Module Focus: Grade 10 – Module 1 and Module 2
Sequence of Sessions
Overarching Objectives of this May 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
●
●
●
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 10 curriculum, A Story of Functions.
Key Points Module 1
•
•
•
•
Module is anchored by the definition of congruence
Emphasis is placed on extending the meaning and use of vocabulary in constructions
There is an explicit recall and application of facts learned over the last few years in unknown angle problems and proofs
Triangle congruence criteria are indicators that a rigid motion exists that maps one triangle to another; each criterion can be proven
to be true with the use of rigid motions.
Key Points Module 2
•
•
Just as rigid motions are used to define congruence, so dilations are used to define similarity.
To understand dilations and their properties, begin with scale drawings and how they are created.
Right triangle similarity is rich in relationships: dividing a right triangle into two similar sub-triangles, trig ratios and their
applications
•
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 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.
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
Introduction
38 min
Conduct an overview of module
structure, lesson types, and lesson
components.
Grade 10 Module 1:
26 min
Examine constructions and
Prepared Resources



Facilitator Preparation
Grade 10 Module 1 and 2
PPT
Grade 10 Module 1 and 2
Facilitator Guide
Review Grade 10 Module 1 and
Module 2
Grade 10 Module 1 and 2
Review Grade 10 Module Topic A
Topics A and B
unknown angles to develop the
basic language of geometry.


Grade 10 Module 1:
Topic C
20 min
Explore congruence in terms of
transformations.


Grade 10 Module 1:
Topic D
24 min
Explore congruence and rigid
motions.


Grade 10 Module 2:
Topic A and Topic B
26 min
Explore scale drawings and
dilations.


Grade 10 Module 2:
Topic C
Grade 10 Module 2:
Topic D
Grade 10 Module 2:
30 min
Examine similarity
transformations.
21 min
Examine the similarity
relationships that arise when an
altitude is drawn from the vertex
of a right triangle to the
hypotenuse, how to use similarity
to prove the Pythagorean
Theorem, and how to simplify
radical expressions, specifically
multiplying and dividing radical
expressions and adding and
subtracting radical expressions
using the Distributive Property.
Exmaine the basic definitions of




PPT
Grade 10 Module 1 and 2
Facilitator Guide
and Topic B
Grade 10 Module 1 and 2
PPT
Grade 10 Module 1 and 2
Facilitator Guide
Review Grade 10 Module Topic C
Grade 10 Module 1 and 2
PPT
Grade 10 Module 1 and 2
Facilitator Guide
Review Grade 10 Module Topic D
Grade 10 Module 1 and 2
PPT
Grade 10 Module 1 and 2
Facilitator Guide
Review Grade 10 Module 2 Topic
A and Topic B
Grade 10 Module 1 and 2
PPT
Grade 10 Module 1 and 2
Facilitator Guide
Review Grade 10 Module 2 Topic
C
Grade 10 Module 1 and 2
PPT
Grade 10 Module 1 and 2
Facilitator Guide
Review Grade 10 Module 2 Topic
D
Grade 10 Module 1 and 2
Review Grade 10 Module 2 Topic
Topic E
sine, cosine, and tangent, as well
as how they are applied in a
variety of settings.

PPT
Grade 10 Module 1 and 2
Facilitator Guide
E
Session Roadmap
Section: Introduction
Time: 38 minutes
In this section, you will explore an overview of the module
structure, lesson types, and lesson components.
Materials used include:
 Grade 10 Module 1 and 2 PPT
 Grade 10 Module 1 and 2 Facilitator Guide
Time Slide # Slide #/ Pic of Slide
3 min
1.
2 min
2.
Script/ Activity directions
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?
GROUP
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
3.
Our objectives for this session are:
• Examination of the development of mathematical
understanding across the module using a focus on Concept
Development within the lessons.
• Examples that demonstrate themes and changes according to
the Common Core State Standards.
2 min
4.
Here is our agenda for the day. If needed, we will start with orienting
ourselves to what the materials consist of.
Overall, I’d like to spend our session discussing the overarching themes of
Modules 1 and 2. The idea is to leave with an understanding of where the
major shifts in the Geometry are and use examples to make sense of those
changes.
(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
5.
(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 that wouldn’t
be fitting to give the students ahead of time, as well as the assessments.
4 min
6.
(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
7.
(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.
8 mins
8.
Module 1, focused on Congruence, (and Module 2, focused on Similarity) are
where some of the most significant changes are occurring in geometry vs.
traditional curriculum. This is due to the way congruence and similarity are
defined under the CCSS, which is with the use of ‘geometric transformation’
as stated in this quote.
Traditionally, we have called two segments for example congruent if they
had the same lengths. It is not untrue to make this statement, however the
CCSS lens allows us to declare figures as congruent in one fell swoop as
opposed to making many individual measurements. Another benefit is that
this approach allows us to broaden the kinds of figures we are able to
compare at all. A ‘complex’ figure with not just straight edges but also
curves is now among the kinds of figures we can compare. Since rigid
motions preserve distances and angles, we have a way to try and manipulate
the figure onto another figure for comparison. Before, by using the lengths
of segments as the means of identifying two figures as congruent, we were
really limited in the types of figures we could compare (we could really only
compare rectilinear figures).
