A Story of Ratios – Part 2
Exemplar Module Analysis: G8-M1
Sequence of Sessions
Overarching Objectives of this May 2013 Network Team Institute
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Participants will be familiar with the structure and components of modules in the A Story of Ratios and A Story of Functions documents
giving them an early sense of comfort with how to use the curricular materials.
Participants will understand the focus of module 1 for each grade level 6-12 and the vertical coherence of its related content within the K12 curriculum, preparing them to describe the coherence of the curriculum to colleagues.
Participants will understand the precise concepts, definitions, and representations for the content in G6-M1 and G7-M1 and be prepared
to deliver the content of these modules (or train others to do the same), acknowledging their alignment with the related Progressions
document, 6-7 Ratios and Proportional Relationships.
Participants will understand the major shifts in instruction of Geometry content of Grades 8 and 10, preparing them to shift their own
instructional strategies and communicate these shifts to their colleagues.
Participants will understand the major shifts in instruction of Algebra and Precalculus content of Grades 9, 11, and 12, preparing them to
shift their own instructional strategies and communicate these shifts to their colleagues.
High-Level Purpose of this Session
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To examine the content of G8-M1 including the key definitions, concepts and representations so that participants can provide instruction
with fidelity to those definitions concepts and representations and train others to do the same.
To examine the differences between the old standards and the new standards and be able to communicate those differences with their
colleagues, preparing them to shift their own instructional strategies.
To examine the coherence between concepts of Topic A of G8-M1 and Topic B of G8-M1 and be prepared to communicate those
connections to their colleagues.
To work an assessment question from one of the modules and score it using the rubric so that participants are prepared to use the rubrics
to grade assessment questions.
Related Learning Experiences
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This NTI was preceded by NTI’s in which participants studied the PARCC Framework.
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This NTI was preceded by NTI’s in which participants studied the PARCC Model Content Framework for Grades 3-11
This Exemplar Module Analysis was preceded by a similar session on G6-M1 and G7-M1.
This Exemplar Module Analysis will be followed by similar sessions on G9-M1, G10-M1, G11-M1 and G12-M1
Key Points
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A precise definition of x^n, plays a critical role in leading students to understand the laws of integer exponents.
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Guided student discovery and classroom discussions play a critical role in students ability to develop a logical progression of
statements.
Session Outcomes
What do we want participants to be able to do as a result of
this session?
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How will we know that they are able to do this?
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Be familiar with the content of Grade 8 Module 1.
Understand the differences between the old standards and the
new standards.
Know the relationship between the concepts in Topic A,
properties of integer exponents, and Topic B, scientific notation.
Know how to use the provided rubrics to grade assessment
questions.
Participants will speak with their colleagues regarding these
important take-aways.
Session Overview
Section
Time
Session Introduction
11:20-11:23
 Review session objectives
•
11:23-12:00
 Examine Module 1
 Review and discuss scientific notations and
integer exponents
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Session PowerPoint
G8-M1 Module Overview and Assessments
12:00-12:15
 Complete a Module 1 Assessment item and score 
it with the rubric
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Session PowerPoint
G8-M1 Module Overview and Assessments
Module Analysis
Overview
Prepared Resources
Facilitator Preparation
•
Review session notes and PowerPoint
presentation
•
Review session notes and PowerPoint
presentation
Review Module Overview and Assessments
Session PowerPoint
•
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Assessment and Closure
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Review session notes and PowerPoint
presentation
Review Module Overview and Assessments
Session Roadmap 11:20-12:15
Section: Session Introduction
Time: 11:20-11:23
[3 minutes] In this section, you will…
• Review the focus of this session
Materials used include:
• Session PowerPoint
Time
Slide #/ Pic of Slide
Script/ Activity directions
GROUP
1 min.
Slide 1
Welcome and introduction.
Grade
band
6-8
2 min.
Slide 2
Here are our objectives for today:
(Click to advance first 2 bullets)
The first thing we want you to be comfortable with is Module 1 in general,
but specifically the differences between what you’ve seen/experienced
before and what is expected now. These differences will be noted
immediately in the slides that follow.
(Click to advance 3rd bullet)
The relationship between Topic A and B will be seen by looking at an
application problem from Topic B (scientific notation) and how students will
be prepared to deal with the problem with skills learned in Topic A (integer
exponents).
(Click to advance 4th bullet)
Finally, participants will actually complete an assessment item and practice
scoring it using the rubric.
Section: Module Analysis
Time: 11:23-12:00
[37 minutes] In this section, you will…
 Explore Module 1 and note differences between past practices and
new ones
 Examine the relationship between integer exponents and scientific
notation
Materials used include:
 Session PowerPoint
 G8-M1 Module Overview and Assessments
Time
Slide #/ Pic of Slide
Script/ Activity directions
4 min.
