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?
How will we know that they are able to do this?
· 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.
Participants will be able to articulate the key points listed
above.
Session Overview
Section
Time
Overview
Prepared Resources
Facilitator Preparation
Introduction
38 min
Conduct an overview of module structure, lesson types, and
lesson components.
· 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: Topics A and B
26 min
Examine constructions and unknown angles to develop the basic
language of geometry.
· Grade 10 Module 1 and 2 PPT
· Grade 10 Module 1 and 2 Facilitator Guide
Review Grade 10 Module Topic A and Topic B
Grade 10 Module 1: Topic C
20 min
Explore congruence in terms of transformations.
· 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: Topic D
24 min
Explore congruence and rigid motions.
· Grade 10 Module 1 and 2 PPT
· Grade 10 Module 1 and 2 Facilitator Guide
Review Grade 10 Module Topic D
Grade 10 Module 2: Topic A and Topic B
26 min
Explore scale drawings and dilations.
· 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 2: Topic C
30 min
Examine similarity transformations.
· 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 2: Topic D
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.
· 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 2: Topic E
Exmaine the basic definitions of sine, cosine, and tangent, as
well as how they are applied in a variety of settings.
· Grade 10 Module 1 and 2 PPT
· Grade 10 Module 1 and 2 Facilitator Guide
Review Grade 10 Module 2 Topic 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
Script/ Activity directions
GROUP
3 min
1.
2 min
2.
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
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 to develop the basic language of geometry.
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
13.
Module 1 is a 45 day module, in fact it is one of two 45-day
modules (Module 2, Similarity is the other).
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.
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.
A circle with center C and radius AB is written as CircleC,
radius CB.
2 min
17.
-- 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
GROUP
3 min
36.
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).
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 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 classwith an offer: "We have to cover several chapters from the
textbook andthere are approximately forty formulas. I may offer you
a deal: you will learnjust four formulas and I will teach you how
to get the rest out of theseformulas." 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
similarity, proof and trigonometry by exploring scale drawings and
dilations.
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
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.
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
GROUP
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.
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.
Take a moment to attempt this question.
-- Allow a few moments for participants to attempt the
question.
2 min
61.
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 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.
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
65.
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.
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.
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 the basic definitions of sine, cosine, and tangent, as
well as how they are applied in a variety of settings.
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
1 min
74.
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.
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
Active learning
Turn and talk
Turnkey Materials Provided
· 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
· How to Implement A Story of Functions
· A Story of Functions Year Long Curriculum Overview
· A Story of Functions CCLS Checklist