.
3. Does the problem genuinely engage students?
Since the problem involves cell phones, students can relate to the context. It engages
students by asking that they assume a role as designers hired by the phone company and share
their mathematical knowledge. In this respect, it is empowering; students are the mathematical
Algebra I Unit Plan 19
authorities and are being held responsible for doing the necessary critical mathematical thinking
and for communicating this thinking.
The design of giving students a role like this is influenced by the work of James Gee
(2007) as discussed in the work of Richard Milner (2010, p. 126). Studying the potential
positive influences of the integration of video gaming into classroom learning, Gee posits that
students who are encouraged to develop an imaginary identity, particularly one that requires
them to assume responsibility, are much more likely to be engaged and naturally motivated to
succeed. In giving students a specific role and entrusting to them the task of presenting a board
of executives with their mathematical expertise, I provide a challenging, relatable, and engaging
task. While it is imaginary, it is not fantastical; therefore, students will be able to participate
with the hopes of someday being given a similar real opportunity to share their mathematical
expertise.
In describing effective real-world problems, Chazan (2000) states that “the notion is that
students will be attracted to problems that implicitly suggest that mathematics is a useful body of
knowledge that allows people to solve problems they face in their lives” (p. 40). The cell phone
problem will help students understand that the mathematics that they have learned will be
valuable not only after the unit but after the course and even after high school have been
completed. Students will be “attracted to” and engaged in the problem because of the likelihood
that they have either dealt with or will soon encounter a similar problem in their lives outside of
school.
4. Does the problem elicit responses that reveal students’ mathematical thinking?
The problem requires that students reveal their mathematical thinking and understanding
in multiple ways. In asking students to make some sort of presentation to the executives that
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would make sense to people that have not been sitting in our classroom all year, I ensure that
students will communicate in more “everyday language” what they understand. This language
will help me determine how they really think about math (whereas simply requiring that they
offer answers heavy on ‘textbook language’ can lead to the danger of allowing students to repeat
what they hear or read without understanding).
Since this task is so open-ended, students must create a lot of the mathematics from
scratch. In other words, in order to create equations and graphs, they must be creative. It is
through this creativity that I will be able to see how they think about the ways that linear
relationships can occur in the real world.
5. Does the context of the task present adequate criteria by which students can judge the
“goodness” of their response? That is, can students make judgments about whether
reasonable progress is being made? Can they judge from the task statements whether
a response is complete?
Since the task is relatable to students, they will be able to determine how reasonable their
responses are. For example, students bring in outside knowledge from having talked to their
parents or perhaps even having helped arrange their own cell phone contracts; they know that it
will not cost $10 per minute. In addition, they bring in their understanding of how often people
talk on the phone. Therefore, creating a plan that would result in an average teenager spending
about $65 per month based on the number of minutes and texts used is a reasonable task.
I provide the rubric with the task, so students know the explicit expectations. This is an
important component of both culturally responsive pedagogy (Milner, Context of Teaching and
Learning, 10 May 2010) and backwards design (Wiggins & McTighe, 2005, p. 142). Since
students have the very same rubric that I will use to assess the task in front of them as they
Algebra I Unit Plan 21
complete the task, they are able to make judgments about the completeness and quality of their
work.
Other evidence:
Homework assignments will combine textbook problems that give students opportunities
to practice procedures with what Wiggins and McTighe (2005) call “academic prompts.” These
slightly more open-ended questions “are ‘open,’ with no single best answer or strategy expected
for solving them,” “typically require an explanation or defense of the answer given and methods
used,” and “involve questions typically only asked of students in school” (p. 153). It is my
intention to give students repeated chances to practice the skills that they acquire and to help
them think more conceptually on a routine basis. In particular, justification and explanation are
priorities of many of these academic prompts. Since students will need to justify and explain
their understanding in the culminating assessment, it is important that they learn to develop solid
justification habits.
Though I have not listed any informal checks for understanding here, there are many
embedded in the unit. Students will sometimes complete ungraded exit slips. Other times, I will
assess by asking questions when students are working individually, in small groups, or in a
whole-class setting. I also intend to observe small group discussions and presentations to check
for students’ understanding.
Homework/classwork exercises not found above or in the textbook are as follows:
(AP/QT) KY/TN State Fair Problem (Shahan, 2009)
1. The Kentucky State Fair charges an entrance fee of $8.00 plus an additional $1.50 per
ride.
Algebra I Unit Plan 22
a. How much would it cost you, altogether, to enter the park and go on 1 ride? How
about 2 rides? How about 10 rides? How about 20 rides?
b. Write an algebraic expression to represent the relationship between the number of
rides (n) and the amount of money (c) you spend at the fair. (Assume you spend
money only on the entrance fee and rides.)
c. Create a graph to show the relationship between the number of rides you go on
and the amount of money you spend, using n as your independent variable and c
as your dependent variable.
d. What does the slope of your line represent? What does the c-intercept represent?
How about the n-intercept?
2. The Tennessee State Fair charges $2.00 per ride. You’ve been on 7 rides and have $50
left in your wallet.
a. Assuming you only spend money on rides and aren’t robbed, how much money
would you have left in your wallet after you’ve been on 8 rides? How about after
