Module Focus: Grade 9 – Module 4
Sequence of Sessions
Overarching Objectives of this February 2014 Network Team Institute

Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate
how these modules contribute to the accomplishment of the major work of the grade.

Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom
teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.

Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding
how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.

Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the
mid-module assessment and end-of-module assessment.
High-Level Purpose of this Session
●
●
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Implementation: Participants will be able to articulate and model the instructional approaches to teaching the content of the first half of the lessons.
Standards alignment and focus: 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.
Coherence: Participants will be able to articulate connections from the content of previous grade levels to the content of this module.
Related Learning Experiences
●
This session is part of a sequence of Module Focus sessions examining the Grade 9 curriculum, A Story of Functions.
Key Points Topic A




Consider having students come up with their own summaries for how they approach factoring /solving / graphing a quadratic.
It’s better to study deeply a given application problem and the analysis of its graph’s features than to do multiple problems.
Introduce concepts like domain, range, increasing, decreasing, average rate of change, etc. by using words that feel natural in the
context, and then repeat the statement or question using the more formal words.
Scaffolds are a critical tool for successful implementation. In addition to those given in the module, consider the ones we explored in
this session. (Take time now to reflect and take note of them.)
Key Points Topic B





Completing the square has a geometric meaning.
A scaffold for completing the square when the leading coefficient is not 1 involves multiplying the equation through first by the
leading coefficient (if not already a perfect square) and then by the factor 4, if the coefficient of the 𝑥 −term is not easily halved.
This same scaffold used with the geometric model provides an alternative to the purely algebraic derivation of the quadratic formula.
The final lesson should include a reflection on the student’s general strategy for graphing quadratic functions.
Lessons 16-21 in Topics B and C provide a second opportunity for students to master transformations of functions.
Key Points Topic C


Comparing features of functions provided in different forms deepens and consolidates student understanding of the relationship
between the structure of expressions and equations, the graphs of equations and functions, and the contexts they model.
Students should walk away from quadratics understanding that a primary use of these functions is in modeling height over time of
projectile objects, that they are naturally related to rectangular area problems, and that there are also used in an early study of
business applications.
Key Points Module 4



Students are called upon to Look for and make use of structure (MP.7) as they choose equivalent forms of quadratics to gain insight
into the function’s behavior and its graph.
Students are called upon to reason abstractly and quantitatively (MP.2) as they decontextualize and work with quadratic equations
representing real-world contexts and then re-contextualize as they analyze and interpret the key features of the function and its
graph in the context of the problem.
Note that the physics contexts have the same coefficients due to the mathematics of objects in motion.
Session Outcomes
What do we want participants to be able to do as a result of this
session?
 Participants will develop a deeper understanding of the sequence of
mathematical concepts within the specified modules and will be able to
articulate how these modules contribute to the accomplishment of the major
work of the grade.
 Participants will be able to articulate and model the instructional approaches
that support implementation of specified modules (both as classroom
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.
teachers and school leaders), including an understanding of how this
instruction exemplifies the shifts called for by the CCLS.
 Participants will be able to articulate connections between the content of the
specified module and content of grades above and below, understanding how
the mathematical concepts that develop in the modules reflect the
connections outlined in the progressions documents.
 Participants will be able to articulate critical aspects of instruction that
prepare students to express reasoning and/or conduct modeling required on
the mid-module assessment and end-of-module assessment.
Session Overview
Section
Time
Overview
Prepared Resources
Facilitator Preparation
Introduction to Module 15 mins
Conduct an overview of module
stucture, lesson types, and lesson
components.


Grade 9 Module 4
Grade 9 Module 4 PPT
Foundations for the
Study of Quadratics
45 mins
Establish a foundation from which
to begin the study of Module 4.


Grade 9 Module 4
Grade 9 Module 4 PPT
Topic A Lesson
100 mins
Examine lessons in Grade 9
Module 4 Topic A.


Grade 9 Module 4
Grade 9 Module 4 PPT
Fluency Exercises
20 mins
Experience rapid white board
exchanges that support fluency in
the skill of factoring .

Personal white boards.
Practice leading a rapid white
board exchange.
Mid-Module
Assessment
Complete a portion of the
assessment, score, and discuss.

45 mins

Grade 9 Module 4 MidModule Assessment
Grade 9 Module 4 PPT
Review assessment, rubric, and
sample solutions.
Topic B Lessons
115 mins
Examine lessons in Grade 9
Module 4 Topic B.