2 min
9.
So, very broadly, what is meant by ‘geometric transformation’? Are
‘transformations’ under the CCSS the same as they were in Regents
Geometry?
Well, yes and no. Yes, because the four transformations studied in this
course are question are translations, reflections, rotations, and dilations; no
because the treatment they get under the CCSS is different. So how they are
introduced, the way in which they are studied, what we use them towards is
different.
Just to remind ourselves of what has traditionally been evaluated on
transformations, here are a few recent Regents questions pertaining to
transformations.
1 min
10.
1 min
11.
These are two recent questions, but we could look through more and see
that 1) transformations have been associated with the coordinate plane and
2) that there are a set of formulas that govern how transformations behave.
4 min
12.
Section: Grade 10 Module 1: Topics A and B
Time: 33 minutes
In this section, you will examine constructions and unknown angles Materials used include:
to develop the basic language of geometry.
 Grade 10 Module 1 and 2 PPT
 Grade 10 Module 1 and 2 Facilitator Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
2 min
Module 1 is a 45 day module, in fact it is one of two 45-day modules (Module
2, Similarity is the other).
13.
The bulk of class time is spent on Topics A-D, so let’s talk a little about the
story that weaves these topics together.
Clearly, Topic C is a focus as it is going to inform how we define what
congruence is. So what are Topics A and B for? Well, the purpose of looking
at constructions and unknown angles is to develop the basic language of
geometry. In other words, we need to know WHAT we will be transforming.
It wouldn’t make sense to talk about transformations before you know what
you’re transforming or before you have a way of talking about what you’re
transforming.
GROUP
4 min
14.
All the constructions in Topic A should look familiar (they have been part of
the NY state standards and were tested on the Geometry Regents).
What is different here is that the focus is not solely about the procedure or
the “how to”, but being about to communicate the “how to” (how to tell
people to ‘how to’). The ability to describe a CONstruction precisely is the
ability to give an INstruction. In this study of constructions we are
extending the meaning of vocabulary and seeing it in action. We are
practicing communicating ideas to one another.
Why do this? Why not just stick to the procedure? If we step back and think
of what it means to understand the content deeply, we could say that Euclid
represents the end of the spectrum of what it means to understand
geometry deeply. He took all his understanding of geometry, started with a
set of assumptions and using a deductive approach, built fact upon fact, and
communicated the ideas effectively and efficiently in the books that make up
Elements.
Students are on the spectrum. And to work their way along the spectrum,
they need to practice communicating these mathematical ideas and give
them meaning beyond the initial level of just “doing” the basic action, e.g. a
construction.
Students do just this right off the bat in Lesson 1.
1 min
15.
We are going to take a look at Lesson 1, and you will have an opportunity to
experience it as students will experience it. The general flow of the lesson is
as follows: [refer to the screen and the four major parts of the lesson]
4 min
16.
-- Allow participants 30 seconds to read/think about the Opening
Exercise prompt, and another 30 seconds to exchange ideas.
Encourage them to consider what students might say. Share out 1-2
responses.
-- Allow participants 30 seconds to read/think about the vocab terms,
another few seconds to exchange answers, and finally, share out
responses.
-- Have a poster ready of circle notation. Instruct participants that they
may want to have such a poster ready, or some allocated space in their
room to show this notation before reaching Example 2.
CB.
2 min
17.
A circle with center C and radius AB is written as CircleC, radius
-- For a 2.5 hour slot, the idea is to introduce participants to what the
flow of the lesson is like, but not to do these individual pieces of the
lesson. Example 2 is where this curriculum really differs from
traditional curriculum, so we are simply showcasing what is going on
up until Example 2.
In Example 1, you will lead students to helping them discover how to use
their compass and straightedges to determine the location of the third cat.
1 min
18.
-- Show solution.
2 min
19.
-- Refer to the Euclid excerpt, Proposition 1. Tell participants that they
can have students annotate the text and use it as a guide to writing
what they believe is a clear set of instructions on how to construct an
equilateral triangle. Tell them to consider using the points from
Example 1 as the points to refer to in the steps.
-- Depending on who is in the audience, consider asking someone who
feels they know the construction to describe the appropriate steps to
the construction (maybe flip back to the solution image of the location
of Mack). Elicit an informal set of steps.
4 min
20.
-- Refer to the Euclid excerpt, Proposition 1. Tell participants that they
can have students annotate the text and use it as a guide to writing
what they believe is a clear set of instructions on how to construct an
equilateral triangle. Tell them to consider using the points from
Example 1 as the points to refer to in the steps.
-- We want to highlight Example 2, as it is a prime example of students
communicating their understanding about why the construction works
vs. strictly completing the procedure.
To recap up until this point- students have been primed to think about
distance between points, have reviewed key vocabulary, have explored how
to perform the construction at hand, and now working on shaping their
thoughts and language to articulate how to perform the construction. This
last part emphasizes deeper understanding, especially as students compare
and try out each others’ steps to constructions, and inherently involves the
Math Practices (MP 3, MP 6).