17 rides? How about after 27 rides?
b. Write an algebraic expression to describe how much money (w) you would have
left in your wallet after you’ve been on m rides.
c. Create a graph to show the relationship between the number of rides you go on
and the amount of money you spend, using m as your independent variable and w
as your dependent variable.
d. What does the slope of your line represent? What does the m-intercept of your
line represent? How about the w-intercept?
Algebra I Unit Plan 23
3. Reflect on the similarities and differences between these two problems. What
information are you given in each? Did you use the same form of linear equation for
each?
(AP/QT) Imagining Lines (Shahan, 2009)
1. If you have a line with a positive y-intercept and a positive slope, what can you conclude
about its x-intercept? Justify your answer. Diagrams or graphs might be helpful in
justifying your answers on this task.
2. The graph of a certain linear function has a negative slope. If its x-intercept is negative,
then its y-intercept must be which of the following? Justify your choice.
a. Positive
b. Negative
c. Non-negative
d. Less than its x-intercept
e. Greater than its x-intercept
3. Try to answer these questions without paper, pencil or calculator. Picture the line y = 3x
+ 5.
a. Which quadrants does the line pass through?
b. Is the x-intercept of this line positive, negative, or zero?
c. In which quadrant would this line intersect the line y = -8?
d. Can you find a line that would intersect this line in the 4th quadrant?
e. As the x value increases by 2, by how much does the y value change?
f. As the x value decreases by 5, by how much does the y value change?
Algebra I Unit Plan 24
Stage 3 – Learning Plan
Glencoe Algebra I suggests the following basic timeline across two units of study:
Day
1
2
3
4
5
6
7
8
9
10
11
Lesson
The Coordinate Plane
Relations
Equations as Relations
Graphing Linear Equations
Functions
Writing Equations from Patterns
UNIT TEST
Slope
Writing Equations in Point-Slope and Standard Forms
Writing Linear Equations in Slope-Intercept Form
Graphing Linear Equations
Parallel and Perpendicular Lines
UNIT TEST
The valuable lessons within the suggested timeline are presented in an order that is bound
to deprive students of important understandings. For instance, on Day 6, students would focus
on writing equations based on provided linear graphs without having looked at the concept of
slope or having learned to graph lines based on an equation. Essentially, students are told to
“decompose” the given line into a series of points and to place these points in a table to facilitate
finding the pattern that occurs. They are then told to find an equation that relates x and y based
on these series of coordinate pairs. The desired result is a linear equation written in slopeintercept form. However, students have not yet defined slope or y-intercept and will not
‘officially’ learn to write a linear equation in slope-intercept form from given information (e.g.
graph, coordinates, etc.) until days later.
Algebra I Unit Plan 25
I propose an edited order which will result in more cohesive flow between lessons. In
addition to changing the order, I have changed and added certain activities within the units. A
general outline of the revised order is listed below:
1
2
3
4
5
6
7
8
9
10
11
The Coordinate Plane
Relations
Relations as Linear Equations – Graphs and Equations
Relations as Linear Equations – Slope and y-intercept
Closer Look at Components of Linear Equations
Forms and Components of Linear Equations
Parallel and Perpendicular Lines
Piecing It All Together
Culminating Assessment I
Culminating Assessment II
Culminating Assessment II
In this sequence and in the Glencoe sequence, students are introduced to linear equations
by building upon their understanding of relations. However, unlike Glencoe’s sequence, this
proposed order offers students the opportunity to begin analyzing linear relationships from the
start. The class will explore relations that result in linear patterns by analyzing both tabular and
graphical representations. Students will use both graphs and tables to study the relationships and
to learn to build linear equations. Instead of teaching students how to graph lines, then teaching
the components of linear equations, then returning to teaching them how to graph linear
equations based on their new knowledge of these components, as is proposed in the Glencoe
sequence, I suggest a timeline which allows students to simultaneously study components of
linearity, learn how to graph lines, and develop the ability to write linear equations based on
given information.
In my revised schedule, students will start from the most basic linear relationship (y = x)
and branch out from a slope of 1 and a y-intercept of 0. Since slope is one of the most important
components of this unit and is an underlying theme of multiple big ideas and essential questions,
Algebra I Unit Plan 26
I plan to introduce it in its most basic form and allow students to build their understanding of it
through meaningful experiences. Students will initially explore slope by changing the steepness
of their original y = x graph; this introduction offers them a chance to construct the meaning of
slope through an experience.
In contrast, Glencoe Algebra I offers the following definition for slope:
The slope m of a line is the ratio of the change in the y-coordinate to the change in the
corresponding x-coordinate.
Shortly after this definition and a few examples of how to determine the slopes of given graphed
lines, the text gives an algorithm to help students know how to determine the slope of a line.
Given the coordinates of two points (x1, y1) and (x2, y2), on a line, the slope m can be
found as follows: m = (y2 – y1) / (x2 – x1), where x1 ≠ x2.
An introduction that encourages students to learn the concept of slope by having them read and
commit to memory a definition and a symbolic formula is bound to inhibit their initial
understanding and provide a weak, meaningless base. Kalchman and Koedinger (2005) note:
[F]or students to understand slope in these definitional and symbolic ways, they must
already have in place a great deal of formal knowledge, including the meaning of ratio,
coordinate graphing, variables, and subscripts, and such skills as solving equations in two
variables and combining arithmetic operations. Knowing algorithms, however, does not
ensure that the general meaning of slope will be understood. (pp. 362-363)
Rather than provide students with a set definition and algorithm, I hope to help them build on
their outside, intuitive knowledge and construct a definition and an algorithm at their own paces
and in their own words.
Algebra I Unit Plan 27
Unit Calendar
Day 1 – The Coordinate Plane
Objectives:

