A Story of Ratios

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NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios

A Story of RatiosGrade 8-Module 1

Integer Exponents and Scientific Notation


N YS C O M M O N CO R E M AT H E M AT I C S C U R R I C U LU M A Story of Ratios



Objectives

• Articulate and model the instructional approaches to teaching the content of the first half of the lessons.

• Examine how the topics and lessons promote mastery of the focus standards and address the major work of the grade.

• Articulate connections from the content of previous grade levels to the content of this module.



Participant Poll

• Classroom teacher• School leader• Principal• District leader• BOCES representative



Agenda• Module Overview Quick Look• Model Exploratory Lesson• Expert Lesson Group Work• Expert Lesson Presentations• Gallery Walk• Summary of Work & Closure



Icebreaker!• Each table needs a poster paper and no more than two markers. • A vocabulary word/phrase will be given. Once you see/hear it, write down

as many words as possible related to the vocabulary word. • The table with the most words wins! (PRIZE!!)• You have 2 minutes to work. • Anyone can write, but only with the two markers provided. • Ready, Set, Go!

LAWS OF EXPONENTS



Module Overview• Topic A and Lesson Titles• Topic B and Lesson Titles• Foundational Standards• Mathematical Practice Standards• New and Familiar Terms



Lesson Types• Problem Set• Teacher and/or students work through a series of examples.

• Exploratory • Students are presented exploratory challenge(s) in the form of activities and/or

exercises. Exploratory challenges comprise the majority of the lesson.• Socratic• Teacher engages students in a discussion leading to a big idea or proof.

• Modeling• Application problem ill/well defined task that students complete. Real world

application of mathematics. (Reserved mainly for high school, but there will be at least 3 modeling tasks throughout the grade 8 curriculum.)



Model Exploratory Lesson• Lesson 4: Numbers Raised to the Zeroth Power• Student Outcomes• Students will know that a number raised to the zeroth power is equal to one. • Students will recognize the need for the definition to preserve the properties of

exponents.



Concept DevelopmentLet us summarize our main conclusions about exponents. For any numbers x

and y, and any positive integers m and n, the following holds:

(1)

(2)

(3)



And if we assume x > 0 in equation (4) and y > 0 in equation (5) below, the we also have:

(4)

(5)



Our goal is to extend these existing properties of exponents for positive integers to all whole numbers. That is, we need to know that these properties still hold when m and/or n is zero.

What should something like be equal to?



Keeping our goal in mind (preserve the existing properties of exponents) let’s see what happens using equation (1) with

(1)

We will let x = 3, and m = 0 (and recall n > 0). Then:



Keeping our goal in mind (preserve the existing properties of exponents) let’s see what happens using equation (1) with

(1)

We will let x = 3, and m = 0 (and recall n > 0). Then:

but



Since and what could it be?

Definition. For any positive number x, we define

Now that we have a definition for let’s check to see if it works with our properties of exponents. Namely equations (1)-(3).

Complete exercise 1.



Exercise 1: List all the possible cases of whole numbers m and n for identity (1). More precisely, when m > 0 and n > 0, we already know that (1) is correct. What are the other possible cases of m and n for which (1) is yet to be verified?

Case A:

Case B:

Case C:

Now that we know what we need to check. Let’s begin that process.



Exercise 2: Check that equation (1) is correct for each of the cases listed in exercise 1.

(1)

Case A:

Yes! It’s true.

Case B:

Yes! It’s true.

Case C:

Yes! It’s true.



Expert Lesson Group Work• Read through the lesson at least once• Do all Exercises• Do the Exit Ticket• Do the Problem Set• Become an Expert! You and your group will be presenting a 10 minute

mini-lesson to the whole group. YOU select which discussion(s), examples, exercises to present.

• Complete the task in 35 min.



Expert Lesson Presentations (audience role)• Take 3-column notes!• First column: Note teaching strategies/concepts that are new to you.• Second column: Parts of lesson that you know will be successful• Third column: Parts of lesson that concern you (may be problematic for students),

questions.• Leave 4-5 lines at bottom of page/section for each lesson.

• You must write at least one thing in each column for each lesson.• Be respectful to the team that is presenting.



Gallery Walk• Share your notes with the group you are walking with.• Record generalizations (at least one in each column) on the poster paper.• You have 3 minutes at each poster to complete the task.



Gallery Walk-Solutions• Take the poster for your lesson.• Read through all three columns.• Focus on the third column, brainstorm solutions/fixes/supplemental work

to overcome obstacles identified and answers to questions.• Be prepared to present solutions to the whole group. (approximately 2-3

minutes per lesson)• You have 15 minutes to complete the task.



Biggest Takeaway• What is the biggest takeaway from today? Think for just one minute.• “Whip Around” the table to share.



Key Points • The Laws of Exponents are dependent upon a clear definition of

exponential notation.• The “rule” for dividing exponential expressions is a consequence of the first.

• There exists a logical sequence that illustrates the coherence of all concepts in Topic A. • Begin with positive integer exponents, move to whole numbers, then extend to

integer exponents.• Definitions, definitions, definitions.



Agenda• Take the Mid-Module Assessment• Table Discussion• Rubric Scoring with Student Exemplars• Summary & Closure



Mid-Module Assessment• Take the Mid-Module Assessment• 20 minutes• No talking, group work, etc.



Mid-Module Assessment Discussion• Table Discussion• Predict the errors that students will make• Identify vocabulary or context that students may struggle with• Discuss strategies to overcome these issues that will support student success



Mid-Module Assessment Scoring• Rubric Scoring• Each table has been provided a set of student exemplars• Use the rubric to score the assessment• After you have scored at least two assessments, compare the scores you gave with

someone else. Discuss any discrepancies.



Summary and Closure• What did you think about the scoring process in general?• Final comments.

A Story of Ratios

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