Slide 3
One quick note as we get started: the title of the module was modified
since the last release of the A Story of Ratios Curriculum Overview – it
was formerly “The Number System and Properties of Exponents”; now it
is “Integer Exponents and Scientific Notation”
Module 1 is all about learning the properties of integer exponents and
applying those rules to perform operations on numbers written in
scientific notation. The preparation students have had with respect to
exponents come from learning in grades 5 through 7.
(Read bullet points listed under Topic A, and then read through the
outcomes listed in the bullets under Topic B.)
GROUP
4 min.
Slide 4
(Read the current NY Grade 8 standard and the common core standard
as it relates to 8.EE.1.)
“Notice that the Common Core standard calls for students to know the
properties where as before students were to develop the properties.
What does it mean to know something as compared to develop?”
(Provide time 1-2 min for table discussion. Allow for participants to
contribute, then summarize with :)
Answer: Develop is defined as “cause to grow or experience” where
know, has a much deeper meaning. Know is defined as “becoming
aware through observation, inquiry, or information.” Specifically in
math, to know something means to be absolutely certain or sure about
something.
5 min.
Slide 5
Read through the questions and discuss your answers with a partner at
your table.
(Allow 2 minutes and then ask for responses from participants; consider
the following might or might not be the case for individual teachers :)
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There was never time to “develop” anything?
Laws were shown, memorized, practiced and assessed simplistically.
Note that Common Core curriculum spends half of Module 1 developing
the laws of exponents so that students truly “know” and “apply” them.
Part of knowing is being able to recreate the properties when forgotten.
4 min.
Slide 6
So, why spend so much time on the laws?
Let’s consider the problem on this slide. Take a moment or two to read
and examine the solution.
(Provide 2 mines for audience to read through the problem on the slide).
To perform operations on numbers written in scientific notation,
students must be proficient with the laws of exponents (current
standard) and fractions.
Step 1 relies on students’ knowledge ratios to write the fraction.
Step 2 requires knowledge of equivalent fractions and the product
formula for complex fractions,
Step 3 requires knowing how to divide numbers written in exponential
notation,
Steps 4 and 5 are computations.
To work with numbers written in scientific notation is a true application
of the learning that takes place prior to grade 8. We can now answer the
question, “Why do I have to learn this?”
5 min.
Slide 7
Skip if time is an issue. (5 min) Second example of work with scientific
notation. *If asked, the reason that we have to use “average distance” is
because the distances vary due to orbits being elliptical.
Ask the audience to talk at their tables to answer the question, “What
do students need to know to answer this?” Give time for audience to
read. Then state that for this problem, students must be fluent in the
first law of exponents (noted in red on the first line), multiplying by
powers of 10 (noted in blue on the second line), the distributive
property in line three, multiplying by powers of 10 again in line 5 to
complete the problem.
2 min.
Slide 8
(Read through the slide. Make clear that the foundation is built with the
aide of the Standards for Mathematical Practice. Definitions = MP4:
Attend to precision, Structure = MP7: Look for and make use of
structure, Logical Progression = MP3: Construct viable arguments and
critique the reasoning of others.)
The slides that follow demonstrate how these standards for
mathematical practice come into play in Module 1.
7 min.
Slide 9
The work begins with a precise definition of exponential that is referred
to throughout the remaining lessons on laws of exponents. (read the
definition aloud)
Students are expected to use the definition. Not just memorize it.
Talk to a partner at your table about how you would respond to this
question?
Answer: The notation is used incorrectly because, as is, the answer is
6
the negative of 15 instead of the product of 6 copies of -15. The base is
(-15), which means the base should be written clearly, through the use
of parentheses.
Talk to a different partner at your table about “What’s the purpose of
such an exercise?”
Answer: It’s twofold, first we want students to begin examining notation
and using it, and related vocabulary, properly. Second, we want students
to be able to self-correct. If they can point out others’ errors, they are
more likely to recognize their own.
4 min.
Slide 10
(Read through the general statement)
Ask tables to discuss why we restrict x in this statement.
Answer: We restrict the exponents to positive integers, for now,
because that is all we know based on the definition we have (attend to
precision, we can’t go about making statements about numbers, i.e.,
negative exponents, if we don’t know anything about them yet).
1 min.
Slide 11
“Structure is linked back to what we know about repeated addition and
the distributive property: mx + nx = (m+n) x. Consider, 5 copies of 4
added to 3 copies of 4. Altogether you have (5+3) copies of 4.”
2 min.
Slide 12
A direct consequence of knowing the first law of exponents leads us to
knowing something about division (make it clear that division is not another
law, it is a consequence of the first law).
(Pause to let audience read the slide.)
Now for a special case of (3^7)/(3^5) to show how to divide expressions.
