© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM Exemplar Module Analysis Grade 8 – Module 1

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Participant Outcomes: 2 Become familiar with content related to 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. Review an assessment item and participate in scoring the item through the use of a rubric.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Integer Exponents and Scientific Notation 3 Topic A: Exponential Notation & Properties of Integer Exponents Know and apply properties of integer exponents to generate equivalent expressions 5.NBT.2: Use whole number exponents to denote positive powers of 10 6.EE.1: Use exponents with bases other than 10 7.G.4 & 7.G.6: Use exponents with area and volume Topic B: Magnitude & Scientific Notation Estimate very large and very small quantities expressed in the form of a single digit times an integer power of ten Perform operations with numbers written in scientific notation Scientific notation relates to place value learned in K-5.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios What’s new? 4 Current NY Grade 8 Standard: Develop and apply the laws of exponents for multiplication and division Common Core: Know and apply properties of integer exponents to generate equivalent expressions  What does it mean to know something as compared to develop?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Table Talk 5 Current NY Grade 8 Standard: Develop and apply the laws of exponents for multiplication and division Common Core: Know and apply properties of integer exponents to generate equivalent expressions  How many days are typically spent on these topics?  How are concepts developed?  How well do students retain information, what do they struggle with? In other words, what do students really know?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Why spend so much time on the laws? 6 A strong foundation in properties of exponents reduces the work with scientific notation to computational exercises. California’s geographic area is 163,696 sq mi, and the geographic area of the U.S. is 3,794,101 sq mi. Let us round off these figures to and Estimate the ratio of the area of California to the area of the U.S.A. The ratio of the area of California to the area of the U.S.A. is 1:23.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Another example: 7 The average distance from Earth to the moon is about km, and in year 2014, the distance from Earth to Mars is approximately km. On this simplistic level, how many more kilometers is it to travel from Earth to Mars than travel from Earth to the moon?  What do students need to know to answer this?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios How is the strong foundation in properties of exponents established? 8 By attending to precision with respect to definitions (MP 6), By looking for and making use of structure (MP 7), And by constructing logical arguments to build a progression of statements to explore the truth of conjectures (MP 3).

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Precise definition of exponential notation: 9 In general, for any number x and any positive integer n, x n is by definition, EX: Josie says that Is she correct? How do you know?  What’s the purpose of such an exercise?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Apply definition to write equivalent expressions: 10 In general, if x is any number and m, n are positive integers, then  Why do we restrict m and n to positive integers?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Apply definition to write equivalent expressions: 11 By analogy, this law relates to what we know about repeated addition and the distributive property:

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Apply definition to write equivalent expressions: 12

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Apply definition to develop another law: 13 In general, if x is any number and m, n are positive integers, then

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Apply definition to develop another rule: 14 Suppose we add 4 copies of 3, thereby getting ( ), and then add 5 copies of the sum. We get:

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Do you “teach” or do you “tell”? x 0 = ? 15 Build logical progression of statements to explore truth of a conjecture. Make a conjecture: A Socratic discussion leads to the need to define as 1 (for positive number x). Verify the conjecture: Students check to see if the first law of exponents, is still valid for whole numbers m and n (rather than just positive exponents in previous lessons). Students develop cases for m and n explore the truth of the conjecture (MP 3) made about is accurate.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Exploring conjectures and extending definitions: 16 Make a conjecture: A new discussion leads to the need to define as for nonzero number x and positive integer n. Verify the conjecture: Again, students see, by exploring a number of cases, that this definition is consistent with for all integers m and n, and therefore a valid extension to the original definition.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Precision means we must be careful to quantify the variable: 17 This work leads students to expanding the existing definition to positive number x and any integer b in Why the restrictions on x and b? How often do you put restrictions on symbols?

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

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

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Key Points: 20 Importance of precise definition of, leading to an understanding of the laws of integer exponents. Support students, through discussion, in developing logical progression of statements.