Unit Planning Matrix
Student & Family
Knowledge
Big Ideas
Skills & Standards
What are the enduring
understandings/
essential questions to be
addressed?
What important
skills/standards will
students learn, practice,
or apply?
How will you draw on
students’ ideas, interests
and experiences to
connect students to the
big ideas?
EQ1. Why would
someone represent a
real life situation as a
system of equations?
Standards
Algebra 9.0 “Students
solve a system of two
linear equations in two
variables algebraically
and are able to interpret
the answer graphically.”
(L1) Pairs activate prior
knowledge of graphing.
Students are familiar
with internet websites.
(L1) Visual and high
interest graphing.
(L2) Hands on activates
prior knowledge &
connects real world
problem to written word
problem to equations.
(L3, L4) New methods
are introduced by
building on prior day’s
application problem.
(L5) Prior knowledge of
multiplying equations is
reviewed not assessed.
(L2, L3, L4) New
problems are chosen for
student ability to relate
(hourly jobs, food
items). Students work
in groups to connect the
problem to equations
and interpret the
solution (big ideas).
(L5, L6) Students
extend and apply
knowledge by creating
and assessing their own
(L1) Motivational
question: “Is it ever the
case that it will be
cheaper to buy a used
car and drive than to
ride the bus? Can
mathematics be used to
make this case?”
EQ2. What does the
solution of a system of
linear equations
represent in “everyday”
terms?
Enduring
understanding: What
does the solution of an
equation(s) represent in
mathematical terms?
The student will be able
to:
Represent real world
applications as systems
of Linear equations.
1. Represent linear
relationships (such as
website visits) as linear
equations.
2. Correctly graph 2
linear equations using
appropriate units and
scales for the x- and yaxes on 1 coordinate
system.
Solve systems of linear
equations and interpret
the solution graphically.
1. Find and describe the
solution by inspecting a
graph.
2. Find and describe the
Mathematics Exemplar -– (Algebra B) Systems of Linear Equations
Assessment
Resources &
(Formative &
Instructional Components
Materials
Summative)
What is meaningful
evidence that students
have understood the big
ideas and reached
proficiency on the
skills/standards?
Formative
1. (L1) Assess by task
prior knowledge of
graphing lines. Pairs
help each other. Check
by Walking Around
(CBWA)
2. (L1) Assess graphs.
Grade on effort; is more
review needed?
3. (L2) Assess problem
solving and multiple
representations prior
knowledge. CBWA &
discussion.
4. (L3) Assess student
connects problem to
equations by discussion.
5. (L1,L3,L4) White
board assess student
able to use algebraic
methods on equations.
6. (L2,L3,L4,L5) Warm
up. Assess retention;
More review needed?
7. (L5) Rubric is used
to evaluate a student
project. Subjectively
assess on Bloom
taxonomy. Connection
of applications to
What instructional practices and
strategies will support students to meet
the standards and grasp the big ideas?
What resources will best
convey the big ideas and
concepts to support skill
attainment?
Lesson one (L1):
[Practiced group work in prior lessons.]
1. (Pairs) graph data from math &
science club websites to review graphing.
2. (class) Q&A identifies equations of
the lines and constructs meaning of point
of intersection (equal visits to both sites).
2. Review graphing lines from slopeintercept. Pairs practice on graph paper
3. Review converting lines into slopeintercept form. White board practice.
4. HW: Convert and graph equations.
Lesson two (L2):
1. Warm up: Graphing. Review answers
2. (Groups of 4) use plastic spiders and
ants to “find how many spiders and ants
are in a box of 12 bugs and 80 bug legs”
by modeling then creating a table.
3. (class) Q&A identifies the equations
involved, focuses on same solution from
multiple representations and constructs
“solution” as “point of intersection.”
4. Introduce “Graph & Check” with bug
example. Work additional examples.
5. Student practice using graph paper.
6. HW: Graph & Check problems.
Lesson three (L3):
1. Warm up: Graph & check. Review.
2. Mastery quiz on graph & check.
3. Introduce substitution using bug
problem and other text examples.
Module materials
Realia: plastic spiders
and ants.
