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The goal of this text is to provide prospective elementary teachers with a deep understanding of the mathematics they will be called on to teach. Through a careful, mathematically precise development of concepts, this text asks that students go beyond simply knowing how to carry out mathematical procedures. Students must also be able to explain why mathematics works the way it does. Being able to explain why is a vital skill for teachers. Through activities, examples and applications, the author expects students to write and solve problems, make sense of the mathematics, and write clear, logical explanations of the mathematical concepts. The accompanying Activities Manual promotes engagement, exploration, and discussion of the material, rather than passive absorption. Both students and instructors should find this material fun, interesting, and rewarding.
1. Problem Solving.
Solving Problems.
Explaining Solutions.
2. Numbers and the Decimal System.
Introduction to the Number Systems.
The Decimal System and Place Value.
Representing Decimal Numbers.
Comparing Sizes of Decimal Numbers.
Rounding Decimal Numbers.
3. Fractions.
The Meaning of Fractions.
Equivalent Fractions.
Fractions as Numbers.
Comparing Sizes of Fractions.
Percent.
4. Addition and Subtraction.
Interpretations of Addition and Subtraction.
The Algorithms for Adding and Subtracting.
Adding and Subtracting Fractions.
When Do We Add Percentages?
Percent Increase and Percent Decrease.
The Commutative and Associative Properties and Mental Math.
5. Multiplication.
The Meaning of Multiplication.
Why Multiplying Decimal Numbers by 10 Is Easy.
Multiplication and Areas of Rectangles.
The Commutative Property of Multiplication.
Multiplication and Volumes of Boxes.
The Associative Property of Multiplication.
The Distributive Property.
Mental Math, Properties of Arithmetic, and Algebra.
Why the Procedure for Multiplying Works.
6. Multiplication of Fractions, Decimals, and Negative Numbers.
Multiplying Fractions.
Powers.
Multiplying Decimals.
Multiplying Negative Numbers.
Scientific Notation.
7. Division.
The Meaning of Division.
Understanding Long Division.
Fractions and Division.
Dividing Fractions.
Dividing Decimals.
Ratio and Proportion.
8. Geometry.
Visualization.
Angles.
Circles and Spheres.
Triangles.
Quadrilaterals and Other Polygons.
Constructions With Straightedge and Compass.
Polyhedra and Other Solid Shapes.
9. Geometry of Motion and Change.
Reflections, Translations, and Rotations.
Symmetry.
Congruence.
Similarity.
10. Measurement.
The Concept of Measurement.
Error and Accuracy in Measurements.
Length, Area, Volume, and Dimension.
Calculating Perimeter, Area, and Volume.
Comparing Sizes of Objects.
Converting From one Unit of Measurement to Another.
11. More About Area and Volume.
The Moving and Combining Principles About Area.
The Pythagorean Theorem.
Approximating Areas of Irregular Shapes.
Cavalieri's Principle About Shearing and Area.
Areas of Triangles.
Areas of Parallelograms.
Areas of Circles and the Number Pi.
Relating the Perimeter and Area of a Shape.
Principles for Determining Volumes.
Volumes of Solid Shapes.
Areas, Volumes, and Scaling.
12. Number Theory.
Factors and Multiples.
Greatest Common Factor and Least Common Multiple.
Prime Numbers.
Even and Odd.
Divisibility Tests.
Rational and Irrational Numbers.
13. Functions and Algebra.
Mathematical Expressions, Formulas, and Equations.
Solving Equations With Pictures and With Algebra.
Sequences.
Series.
Functions.
Linear Functions.
14. Statistics.
Designing Investigations and Gathering Data.
Displaying Data and Interpreting Data Displays.
The Center of Data: Mean, Median and Mode.
The Spread of Data: Percentiles.
15. Probability.
Basic Principles and Calculation Methods of Probability.
Calculating Probabilities by considering the Ideal Outcome.