Throughout Topic A, students will practice not only the constructions, but
paying careful attention to the instructions that determine the construction.
We want students to understand that a construction may not ever be
completely perfect, due to human error, but the instructions that describe
how to perform a construction do describe a perfect construction.
2 min
21.
The motivation in Topic B is similar to that in Topic A: there is an actual task
and then a step towards being able to discuss or explain the task. With
unknown angle problems, students are actually solving a problem but also
beginning to use justifications from their knowledge of facts related to
angles. With unknown angle proofs, numbers are replaced by variables, and
the focus on geometric relationships is sharper, and requires students to be
attentive to the reasons that allow them to take the next step or draw the
next conclusion in a problem. The practice with numeric problems makes
for a smooth transition to reasoning without numbers.
Work with angles, even single step unknown angle problems (i.e. solving for
the missing angle), began in Grade 4. Students see an especially closer look
at unknown angle problems in Grade 7, Module 3.
It is worth noting here that as part of the transition from Topics A to B, an
extensive chart of facts associated with angles learned since Grade 4
appears in the Problem Set of Lesson 5 (the last lesson in Topic A). This
primes students with the review of needed facts to (1) solve problems and
(2) explain why they take the algebraic steps they take in a problem.
4 min
22.
Remember that Lessons 6-8 are unknown angle problems, therefore the
first example is about solving for the measure of a and providing a
justification for how you arrived at that answer.
-- Allow participants the opportunity to answer the question and cite a
reason.
-- Step through to show the second example.
What do you notice about the second example relative to the first example?
-- Elicit that it is non-numeric and that the goal here is to establish a
relationship of some sort.
You might want to say, well I still know that the sum of two remote interior
angles is equal to the measure of the respective exterior angle. But is it
possible to actually show this with the use of any other facts?
3 min
23.
-- Take participant suggestions about establishing a relationship
between x, y, and z before reviewing the solution.
The unknown angle proofs require students to really break down
relationships, including ones they know to be true; they call on simpler facts
to explain newer ones. The process is deductive.
4 min
24.
-- This is an additional problem, something to highlight another
Unknown Angle Proof lesson. Depending on time, step participants
through the problem and solution.
Section: Module 1: Topic C
Time: 20 minutes
In this section, you will explores congruence in terms of
transformations.
Materials used include:
 Grade 10 Module 1 and 2 PPT
 Grade 10 Module 1 and 2 Facilitator Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
2 min
25.
We have laid the ground work in Topics A and B: we have reviewed both
precise vocabulary and called explicit attention to facts that we have been
learning over several years, and now we are ready to tackle
transformations, specifically rigid motions.
If Topics A and B represent the “what” in geometry, figures we study,
extending their meaning by using them in constructions, then in Topic C,
transformations tell us how things relate or compare to one another, they
help us establish what the relationship is between two things. We
eventually use this for a very particular relationship, one where the rigid
motions help establish whether two things are the same or not.
Specifically, students will learn the formal definitions of each rigid motion,
and how to apply them to figures, how each transformation is tied to
constructions, what compositions of rigid motions look like, and toward the
end of Topic C, we explain congruence in terms of transformations.
1 min
26.
Students study rigid motions in Grade 8, chiefly in a hands-on and
descriptive approach to build intuitive understanding of how the rigid
motions behave.
3 min
27.
In grade 8 (M2, L4), reflections are introduced in the following hands-on
way. Students have a piece of paper with a point Q on line L represented in
black ink.
-- Summarize the Grade 8 experience, the following doesn’t need to be
said verbatim:
- Let L be a vertical line and let P and A be two points not on L as shown
below. Also, let Q be a point on L. (The black rectangle indicates the border
of the paper.)
The following is a description of how the reflection moves the points P, Q,
and A by making use of the transparency:
- Trace the line L and three points onto the transparency exactly, using
red. (Be sure to use a transparency that is the same size as the paper.)
- Keeping the paper fixed, flip the transparency across the vertical line
(interchanging left and right) while keeping the vertical line and point
Q on top of their black images.
- The position of the red figures on the transparency now represents the
reflection of the original figure. Reflection(P) is the point represented
by the red dot to the left of L, Reflection (A) is the red dot to the right of
L, and point Reflection(Q) is point Q itself. Note that point Q is
unchanged by the reflection.
- The red rectangle in the picture on the next page represents the
border of the transparency.
- In the picture above, you see that the reflected image of the points is
noted similar to how we represented translated images in Lesson 2.
That is, the reflected point P is P'. More importantly, note that the
line L and point Q have reflected images in exactly the same location
as the original, hence ReflectionL=L and Reflection(Q)=Q, respectively.
Pictorially, reflection moves all of the points in the plane by “reflecting”
them across L as if L were a mirror. The line L is called the line of reflection.
A reflection across line L may also be noted as Reflection(L).
1 min
28.
Lesson 14 begins with a tie to constructions. Students discover that the line
of reflection coincides with the perpendicular bisector of each segment
determined by a pair of corresponding vertices in the figure. For segments
AA’, BB’, and CC, the perpendicular bisector of each coincides with the line of
reflection DE.