Grade 9 Module 4
Grade 9 Module 4 PPT
Topic C Lessons
65 mins
Examine lessons in Grade 9

Grade 9 Module 4
Review Grade 9 Module 4
Module 4 Topic C.
Module Summary
End-of-Module
Assessment
25 mins
60 mins
Complete a portion of the
assessment, score, and discuss.

Grade 9 Module 4 PPT


Grade 9 Module 4
Grade 9 Module 4 PPT

Grade 9 Module 4 End-of
Module Assessment
Grade 9 Module 4 PPT

Review assessment, rubric, and
sample solutions.
Session Roadmap
Section: Grade 9 Module 4
Time: 512 minutes
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
3 min
1.
GROUP
5 min
2.
Give participants 5 min to do the opening exercise.
2 min
3.
During this session you will be actively engaged in
unpacking the content of Grade 9 Module 4. You will be
asked to interact with the materials from both the student’s
and teacher’s perspective at various times during the
session to deeply understand the content of the module.
We will revisit the opening exercise shortly. First let’s get to
know each other a bit.
2 min
4.
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
5.
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
6.
Here is our agenda for the day. If needed, we will start with
orienting ourselves to what the materials consist of. Next I
will walk you through some exercises I find lay an
important foundation for teachers embarking on a module
studying quadratics. After that brief introduction to
quadratics, we’ll begin examining and experiencing some
excerpts from lessons of each topic. After topic A, I’ll take
time to model for you an fluency strategy that you can use
as-needed while your students work their way through the
module. At the mid-way point, we’ll stop and take a portion
of the mid-module assessment, and then at the end we’ll
take a portion of the end of module assessment.
(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
7.
(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-ofmodule 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
8.
(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
9.
(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
10.
That concludes the materials orientation. Let’s get started
with our study of quadratics.
15
min
11.
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 quadratics module.
Reflecting back at our experience with sequences from
module 3. Consider the sequence that begins with, … What
did your study in module 3 tell you about the next number
in this sequence? (Some participants may answer X,
hopefully someone will also suggest that one isn’t sure what
it is unless one is given more information, like an explicit or
recursive formula for the sequence, or perhaps one is told
that it is arithmetic). So we learned not to always trust
patterns, and we also spend a good deal of time studying
arithmetic and geometric sequences and comparing and
contrasting those particular types of sequences. So a valid
conjecture for the next term in this sequence might be
what? (X)
10
min
12.
As usual, as presenter, my interactions with you will bounce
back and forth between speaking one professional to
another, and speaking as though I were a teacher and you
are the student. Please indulge me in those times where I
am modeling my own delivery of this in a classroom.
•Do heavier objects fall through the air faster than lighter
objects of the same shape and size? Consider a real
elephant and a life-sized paper Mache model of an elephant.
Which do you suppose would hit the ground first? How do
you think your students would answer this question?
•What Galileo was thinking is that they would land at the
same exact time. He recognized that as they fell their speed
would be increasing all the while. Depending on your
student population you may need to scaffold their
understanding. Always through questioning, not through
telling. So we might say, how fast do you think the
elephants are moving when they first get pushed off the
edge? Give me an estimate? 5 mph? 20 mph? Do they keep
going that same speed as they fall? What is causing the
speed of the elephants to increase? Galileo was thinking
that the speed was getting faster and faster as time passed,
and he was thinking that they would land at the same exact
time. Do you agree?
•Imagine what experiment / data he would need to do to
find out.
•Suppose he got the following data, would this confirm or
contradict his thinking? (0 sec, 100 m) (1 sec, 95 m) (2 sec,
80 m) (3 sec, 55m)
•(4 sec, 20 m)
•Since measuring the height of the elephant moving through
the air over constant time intervals will give us speed, and
measuring the change in speed over constant time intervals
will give us acceleration, Galileo is suggesting that the
formula that the formula that describes the height over time
could be given by something of degree 2; a formula of the
form: at2+bt+c
So why such fascination? The force of gravity is a huge part
of life on earth! Quadratic formulas predict elevation over
time of projectiles.
20
min
13.
Each table has a chain of sorts, an extra large sheet of graph
paper with an x and y- axis drawn on it, and some tape.
Hold up the graph paper and tape the ends of the chain so
that the bottom of the chain hangs down nicely at the origin
of the graph. As a group, identify points that the chain goes
