Module Focus: Algebra II – Module 1
Sequence of Sessions
Overarching Objectives of this July 2014 Network Team Institute

Participants will be able to identify, practice, and use best instructional moves and scaffolds for chosen common core standards.
High-Level Purpose of this Session


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Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within this
module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same.
Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work
of the grade in order to fully implement the curriculum.
Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while
maintaining the balance of rigor that is built into the curriculum.
Related Learning Experiences
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This session is part of a sequence of Module Focus sessions examining the Algebra II curriculum in A Story of Functions.
Key Points
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Topic A
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Topic B
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Rational expressions are analogous to rational numbers.
When solving a rational or radical equation, it is necessary to check for extraneous solutions because steps may have been taken that are not
guaranteed to preserve the solution set.
A parabola is a specific type of u-shaped curve.
There is a connection between algebra, geometry, and functions.
Topic D
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Polynomial functions are easy to graph when written in factored form (though not always easy to get into factored form).
Polynomials can still be divided just like integers even if there is a remainder.
When a polynomial is divided by (𝑥 −𝑎), the remainder is the value of the polynomial at x = 𝑎.
When 𝑎 is a zero of polynomial p, then (𝑥 − 𝑎) is a factor of p.
Topic C
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Seeing structure in expressions requires a dynamic view in which potential rearrangements and manipulations are possible.
Algebra is a powerful and useful tool in a variety of situations.
Multiplication and division of polynomials follows the same principles as multiplication and division of integers.
Complex numbers are neither imaginary nor complex.
Complex numbers have a geometric meaning.
The inclusion of complex numbers in our number system means that every polynomial of degree n can be written in terms of n linear factors.
End of Module Assessment
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End of Module assessment are designed to assess all standards of the module (at least at the cluster level) with an emphasis on assessing thoroughly
those presented in the second half of the module.
Recall, as much as possible, assessment items are designed to assess the standards while emulating PARCC Type 2 and Type 3 tasks.
Recall, rubrics are designed to inform each district / school / teacher as they make decisions about the use of assessments in the assignment of grades.
Session Outcomes
What do we want participants to be able to do as a result of this
session?



Participants will draw connections between the progression documents
and the careful sequence of mathematical concepts that develop within
this module, thereby enabling participants to enact cross- grade
coherence in their classrooms and support their colleagues to do the
same.
Participants will be able to articulate how the topics and lessons promote
mastery of the focus standards and how the module addresses the major
work of the grade in order to fully implement the curriculum.
Participants will be prepared to implement the modules and to make
appropriate instructional choices to meet the needs of their students
while maintaining the balance of rigor that is built into the curriculum.
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.
Session Overview
Section
Time
Overview
Prepared Resources
Facilitator Preparation
Introduction
37 min
Introduces Algebra II and module
study.


Algebra II PPT
Facilitator Guide
Review Algebra II Module 1
Overview
Topic A: Polynomial
– From base 10 to
base X
101 min
Explores the foundations for the
study of Polynomials.


Algebra II PPT
Facilitator Guide
Review Topic A
Topic B: Factoring –
Its Uses and Its
Obstacles
87 min
Explores factoring and obstacles
related to factoring.


Algebra II PPT
Facilitator Guide
Review Topic B
Mid-Module
35 min
Explores the Mid-Module

Algebra II PPT
Mid-module Assessment
Assessment in depth.

Facilitator Guide
137 min
Explores solving polynomial,
rational, and radical equations.


Algebra II PPT
Facilitator Guide
Review Topic C
Topic D: A Surprise
from Geometry –
Complex Numbers
Overcome All
Obstacles
54 min
Explores how complex numbers
and their connection to concepts
from Geometry.


Algebra II PPT
Facilitator Guide
Review Topic D
End-of-Module
Assessment
37 min
Examines the End-of-Module
Assessment.


Algebra II PPT
Facilitator Guide
End-of-Module Assessment
Conclusion
40 min
Concludes study of Module 1 and
look ahead to remaining modules
in Algebra II and in Pre-calculus.