Recall that students have studied the construction of the perpendicular
bisector in Lesson 4 (see next slide for reference).
2 min
29.
In fact, students first learned the construction for an angle bisector, and how
observations on the verification of an angle bisector construction led to the
perpendicular bisector construction. Students verified an angle bisector
construction by folding along the angle bisector– a look at the segment EG
demonstrated whether the construction was done correctly. When the
angle is folded along the bisector AJ, E should coincide with G, in fact a
correct construction showed that E was as far from F as G was, or F was the
midpoint of EG. Furthermore, by folding along the bisector of a correctly
done construction meant that angles EFJ and GFJ should coincide, have the
same measure, and since they lie on a straight line, should each have a
measure of 90˚.
With this understanding, and with the knowledge of the steps of the angle
angle bisector construction, students determine steps of the perpendicular
bisector and have an understanding of the implications of a perpendicular
bisector.
2 min
30.
2 min
They use this understanding to determine a line of reflection between two
figures in Example 1.
Based on the Exploratory Challenge, how is the line of reflection
determined?
-- Allow the audience an attempt at the response.
Students must join any two corresponding points between the figures and
construct the perpendicular bisector in order to determine the line of
reflection.
1 min
31.
This solution shows the perpendicular bisector of CC’ constructed; any two
of the corresponding points used would be acceptable and would have
yielded the same results.
2 min
32.
These exercises lead to the formal definition of reflection.
Just to review-- this definition comes after an intuitive understanding is
developed and student have had experience manipulating reflections and
testing out their properties with transparencies (G8), and after making a
connection with constructions from the beginning of Module 1. Students
have an understanding of how reflections behave and know, by
construction, why they work the way they work.
1 min
33.
In addition to however you want to break the definition down with students,
the lesson provides an examination of each part of the definition.
2 min
34.
These are exit ticket questions for this lesson. Students are able to (1)
determine the line of reflection between a figure and it’s reflected image and
(2) reflect a figure across a line of reflection.
Each of the three transformation lessons builds to the formal definition and
then works on the application of the transformation to figures. Each one
also makes use of constructions. For example, in studying rotations,
students discover that the center of rotation between a figure and its rotated
image can be found with the help of perpendicular bisectors. Applying a
translation to a figure involves the construction of parallel lines. So we see
the progression of the topics beginning to come together.
3 min
35.
Congruence is defined in L19 after a study of each basic rigid motion, as well
as a look at symmetry (as a rigid motion) in Lesson 15, and the Parallel
Postulate in Lesson 18.
- The idea of “Same size, same shape” only paints a mental picture; not
specific enough:
Just as it is not enough to say, “Hey he looks like a sneaky, bad guy who
deserves to be in jail”, it is not enough to say that two figures are congruent
if they have the same size and same shape- it lacks specificity needed in a
mathematical argument.
- It is also not enough to say that two figures are alike in all respects except
position in the plane.
In defining congruence as a finite composition of basic rigid motions that
maps one figure onto another, we need to be able to refer and describe this
rigid motion. Additionally, a congruence by one rigid motion and a
congruence by a different rigid motions are two separate things. Specifying
one of many possible rigid motions may be important.
- A congruence gives rise to a correspondence.
A correspondence between two figures is a function from the parts of one
figure to the parts of the other, with no conditions concerning same measure
or existence of rigid motions. In other words, two figures do not have to be
congruent for a correspondence to exist, but a congruence always yields a
correspondence. If there exists a rigid motion T that takes one figure to
another, then a natural correspondence results between the parts. For
example, if a figure contains a segment AB, then a congruent figure includes
a corresponding segment T(A)T(B).
Section: Module 1: Topic D
Time: 24 minutes
In this section, you will explore congruence and rigid motions.
Materials used include:
 Grade 10 Module 1 and 2 PPT
 Grade 10 Module 1 and 2 Facilitator Guide
 Grade 10 Module 1 Mid-Module Assessment
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
3 min
In Grade 7, students discovered that deciding whether two triangles are
identical to each other does not mean that 6 measurements (3 side lengths
and 3 angle measures) of both triangles had to be known. For example, they
discovered that if only 3 side lengths of each triangle were known, they
could predict whether the triangles would be identical or not (if the
triangles had the same side length measures, they would indeed be identical
because there was only one way those lengths could be put together to form
a triangle; the three sides condition).
36.
However in Grade 10, the goal is to explicitly prove why these “shortcuts”
work; prior to the CCSS, we used these patterns to forecast that if the parts
in the shortcut (e.g. A-S-A) were the same for both triangles, then all three
side lengths and all three angle measures were the same for both triangles.
Even if we know this information, we are now defining figures to be
congruent if a rigid motions maps one figure to another. Students will see
that the shortcuts are indicators that a rigid motion does exist in the cases
those indicators hold true, but there are ways to describe the actual
GROUP
sequence of rigid motions that maps one figure onto a another figure.
1 min
37.
We are going to show that the SAS criteria for triangles indicates that the
two triangles ABC and A’B’C’ are congruent because there exists a sequence
of rigid motions that will map one triangle onto the other.