through, and then come up with a quadratic formula to
model the curve you’ve created.
(Give the participants time to work on this and then ask
them to report their experiences. They will find that there
is not a nice quadratic equation to model a hanging chain,
and that not all u-shaped curves are representative of a
quadratic function.)
3 min
14.
(total elapsed time at the end of this slide is 1 hour)
We’re ready to begin examination of Topic A. First let’s
read the standards we’ll be covering in this module. (Give
participants time to read the standards listed on pages 4-7
of the teacher materials.)
Now let’s go through an overview of the flow of the module
as it addresses those standards.
2 min
15.
(Go through the bullets to give an overview of the
progression or flow of each topic and the module as a
whole.)
2 min
16.
(Go through the bullets to give an overview of the
progression or flow of each topic and the module as a
whole.)
2 min
17.
(Go through the bullets to give an overview of the
progression or flow of each topic and the module as a
whole.)
3 min
18.
(Review the bullet points with participants to remind them
of the background students are coming in to this module
with.)
15
min
19.
The study of quadratic expressions, equations and functions
is a common part of most any algebra course. In today’s
session we will be focused on the following key ideas:
-Studying quadratics with the shifts in mind:
Understanding, Application and Fluency
-Scaffolding to meet the needs of your specific students
whether they be at grade level, below grade level, or above
grade level.
-Engaging students through questioning not telling.
Let’s get started just as the students would with the opening
exercise of Lesson 1.
Now go ahead and work Example 1. Here is a possible
extension for advanced students. (see bullet)
Page 19 contains mathematically accurate descriptions that
lead up to describing what it means for a polynomial
expression to be prime or irreducible over the integers.
Let’s examine some scaffolds for making these descriptions
digestible. First read through the page and give it some
thought.
Students who find the word factor too abstract, can benefit
from thinking about what it means to count by 2’s, county
by 3’s, count by 7’s. So 7 is prime because you can’t get to 7
when counting by any other number except 7 and 1. 4 is
not prime because you can get to 4 when counting by 2’s, as
well as by 1’s and 4’s.
So next we need to ask, what do you think it means when
my polynomial expression is prime? Let the students
explore coming up with what it means to be prime in this
new context of polynomials.
In tackling the description given, have students come up
with their own binomials and test it out with the criteria for
prime. Try to write it as a product of two other polynomials
with integer coefficients, get confident that there are some
for which it can’t be done. Get the class to agree on some
examples of prime polynomials.
Students spend the remainder of the lesson multiplying and
then factoring a binomial of degree 1 times another
binomial of degree 1, spending the bulk of their time
practicing with the difference of two squares and then
contrasting it with the product of (a + b)2.
10
min
20.
Lesson 2 cites the term quadratic expression. Why are they
called quadratic expressions? Why not something
indicating that the degree is 2. We call a polynomial of
degree 3, cubic, and a polynomial of degree 4 a quartic, and
one of degree 5 a quintic, degree 6 a sextic or hexic. Why
quadratic for degree 2?
The usefulness of quadratics in solving problems involving
quadrangles, and the relationship between squares and
solving quadratics is at the heart of this association. Hence
the name of the Topic, and the term ‘completing the square’.
Note the scaffold box at the top of page 31, I strongly
recommend using the tabular method to reinforce proper
use of the distributive property and to help students’
thinking about factoring. Let’s work through that process
together with Exercise 7. Then try the same process for
Exercise 8.
15
min
21.
Lesson 3 opens up with an application problem. Work the
opening exercise now.
Continue to use the tabular method as needed.
Encourage students to verbalize their process of finding
factors that work. This is preferable over telling them a
procedure that isn’t originating from their own thinking.
Work problem 3 from the problem set in lesson 4.
15
min
22.
Lesson 5’s opening exercise and exercises 1-4 lead students
to know and apply the zero factor property without real
world context. Let’s work now on Example 1 which
provides context for its application.
This capacity to discern and recall their new-found power
to solve quadratic equations might be overlooked or
underappreciated by students at this time, but it should be
celebrated as huge. One suggestion for emphasis is to open
the exercise with the challenge to solve an general quadratic
equation (one that could be set to zero, factored, and solved
with integer answers). Students may be able to use trial
and error to find one or both solutions from the get go, but
they might appreciate their newfound skill more readily as
they contrast their trial and error approach to this one.