Students will be introduced to/ review their knowledge of the coordinate plane.

Students will learn to plot points based on given coordinate pairs.

Students will learn to identify the coordinate pair that describes a graphed point.
This first lesson will be a review for many (if not all) students. It is important that before
students begin to graph and analyze linear equations, they have a solid understanding of the
coordinate plane.
A main goal that I have in this lesson is to stress that the coordinate plane is infinite and
that points need not be comprised of integers. For example, (0.5, 1.5), (1/17, 6.896), and
(1,284.44, 9.0000002) are all points on the plane. Students often struggle to understand that a
point does not have to be “neat.” This misconception can influence their understanding of what
a line represents; students see that (1,4) and (2,8) are points on the line but do not understand
how (1.5,6), (1.25,5), (1.125, 4.5), etc. are all points on this line as well. It is my intention to
build a solid basic understanding of the coordinate plane and its infiniteness.
Tasks/Activities:
I will begin by directing students’ attention to the map of the United States projected at
the front of the classroom. The map will have clearly labeled lines of longitude and latitude. I
will ask students whether they have ever seen a map with these lines and whether they have ever
heard the terms “latitude” and “longitude” before. We will have a brief discussion about the way
that latitude and longitude work, and I will assess by asking students to identify the coordinates
Algebra I Unit Plan 28
of certain cities and to tell me the cities that lie at certain coordinates. I will purposefully ask
about cities that fall right on two lines (e.g. New Orleans at 90°, 30°) and cities that fall in
between the lines (e.g. Nashville at 86.5°, 37°) in order to build towards students’ awareness that
points occur in between the identified gridlines. This activity is based on the “real-world
application and problem solving” question (#41 on page 258) in the Glencoe text. I have chosen
to use it as the introduction instead of a practice problem in order to build upon what students
might be more familiar with and then help them transfer this understanding to a less familiar
context.
I will then show students a traditional Cartesian coordinate plane and explain the
similarities. As it is likely that students have seen the coordinate plane in Pre-Algebra classes or
before, I will try to let them contribute as much as possible by asking such questions as “does
anyone know what is meant by ‘y-axis?’” and “what does an ordered pair look like?” We will
spend some time going over terms, with the goal of letting students do as much of the defining as
possible. I hope to stimulate students’ thinking by asking such questions as addressed by “the
completeness property for points in the plane” on page 256 of the text (e.g. Do you think an
ordered pair can ever refer to more than one point? Are there ever two ways of naming a point
on the plane?).
Towards the end of the lesson, I will show students the Culminating Assessments I and II
and explain to them that by the end of this unit, they will be able to complete every task on both
of these assignments. It is important for students to be aware of the learning goals of the unit
from the beginning; there should not be an element of “mystery…about either intended learning
outcomes or what success in achieving those outcomes will look like” (Tomlinson & McTighe,
2006, p. 86). In making students aware of the expectations on the culminating assessment, I
Algebra I Unit Plan 29
respond to the naturally occurring but often unvoiced question that students often ponder: “how
does this relate to me now and in the future?” (Milner, Context of Teaching and Learning, 10
May 2011). I am making the expectations clear, and I am helping students understand how
plotting points on a coordinate plane and identifying the slope of a linear equation will help them
achieve the more socially relevant goals of analyzing and creating cell phone plans.
Assessment:
Students will be given an exit slip that asks them to identify the points at certain
coordinates. They will also be asked to list any questions that they have about the coordinate
plane so that I can try to address the issues in the next lesson.
Many of the lessons will conclude with an exit slip. These un-graded assessments serve
dual purposes. When students are given time to individually reflect upon what they have learned
and what they are still struggling to understand, they are engaging in the very important practice
of self-assessment. Students need to be aware of their own learning, and informal and formal
self-assessments are important components of the development of metacognition (Bransford,
Brown, & Cocking, 2000, p. 140). In addition to providing students with structured
opportunities for self-assessment, exit slips and other informal checks at the end of lessons will
give me an idea about how to modify my plans for the following day to build on students’
strengths and address their questions and weaknesses more specifically.
Homework:
Students will complete exercises 13, 14, 16, 19, 20, 22, 23, 24, 25, 28, 30, 34, 42 on page
257 in the textbook.
Day 2 – Relations
Algebra I Unit Plan 30
Objectives:

Students will be able to identify the domain, range, and inverse of a relation.