The two triangles are distinct. We are trying to determine if one triangle
will map to the other, or coincide with the other. If the goal is to see what
happens when they coincide, we must use a transformation that will bring
them together.
1 min
38.
A translation by vector A’A will result with the two triangles coinciding at A.
Could we have translated by say B’B or C’C? No, because we are working
with parts we know to be equal in measure, and angle A and angle A’ are of
equal measure.
1 min
39.
Continuing in our goal to try and make the triangles coincide, we can use a
rotation of d degrees about A to bring AC’ to AC. We know C’ will map to C
because by the assumption AC and AC’ are the same length and a rotation is
distance preserving, so there is no change in any length in the
transformation.
5 min
40.
Again, in the pursuit to see if the triangles coincide, we see B and B’’’ are on
opposite sides of AC. We use a reflection over the line that contains AC so
that B’’’ maps to B. How do we know that B’’’ definitely maps to B? Because
rigid motions angle measures, we know angle B’’’AC=BAC and therefore ray
AB’’’ maps to AB. Additionally, by assumption we know that AB=AB’’’, so
between these two facts, B’’’ definitely maps to B.
This sequence of rigid motions takes triangle A’B’C’ to triangle ABC with the
use of just three known criteria from each triangle: two side lengths and the
included angle measure. We can generalize this argument for any two
triangles with the same criteria (distinct or coinciding along some part of
the triangle). Hence we have proven that when the SAS criteria is satisfied
between two triangles, there always exists a rigid motion that will map the
one triangle onto the other.
Students will complete proofs for all the triangle congruence criteria using
rigid motions, as well as the otherwise accepted fact that base angles of an
isosceles triangle are equal in measure. Once the the congruence criteria
are proved (and any known fact is proved), we are free to call on them in
problems.
3 min
41.
-- Anecdote to share:
Many years ago I taught calculus for business majors. I started my class
with an offer: "We have to cover several chapters from the textbook and
there are approximately forty formulas. I may offer you a deal: you will learn
just four formulas and I will teach you how to get the rest out of these
formulas." The students gladly agreed.
5 min
42.
This is an example from the Mid-Module Assessment (that also pertains to
what has been discussed in this presentation). I know we did not have an
in-depth look at rigid motions as the students will have once they arrive at
this question, but please take a few moments and formulate a response.
-- Allow participants 2 minutes to consider and discuss amongst
themselves.
3 min
43.
-- Review solutions.
2 min
44.
That wraps up the portion of our session on Module 1.
-- Review all four bullet points; these are the key themes in the module.
Section: Module 2: Topic A and Topic B
Time: 26 minutes
In this section, you will begin exploring Module 2’s focus on
Materials used include:
similarity, proof and trigonometry by exploring scale drawings and
 Grade 10 Module 1 and 2 PPT
dilations.
 Grade 10 Module 1 and 2 Facilitator Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
1 min
45.
Module 2 is the other 45 day module in the curriculum. It too is a place
where the course sees a significant shift versus traditional curriculum, this
time because of the way similar is defined under the CCSS. Just as
congruence is defined through the lens of transformations, so is similarity.
In addition to the need for rigid motions, similarity requires the discussion
of another transformation, dilation.
2 min
46.
Topic A, Scale Drawings, is a natural place to begin the discussion that leads
to dilations, as they are a concept familiar to students (G7, M1). They know
a scale drawing of a figure is a magnification or reduction of a figure, and it
has side lengths that are in constant proportion to the corresponding side
lengths of the original figure, and have angle measures that are equal to the
corresponding angle measures of the original figure. Work in grade 7 is
more heavily observational and students verify these properties by
calculations and measurements.
Here, students are creating scale drawings. They two formal methods of
creating scale drawings. In the last two lessons of Topic A, the use the two
different methods to draw scale drawings yields two theorems that will be
used repeatedly to prove the properties of dilations in Topic B.
2 min
47.
To use the Ratio Method requires an understanding of how to dilate points,
which is first addressed in Grade 8, Module 3.
-- Review definition of dilation.
GROUP
2 min
48.
By the Ratio Method of drawing scale factors, key vertices are dilated about
a center O and scale factor r.
A ray is drawn from the center of dilation through each vertex. Then,
according to the scale factor, an appropriate distance is measured and
marked as the dilated vertex. Once key vertices are located under the
dilation, the dilated points are joined.
5 min
49.
Use the handout and ruler to scale the following figure (Lesson 2, Example
2) according to the Ratio Method.
-- Allow participants time to create the scale drawing.
Step 1. Draw a ray beginning at O through each vertex of the figure .
Step 2. Use your ruler to determine the location of A' on OA ; A' should be 3
times as far from O, as A. Determine the locations of B' and C' in the same
way along the respective rays.
Step 3. Draw the corresponding line segments, e.g., segment A'B'
corresponds to segment AB.
2 min
50.
By the Parallel Method of drawing scale factors, an initial dilated vertex is
provided or must be located (in the same way as by the way points are
determined by the Ratio Method). Then a set square and ruler are used to
draw segments parallel to the segments of the initial figure using the initial
point. Students first use set squares in Grade 7; there is a refresher exercise
at the beginning of this lesson (L3).
