Measuring the World Mathematics for Elementary and Middle School Teachers by Susan Addington and David Dennis Table of Contents Introduction Chapter 1. Measurement without numbers 1.1. Comparing areas A. What is an attribute? What is a quantity? Greater than, less than, equal to. B. What is area? Rules for area C. Dissection 1.2. Comparing lengths A. What is length? B. Methods of comparing lengths; adding lengths 1.3. Volume A. Volume and some types of 3dimensional objects B. Adding and subtracting volumes 1.4. The size of a set A. Discrete sets B. Counting without numbers Onetoone correspondence C. Adding the sizes of sets 1.5. Weight and mass A. Weight, mass, and measuring devices 1.6. Culminating activity (and introduction to Ch. 2) A. Units of measure for length, area, volume B. Volumes of boxes
Chapter 2. Numbers and operations 2.1. Units and numbers A. Quantity with unit > number B. Other quantities and other uses of numbers C. Concrete and abstract numbers D. Standard and nonstandard units for quantities E. The number line, real numbers and integers F. Systems of units for the same quantity, converting units G. Indirect measurement 2.2. Fractions with the length model A. Fractional units and unit fractions B. Systems of fractions using the length model Binary fractions: definitions, comparing, simplifying Common fractions: definitions, comparing, simplifying Decimal fractions: definitions, comparing, simplifying C. Comparing fractions; benchmark fractions D. Rational numbers 2.3. Fractions, continued A. Other models for numbers B. Length and length manipulatives C. Area and area manipulatives D. Volume and volume manipulatives E. Discrete sets and discrete manipulatives F. Unitizing and reunitizing G. Other uses for fractions: functions and operators 2.4. Addition A. Ideas of addition B. Length models of addition Cuisenaire rods Number line with vectors Vectors in the plane and space C. An area model of addition of fractions E. Functions, tables, graphs F. Properties of addition G. Addition and calculators 2.5. Subtraction A. Ideas of subtraction Take away Missing addend Comparison, difference, change Cancelling Adding the opposite B. Length models for subtraction C. Properties (and nonproperties) of subtraction D. Applications of subtraction Measuring volumes by subtraction Measuring areas by subtraction on a grid Approximate numbers, tolerances and error in measurement E. Vectors, addition and subtraction F. Subtraction and calculators 2.6. Division A. Ideas of division Sharing; dealing out Measurement; repeated subtraction Division as a fraction Division as a ratio B. Division and units C. Dividing fractions D. More examples E. Division with 0 F. Division and calculators 2.7. Multiplication A. Ideas of multiplication Multiplication as scaling: stretching or shrinking Multiplication as change of unit Multiplication as the area of a rectangle Multiplication as repeated addition Multiplication as counting sets of sets, and the array model Multiplication as a function or operator B. Multiplication and units C. Multiplying fractions D. Properties of multiplication E. Multiplication on a calculator
Chapter 3. Algorithms for arithmetic 3.1. Systems of units and place value A. Measuring with combinations of units B. The base 10 place value system for whole numbers C. Notation systems in other cultures Roman numerals and counting boards The abacus Mayan numeration (optional) Babylonian numeration D. Fractions and place value: decimals and sexagesimals 3.2. Algorithms for addition and subtraction A. Algorithms for adding whole numbers Ingredients: Combining like units Regrouping Learning the 1digit addition facts Right to left addition Left to right addition Partial sums Add on B. Algorithms for subtracting whole numbers Right to left subtraction Left to right subtraction Partial differences Add up C. Adding and subtracting decimal fractions 3.3. Algorithms for multiplication A. Ingredients The distributive property The commutative property of multiplication The associative property of multiplication Multiplying by 10 in the base 10 system Learning the 1digit multiplication facts B. Five methods for multiplying whole numbers Area/rectangle method Grid table method Partial products method Standard U.S. method Lattice method C. Multiplying decimals 3.4. Algorithms for division A. Repeated subtraction algorithms B. Transition to long division C. Long division with decimals D. Decimal representations of common fractions 3.5. Approximate numbers A. Significant figures B. Rounding C. Arithmetic with approximate numbers Addition and subtraction Multiplication and division Square roots 3.6. Signed numbers A. Interpretations of negative numbers B. Addition and subtraction C. Multiplication and division D. Patterns and properties of operations with signed numbers 3.7. Algorithms for fraction arithmetic [Review and tying ideas together] Chapter 4. Multiplicative thinking with real numbers 4.1. Indirect measurement with multiplication and division [Introductory activities] Managing units Area and volume Ratio quantities Measuring other quantities using weight/mass The principle of the lever 4.2. The syntax of multiplication and division A. Syntax and symbols B. More ideas of division C. Division and multiplication as inverse functions D. Chains of several multiplications and divisions E. Factoring F. Division as missingfactor multiplication G. Properties of multiplication and division, order of operations H. Canceling in fractions I. The syntax of units J. Further calculator and noncalculator techniques 4.3. Ratios A. Definitions and examples of ratios Part/whole ratios Part/part ratios B. Representations of ratios Pictures Parallel number lines Tables Graphs and charts Algebraic expressions and equations C. Additive methods for working with ratios (for beginners) D. Multiplicative methods for working with ratios E. Algebraic methods for working with ratios 4.4. Percent A. Percent definitions and examples B. Representations of percent and computational methods C. Percent of, percent more than, percent less than D. Percent as comparison 4.5. Linear functions and graphs A. Cartesian graphs B. Slope of a graph C. Comparing ratios A, Interpreting graphs B. Making graphs with and without technology C. Problem solving: two functions of one variable