Algebra II PPT
Facilitator Guide
Review Module 1
Assessment
Topic C: Solving and
Applying Equations –
Polynomial, Rational,
and Radical
Session Roadmap
Section: Introduction
Time: 37 minutes
In this section, you will be introduced to Algebra II and module
study.
Materials used include:
 Algebra II Module 1 PPT
 Algebra II Module 1 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
2 min
1.
[Note: This session is 9 hours in length.]
Introduce myself and talk about the session.
Mostly we will be working on the student pages located at the front of your
binder. Avoid looking at the teacher pages for the most part. I mainly
want you to experience the module from the student perspective.
5 min
2.
Start here. Then back up to opening slide.
Give participants 5 min to do the opening exercise.
2 min
3.
In order for us to better address your individual needs, it is helpful to
know a little bit about you collectively.
Pick one of these categories that you most identify with. As we go through
these, feel free to look around the room and identify other folks in your
same role that you may want to exchange ideas with over lunch or at
breaks.
By a show of hands who in the room is a classroom teacher?
Math trainer?
Principal or school-level leader
District-level leader?
And who among you feel like none of these categories really fit for you.
(Perhaps ask a few of these folks what their role is).
Regardless of your role, what you all have in common is the need to
understand this curriculum well enough to make good decisions about
implementing it. A good part of that will happen through experiencing
pieces of this curriculum and then hearing the commentary that comes
from the classroom teachers and others in the group.
2 min
4.
We have three main objectives for this mornings work. Our main task will
be experiencing lessons and assessments. As a secondary objective, you
should walk away from the study of module 4 being able to articulate how
these lessons promote mastery of the standards and how they address the
major work of the grade. Lastly, you should be able to get a sense for the
coherent connections to the content of earlier grade levels.
2 min
5.
Here is our agenda for the day. If needed, we will start with orienting
ourselves to what the materials consist of.
We will spend most of our time on G11 M1. As we go through the module,
I will talk about foundational skills developed in prior grades, particularly
those developed in G9. We will discuss some fluency drills and other
scaffolds that can be used to address possible gaps in content knowledge.
At the end of the session, we will preview the rest of the G11 curriculum
and the beginning of G12.
(Click to advance animation.) Let’s begin with an orientation to the
materials for those that are new to the materials (Skip if participants are
already familiar with the materials).
4 min
6.
(Not accounted for in the timing – these slides are optional if participants
are new to the materials.)
Each module will be delivered in 3 main files per module. The teacher
materials, the student materials and a pack of copy ready materials.
Teacher materials include a module overview, and topic overviews, along
with daily lessons and a mid- and end-of-module assessment. (Note that
shorter modules of 20 days or less do not include a mid-module
assessment.)
Student materials are simply a package of daily lessons. Each daily lesson
includes any materials the student needs for the classroom exercises and
examples as well as a problem set that the teacher can select from for
homework assignments.
The copy ready materials are a single file that one can easily pull from to
make the necessary copies for the day of items like exit tickets, or fluency
worksheets that wouldn’t be fitting to give the students ahead of time, as
well as the assessments.
4 min
7.
(Not accounted for in the timing – these slides are optional if participants
are new to the materials.)
There are 4 general types of lessons in the 6-12 curriculum. There is no
set formula for how many of each lesson type we included, we always use
whichever type we feel is most appropriate for the content of the lesson.
The types are merely a way of communicating to the teacher, what to
expect from this lesson – nothing more. There are not rules or restrictions
about what we put in a lesson based on the types, we’re just
communicating a basic idea about the structure of the lesson.
Problem Set Lesson – Teacher and students work through a sequence of 4
to 7 examples and exercises to develop or reinforce a concept. Mostly
teacher directed. Students work on exercises individually or in pairs in
short time periods. The majority of time is spent alternating between the
teacher working through examples with the students and the students
completing exercises.
Exploration Lesson – Students are given 20 – 30 minutes to work