Graph paper or graphing
calculators if available.
Large white board or
power point slides to
work teacher examples.
Write up of “bug”
problem (original).
Identified problems for
whiteboard practice
(from text)
Identified word
problems for group
work (from text).
Sample project (write
up of bug problem) and
rubric.
Texts
Larson, Boswell,
Kanold and Stiff (2004)
Algebra 1 Concepts and
Skills, McDougal
1
solution algebraically
using the methods of
substitution and linear
combination.
.
application problem.
equations made?
(L6) Assess what
students chose to assess
as further evidence of
progress on taxonomy,
connection to big ideas.
Summative
1. (L1, L2, L3, L4).
HW assesses student
readiness to proceed to
new lesson. Correction
provides feedback.
2. (L3,L4,L5). A
standardized (each
Algebra B teacher gives
the same) mastery quiz
is given. Timing, but
not content, can be
adjusted for student
readiness. Mastery
retakes are permitted
throughout the semester.
‘Missing’ masteries
result in an F for the
semester.
3. (L6) Standardized
unit test is given that
emphasizes testing the
standard “algebraic
methods.” Neither
timing nor content can
be adjusted.
4. (Pairs) substitute to solve word
problem (work 2 jobs at 2 rates for 20
hours) then check by graphing.
5. White board practice of substitution.
6. HW: Substitution problems.
Lesson four (L4):
1. Warm Up: substitution. Review.
2. Mastery quiz: Use of substitution.
3. Review multiplication of equations by
a number.
3. Introduce linear combinations using
jobs problem and other text examples.
4. Student pairs solve a word problem
(beef vs chicken on menu) by linear
combination then check by graphing.
5. White board practice
6. HW: Linear combination problems.
Lesson five (L5):
1. Warm Up: Combinations. Review.
2. Mastery quiz of Linear combination.
3. Discuss types of applications suited to
solution as a system of equations.
4. Pairs solve motivational problem (bus
vs buy a used car). Using method of
their choice.
6. Students write an original problem to
be exchanged with another group. They
submit the solution (using any of the 3
methods studied) to the teacher. They
create rubric and evaluate peer solution.
Littell: Boston.
Lesson six (L6):
1. Warm Up: Mixed problems. Review.
2. White board, graph paper and pair
work (applications) review for unit test.
3. Unit test
4. Exchange project problems and solve.
5. Assess peer solutions.
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Unit Planning Matrix - Rationale
The essential questions to be addressed in this unit are (1) “Why would someone
represent a real life situation as a system of equations?” and (2) “What does the solution of a
system of linear equations represent in ‘everyday’ terms?” On pages 78 - 79 Wiggins identifies
six features of an essential question. I cannot apply the “no obvious answer” to my essential
questions without really stretching its meaning. Perhaps my questions can be answered in a
variety of ways, but they do not require testing of hypotheses or review of sanctioned views. My
essential questions are “higher order” in Bloom’s taxonomy. As my lessons connect to these
essential questions, the students will be able to assess whether a given application is well suited
to solution using systems of equations. Further, they will create an original problem and assess
its solution by a peer. The use of real world problems emphasizes the fact that value of applying
these techniques recur naturally throughout one’s life learning, not just during their mathematical
training. The motivational question used to kick off this unit is, “Is it ever the case that it is
cheaper to drive your own car than it is to take the bus? Can we use mathematics to create a
convincing argument?” Students are always interested in a new angle to convince parents that
something, like a new car, is actually practical. Student generally express surprise that
mathematics could be used to construct (not refute) this argument. This meets Wiggins’ fifth
criteria of ‘provoke and sustain student interest’. My lessons also include use of realia and visual
graphing to address learning styles. I have linked my essential questions to every day
experiences (web sites, restaurants), but have not linked them to additional disciplines such as
science and economics. The closest I have come to meeting the sixth criteria ‘link to other
essential questions’ is by asking students to identify their own application problems.
The enduring understanding that I want my students to take away is that the solution to a
system of linear equations represents a point of intersection of two lines, not just “some
numbers.” This connection of a solution to the intersection of curves and lines is a recurring
concept throughout the rest of this course and throughout mathematics. For example, the
solution to a quadratic equation is the two (or one) points where a parabola intersects the x-axis.