Students will be able to express relations as sets of ordered pairs, tables, and graphs.
Tasks/Activities:
After giving students the answers to the homework exercises and discussing any major
questions that students have, I will direct their attention to the “U.S. Unemployment Rate” chart
at the front of the classroom. This activity is based on one given on page 266 of the Glencoe
text; however, instead of using it as a practice problem, I will use it to introduce the topic of
relations. We will define ‘relation’ in our own words and then compare with the definition
provided in the text. We will look at the different ways of expressing relations and talk about
when certain ones are most useful. We will also talk about domain, range, and inverse. Most of
the discussion will be in whole-class format, and I will continually remind students to define
terms in their words and to compare these definitions with the more standard textbook
definitions.
This second lesson in the unit is not one in which students have the opportunity to spend
significant time in groups. However, since most of what they will be seeing in this lesson is a
likely to be a review for many of them, and since students will be working in pairs and groups in
the next few lessons, it is appropriate that they have some time to work independently and in a
large-group setting to define these base terms. Since the format will be whole-class discussion,
there will be an element of collaboration as the students work together to settle on acceptable
definitions of the various terms.
Assessment:
Algebra I Unit Plan 31
Students will complete an exit slip that asks them to state the domain and range of given
relation. It will also have a space for them to list questions about relations that went unanswered
during class.
Homework:
Students will complete exercises 16-22, 26-31, and 43 on page 267 in the textbook.
Day 3 – Relations as Linear Equations
Objectives:

Students will understand the relationship that exists between independent and dependent
variables.

Students will be able to write and understand the linear equation y = x in both algebraic
terms and in their own words.

Students will graph the line y = x.

Students will be introduced to slope.
Tasks/Activities:
The “walk-a-thon” task is borrowed from the ideas of Kalchman and Koedinger (2005) as
expressed in “Teaching and Learning Functions.” Though the goals of this unit on linear
equations are somewhat different from those in the functions unit that the authors discuss, there
are significant opportunities for students to develop a solid base for their understanding of slope
and dependence.
Launch:
Algebra I Unit Plan 32
To ensure that all students have access to the context of a walk-a-thon, I will begin by
showing pictures and leading a discussion about the event. It is important that students are not
distracted by uncertainties about the context of a mathematical problem; a teacher must launch
the task in a way that invites all students to “get in the game” (Cobb, Launching Tasks, 15
November 2011).
After the context has been set, I will introduce students to the problem of someone being
paid $1 for every mile he walks. We will set this up as a table and then as a graph, and I will ask
students to come up with some sort of “rule” that explains how much money is being made each
mile. Students can write this in any “language” that they like (e.g. money = miles x 1). As
pointed out by Kalchman and Koedinger (2005), integrating students’ informal language about
mathematics into the development of algorithms, functions, and definitions is a valuable tool for
fostering conceptual understanding (p. 363).
I will lead students in a discussion about the infiniteness of this linear equation; that is, I
want them to be aware of the fact that it goes on forever in both directions. I will ask questions
such as “what do you think would happen if I walked 3.5 miles?” and “what about if I just kept
walking for days on end and walked 100 miles?” Eventually, I would intend to lead students to
engage in what Mark Driscoll (1999) calls “abstracting from computation;” that is, students
should begin to “think about computations freed from the particular numbers they are tied to in
arithmetic” (p. 2). Though this will probably not be the first time that most of the students in the
class work to go from a concrete situation to an abstracted formula, it could be the first time that
they formally do so in the context of a linear equation.
We will work together towards translating students’ informal equations to “y = x.” This
must be done very carefully and through students’ construction rather than by my telling them
Algebra I Unit Plan 33
that “‘y = x’ is a better way to represent the relationship than ‘money = miles x 1.’” Students
often struggle to understand the necessity of literal symbols and confuse the different ways that
letters are used to represent variable and constant quantities in mathematics (Philipp, 1992, p.
157). Students must continue to develop their “abstracting from computation” habit of mind
through their own creative thinking; it would be detrimental to their understanding of this
particular concept (as well as their general abilities to think about mathematics abstractly) if I
were to simply present them with the equation.
It is noteworthy, again, that I choose to introduce graphing lines and writing linear
equations with the base equation y = x. Glencoe Algebra I (and many other standard texts and
curricula) often begin by giving students an equation such as y = -3x + 4 and identifying -3 as the
slope and 4 as the y-intercept. Students then “work backwards” towards lines such as y = 4x (yintercept of 0) and y = x + 2 (slope of 1). However, after repeated work with slopes other than 1
and y-intercepts other than 0, students are often confused when they encounter the line y = x.
Instead of treating the line y = x as some sort of “special case” of line that is void of the
components that lines are “supposed to have,” I hope to help my students treat it as the base upon
which every other line is built.
After we have discussed in detail as a class the base equation y = x, I will guide students
into thinking about slope. The questions that I pose will have a significant impact on the
direction that the discussion takes; it therefore important to pay close attention to timing and
wording. I would continue to frame everything in the context of the walkathon. In their study
and discussion of teacher questioning practices, Martino and Maher (1999) advise that “the type
of question asked by the teacher must be connected to the student’s present thinking about a
solution” (p. 56). At this point in the lesson, students see the line y = x within the context of
Algebra I Unit Plan 34
someone being paid a dollar for every mile walked, so questions must continue to be framed
within this scenario. In the next few lessons, it will be important to change the context so that
students have the necessary chance to transfer their learning to unfamiliar situations (Bransford,
Brown, & Cocking, 2000, p. 51). It is important, however, for their initial learning to be solid
before they are forced to transfer into new contexts.
Exploring Slope:
I would begin by asking students if they could change the “dollar for every mile” rule to
make the line steeper. I would not use the word slope, nor would I start by discussing “a change
in the rise over run ratio;” I want students to continue to construct their own definitions. After
we arrived as a class at an equation such as y = 2x or y = 4x (graphed and expressed in student
and formal algebraic terms), I would then guide students into thinking about slope or steepness
as a more abstract concept. We would work together to formalize a definition and compare it
with the definition provided in the textbook.
Assessment:
In this lesson, I do not have students turn in an exit slip for a few reasons. The class is so
discussion-heavy that I will be able to gain a decent understanding of what students are learning
through their questions and responses. The homework assignment, which will eventually be
turned in, will also tell me about their understanding of slope. When they engage in the activity
that uses their work from the homework assignment during the next lesson, they will have the
chance to evaluate their own understanding of the concepts.
Homework:
I will instruct students to come up with three different sponsor amounts of their choice.
They should write equations and on a separate piece of paper graph the equations without
Algebra I Unit Plan 35
labeling them. In other words, they should have one sheet of paper with three equations and one
piece of paper with three corresponding but unlabeled graphs.
I do not use any exercises from the book in this lesson because the authors take such a
different approach to teaching linear equations. When the Glencoe text introduces linear
equations, they begin with those that have slopes other than 1 and y-intercepts other than 0, so
there is no appropriate set of exercises that would give students “practice” with what they
learned during this lesson.
Day 4 – Relations as Linear Equations: Slope and y-intercept
Objectives:

Students will continue to explore slope and solidify understanding of the concept of rate
of change.

Students will learn to identify y-intercept from a graph or an equation and understand its
role in a line.
Tasks/Activities:
Students will begin by exchanging unlabeled graphs with a partner. Each student will
then determine the corresponding linear equations and check their solutions with the original
authors of the graphs. This will serve as a homework review and a bit of a warm-up to get
students back in the mindset of thinking about slope.
We will return in a class discussion to the situation discussed during the previous lesson
in which someone is making more than $1 for each mile walked. We will discuss the
possibilities of making more than that (slope >1) and less than that (slope <1) and what it does to
Algebra I Unit Plan 36
the linear equation. I will also use this opportunity to emphasize that slope does not have to be a
whole number. For example, I will ask students what it would mean if I were making $2.50 per
mile and would hope to transition the conversation to the notion that a slope of 2.5 is equal to a
slope of 5/2 and this is often how it can be identified on a graph (i.e. “up 5, over 2”).
The concept of y-intercept will be introduced through a question that asks how the line
would change if the walker were paid an initial amount before even starting. Here I continue to
borrow from Kalchman and Koedinger’s activity (2005). The discussion would follow a similar
format as the one in the previous lesson; I want students to describe the situation more than I do,
and I would like for us to come up with definitions in their language before turning to the
textbook definitions and comparing. We would work to integrate this new component into our
linear equations (e.g. walker is paid an initial amount of $4 and paid $2 for each mile walked
corresponds to y = 4 + 2x).
During the discussion, I hope to be able to begin to address the concept of negative yintercept. I would ask students to come up with situations that might result in a negative yintercept and write the corresponding equations and draw the graphs.
Students would be broken up into groups of three and asked to come up with two
different ways that a walker could make $134 in the walk-a-thon. Students would be given large
pieces of graph paper. I would ask that groups be creative; that is, offer some sort of reason for
the initial payment and the per-mile amount. I want to continue to keep the problem embedded
in its context. In the last few minutes of class, I would ask groups to (informally) present their
situations and graphs to the entire class.
Assessment:
Algebra I Unit Plan 37
The informal presentations at the end of the class, as well as observations of students’
conversations during the group work time, will provide me with ideas about what students
understand and what I should emphasize in the next lessons.
Homework:
Students will complete exercises 15-26 on page 300 in the textbook. In addition to
writing the equation for each relation, students will graph the equations. I want to make sure that
students have practice with identifying the slopes and y-intercepts of lines and working them into
slope-intercept equations.
Day 5 – Components of Linear Equations
Objectives:

Students will solidify understanding of slope, y-intercept, and x-intercepts.