independently or in small groups on one or more exploratory challenges
followed by a debrief. This is typically a challenging problem or question
that requires students to collaborate (in pairs or groups) but can be done
individually. The lesson would normally conclude with a class discussion
on the problem to draw conclusions and consolidate understandings.
Socratic Lesson – Teacher leads students in a conversation with the aim of
developing a specific concept or proof. This lesson type is useful when
conveying ideas that students cannot learn/discover on their own. The
teacher asks guiding questions to make their point and engage students.
Modeling Cycle Lesson --Students are involved in practicing all or part of
the modeling cycle (see p. 62 of the CCLS, or 72 of the CCSSM). The
problem students are working on is either a real-world or mathematical
problem that could be described as an ill-defined task, that is, students will
have to make some assumptions and document those assumptions as they
work on the problem. Students are likely to work in groups on these types
of problems, but teachers may want students to work for a period of time
individually before collaborating with others.
5 min
8.
(Not accounted for in the timing – these slides are optional if participants
are new to the materials.)
Follow along with a lesson from the materials in your packet.
The teacher materials of each lesson all begin with the designation of the
lesson type, lesson name, and then 1 or more student outcomes. Lesson
notes are provided when appropriate, just after the student outcomes.
Classwork includes general guidance for leading students through the
various examples, exercises, or explorations of the day, along with
important discussion questions, each of which are designated by a solid
square bullet. Anticipated student responses are included when relevant –
these responses are below the questions; they use an empty square bullet
and are italicized. Snapshots of the student materials are provided
throughout the lesson along with solutions or expected responses. The
snap shots appear in a box and are bold in font. Most lessons include a
closing of some kind – typically a short discussion. Virtually every lesson
includes a lesson ticket and a problem set.
What you won’t see is a standard associated with each lesson. Standards
are identified at the topic level, and often times are covered in more than
one topic or even more than one module… the curriculum is designed to
make coherent connections between standards, rather than following the
notion that the standards are a checklist of items to cover.
Student materials for each lesson are broken into two sections, the
classwork, which allows space for the student to work right there in the
materials, and the problem set which does not include space – those are
intended to be done on a separate sheet so they can be turned in. Some
lessons also include a lesson summary that may serve to remind students
of a definition or concept from the lesson.
2 min
9.
(no time allotted)
5 min
10.
Display and have participants read the Module overview, scan the
standards and the new terminology.
2 min
11.
(Go through the bullets to give an overview of the progression or flow of
each topic and the module as a whole.)
2 min
12.
(Review the bullet points with participants to remind them of the
background students are coming in to this module with.)
We will be discussing in more detail as we go through the module what
students’ previous experiences have been.
Section: Topic A: Polynomial – From base 10 to base X
Time: 101 minutes
In this section, you will explore the foundations for the study of
Polynomials.
Materials used include:
 Algebra II Module 1 PPT
 Algebra II Module 1 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
2 min
13.
(no time allotted)
5 min
14.
Read the topic A opener (p. 12 – 13)
2 min
15.
GROUP
5 min
16.
5 min
17.
In this way, we can think of a polynomial as a number in base x where the x
is a placeholder for some number yet to be determined.
Is it ok to have a coefficient of 7 if we decide we are in base 5?
5 min
18.
These next 3 slides were directly informed and inspired by the work of Dr.
James Tanton and his posted course on quadratics found at
www.Gdaymath.com. I find these exercises a highly desirable experience
base for teachers embarking on a polynomial module.
Students studied sequences in grade 9 (M3). Based on the apparent
pattern, what might the next number in the sequence be?
How did you decide it would be 19?
You noticed that the difference between each term was constant; it was 3.
Some people describe a sequence as linear if the difference from term to
term is constant; if it has a constant set of first differences.
If we imagine our sequence as a set of data values and plot them on a graph