Lesson 6 encourages students to recall that in module 1,
they used reasoning to solve some quadratic equations. For
example, given 6x2 = 24, students reasoned that x2 = 4 and
that x could equal 2 or -2. This is still a viable option!
15
min
23.
In working the Extension question, remind participants that
average rate of change is defined on closed intervals only,
encourage participants to be consistent with the closed
interval requirement when presenting such questions to
students.
15
min
24.
Lesson 9 is on graphing quadratic equations from factored
form. Let’s go through the opening exercise from the
students perspective.
Example 2 – reading example 2 asks students to take leap.
Pose the question, how do you suppose they came up with
this formula? Does it seem reasonable that the science class
was able to create this formula?
So from here they get to use the entire graph to help make
sense of some questions about the context including
questions about increasing and decreasing and domain and
maximum height and such. So these exercises really dig in
to the graph and identifying its key features.
Scaffold in formal terms by using contextual everyday
language and then repeating with more formal words. For
example, ask over what time period is this graph relevant?
What’s the domain of this function in this context?
Lesson 10: Interpreting quadratic functions from graphs
and tables.
Dolphin height vs time graph. This graph bears discussion,
is it reasonable that a single force sets the vertical motion in
place after the dolphin has descended to the lowest point?
Is it reasonable that gravity is the main force once the
dolphin hits the water? Do we think the dolphin might also
be engaging in a swimming motion? Should the curve
appear steeper or wider under water than above water.
Example 2 uses data from a table and asks the leading
question of “What kind of function do you think this table
represents and how do you know?” Really spend some time
here… how do you know?
6 min
25.
(Total elapsed time 2 hour and 40 min + 20 min of time
budgeted for Q&A at any point during the process = 3
hours.)
(Go through each point listed.)
2 min
26.
Let’s experience now an exercise to build students fluency
with the skills in this module.
15
min
27.
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.
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.
2 min
28.
We’re ready to have a look at the mid-module assessment
25
min
29.
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.
20
min
30.
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 6 minutes or so have passed, call the group together
and address any questions or concerns that participants
noted on their post-it notes.
2 min
31.
We’re ready to move on to Topic B.
20
min
32.
Topic B covers the method of completing the square and the
derivation of the quadratic formula.
A valuable scaffold even with the opening exercise is to use
a geometric model of a square. Knowing that that you are
attempting to factor it such that you are creating a perfect
square develops students capacity to do so.
Let’s work Example 1 together using the geometric model.
…
Getting to the last example. So my model says (x + 4)(x + 4)
produces x2 + 8x + 16. But what I wanted was x2 + 8x + 3.
Let’s just make it so, If what I want is x2 + 8x + 3, I can use
my model and make an adjustment. I’ll write my x2 + 8x +
16 then I’ll subtract 13 so that I have the + 3 that I want.
Then I can write (x+4)2 – 13. Is this useful? Why was I
bothering to re-write these things anyhow? Remember all
that work I made us do with factoring… I wanted to do it to
help solve quadratic equations.
Is this form useful in solving quadratic equations? If I have
the equation x2 + 8x + 3 and I write it as (x+4)2 – 13 = 0, is
that helpful for solving the equation? (Participants are
likely to see this readily, but ask them, ) How are your
students likely to respond to my question?
Notice the Lesson Summary suggests exactly what we’ve
indicated, “Just as factoring a quadratic expression can be
useful for solving a quadratic equation, completing the
square also provides a form that facilitates solving a
quadratic equation.” (See the bottom of page 130.)
If your students are not likely to see the possibility here.
You might scaffold this lesson by starting right off the bat
solving simple quadratics by inspection.
•Solve by inspection: �2 = 16, �2 – 25 = 0, (� – 1)2 = 9,
(� + 2)2 −36 = 0
Discourage use of the ± sign. Instead model less abstract
‘or’ that emulates our thinking. “If something squared is 9,
then either that something equals 3 or that something
equals -3. “ This scaffold helps students follow their own
thinking all the way through to a final answer.
It is vital that throughout this process, students feel a sense
of why am I doing these things. It is far too easy to get lost
in a procedural shuffle here, so draw them back to purpose
whenever you sense they are losing connection.
35
min
33.
So what happens when things get messy? What are some
strategies we use to complete the square for something like
2�2 + 16� + 3.
(Participants will likely suggest factoring out the GCF.)
Let’s try Lesson 12 Example 1, then. (Take time to make
sure participants understand the mechanics of this