Notice that the “rule” for division requires a direct use of what we know about
how to multiply numbers of the same base that are written in exponential
notation. This is a use again of definitions, structure, and a logical progression
of statements.
How does this problem demonstrate a need to know from the first law of
exponents? Take 1 minute to discuss with a partner at your table.
(Allow for 1 minute of discussion and then ask participant(s) to share.)
2 min.
Slide 13
“Think about the mathematical practices that apply here.” (Allow 1 min for
participants to read through this law of exponents.
State that again, “this is a direct use of definitions and structure. We can take
m
powers of powers because we have a definition for x that we can use to make
sense of what’s happening. “
2 min.
Slide 14
(Skip this slide if time is an issue.)
Structure for this law of exponents goes back to the associative property
for multiplication. Example: Add 4 copies of 3, and then 5 copies of the
sum is equal to adding 5 x 4 copies of 3. By definition we can rewrite
repeated addition as multiplication, and by repeated used of associative
and commutative properties we can write an equivalent expression.
Symbolically, (5 x 4) x 3.
3 min.
Slide 15
(Click to advance 1st bullet)
We want students to understand that math is a logical progression of
statements. Further, we want to demonstrate how those statements
come to be. To do so, we must model mathematical thinking and show
students how to explore the truth of a conjecture.
(Click to advance 2nd bullet)
Once laws of exponents are developed for positive integers, the next
step is to expand those laws to whole numbers (include zero in our
definitions). Students first wrestle with what a number raised to the
zero power should be equal to.
(Click to advance 3rd bullet)
Then they develop cases to check the first law of exponents. Take a
moment to think of those cases: What are they?
(Click to advance 4th bullet)
Cases are:
m = 0, n >0
m > 0, n = 0
m, n = 0
Finally students realize the truth of their conjecture because of the
logical progression of statements made (and the desire to maintain the
first law of exponents).
2 min.
Slide 16
(Click to advance first 2 bullets)
Now students want to expand knowledge of exponents to integer
exponents. Students begin, as they did with the zero power, by
-n
discussing possible definitions for x . Then students check the validity
of their assumption about negative exponents with the first law of
exponents.
Section: Fluency and Assessments
Time: 12:00-12:15
[15minutes] In this section, you will…
 Review use of the sprint for building fluency
 Complete 2 sample assessments and use the rubric
 Review key points.
Materials used include:
 Session PowerPoint
 G8-M1 Overview & Assessments
Time
Slide #/ Pic of Slide
Script/ Activity directions
2 min
Slide 17
Bullet 1: “It is the definition we just discussed that leads to the proof
that x^-b = (1/(x^b)) for positive number x and any integer b.”
Ask to share with a partner: Why the restrictions on x and b? How often
do you put restrictions on symbols?
Answer: We must restrict x and b, to prevent the chance of producing
complex or irrational numbers. We (teachers) frequently say that x can
represent any number, but have we really thought about that
completely?
GROUP
12 min.
or
2 min.
Slide 18
If sprints are new to the audience, then have participants actually do a
sprint. Then follow with the discussion below.
Sprints are one type of fluency exercise. Sprints take about 10 minutes
to complete. First, students answer as many as possible in a limited
time, 1 min, then the teacher provides answers while students check
their work, finally, students work to finish any skipped problems with a
partner if needed. After 5 minutes total have passed, the process begins
again with a second version of the sprint.” “Teachers can insert
incentives into fluencies if needed. Some ideas may be keeping track of
the student in each class who finished the most correctly, or having
students stand as the teacher says “5 right, 6 right, 7 right” and sit when
the number said by the teacher surpasses the number they got right
(last man standing), etc.
If sprints are familiar to the audience, ask participants to share at their
tables experiences with sprints either doing them or having students do
them. What were students’ reactions? Motivating?
7 min.
Slide 19
Let’s do a ‘Then and Now’ comparison of an assessment item. The
‘Then’ in this slide is a sample NY state exam test item. The ‘Now’ is one
of the common core assessment items.
Compare the knowledge required to answer the “then” question with
what’s required to answer the “now” question.
(Allow for 1 minute of reflection.)
The “then” question requires minimum depth of knowledge… nothing
more than the memorization of a rule.
8 min.
Slide 20
The “now” question requires maximum depth of knowledge... students
must apply several rules with this one item.
So, we’ve seen what G8-M1 is all about. Let’s summarize some of the
key take away’s from our review of this module.
Would anyone like to share their observations before we go through the
list that I prepared?
(Allow for participants to contribute.)
Thank you all for sharing. Here are the ones I made note of:
Content-wise, it will be made clear how the use of definitions and
development of logical arguments lead to an understanding of the laws
of integer exponents.
It is important to support students development of a logical progression
of statements through classroom discussion.
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
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PPT
Facilitators Guide
G8-M1 Module Overview and Assessments