we see why they might call this a linear sequence.
How about the next sequence. What might you be inclined to believe is the
next number in this sequence? (40).
How did you get that?
Did anyone notice that the second differences are constant?
Show that for the following sequence, the 3rd differences are constant.
How many differences must one complete in the sequence below, the
powers of two, to first see a row of constant differences?
The square numbers begin, 1, 4, 9, 16, … is there a row of their difference
table that is constant?
10 min
19.
Switch to document camera and do an example where given leading
diagonal.
Write out lead diagonals for 1, n, n^2, n^ 3.
Do you think the leading diagonal for n^2 + n is the sum of the leading
diagonal of n^2 and n? Let’s see.
5 min
20.
Document camera
Give them a leading diagonal and ask them to write an explicit formula for
the sequence.
10 min
21.
Give participants time to work and then address any questions or concerns.
The exit ticket is in the teacher pages.
Note that the exit ticket is material students would be familiar with from
grade 9 (m4)
5 min
22.
Review of some vocabulary. A variable should be thought of as a place
holder for a number whose value has not been determined.
5 min
23.
Students use the area model to multiply in elementary school. We continue
using this idea is high school but switch the terminology to the “tabular
method” to allow the inclusion of negative values.
Students learned both ways of multiplying polynomials in grade 9 (m1)
Is everyone familiar with the area model or tabular method? Do we need to
see another example?
10 min
24.
Students learn a variety of ways of dividing polynomials within the module
– reverse tabular method, long division, reducing a common factor,
inspection.
Complete the example on the ppt. Switch to document camera and work
another example using the reverse tabular method.
Work on L3 problem set #1 and 12
The long division algorithm to divide polynomials is analogous to the long
division algorithm for integers. The long division algorithm to divide
polynomials produces the same results as the reverse tabular method.
L4 Example 1 – We use a base 10 number system. A polynomial can be
thought of as a number in “base x” an idea we explored in grade 9.
10 min
25.
Give students the freedom here to select the appropriate method for finding
the quotients.
If students see the pattern (MP 8) here, it will be much easier for them to
remember how to factor various polynomials based on its structure (MP 7).
5 min
26.
The opening exercise came from lesson 7 on mental math.
Add question 4. Ask for participant responses. – Do we believe in patterns?
(a theme in G9)
Questions on any of the other exercises?
5 min
27.
This is a very interesting topic that has an abundance of information that
can be found. Spark student curiosity by having them research this topic or
by showing the video.
10 min
28.
Connects Algebra to Geometry.
Work through examples 1 and 2. Discuss. Share responses.
2 min
29.
Total time elapsed: 121 min + 9 min for questions/discussion = 130 min (2
hr 10 min)
(Go through each point listed.)
Section: Topic B: Factoring – Its Uses and Its Obstacles
Time: 87 minutes
In this section, you will explore factoring and obstacles related to
factoring.
Materials used include:
 Algebra II Module 1 PPT
 Algebra II Module 1 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
0 min
30.
15 min
31.
Script/ Activity directions
Read through the topic opener for topic B (p. 128 in the teacher pages)
I want to examine students’ experiences from grade 9. You may have to
provide some scaffolding for next year’s students.
Switch over to the document camera and show factoring (tabular method)
and completing the square from grade 9. Show both the technique of
working on one side of the equation and also multiplying by 4 or a multiple
of 4.
15 min
32.
Supplies: personal board, dry erase markers, felt
Factoring is a skill that students need to develop fluency with. It is a great
example of a skill for which a rapid white board exchange is a fitting fluency
exercise.
How to conduct a white board exchange:
All students will need a personal white board, white board marker, and a
means of erasing their work. An economical recommendation is to place
card stock inside sheet protectors to use as the personal white boards, and
to cut sheets of felt into small squares to use as erasers. You have these
materials at your tables today.
GROUP
It is best to prepare the questions in a way that allows you to reveal them to
the class one at a time. A flip chart, or Powerpoint presentation can be
used, or one can write the problems on the board and either cover some
with paper or simply write only one on the board at a time.
Prepare 10-15 problems that progress in difficulty. The best way to get the