5 min
51.
Use the handout and ruler to scale the following figure (Lesson 3, Example
1) according to the Parallel Method.
Use the figure below, center O, a scale factor of r = 2 and the following steps
to create a scale drawing using the parallel method.
Step 1. Draw a ray beginning at O through each vertex of the figure.
Step 2. Select one vertex of the scale drawing to locate; we have selected A'.
Locate A' on ray OA so that OA'=2OA.
Step 3. Align the setsquare and ruler as in the image below; one leg of the
setsquare should line up with side AB and the perpendicular leg should be
flush against the ruler .
Step 4. Slide the setsquare along the ruler until the edge of the setsquare
passes through A'. Then, along the perpendicular leg of the setsquare, draw
the segment through A' that is parallel to AB until it intersects with OB; label
this point B'.
2 min
52.
Once students are familiar with the Ratio and Parallel Methods of creating
scale drawings, they establish that both methods in fact create the same
scale drawing of a given figure. Proving this yields the Triangle Side Splitter
Theorem, which in turn is used in the following lesson to establish the
Dilation Theorem. The proof for the Dilation Theorem explains why it is
that the Ratio and Parallel Methods yield scaled or enlarged/reduced
versions of a given figure.
These two theorems are used over and over again to establish properties of
dilations in Topic B lessons.
1 min
53.
Topic B explores dilations inside and out, from the definition to the explicit
proof of why the properties of dilations (such as why it is that a dilation of a
segment maps to another segment).
4 min
54.
Students prove that a dilation maps a segment to another segment (in this
case a segment, but the proof is generalized later for a line). This proof is an
example of the step by step reasoning students have to apply to reach a
conclusion.
They use the definition of a dilation to establish that the dilated segment
P’Q’ splits the sides PQ and OQ proportionally, after which they call on the
Triangle Side Splitter Theorem to establish that the lines that contain PQ
and P’Q’ are parallel. Then they have to try and establish why it is that all
the points between P and Q map to all the points between P’ and Q’.
Students are provided time to come up with their own ideas before being
provided a “model” proof, which makes use of a construction of a ray drawn
from O so that it intersects PQ at R, and consequently it intersects P’Q’ at R’ .
Since R' belongs to P'Q' by construction, and we already know that P'Q' is
parallel to PQ, then P'R' must be parallel to PR. This is an instance where the
Triangle Side Splitter Theorem is called again on to show that P’R’ splits
triangle OPR proportionally. Finally the definition of a dilation is called on a
second time to show that the arbitrary point R is sent to R’; we can use this
reasoning to account for every point between P and Q being sent to a point
between P’ and Q’. The argument is a great example of MP 1, 3, and 7.
Similar arguments are used for…
Section: Module 2: Topic C
Time: 30 minutes
In this section, you will examine similarity transformations.
Materials used include:
 Grade 10 Module 1 and 2 PPT
 Grade 10 Module 1 and 2 Facilitator Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
2 min
55.
With a thorough understanding of dilations, we then define a similarity
transformation in Topic C as the composition of a finite number of dilations
and/or rigid motions, and we say two figures are similar if there exists a
similarity transformation that maps one figure onto another. Just as the use
of rigid motions allowed us to compare figures other than rectilinear figures,
so too does the idea of a similarity transformation allow us to compare and
determine whether any two figures are similar to each other.
4 min
56.
Traditionally, when you open to the similarity chapter of a textbook, it
quickly focuses on triangles as the chief shape to study, so much so that
again, like congruence, when you ask students about what they remember
regarding congruence and similarity, they may be able to recite the
‘shortcuts’ (SSS, ASA, SAS for similarity, etc), but the understanding ends
with the abbreviations. You’ll notice that the first lessons in Topic C use
curvilinear images to drive home the point that we can compare any kind of
figure with the aid of similarity transformations.
Having said that, there is still much to be said about triangles and similarity,
and the topic addresses the criteria that identifies two figures as similar (i.e.
AA, SAS, and SSS criteria).
Step 1. The dilation will have a scale factor of r<1 since Z' is smaller than Z.
Step 2: Notice that Z' is flipped from Z1. So take a reflection of Z1 to get Z2
over a line l.
Step 3: Take a translation that takes a point of Z2to a corresponding point in
Z'. Call the new figure Z3.
GROUP
Step 4: Rotate until Z3 coincides with Z'.
5 min
57.
The AA criterion for similar traingles is covered in Lesson 15, and the SAS
and SSS criteria for similar triangles is covered in Lesson 17.
Students prove that a similarity transformation exists for two triangles with
two pairs of angles of equal measure.
Take a moment to sketch a rough outline of what this proof might look like.
Hint: Use a dilation with center A (r < 1…but you can be specific about
this)….you will need triangle congruence indicators as well.
4 min
58.
In a nutshell:
- Dilate about A with r < 1…in fact specifically so that r = DE/AB , so
that B goes to B’ and C goes to C.
