A Story of Ratios – Part 2 Exemplar Module Analysis: G8-M1 Sequence of Sessions Overarching Objectives of this May 2013 Network Team Institute 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 K-12 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 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.
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A Story of Ratios – Part 2
Exemplar Module Analysis: G8-M1
Sequence of SessionsOverarching Objectives of this May 2013 Network Team Institute
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 K-12 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 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 This NTI was preceded by NTI’s in which participants studied the PARCC Framework. 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 A precise definition of x^n, plays a critical role in leading students to understand the laws of integer exponents.
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?
How will we know that they are able to do this?
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 Overview Prepared Resources Facilitator Preparation
Session Introduction 11:20-11:23 Review session objectives• Session PowerPoint
• Review session notes and PowerPoint presentation
Module Analysis 11:23-12:00 Examine Module 1 Review and discuss scientific notations and
integer exponents
Session PowerPoint G8-M1 Module Overview and Assessments
• Review session notes and PowerPoint presentation
• Review Module Overview and Assessments
Assessment and Closure 12:00-12:15 Complete a Module 1 Assessment item and score it with the rubric
Session PowerPoint G8-M1 Module Overview and Assessments
Review session notes and PowerPoint presentation
Review Module Overview and Assessments
Session Roadmap 11:20-12:15Section: 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 band6-8
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 GROUP
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.)
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 :)
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 the negative of 156 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
powers of powers because we have a definition for xm 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 >0m > 0, n = 0m, n = 0Finally 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 discussing possible definitions for x-n. 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 GROUP
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?
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. The “now” question requires maximum depth of knowledge... students must apply several rules with this one item.
8 min. Slide 20 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 PPT Facilitators Guide G8-M1 Module Overview and Assessments
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