feel is for us to do one ourselves. I’ll reveal the problem, you work it as fast
as you can and still do accurate work and then hold it up for me to see.
(Reveal the first problem in the list and announce, “Go”. Give immediate
feedback to each participant, pointing and/or making eye contact with the
participant and responding with an affirmation for correct work such as,
“Good job!”, “Yes!”, or “Correct!”, or guidance for incorrect work such as
“Look again,” “Try again,” “Check your work,” etc. Do several to
demonstrate the progression of problems.)
If many students have struggled to get the answer correct, go through the
solution of that problem as a class before moving on to the next problem in
the sequence. Fluency in the skill has been established when the class is
able to go through each problem in quick succession without pausing to go
through the solution of each problem individually. If only one or two
students have not been able to get a given problem correct when the rest of
the students are finished, it is appropriate to move the class forward to the
next problem without further delay; in this case find a time to provide
remediation to that student before the next fluency exercise on this skill is
given.
10 min
33.
Try some of the techniques I’ve shown you to work exercise 1.
Switch to document camera.
For example 2, let students be stumped for a minute. Factoring is not
always clear or obvious (or possible). What if I told you that one factor was
x – 3?
Would that be helpful? How? Then, work it by grouping.
10 min
34.
It is not correct to refer to the graph of f(x) or the zeros of f(x). It is ok to
say the graph of f or the graph of the equation y = f(x).
Work the example out on chart paper.
How could we know which direction the graph should go? Use test values
How do we know when the graph changes directions? We don’t exactly.
We really need Calculus to find the points where the graph changes
directions.
In lesson 15, students will study the structure of the graphs more closely
and look at end behavior,
Look at the Discussion (S.71)
10 min
35.
Ask about familiarity with progressions documents. If you have not read
thru the algebra (and functions) progessions document, I recommend you
do so. It really helps to clarify how students skills should develop
throughout high school.
With this idea in mind, look at the opening exercise. Work out the opening
exercise on chart paper or document camera.
How does understanding this problem help to lead students to the idea of
example 1 (b) where we have a remainder for the first time?
5 min
36.
Look at the standard.
Work through a selection of the exercises from the lesson. Notice students
are using inspection, reverse tabular system, and long division. Also notice
what is missing…synthetic division.
5 min
37.
Read through the excerpt. The remainder theorem is a hugely important
topic. Students will apply it in a later lesson to a modeling problem. We
will use it again to develop the FTA at the end of the module.
5 min
38.
It is easy to get students to understand that if p(a) = 0, then x = a is a zero of
p, and x – a is a factor. Make sure you don’t leave off the next part.
10 min
39.
This is the part that we sometimes don’t emphasize or that we lose kids
with.
Look at exercise 1. Discuss the analogy.
Work exercise 6.
Topic B closes with a 2 day modeling problem pertaining to a river bed. It
incorporates the Remainder Theorem and the other topics covered in this
part of the module.
2 min
40.
(Go through each point listed.)
Section: Mid-Module Assessment
Time: 35 minutes
In this section, you will explore the Mid-Module Assessment in
depth.
Materials used include:
 Algebra II Module 1 PPT
 Algebra II Module 1 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
0 min
41.
Script/ Activity directions
GROUP
20 min
42.
Have participants locate the assessment. Give them approximately 25 min
to take the assessment with their partner. After 20 minutes have passed
give a verbal warning for them to scan any remaining questions that they
have not yet attempted. If everyone finishes early, stop this part and start
the next portion of this session.
15 min
43.
Total time elapsed: 252 min + 8 min questions / discussion = 260 min (4 hr
20 min)
Again, work with a partner to examine your work against the rubric and
exemplar. If you have any questions or concerns, jot them down on a post-it
note and we will address those before we move on.
After 10 minutes or so have passed, call the group together and address any
questions or concerns that participants noted on their post-it notes.
0 min
44.
(no time allotted)
Section: Topic C: Solving and Applying Equations –
Polynomial, Rational, and Radical
Time: 137 minutes
In this section, you will explore solving polynomial, rational, and
radical equations.
Materials used include:
 Algebra II Module 1 PPT