approach, working it together if needed, or asking folks to
share with their neighbor.)
Now let’s do Lesson 12 Example 2. (Give participants time
to work it through on their own and then discuss.)
Do your students struggle with this approach?
So here is an uncomfortable fork in the road. For students
ready to do this work of reasoning with abstract algebraic
equations, this approach of completing the square without
the context of an equation has advantages.
I’m going to take you through a scaffold that relies on the
advantage of putting this into the perspective of an
equation. This an alternative approach that helps us make
the best of the geometric model and what students already
know about equations. But, be forewarned it does not get
us all the way to our goal of completing the square in the
context of a quadratic function.
Suppose consider that we are solving the equation 2x2 + 16
x+3=0
I would like to make a perfect square from this quadratic
expression that will help me solve my equation.
That 2 in front is making things difficult. Would I change
my solution set if I multiplied both sides of my equation
through by 2?
So now I have 4 x2 + 32x + 6 = 0, which I can solve by
creating a perfect square expression. Here’s where I think
the geometric model is so helpful in making sure a
struggling student keeps things straight. (2x + ___)(2x + ___)
What will go in these blanks if I want these two to combine
to give me 32? 8. Let’s double check. To make the scaffold
of the geometric method be a true help to your students you
must establish a routine of always checking. Yep, I get 4x2 +
32 x + 64. Hmmm, that gives me a perfect square but it’s
not exactly what I want, I wanted + 6. So I’ll subtract off 58
to get exactly what I want. (2x + 8)2 -58 = 0.
Now suppose I have something like 3x2 + 5x – 12 = 0. I can
multiply through by 3.
This scaffold option I’ve gone through with you here
requires a significant deviation from the basic layout of
these lessons. So I encourage you think carefully about this
choice. You know your students best. What approach will
work the best for them. Make the decision ahead of time.
This scaffold has the added advantage of providing a
substantial scaffold to deriving the quadratic formula as we
shall see. But it has a downside as well.
Eventually your students will need to be able to take an
equation in 2 variables (or a function) of the form f(x) = ax2
+ bx + c and reason through getting it into vertex form. One
doesn’t have the luxury of simply multiplying through by
constants now.
So if you choose to take this particular scaffolded approach,
you won’t be able to rely on completing the square as an
approach to putting something into vertex form. What can
students do instead? In topic A, they learned how to graph
the quadratic function by finding the intercepts and then
using symmetry to identify the vertex. Their understanding
of transformations of functions can get them into vertex
form. Let’s continue to reflect on this as we progress
through the module.
15
min
34.
(Depending on the audience either allow them to work this
through independently, walk them through it together, or
agree that neither is necessary, then demonstrate using the
square geometric model to get to the same place.)
20
min
35.
Lesson 15: Look with me at Exercises 1-5 and the
discussion that follows (allow a couple of minutes, or
depending on the needs of the participants ask them to do
the exercises).
Using the quadratic formula, students solve using the
quadratic formula and find that sometimes there might be
two solutions, one solution, or no (real) solutions, and that
sometimes there is a radical and sometimes not. The term
discriminant is introduced. Students use the discriminant
to categorize the solutions to a quadratic without finding
them, they relate their findings to what the graphs must
look like.
Lesson 16: Graphing quadratic functions from the vertex
form: students graph 3 equations with horizontal shifts and
are asked to make a conjecture about where a vertex is for
another such function without graphing; then students are
asked to create the equation with a given vertex (jumped
here to include vertical shifts – not such a big jump though
because of module 3 exposure). Then a pretty big jump that
in a discussion just before the closing, students are asked
what is the effect on the graph to have a leading coefficient
other than 1. Though the lesson does not include a≠1
examples, the exit ticket does. Consider the scaffold on the
bottom of page 172 and explore this deeply if students still
need time to study the effect of � on the ‘steepness’ of the
graph.
20
min
36.
Lesson 17: Graphing from standard form. This is a put it all
together, examine these forms in context of another gravity
(baseball) situation. Let’s work this Opening Exericse.
Example 1 provides another opportunity to ask, ‘How do
you suppose the math class was able to determine this
formula?’
It is not explicitly asked or stated, but is suggested, ‘How
can I put a function into vertex form?’ What strategies do
students have for doing that?
•Using intercepts or point plotting and symmetry to find the
vertex.
•How do they identify the leading coefficient a? Will it be
the same as if it were in general form?
