CHAPTER 23 Developing Concepts of Exponents, Integers, and Real Numbers Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville

23-2 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Big Ideas 1.Our number system includes whole numbers, fractions, decimals and integers, all of which are rational numbers. 2.Integers are the negative and positive counting numbers and 0. 3.Exponential notation is a way to express repeated products of the same number. 4.Many numbers are not rational; the irrationals can only be expressed symbolically or approximately using a close rational number.

23-3 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Exponents Exponential notation is more efficient for conveying numeric or quantitative information. Exponent indicated the number of times the base is used as a factor.

23-4 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Order of operations Order of operations is not just a convention. Based on the meaning of the operations. Mnemonics- PEMDAS is sometimes used to help remember the order.

23-5 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Apply order of operations Try this one Activity 23.2 Guess My Number

23-6 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Integer Exponents Explore power of 10 changes Negative exponent is reciprocal of the value it would be without negative sign. Explore negative exponents with a calculator. Use fraction-to-decimal conversion feature.

23-7 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Context for Very Large Numbers State lottery- 44 numbers from which to pick 6 (7 million possible combinations) Estimated size of the universe of 40 billion light- years Human body has 100 billion cells Distance to the sun is about 150 million kilometers Population in the world in 2011 was about 6.96 billion Connect large numbers to meaningful points of reference.

23-8 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Context for Very Small Numbers Length of DNA strand in a cell Human hair growth rate Chance of winning the state lottery Mass of one atom of hydrogen Sound to travel the length of a football field Finding real data that are very, very, small can build meaning.

23-9 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Contexts for Exploring Positive and Negative Numbers Quantity contexts Golf scores Money: Payments and Deposits Linear Contexts Temperature Altitude Timelines Football

23-10 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Models for Teaching Positive and Negative Numbers Two color counters Number Lines Integers involve two concepts- magnitude (counters or length of arrows) and direction (different colors or directions)

23-11 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Addition with Integers

23-12 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Subtraction with Integers

23-13 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Connecting Visual Representations with Equations Notate number line illustrations

23-14 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Multiplication with Integers

23-15 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Division with Integers

23-16 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Real Numbers Rational numbers are all numbers that can be represented as a fraction (irrational numbers cannot be represented as a fraction) Integers are whole numbers and their negatives or opposites

23-17 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Try this one Activity Repeater or Terminator?

23-18 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Student example:

23-19 Van de Walle, Karp, Bay-Williams, Elementary and Middle School Mathematics: Teaching Developmentally, 9/e, © 2016, 2013, 2010, 2007, 2004 by Pearson Education, Inc. All Rights Reserved. Roots and Cubes Try this one Activity Edges of Squares and Cubes