The goal of this text is to provide prospective elementary teachers with a deep understanding of the mathematics they will be called on to teach. Through a careful, mathematically precise development of concepts, this text asks that students go beyond simply knowing how to carry out mathematical procedures. Students must also be able to explain why mathematics works the way it does. Being able to explain why is a vital skill for teachers. Through activities, examples and applications, the author expects students to write and solve problems, make sense of the mathematics, and write clear, logical explanations of the mathematical concepts. The accompanying Activities Manual promotes engagement, exploration, and discussion of the material, rather than passive absorption. Both students and instructors should find this material fun, interesting, and rewarding.
Table of Contents
1. Problem Solving.
Solving Problems.
Explaining Solutions.
2. Numbers and the Decimal System.
Introduction to the Number Systems.
The Decimal System and Place Value.
Representing Decimal Numbers.
Comparing Sizes of Decimal Numbers.
Rounding Decimal Numbers.
3. Fractions.
The Meaning of Fractions.
Equivalent Fractions.
Fractions as Numbers.
Comparing Sizes of Fractions.
Percent.
4. Addition and Subtraction.
Interpretations of Addition and Subtraction.
The Algorithms for Adding and Subtracting.
Adding and Subtracting Fractions.
When Do We Add Percentages?
Percent Increase and Percent Decrease.
The Commutative and Associative Properties and Mental Math.
5. Multiplication.
The Meaning of Multiplication.
Why Multiplying Decimal Numbers by 10 Is Easy.
Multiplication and Areas of Rectangles.
The Commutative Property of Multiplication.
Multiplication and Volumes of Boxes.
The Associative Property of Multiplication.
The Distributive Property.
Mental Math, Properties of Arithmetic, and Algebra.
Why the Procedure for Multiplying Works.
6. Multiplication of Fractions, Decimals, and Negative Numbers.
Multiplying Fractions.
Powers.
Multiplying Decimals.
Multiplying Negative Numbers.
Scientific Notation.
7. Division.
The Meaning of Division.
Understanding Long Division.
Fractions and Division.
Dividing Fractions.
Dividing Decimals.
Ratio and Proportion.
8. Geometry.
Visualization.
Angles.
Circles and Spheres.
Triangles.
Quadrilaterals and Other Polygons.
Constructions With Straightedge and Compass.
Polyhedra and Other Solid Shapes.
9. Geometry of Motion and Change.
Reflections, Translations, and Rotations.
Symmetry.
Congruence.
Similarity.
10. Measurement.
The Concept of Measurement.
Error and Accuracy in Measurements.
Length, Area, Volume, and Dimension.
Calculating Perimeter, Area, and Volume.
Comparing Sizes of Objects.
Converting From one Unit of Measurement to Another.
11. More About Area and Volume.
The Moving and Combining Principles About Area.
The Pythagorean Theorem.
Approximating Areas of Irregular Shapes.
Cavalieri's Principle About Shearing and Area.
Areas of Triangles.
Areas of Parallelograms.
Areas of Circles and the Number Pi.
Relating the Perimeter and Area of a Shape.
Principles for Determining Volumes.
Volumes of Solid Shapes.
Areas, Volumes, and Scaling.
12. Number Theory.
Factors and Multiples.
Greatest Common Factor and Least Common Multiple.
Prime Numbers.
Even and Odd.
Divisibility Tests.
Rational and Irrational Numbers.
13. Functions and Algebra.
Mathematical Expressions, Formulas, and Equations.
Solving Equations With Pictures and With Algebra.
Sequences.
Series.
Functions.
Linear Functions.
14. Statistics.
Designing Investigations and Gathering Data.
Displaying Data and Interpreting Data Displays.
The Center of Data: Mean, Median and Mode.
The Spread of Data: Percentiles.
15. Probability.
Basic Principles and Calculation Methods of Probability.
Calculating Probabilities by considering the Ideal Outcome.