Algebra II Module 1 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
5 min
45.
Read Topic Opener for Topic C (p. 237 in teacher pages)
5 min
46.
In grade 12, we will learn that the graphs of y = (x^2-4)/(x – 2) and y = x + 2
look identical except at x = 2. In calculus, we will call that a removable
discontinuity.
The next 2 lessons are fairly straightforward, students are working with
operations on rational expressions.
5 min
47.
Applying the properties of Equality is guaranteed to preserve the solution
set.
Applying the Distributive, Associative, and Commutative Properties or the
properties of rational exponents to either side is also guaranteed to
preserve the solution set.
You can do anything that’s useful, but it is not guaranteed to preserve your
solution set! So if we do anything outside of these properties, we MUST
check the solutions!
GROUP
Give participants time to work on the problems and then discuss.
15 min
48.
Applying the properties of Equality is guaranteed to preserve the solution
set.
Applying the Distributive, Associative, and Commutative Properties or the
properties of rational exponents to either side is also guaranteed to
preserve the solution set.
You can do anything that’s useful, but it is not guaranteed to preserve your
solution set! So if we do anything outside of these properties, we MUST
check the solutions!
Give participants time to work on the problems and then discuss.
25 min
49.
Supplies: Sprints A and B
A sprint is another fluency exercise that can help to bridge gaps in
knowledge and increase automaticity of a skill. They are fast-paced and
don’t take up too much class time.
Students worked with radicals to some extent in G9 and G10. They also
covered them in Lesson 9 of this module. But this is still an area where
there could be potential gaps in content knowledge. This might be a good
place to do a rapid white board exchange as we saw earlier or a sprint. We
are going to do a sprint now.
Conduct sprint.
10 min
50.
As we saw with rational eqns, the idea of potential extraneous solutions
should be highlighted.
Scan through the problem set and work a few of the problems.
5 min
51.
Students were introduced to this idea to some degree in G9 when writing
equations of quadratics. However, they were always given the y-intercept
as one of the points. In Algebra II, students must solve systems in three
variables.
10 min
52.
Students should be able to find the intersection point(s) between a line and
parabola or a line and a circle both algebraically and graphically. In grade 9,
students graph parabolas. In grade 10, students graph circles (including
completing the square to write the equation in standard form). These skills
are reviewed in this lesson and the lesson 32
Work on the exit ticket (Teacher p. 336)
Stop here day 1?
15 min
53.
Supplies: chains, chart paper, ruler, tape
Had participants come up with a sequence using a ruler. For equal changes
in x, collect data. Measure length of the y-value. Made a table of values.
Analyze the data together looking at first and second differences.
So…are all u-shaped curves quadratics? Hanging chains are not modeled by
quadratics. They are catenary curves.
5 min
54.
Catenary is derived from the latin word chain. A free hanging chain or wire
forms a catenary curve not a parabola. Nice to be prepared for student
questions. May spark their curiosity.
While students are not responsible for knowing what a catenary curve is,
they should realize that not all u shaped graphs can be represented by a
quadratic. A quadratic produces one special type of u shaped graph called a
parabola.
15 min
55.
Supply: ruler
This is the only geometry standard in G11. Ask about familiarity with the
definition of a parabola. Discuss the definition of a parabola.
Complete Exercise 1. Work through Exercise 2 together if necessary. Do we
need to see another example? If yes, work problem set #12.
10 min
56.
Supplies: Graph paper, transparencies, dry erase markers
Students learned these facts in grade 8 and solidified this knowledge in
grade 10. In grade 8 students perform rigid motions and in grade 9
students apply these rigid motions to transformations of graphs of
functions on the coordinate plane.
Demonstrate with transparencies.
10 min
57.
Supplies: Graph paper, transparencies, dry erase markers
Students have performed dilations about a point in grade 8 and 10.
dilations about a line in grade 9 (stretching/shrinking the graph of a
function on a coordinate plane)
Demonstrate with personal board and transparencies.
Work on exit ticket. (Teacher p. 398)
2 min
58.
Total time elapsed: 387 min + 8 min questions / discussion = 395 min (6 hr
35 min)
(Review the points outlined.)
Section: Topic D: A Surprise from Geometry – Complex
Numbers Overcome All Obstacles
Time: 54 minutes
In this section, you will explore how complex numbers and their
connection to concepts from Geometry.
Materials used include:
 Algebra II Module 1 PPT