Students will be able to identify slope and intercepts given an equation.
Tasks/Activities:
Students will be given the solutions to the homework and a few minutes to ask questions.
After we have gone over the homework, we will review the vocabulary that we have already
defined and will address a few new words (namely, independent variable and dependent
variable). I will ensure that all students have been exposed to and have written down formal
definitions as well as informal “in your own words” definitions.
I will pass out the “Kentucky State Fair” activity (Shahan, 2009). Students will work in
groups of four to complete the questions. After fifteen minutes, we will reconvene and go over
the answers to the questions as a group. I will ask groups to volunteer to present their findings
Algebra I Unit Plan 38
for a question at the front of the class, and I will attempt to direct all student questions back at
either the presenting group or other students in the class. Martino and Maher (1999) state that
“the teacher who also backs away strategically when communication and reasoning flourish,
allows students to play more active roles in their own and in each other’s learning…” (p. 75). I
want to place most of the responsibility on the students so that they learn to take ownership of
what they are learning and can construct their ideas based on their own interpretations.
In this discussion (as in most discussions), I am also attempting to do what Kazemi
(1998) identifies as “pressing for learning;” that is, emphasizing student effort, focusing on
learning and understanding, supporting students’ autonomy, and emphasizing reasoning more
than the production of correct answers (p. 410). I am much more concerned with how students
think about the concepts than the answers they produce. If students do not ask enough questions
or challenging questions of each other, I will try to press for learning by probing. Eventually, I
want students to think about why the line should or should not be graphed to the left of the yaxis.
Assessment:
Students will complete an exit slip that asks a similar “state fair question” with different
monetary amounts. I will ask them to graph the situation and to explain their graphs in detail.
Here, I am looking to see how much attention students pay to graphing to the left of x = 0 and
if/how they justify this. This will help me understand how effective the related discussion was
and how we should return to it in the next lesson.
Homework:
Students will complete the “Tennessee State Fair” activity on their own. I want to make
sure that students have opportunities to work collaboratively and independently. This is a similar
Algebra I Unit Plan 39
activity with a few minor differences, and a prompt actually asks students to compare the two
problems.
In addition to completing this similar assignment, I will give students the following
prompts to think/write about: What do you think is the slope of a horizontal line? What about a
vertical line? Justify your answers! What do equations of these types of lines look like?
Day 6 – Forms of Linear Equations
Objectives:

Students will solidify understanding of slope, y-intercept, and x-intercepts.

Students will be able to identify slope and intercepts given an equation.

Students will be able to graph and algebraically transition between lines in standard,
slope-intercept, and point-slope forms.
Tasks/Activities:
We will discuss the work that students did on the Tennessee State Fair problem. We will
spend less time actually going over the answers and more time talking about how the two
problems are similar and different.
We will then have a discussion about horizontal and vertical lines, drawing on students’
answers and justifications from their homework. I want the discussion to primarily be focused
on what students have come up with, but eventually I will ensure that the “rules” we develop are
in line with what is presented in their textbook and in the Common Core Standards.
After having completed the homework, I will introduce example 1 from page 333 in
textbook. I will give students the point (-3, 5) and the slope -3/4 and ask them to write an
Algebra I Unit Plan 40
equation. They have not been introduced to point-slope, so I imagine that most of them would
graph the line, determine the y-intercept, and write an equation in slope-intercept form. This is
actually what I would like for students to do so that when I do introduce point-slope form
immediately after that, they can see that the two equations represent the same line. I will show
them through the example how to write an equation in point-slope form. I want them to come up
with their own “definition” of a point-slope equation before we compare it to the one in the
textbook, which is very symbol-heavy. I will then introduce standard form in a similar way,
encouraging them to come up with their own definition before we compare with the textbook
definition. We will work through examples 1-3 before students are given some time to work on
exercises 18-38. During this time they will work primarily independently but will also be able to
ask each other questions and to elicit my help when needed.
This lesson obviously does not lend itself to significant cultural or social relevance, nor
does it allow for rich conversation and thought-provoking questions. However, the ability to
write equations in point-slope and standard forms and to convert between the three forms of
linear equations is essential. Standards and standardized testing call for lines to be written in all
three forms and converted from one to another. This lesson is purposefully devoted to giving
students the opportunity to practice these procedures. Unfortunately, the lesson is lacking in “big
ideas;” however, the skills that students are spending time developing (with assistance from each
other and myself) are vital components of some of the more overarching goals of the unit, like
the ability to communicate an understanding of linearity in different ways.
Assessment:
I will assess informally by circulating the room while students work independently.
Homework:
Algebra I Unit Plan 41
Students will complete Imagining Lines worksheet (Shahan, 2009).
Day 7 – Parallel and Perpendicular Lines
Objectives:

Students will be able to identify lines as parallel or perpendicular by comparing slopes.