- Since we have dilated B and C by the same scale factor, B’C’ is a
proportional side splitter; by the Triangle Side Splitter Theorem, we
know that B'C'||BC.
- Since B'C'||BC, then m∠AB'C'=m∠ABC because corresponding angles
of parallel lines are equal in measure.
- Then △AB'C'≅△DEF by ASA.
- Thus, a similarity transformation exists that takes △ABC to triangle
△DEF; triangle △ABC is similar to △DEF.
- Since the triangles are similar, we can confirm that the Angle-Angle
criterion between two triangles guarantees that the triangles are
similar.
Once the proof is established, students can use what they know about the
length relationships between similar triangles to solve for unknown sides,
much like problems found in current text in similarity units.
3 min
59.
The AA criterion for similar traingles is covered in Lesson 15, and the SAS
and SSS criteria for similar triangles is covered in Lesson 17.
Students prove that a similarity transformation exists for two triangles with
two pairs of angles of equal measure. In a nutshell, they will show that a
dilation of a triangle about one vertex can create a triangle congruent to the
partner triangle. Then students can call on a triangle congruence indicator
to establish that a transformation exists that will map one triangle to the
other.
Once the proof is established, students can use what they know about the
length relationships between similar triangles to solve for unknown sides,
much like problems found in current text in similarity units.
2 min
60.
2 min
61.
Take a moment to attempt this question.
-- Allow a few moments for participants to attempt the question.
4 min
62.
The SAS and SSS criteria for two triangles to be similar is covered in Lesson
17.
Take a moment to sketch a rough outline of what this proof of why the SAS
criterion is enough to determine that two triangles are similar. Hint: Use a
dilation about A.
-- Allow participants to consider for a few moments.
The proof of this theorem is simply to take any dilation with scale factor r =
A'B’/AB = A'C/'AC. This dilation maps △ABC to a triangle that is congruent to
△A'B'C' by the Side-Angle-Side Congruence Criterion.
3 min
63.
Take a moment to sketch a rough outline of what this proof of why the SSS
criterion is enough to determine that two triangles are similar. Hint: Use a
dilation about A.
-- Allow participants to consider for a few moments.
The proof of this theorem is simply to take any dilation with scale factor r =
A'B’/AB = B'C’/BC = A'C’/AC. This dilation maps △ABC to a triangle that is
congruent to △A'B'C' by the Side-Side-Side Congruence Criterion.
1 min
64.
A fitting use of similarity is seen in the ancient Greek, Eratosthenes’,
calculation of the circumference of the Earth. With an impressive use of
observation, measurement, and understanding of similarity, he calculated a
close approximation of the Earth’s circumference.
Roughly around the same time (approximately 250 BC), another ancient
Greek named Aristarchus approximated the distance from the Earth to the
moon.
Both of these calculations are presented to students in Lessons 19 and 20.
Section: Module 2: Topic D
Time: 21 minutes
In this section, you will examine the similarity relationships that
Materials used include:
arise when an altitude is drawn from the vertex of a right triangle to
 Grade 10 Module 1 and 2 PPT
the hypotenuse, how to use similarity to prove the Pythagorean
 Grade 10 Module 1 and 2 Facilitator Guide
Theorem, and how to simplify radical expressions, specifically
multiplying and dividing radical expressions and adding and
subtracting radical expressions using the Distributive Property.
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
2 min
With an understanding of what a similarity transformation is, what it means
for two figures to be similar, and an understanding of the triangle similarity
criteria, students then study similarity within the scope of right triangles.
65.
The major concepts studied are the similarity relationships that arise when
an altitude is drawn from the vertex of a right triangle to the hypotenuse,
how to use similarity to prove the Pythagorean Theorem, and how to
simplify radical expressions, specifically multiplying and dividing radical
expressions and adding and subtracting radical expressions using the
Distributive Property.
3 min
66.
Students understand that the altitude of a right triangle from the vertex of
the right angle to the hypotenuse divides the triangle into two similar right
triangles that are also similar to the original right triangle. They initially
study this in Grade, Module 3 (?) but revisit it here as a way to help prepare
for trigonometry.
How can we show that △ABC~△BDC~△ADB?
-- Allow the audience a minute to discuss among themselves?
The altitude of a right triangle splits the triangle into two right triangles,
each of which shares a common acute angle with the original triangle. By the
AA criterion, △ABC and △BDC are similar and △ABC and △ADB are similar
and since similarity is transitive we can conclude that △ABC~△BDC~△ADB.
GROUP
1 min
67.
We want to emphasize the use of the equal values of corresponding ratios to
solve for the unknown values. Eventually this focus will be shifted to how
the ratio is dependent on a given acute angle of the right triangle.
We highlight the importance of this concept by having students write out
ratios of lengths within each triangle, for each of the three triangles.
4 min
68.
Take a moment and find the appropriate ratios for each table and triangle.
-- Allow time to complete the tables.
2 min
69.
What would be one way to solve for x, using these ratios?
x/5 = 5/13; x = 25/13 or 1 12/13
Why can we use this equation to solve for x in this way? Because
corresponding ratios of sides are equal in value.