Algebra II Module 1 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
2 min
59.
(no time allotted)
5 min
60.
Read Topic D Topic Opener (teacher p. 403)
15 min
61.
Supply: Graph paper
Students are immediately suspicious of complex numbers mainly because of
the unfortunate terms of “complex” and “imaginary.” Historically people
have always been suspicious about numbers they don’t understand.
Negative numbers were not fully embraced until the 18th century. The
Greeks did not believe x^2 = 2 had a solution.
We define numbers as we need them. For example, we need negative
numbers to explain the solution of x + a = b when a > b.
GROUP
We need an irrational number to explain the hypotenuse of an isosceles
right triangle.
We need a complex number to explain the solution of x^2 = -1 so i^2 is
defined to be -1.
By the way, the unfortunate term imaginary with exactly such a connotation
has been coined by Descartes. He also called the negative roots of an
equation false which, fortunately, did not stuck.
In Grade 12, students will see how complex numbers and trigonometry are
connected.
Switch to document camera and demonstrate.
Work on problem set 5, 6, and 11.
15 min
62.
Work opening exercise.
Discuss each bullet point.
Work through example 2.
5 min
63.
Why would this knowledge be considered “fundamental” to
mathematicians?
• The Fundamental Theorem says that the complex number system
contains every zero of every polynomial function. We do not need to
look anywhere else to find zeros to these types of function
• The Fundamental Theorem of Algebra ensures that there are as many
zeros as we’d expect for a polynomial function, and that factoring will
always (in theory) work to find solutions to polynomial equations.
This is not trivial. The FTA was published in 1799. That is about 100 years
after the Fundamental Theorem of Calculus. One reason we spend so much
time studying polynomials in algebra is because mathematicians love to
work with polynomials. They are predictable, continuous, and easy to work
with. In Calculus II, students will learn to write approximate different types
of functions as a polynomial function (Taylor series).
2 min
64.
Go through each point.
10 min
65.
Ask participants to list key points or ideas from module 1.
Ex.
•
•
•
Students are called upon to Look for and make use of structure
(MP.7) as they work with polynomials to gain insight into the
function’s behavior and its graph.
Polynomials are analogous to integers; rational expressions are
analogous to rational numbers.
Students see algebra as a powerful tool that can assist in solving a
wide range of mathematical problems.
Take a moment now to re-read the standards that this module covers… Can
you think back to moments in the lessons that get students to arrive at
those understandings? What things stand out to you now that did not stand
out early on?
Section: End-of-Module Assessment
Time: 37 minutes
In this section, you will examine the End-of-Module Assessment.
Materials used include:
 Algebra II Module 1 PPT
 Algebra II Module 1 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
0 min
66.
(no time allotted)
20 min
67.
Have participants locate the assessment. Give them approximately 25 min
to take the assessment with their partner. After 20 minutes have passed
give a verbal warning for them to scan any remaining questions that they
have not yet attempted. If everyone finishes early, stop this part and start
the next portion of this session.
15 min
68.
Again, work with a partner to examine your work against the rubric and
exemplar. If you have any questions or concerns, jot them down on a post-it
note and we will address those before we move on.
After 10 minutes or so have passed, call the group together and address any
questions or concerns that participants noted on their post-it notes.
GROUP
2 min
69.
Total time elapsed: 484 min + 11 min questions / discussion = 495 min (8
hr 15 min)
(Review each key point one at a time.)
Section: Conclusion
Time: 45 minutes
In this section, you will conclude study of Module 1 and look ahead
to remaining modules in Algebra II and in Pre-calculus.
Materials used include:
 Algebra II Module 1 PPT
 Algebra II Module 1 Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
0 min
(no time allotted)
70.
GROUP
10 min
71.
10 min
72.
10 min
73.
10 min
74.
Discuss what else is covered in G12?
5 min
75.
Take a few minutes to reflect on this session. You can jot your thoughts on
your copy of the powerpoint. What are your biggest takeaways? (pause
while participants reflect then click to advance to the next question). Now,
consider specifically how you can support successful implementation of
these materials at your schools given your role as a teacher, school leader,
administrator or other representative.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Turnkey Materials Provided
●
●
Algebra II Module 1 PPT
Algebra II Module 1 Facilitator’s Guide
Active learning
Turn and talk
Additional Suggested Resources
●
●
A Story of Functions Year Long Curriculum Overview
A Story of Functions CCLS Checklist