Students will understand and be able to argue the reasons that opposite reciprocal slopes
result in perpendicular lines and that similar slopes result in parallel lines.
Tasks/Activities:
We will spend fifteen-twenty minutes discussing the worksheet that students did for
homework. I wanted them to complete this independently; it is so open-ended and I knew that in
sharing work as a class, we would see a wide variety of interpretations (not necessarily in
solutions but in justifications). Martino and Maher (1999) point out that a task that will provide
for a rich discussion involving justification “should provide the opportunity for student
reinvention of mathematical ideas through both exploration and the refining of earlier ideas” (p.
54). Nothing that students see in this worksheet addresses a completely new concept, but the
words and symbols involved might be different than what they have seen. I will try to have
students re-word other students’ thoughts to ensure that everyone understands. The conversation
that will follow this activity will hopefully reveal many different interpretations and diverse
constructions of reasoning and solutions.
I will then give the students the following prompt for them to think about individually
and then to discuss in pairs: “What do you think is special about lines that have the same slope?”
I want them to approach it however they would like (through examples, sketching graphs,
Algebra I Unit Plan 42
writing equations, etc.) but when we discuss as a class, I will push them to justify their ideas. I
will then ask them to return to working individually to determine if they think there is anything
special about the slopes of perpendicular lines and, if so, what is the property? After giving
students a few minutes to think individually I will ask that they return to their pairs and discuss
and, as before, prepare to share their ideas with the class. Again, I am primarily concerned with
how students justify their ideas, so my questions and conversation highlights will mainly address
justification issues.
To stress the infiniteness of the plane, I will give students the following question for them
to discuss in small groups of four: How many lines are parallel to the line y = 3x + 3? As groups
conclude that there are infinitely many, I will further probe with questions about how many lines
parallel to y = 3x + 3 exist between x = 0 and x = 10 and how many exist between x = 1 and x =
2. I want students to understand that there are infinitely many lines on the plane and infinitely
many lines with the same slope, even within a “small domain.”
Assessment:
Students will individually complete the exit slip asking them a question similar to the
“how many parallel lines” question, but in this case it will address perpendicular lines. I will
stress that students should argue their conclusion, and I am looking to see that students have an
understanding of the infiniteness of the plane.
Homework:
Students will complete exercises 18-27, 38-46 on page 367 in the textbook. The exercises
will give students the opportunity to identify slopes and parallel and perpendicular lines and to
continue practicing changing lines from standard and point-slope forms to slope-intercept form.
Algebra I Unit Plan 43
I will also stress to students that during the following lesson, they will have the
opportunities to ask questions to either their peers or to me in preparation for the culminating
assessments. They should go back over homework assignments and look for questions that they
had about specific problems or about overarching concepts and come ready to address them to
the class.
Day 8 – Piecing It All Together/ Review Day
Objectives:

Students will solidify recently developed understandings about slope and x- and yintercepts.

Students will review understanding of relationships explored in the unit.

Students will prepare for a skill assessment.
Tasks/Activities:
We will spend time going over the homework, paying particular attention to question 46.
I want students to talk about how they arrived at their answers and how they know they are
correct in their thinking.
I will then ask students to address any questions they have about topics from the unit.
This is a time for procedural questions as well as more overarching conceptual questions.
Generally, I would like for students to answer questions; I will serve as facilitator of the class
discussion.
Algebra I Unit Plan 44
Students will complete Aunt Bee’s Phone Bill assignment in small groups. This will
serve to give them practice solving linear equations and prepare them for the context of the
Culminating Assessment II.
Day 9 – Culminating Assessment I
Objectives:

Students will formally and independently demonstrate their proficiency with various
skills developed throughout the unit.
Tasks/Activities:
Students will complete the Culminating Assessment I – Skills Test. Students will work
independently and will have the entire class period to work.
Day 10 – Culminating Assessment II
Objectives:

Students will demonstrate their understanding of the concept of linearity and their
abilities to appropriately and efficiently communicate linear relationships to a wide
variety of audiences.

Students will continue to learn how to share responsibilities with group members.
Tasks/Activities:
I will introduce the assignment and tell students that they will have two days in which to
complete it. I will ensure that all students understand the context of cell phone plans, asking for
Algebra I Unit Plan 45
examples of charges and how they are often rated. I will provide the rubric and go over it in
detail before students begin working. Students will work in small groups of four to complete the
Cell Phone Plan assessment.
Day 11 – Culminating Assessment II
Objectives:

Students will demonstrate their understanding of the concept of linearity and their
abilities to appropriately and efficiently communicate linear relationships to a wide
variety of audiences.