•Completing the square is an option for students ready.
Have students consolidate their thinking by asking them to
come up with a general approach to graphing a quadratic
equation on their own before considering what is outlined
for them.
5 min
37.
(Review the points outlined.)
2 min
38.
Now let’s have a look at Topic C.
20
min
39.
Allow participants to complete exercises 1 – 3. Then
challenge them to describe privately with a neighbor how
the graphs of these functions in Exercise 2 and in Exercise 3
relate to each other.
Discuss the suggestion. This is a real opportunity for
reflection, coherence (bringing in their experiences in Grade
8) and students practicing their articulation of abstract
ideas.
We use the term symmetric about the origin once in this
lesson, but this idea should not be forced on students who
are not ready, reserve it only as an extension for high level
students.
20
min
40.
Lesson 20: Stretching and shrinking functions – who can
identify what is wrong with the name of this lesson?
Lesson 21: Transformations of the quadratic parent
function. Did we introduce the term “parent function?” we
just start using it and define it in that first instance, like “the
quadratic parent function f(x) = x^2
Lesson 22: Comparing Quadratic, Square root and cube root
functions represented in different ways. Students answer
questions that compare two functions where for one we
only have a graph and for another we only have an
equation, then in another case one has a table and one has a
graph.
Lesson 23: Modeling with Quadratic Functions – Students
and teachers both get notes about objects in motion. Makes
students capacity so very deliberate to decipher out the
physics and relationships and units and such. Then works
with business applications for both teacher and students.
Lesson 24: L23 continued. Given 2 points, how many
quadratic graphs can you draw? (this raises the question
how do I know that the graph I’ve drawn is even quadratic).
There are infinitely many. Given 3 points how many
quadratic graphs can you draw (that go through the
points)? Check out that scaffold box on page 247. Gives
data and asks, how do we know this data “will” (uggh) be
represented by a quadratic function? Students use 3 points
to come up with the quadratic function that models the
data. But you have to know!
20
min
41.
Lesson 22: Comparing Quadratic, Square root and cube root
functions represented in different ways. In these exercises
students answer questions that compare two functions
where for one we only have a graph and for another we only
have an equation, then in another case one has a table and
one has a graph. Let’s do these exercises now.
Lesson 23 opens with a discussion of the mathematics of
objects in motion. Note the following points. (Go through
the bullet points). One can not make up ones own quadratic
equation to use in these situations. The coefficient must be
one of the two options or an equivalent option given the
unit of measurement used for height.
Lesson 23: Try Example 1.
The physics-based exercises throughout this module makes
students’ capacity to make sense of these contexts a
beneficial and inseparable outcome of their study of
quadratics.
Lesson 23 Example 2 then works with business applications
with notes for both the teacher and students, let’s do this
example now.
Lesson 24: Opening Exercise: Given 2 points, how many
quadratic graphs can you draw? (this raises the question
how do I know that the graph I’ve drawn is even quadratic).
There are infinitely many. Given 3 points how many
quadratic graphs can you draw (that go through the
points)? Notice the scaffold box on page 247. Given data,
how do we know this data will be represented by a
quadratic function? Students use 3 points to come up with
the quadratic function that models the data. But you don’t
overlook the fact that 3 points does not assume in and of
itself a quadratic graph, there could be a hexic function
going through those points, or something not even
polynomial!
4 min
42.
Go through each point
5 min
43.
Let’s revisit the answers to the questions in the opening
exercise.
Students study quadratics because of their relevance to
science, specifically the physics of projectiles. There are
also business applications and applications in geometric
problems involving quadrangles.
The word quadratic is derived from the Latin
word quadratus for square.
Not all u-shaped graphs represent parabolas / quadratic
functions. The classic counter example is that of the
catenary curve.
20
min
44.
Review each key point one at a time.
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?
2 min
45.
Finally let’s examine the end-of-module assessment for G9M4.
25
min
46.
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.
20
min
47.
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 6 minutes or so have passed, call the group together
and address any questions or concerns that participants
noted on their post-it notes.
4 min
48.
(Review each key point one at a time.)
10
min
49.
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
Active learning
Turn and talk
Turnkey Materials Provided


Grade 9 Module 4
Grade 9 Module 4 PPT
Additional Suggested Resources

A Story of Functions Curriculum Overview