And so we are beginning to steer students towards the idea of special ratios
within right triangles and how they are a function of the acute angle within
the right triangle, regardless of the magnitude of the triangle.
2 min
70.
The use of the ratios of sides within right triangles continues in Lesson 22 as
one way of proving the Pythagorean Theorem.
Take a few moments to separately arrange and orient the three triangles
within triangle ABC.
-- Provide 2 minutes to complete this.
3 min
71.
In order to prove the Pythagorean Theorem, which ratios can be used to
isolate and express a, b, or c in terms of other variables?
-- Provide 2 minutes to complete this.
2 min
72.
We have three sets of ratios to choose from: shorter leg:hypotenuse, longer
leg:hypotenuse or shorter leg:longer leg
By using the longer leg:hypotenuse and shorter leg:hypotenuse ratios, we
have a way to re-express a2 and b2, which by substitution is c2.
2 min
73.
With this latest look at the Pythagorean Theorem, Lessons 23 and 24 focus
on working with expressions with radicals.
In grade 8 students learned the notation related to square roots and
understood that the square root symbol automatically denotes the positive
root (Grade 8 Module 7). In grade 9, students used both the positive and
negative roots of a number to find the location of the roots of a quadratic
function. We review what we know about roots learned in Grade 8, Module
7, Lesson 4, now because of the upcoming work with special triangles in this
module.
In Lesson 23, students multiply and divide expressions that contain radicals
to simplify their answers. In Lesson 24, the work with radicals continues
with adding and subtracting expressions with radicals.
Section: Module 2: Topic E
Time: 22 minutes
In this section, you will begin exploring Trigonometry by examining Materials used include:
the basic definitions of sine, cosine, and tangent, as well as how they
 Grade 10 Module 1 and 2 PPT
are applied in a variety of settings.
 Grade 10 Module 1 and 2 Facilitator Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
1 min
The final topic in Module 2 is Trigonometry. Here the basic definitions of
sine, cosine, and tangent are addressed, as well as how they are applied in a
variety of settings. Relative to changes in earlier in the module and to
Module 1, the lessons on trigonometry will appear relatively familiar.
74.
GROUP
3 min
75.
In this last lesson before defining the trig ratios sine, cosine, and tangent, we
again highlight the ratios within right triangles. Students are now equipped
with the language that describes a given side of a triangle with respect to a
given acute angle in the right triangle.
In the lesson, two groups take measurements of a given acute angle for a set
of right triangles and the side lengths of the triangles. They calculate the
adj/hyp ratio and the opp/hyp ratio. Eventually, they realize that the two
groups have two sets of similar triangles once the angle measurements of
the triangles are shared out loud. The quote above is suggested guidance for
the teacher, who is slowly guiding them to see that the ratios that are
constant in value between similar triangles is a function of the acute angle in
question.
Therefore, though the use of these ‘internal’ ratios is not new, the lesson
shines light on the fact that ratios depend on the acute angle that the sides
are labeled by and not the magnitude of the triangle. The explicit
connection is made in Lesson 26 with the formal definition of the sine,
cosine, and tangent ratios.
1 min
76.
Students have observed a similar image in L26 with numbers, and predict
the pattern there, but formalize what is actually happening in L27.
The ratios for sin α and cos β are the same, so sin α=cos β and ratios for cos α
and sin β are the same, so cos α=sin β. The sine of an angle is equal to the
cosine of its complementary angle and the cosine of an angle is equal to the
sine of its complementary angle.
2 min
77.
Students are prefaced to the values of the ratios of the special angles with a
discussion on how the values of sine and cosine change as theta changes
between 0˚ and 90˚.
As θ gets closer to 0,
- a decreases. Since sinθ=a/1, the value of sinθ is also
approaching 0.
- b increases and becomes closer to 1. Since cosθ=b/1, the value
of cosθ is approaching 1.
As θ gets closer to 90,
- a increases and becomes closer to 1. Since sinθ=a/1, the value
of sinθ is also approaching 1.
- b decreases and becomes closer to 0. Since cosθ=b/1, the value
of cosθ is approaching 1.
2 min
78.
Students are introduced to the handful of special angle measures (and
special triangles) that appear frequently in trigonometry.
Students are presented with a table of the most common angle measures (0,
30, 45, 60, 90) and then use the following triangles to actually show how
those values are found.
5 min
79.
In Lessons 28-29 students will use trig ratios to solve missing value type of
problems. Please take some time to try the two selected examples from
these lessons.
1 min
80.
-- Take responses from participants; review solution.
2 min
81.
-- Take responses from participants; review solution.
2 min
82.
Lessons 30-33 involve the application of the trig ratios, including how to use
trigonometry to calculate area and the Laws of Sines and Cosines.
-- Review key themes of M2.
3 min
83.
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?
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
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Grade 10 Module 1-Module 2 PPT
Grade 10 Module 1-Module 2 Facilitator Guides
Grade 10 Module 1 Mid-Module Assessment
Additional Suggested Resources
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How to Implement A Story of Functions
A Story of Functions Year Long Curriculum Overview
A Story of Functions CCLS Checklist
Active learning
Turn and talk