Students will continue to learn how to share responsibilities with group members.
Tasks/Activities:
Students will continue to work on the Cell Phone Plan assignment. Each group will have
a few minutes to present their work, but they should include a much more detailed summary of
the sales pitch in their final work that they hand in.
Students will be given a worksheet involving a system of linear equations as a “take
home to think about” to complete for homework. I will explain that this worksheet can tell them
a lot about what will come next.
Conclusion
The linear equations unit plan that I have developed is designed to be implemented in
conjunction with the Glencoe Algebra I textbook. However, minor changes could be made to the
homework assignments and in-class activities to integrate another similar traditional textbook as
Algebra I Unit Plan 46
needed. The purpose of the unit calendar is to provide a general outline of an effective lesson
sequence, but it should not be treated as a rigid prescription. I designed the assessments, lessons,
and calendar in the way that I would most likely implement the unit, but another teacher should
make minor changes and adjustments based on differing contextual aspects (e.g. if this unit were
taught to seventh grade students instead of ninth grade students, perhaps a teacher would want to
make changes to the context or wording of the Culminating Assessment part II). The big ideas,
essential questions, and student goals should not be changed, but the ways in which a teacher
helps certain students arrive at these can and should be changed as needed.
I purposefully did not design the unit calendar with too much detail in terms of time spent
on activities and questions that I would ask, as I am aware of the need to constantly adjust as a
class progresses. In other words, rather than stating that I would spend exactly fifteen minutes
on an activity and that I would ask five specific questions during the discussion, I have offered
suggestions about how to facilitate learning that can supplement my (or another teacher’s) onthe-spot judgment. In addition, it is impossible to know exactly what students will be ready for
on Day Five of a unit before they have been through Day One. Therefore, it is absolutely
essential that I (or again, another teacher using this unit plan) constantly return to the unit
calendar after each lesson to modify the next lesson according to which goals were accomplished
and which goals remain unachieved.
Algebra I Unit Plan 47
References
Bransford, J., Brown, A., & Cocking, R. (2000). How people learn: Brain, mind, experience, and
school. Washington, D.C.: National Academy Press.
Chazan, D. (2000). Beyond formulas in mathematics and teaching: Dynamics of the high school
algebra classroom. New York, NY: Teachers College Press.
Collins, W., Cuevas, G., Foster, A.G., Gordon, B., Moore-Harris, B., Rath, J., Swart, D., &
Winters, L.J. (2001). Glencoe Algebra I: Integrations, Applications, Connections.
Columbus, OH: The McGraw-Hill Companies.
Common Core Standards: Mathematics – High School – Algebra
Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6-10.
Portsmouth, NH: Heinemann.
Kalchman, M. & Koedinger, K. R. (2005). Teaching and learning functions. In M.S. Donovan
& J.D. Bransford (Eds.), How students learn: history, mathematics, and science in the
classroom (pp. 351-393). Washington: The National Academies Press.
Kazemi, E. (1998). Discourse that promotes conceptual understanding. Teaching Children
Mathematics, March, pp. 410-414.
Martino, A. M. & Maher, C. (1999). Teacher questioning to promote justification and
generalization in mathematics: What research practice has taught us. Journal of
Mathematics Behavior, 18(1), 53-78.
Milner, H.R. (10 May 2011). Context of Teaching and Learning, Education 3900: Culturally
Responsive Pedagogy. Class Lecture. Vanderbilt University: Peabody College.
Algebra I Unit Plan 48
Milner, H.R. (2010). Start where you are but don’t stay there: Understanding diversity,
opportunity gaps, and teaching in today’s classrooms. Cambridge: Harvard Education
Press.
Moses, R.P. & Cobb, C.E. (2001). Radical equations: Civil rights from Mississippi to the
Algebra Project. Boston: Beacon Press.
National Council of Teachers of Mathematics. 1991, 2000. Principles and Standards for School
Mathematics.
National Research Council (2001). Adding it up: Helping children learn mathematics. J.
Kilpatrick, J. Swafford, and B. Findell (Eds.) Mathematics Learning Study Committee,
Center for Education, Division of Behavioral and Social Sciences and Education,
Washington, DC: National Academy Press.
Philipp, R. A. (1992). The many uses of algebraic variables. Mathematics Teacher, 85 (7), 55761.
Shahan, E. (2009). “State Fairs” Activity.
Shahan, E. (2009). “Imagining Lines” Activity.
Thompson, A. G., Philip, R. A., Thompson, P. W., & Boyd, B. A. (1994). Calculational and
conceptual orientations in teaching mathematics. In A. Coxford (Ed.), 1994 Yearbook of
the NCTM (pp. 79-92). Reston, VA: NCTM.
Tomlinson, C. A. & McTighe, J. (2006). Integrating differentiated instruction and
understanding by design. Alexandria, VA: Association for Supervision and Curriculum
Development.
Wiggins, G., & McTighe, J. (2005). Understanding by design (Expanded second edition).
Alexandria, VA: Association for Supervision and Curriculum Development.
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Zawojewski, J. (1996). Polishing a data task: Seeking better assessment. Teaching Children
Mathematics, February 1996, 372-78.