1. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Grade 8-Module 1 Integer Exponents and Scientific Notation

2. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios

3. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM 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.

4. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Participant Poll Classroom teacher School leader Principal District leader BOCES representative

5. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Agenda Module Overview Quick Look Model Exploratory Lesson Expert Lesson Group Work Expert Lesson Presentations Gallery Walk Summary of Work & Closure

6. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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

7. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Module Overview Topic A and Lesson Titles Topic B and Lesson Titles Foundational Standards Mathematical Practice Standards New and Familiar Terms

8. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.)

9. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

10. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Concept Development Let 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)

11. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios And if we assume x > 0 in equation (4) and y > 0 in equation (5) below, the we also have: (4) (5)

12. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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?

13. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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:

14. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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

15. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

16. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

17. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

18. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

19. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

20. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

21. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

22. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Biggest Takeaway What is the biggest takeaway from today? Think for just one minute. “Whip Around” the table to share.

23. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

24. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Agenda Take the Mid-Module Assessment Table Discussion Rubric Scoring with Student Exemplars Summary & Closure

25. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Mid-Module Assessment Take the Mid-Module Assessment 20 minutes No talking, group work, etc.

26. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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

27. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 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.

28. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Summary and Closure What did you think about the scoring process in general? Final comments.