Too often, students are taught procedures for solving systems of equations which they practice
on abstract (not connected to real world problems) and therefore seemingly arbitrary equations.
Armed with the enduring understanding that a solution is a point of intersection, students can
readily recreate the procedures needed to solve any system of equations they encounter rather
than relying on rote memorization. They will also be able to transfer this understanding to solve
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problems written using different constructs than those seen in math textbooks, for example
problems from history, science, and economics.
At the core of my unit plan is a recurring cycle of instruction designed to engage and
assess student learning and lesson effectiveness (Instruction). Students are assessed on prior
knowledge by working on a familiar, high interest activity or application (Students). The prior
application is then extended to introduce the new concepts. Students practice the underlying
algebraic methods on whiteboards and solve new applications (Curriculum). Students are
assessed throughout the unit using both formative and summative assessment techniques
(Assessment) such as group activities, class discussion, white boarding, standardized testing and
a student project.
The algebra standard addressed by this unit plan is Algebra 9.0. This unit will only
address the first half of the standard. “Solving and graphing inequalities” is the second half (not
identified in the matrix) and is addressed in the subsequent unit. The standard does not directly
link to the essential questions I have identified. The standard emphasizes the need for students to
master the tools (graphing, substitution, linear combinations) of solving systems of equations. It
does not address a student’s ability to recognize which real world application problems can be
expressed as systems of equations or how to set the equations up. It merely dictates that students
must know how to solve problems that are already expressed as systems of equations. The
standard does not require working with application problems but certainly does not preclude it.
An additional standard (5.0) indicates “students solve multistep problems, including word
problems, involving linear equations and linear inequalities in one variable and provide
justification for each step” that could also apply to this lesson. As a result, I have chosen
objectives that are tied directly to applications.
My objectives target Bloom’s Comprehension and Application levels. “SWBAT
represent real world applications as systems of linear equations” targets Application. “SWBAT
correctly graph 2 linear equations….” targets Comprehension. The remaining objectives,
collected under “SWBAT solve systems of linear equations and interpret the solution
graphically” are tied directly to the standard, and I believe, target Knowledge. My student
project targets Analysis and Synthesis as students distinguish which applications are best solved
by systems of equations and then create a problem and evaluate its peer solution. I did not
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include these as unit objectives because these are extensions to what is expected of a remedial
algebra class.
To address student family knowledge, I consciously choose relevant word problems for
students to study. For example, the textbook includes a question about backpacking. I suspect
several of my students have been backpacking but most have not. I therefore did not include this
problem, but instead chose problems (from the text) that all students have some experience with
at my school (web sites, cell phones and cafeteria menus).
I have placed greater emphasis on assessing student existing prerequisite mathematical
knowledge. I chose high interest activities (bugs, cars, etc) that activate this math background
knowledge (implicitly require use of these skills) so I can assess prerequisite algebraic skills
(graphing, solving single equations, etc) as well as student problem solving strategies, without
students experiencing the anxiety of being tested. Math anxiety is prevalent in my remedial
classes. Student lack of confidence in their own mathematical ability is as likely a cause of a low
score on traditional testing methods as is actual lack of understanding. As students engage with
the plastic spiders and ants they naturally share strategies with each other. They immediately
intuit the strategy of “replacing a spider with an ant” if there are “too many legs” and reversing
this process if there are too few legs. In other words, students engage in mathematical reasoning
and communication with the same enthusiasm they would discuss sports or fashion. In each
subsequent lesson, I build on the newly-activated knowledge of the prior lesson. For example, I
transition to teaching substitution by working with students to solve the bug problem using
substitution, re-enforcing the real world application solving thread of my unit at the same time I
work on the standard’s goal of teaching students to solve problems algebraically.
I am a strong advocate of formative assessments, especially in a remedial classroom.
