CHAPTER 23 Developing Concepts of Exponents,

Integers, and Real Numbers

Elementary and Middle School Mathematics Teaching Developmentally

Ninth EditionVan de Walle, Karp and Bay-Williams

Developed by E. Todd Brown /Professor Emeritus University of Louisville

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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.

23-3Van 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.

ExponentsExponential notation is more efficient for conveying numeric

or quantitative information.

Exponent indicated the

number of times the base is

used as a factor.

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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.

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Apply order of operationsTry this oneActivity 23.2 Guess My Number

23-6Van 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 ExponentsExplore 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.

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Context for Very Large NumbersState 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.

23-8Van 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.

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Contexts for Exploring Positive and Negative Numbers

Quantity contexts

Golf scores

Money: Payments and Deposits

Linear Contexts

Temperature

Altitude

Timelines

Football

23-10Van 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)

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Addition with Integers

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Subtraction with Integers

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Connecting Visual Representations with Equations

Notate number line illustrations

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Multiplication with Integers

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Division with Integers

23-16Van 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

23-17Van 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 oneActivity 23.14 Repeater or Terminator?

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Student example:

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Roots and Cubes Try this one

Activity 23.15 Edges of Squares and Cubes