Sybilla Beckmann earned her undergraduate degree in mathematics from Brown University and her PhD in mathematics from the University of Pennsylvania. For two years, she researched and taught mathematics as the J.W. Gibbs Instructor of Mathematics at Yale University. Since then, she has been at the University of Georgia, where she was recently awarded the Josiah Meigs Distinguished Teaching Professor of Mathematics, and directs the Mathematicians Educating Future Teachers (MEFT) component of the University of Georgia math department’s VIGRE II grant.

Sybilla is actively involved in helping prospective teachers understand and appreciate the mathematics they will teach. She was a member of the writing team of NCTM’s Curriculum Focal Points for pre-kindergarten through 8^th grade mathematics. She was also a member of the Committee on Early Childhood Mathematics of the National Research Council, and co-author of its report, Mathematics Learning in Early Childhood: Path toward Excellence and Equity; she has worked on the development of several state mathematics standards; and she was a member of the Mathematics Writing Team for the Common Core State Standards Initiative. Several years ago, Dr. Beckmann taught daily at an average sixth grade math class at a local public school in order to better understand teaching mathematics at the school level.

1.1 The Counting Numbers

1.2 Decimals and Negative Numbers

1.3 Comparing Numbers in Base-Ten

1.4 Rounding Numbers

Chapter Summary

2. Fractions and Problem-Solving

2.1 Solving Problems and Explaining Solutions

2.2 Defining and Reasoning About

2.3 Equivalent Fractions

2.4 Comparing Fractions

2.5 Percent

Chapter Summary

3. Addition and Subtraction

3.1 Interpretations of Addition and Subtraction

3.2 The Commutative and Associative Properties of Addition, Mental Math, and Single-Digit Facts

3.3 Why the Common Algorithms for Adding and Subtracting Numbers in the Base-Ten System Work

3.4 Adding and Subtracting Fractions

3.5 Adding and Subtracting Negative Numbers

Chapter Summary

4. Multiplication

4.1 Interpretations of Multiplication

4.2 Why Multiplying by 10 is Special in Base-Ten

4.3 The Commutative and Associate Properties of Multiplication, Areas of Rectangles, and Volumes of Boxes

4.4 The Distributive Property

4.5 Properties of Arithmetic, Mental Math, and Single-Digit Multiplication Facts

4.6 Why Algorithms for Multiplying Whole Numbers Work

Chapter Summary

5. Multiplication of Fractions, Decimals, and Negative Numbers

5.1 Multiplying Fractions

5.2 Multiplying Decimals

5.3 Multiplying Negative Numbers

5.4 Powers and Scientific Notation

Chapter Summary

6. Division

6.1 Interpretations of Division

6.2 Division and Fractions and Division with Remainder

6.3 Why Division Algorithms Work

6.4 Fraction Division from the "How Many Groups?" Perspective

6.5 Fraction Division from the "How Many in One Group?" Perspective

6.6 Dividing Decimals

Chapter Summary

7. Ratio and Proportional Relationships

7.1 Motivating and Defining Ratio and Proportional Relationships

7.2 Solving Proportion Problems by Reasoning with Multiplication and Division

7.3 Unit Rates and the Values of a Ratio

7.4 Proportional Relationships Versus Inversely Proportional Relationships

7.5 Percent Revisited: Percent Increase and Decrease

Chapter Summary

8. Number Theory

8.1 Factors and Multiples

8.2 Even and Odd

8.3 Divisibility Tests

8.4 Prime Numbers

8.5 Greatest Common Factor and Least Common Multiple

8.6 Rational and Irrational Numbers

8.7 Looking Back at the Number Systems

Chapter Summary

9. Algebra

9.1 Numerical Expressions

9.2 Numerical Expressions with Variables

9.3 Equations for Different Purposes

9.4 Solving Equations

9.5 Solving Algebra Word Problems with Strip Diagrams and with Algebra

9.6 Sequences

9.7 Functions

9.8 Linear Functions

Chapter Summary

10. Geometry

10.1 Visualization

10.2 Angles

10.3 Angles and Phenomena in the World

10.4 Circles and Spheres

10.5 Quadrilaterals, Triangles, and Polygons

Chapter Summary

11. Measurement

11.1 Fundamentals of Measurement

11.2 Length, Area, Volume, and Dimension

11.3 Error and Precision in Measurements

11.4 Converting from One Unit of Measurement to Another

Chapter Summary

12. Area of Shapes

12.1 Areas of Rectangles Revisited

12.2 Moving and Additivity Principles About Area

12.3 Areas of Triangles

12.4 Areas of Paralellograms and Other Polygons

12.5 Shearing: Changing Shapes Without Changing Area

12.6 Areas of Circles and the Number Pi

12.7 Approximating Areas of Irregular Shapes

12.8 Contrasting and Relating the Perimeter and Area of Shapes

12.9 Using Moving and Additivity Principles to Prove the Pythagorean Theorem

Chapter Summary

13. Solid Shapes and Their Volume and Surface Area

13.1 Polyhedra and Other Solid Shapes

13.2 Patterns and Surface Area

13.3 Volumes of Solid Shapes

13.4 Volume of Submersed Objects versus Weight of Floating Objects

Chapter Summary

14. Geometry of Motion and Change

14.1 Reflections, Translations, and Rotations

14.2 Symmetry

14.3 Congruence

14.4 Constructions with Straightedge and Compass

14.5 Similarity

14.6 Areas, Volumes, and Scaling

Chapter Summary

15. Statistics

15.1 Formulating Statistical Questions, Gathering Data, and Using Samples

15.2 Displaying Data and Interpreting Data Displays

15.3 The Center of Data: Mean, Median, and Mode

15.4 Summarizing, Describing, and Comparing Data Distributions

Chapter Summary

16. Probability

16.1 Basic Principles of Probability

16.2 Counting the Number of Outcomes

16.3 Calculating Probabilities in Multi-Stage Experiments

16.4 Using Fraction Arithmetic to Calculate Probabilities

Chapter Summary

Bibliography

Index

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