Faced with a test, students will too often disengage, preferring to fail for not trying rather than
trying and failing yet again. On the other hand, when students are asked to practice the
substitution and/or linear combination methods using individual white boards, they devise
competitions to see who can solve an equation the fastest, chide each other with, “I told you it
was negative, Fool” and surreptitiously watch and copy their neighbor’s work with a fascination
they never exhibit when asked to “watch and copy” the work of the teacher. As I Check by
Walking Around (CBWA), I am able to holistically assess a student’s mathematical abilities and
interest in addition to their emerging mastery of the standard’s “algebraic methods.”
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I also use formative assessments to adjust my lesson plans. The daily warm up serves to
re-activate the prior day’s knowledge, but more importantly, it allows me to assess student
readiness to go on to new material or their need for additional review. Based on my observations
as I CBWA, I either extend my review of the warm up problems, select students to write their
answers on the board for others to see (again students are more interested in watching another
student write mathematically than in watching me) or merely ask if there is a particular problem
that someone wishes to see.
I also assess student homework, giving students full credit for good effort, but also
marking answers that are incorrect. This provides important feedback to the student (who often
confidently completes every problem incorrectly) and to me. After grading 90+ papers in a
single sitting, patterns of misconceptions are easily recognized. I reference the results I observed
in the homework during the next lesson as I explain (as necessary) that we will be reviewing the
prior lesson or “taking a different approach” to looking at linear combinations, for example. I
want the students to know I am responding to their academic needs, not just trundling them
through the system. I chose to classify homework under summative assessments even though I
do use it in a formative manner. I did this because unfortunately, the standing wisdom is that we
must “move on” when at least 50% of the students have grasped a concept. Thus, its summative
value outweighs its formative value. I do have the flexibility to inject mini lessons into future
units and do so. Overall, my formative assessments are designed to give me feedback, to
increase student confidence in their own abilities and confirm their readiness for “THE TEST.”
The paper and pencil tests that I have included in my unit as required-by-department
mastery tests and the standardized unit (chapter) test, do not adequately assess a student’s ability
to tackle real world problems, choose a method of solution, correctly solve the problem or to
persuasively discuss the meaning of the solution. I do respect the need to test students on
mechanics, ie, the basic toolkit of algebra. The mastery and unit tests accomplish this. I believe
such testing is necessary, but not sufficient. As a result, I have included a project that more
effectively assesses these skills. Students demonstrate their understanding of the meaning of
systems of linear equations, and the connection to real world applications by applying their new
knowledge to identify, define and peer assess a problem they create.
To prepare students for the unit test and the standards (CST) testing as well, I chose my
applications (other than the bug problem which I made up) from the text. Textbook problems
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give my students (many of whom are English Language Learners) practice reading the textual
style of math problems. It is also not practical, or even sensible, to create all my own problems.
The practice white board examples are also taken directly from the text. Some of them are the
same problems they will see on their homework. All are similar to what will appear on the
standardized tests and masteries.
The one technique I employ that may be controversial is the review of material directly
preceding administration of masteries and the unit test. My purpose in doing this is to lessen the
math anxiety that most remedial students experience when faced with a “high stakes” (which the
masteries certainly are, given they result in an overall “F” if any one is not passed) test. I value
the enduring knowledge that is demonstrated by the student project. As I indicated, this deeper
understanding of a problem (beyond rote memorization of process) allows students to recreate
the necessary procedures (as needed) when they face previously unseen problems and/or proceed
to higher levels of mathematics. I do not provide assistance as students complete the tests, nor
would I consider supplying any answer before or during the examination.
My assessment tools are mixed in terms of RSVP. The mastery and unit tests are
standardized, reliable (there is only 1 correct answer) and practical (they are easily “keyed”). I
believe they are considered valid because they do not assess something I do not teach. However,
they only assess a portion (the algebraic methods) that I teach. They do not assess the
connection to the real world applications (big questions). The project I included is less
standardized, reliable (some subjective criteria is included in the rubric) and practical, but I
believe more valid because it assesses both the applications and methods I have taught in a more
realistic (real world) environment.
References:
1. Wiggins, G. Essential Questions and Curriculum Template Excerpts from Educative
Assessment, 1998 Josey-Bass, San Francisco pp. 77-84 as excerpted in course reader.
2. Bloom’s Taxonomy, source unknown, as published in student reader, page 119.
3. Class lecture notes.
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