4.6. Arithmetic revisited A. The distributive property in arithmetic B. Mental arithmetic: shortcuts using multiplicative thinking
Chapter 5: Multiplicative thinking with whole numbers 5.1. Counting without numbers Rhythm and dance activities Art activities Venn diagrams 5.2. Primes and factoring A. Terminology B. Models of factoring C. Methods for factoring Repeated gozintas Tree diagrams Number cards D. Unique factorization E. The Primes Game F. Mental arithmetic and technology in number theory 5.3. Multiples A. Representing multiples B. The Sieve of Eratosthenes C. Common multiples of several numbers D. Multiples and prime factorizations E. Mental arithmetic and multiples F. Arithmetic and the least common multiple 5.4. Divisors A. Efficient factoring methods B. The greatest common divisor of several numbers Selfexplanatory method Prime factorization method Relation between LCM and GCD The GCD in arithmetic and algebra C. Geometry and GCDs The area model, again The Euclidean algorithm D. General facts about divisors E. Technology for divisors 5.5. Divisibility tests A. 10s and 2s B. 3s and 9s C. 11s D. 7s (optional) E. Even more efficient factoring methods 5.6. Arithmetic revisited A. Equivalent fractions B. Efficient algorithms for fraction arithmetic Chapter 6. Geometry 6.1. Introductory activity: regular polyhedra 6.2. Plane geometry without numbers A. Polygons B. Diagrams for organizing concepts Concept maps Venn diagrams C. Geometric constructions Graph paper or lined paper Paper folding (with and without cutting) D. Transformations and Symmetry E. Software for plane geometry 6.3. Angles A. Ideas of angle Rotation Opening between two rays B. Units for measuring angles C. Comparing, adding, and subtracting angles D. Angles in polygons E. Angles in navigation vs. angles in geometry F. The Logo computer language 6.4. Circles A. Definitions B. Geometric constructions with straightedge and compass C. Circumference of a circle D. Area of a circle E. Fractions and percents with circles F. Other facts about circles Figures inscribed in circles Tangents and secants Symmetry 6.5. Geometry in three dimensions A. Types of 3dimensional objects B. Hollow objects from nets C. Solid objects Building figures out of smaller solid objects Prisms by layers or extrusion Solids of revolution Software for 3D geometry D. Other topics in 3dimensional geometry Conic sections Symmetry of 3dimensional objects 6.6. Scaling: stretching and shrinking A. Examples of scaling B. Scaling without numbers Scaling with a projector Judging scaling by eye C. Scaling by changing lengths Dilations: scaling geometric objects The Picture of the Day D. Scaling by changing units 6.6. Similarity A. Similar figures B. Scale factors, area factors, and volume factors C. Similar triangles 6.7. The Pythagorean theorem A. Squares and square roots B. The Pythagorean Theorem As a statement about areas As a statement about lengths in a right triangle As a statement about the distance between points with coordinates C. Proof of the Pythagorean Theorem D. Converse of the Pythagorean Theorem 6.8. Measuring using formulas A. Formulas from dissection B. Base times height and related formulas C. Formulas and units D. Formulas and transformations Scaling: uniform scaling, nonuniform scaling Shears and Cavalieri’s principle Chapter 7. Algebraic Thinking 7.1. Four representations for thinking about functions A. Tables Functions of one variable Problem solving by guess and check Grid tables: functions of two variables B. Graphs Functions of one variable; problem solving with graphs Graphs from grid tables: functions of two variables C. Expressions and equations What is a variable? Writing expressions from guess and check tables D. Flow charts Problem solving with flow charts 7.2. Linear functions in two variables A. Grid tables B. Equations from grid tables C. Graphs from grid tables 3dimensional bar graphs Surface graphs Contour graphs D. Functions that are like the addition table E. Functions that are like the multiplication table F. Problem solving: two equations, two unknowns; bilinear functions 7.3. Inverse proportions and the 1/x function A. Ratio relations, inside out B. Graphs C. Problem solving 7.4. Exponential patterns and functions A. Place value and the metric system, revisited Multiplying and dividing by powers of 10 Scientific notation B. Exponential functions C. Percents as multipliers D. Noninteger bases: growth and decay E. Negative exponents F. Fractional exponents and interpolation G. Guess and check H. Exponential functions and technology 7.7. Power functions and root functions A. Definitions; Exponential vs. power functions B. Differences in tables and graphs C. Higher order differences D. Inverse functions Definitions, flow charts Square root algorithms 7.8. Polynomials A. Review: length and area models for numbers and operations B. Algebra tiles and symbolic algebra C. Adding, multiplying, and factoring with algebra tiles D. Negatives and subtraction with algebra tiles E. Completing the square with algebra tiles 7.9. Symbolic algebra A. Equations without numbers B. Equivalent expressions with a spreadsheet B. Equivalent expressions with algebra tiles C. Writing expressions D. Simplifying expressions E. Solving equations 7.10 Patterns and equations: recursive and closed form formulas A. Recursive and explicit descriptions for sequences and functions B. Sequences from geometry
Chapter 8. Measuring chance: probability and statistics 8.1. Measuring chance A. Events B. Sampling and proportion 8.2. Representing data A. Graphs without numbers Pictographs Bean graphs Venn diagrams B. Distributions 8.3. Decision trees and probability A. The syntax of the measurement of chance B. Chains of choices C. The laws of probability 8.4. Counting A. Counting by grouping The addition principle The multiplication principle Complements B. Counting and diagrams Organized list Venn diagram Grid table Tree C. Compensating for overcounting The addition principle and subtraction The multiplication principle and division 8.5. Combinations and permutations A. Permutations B. Combinations C. Pascal’s triangle Pascal’s triangle The binomial theorem 8.6. Making decisions based on incomplete information Expectation 8.7. Margins of error: What’s the chance your measurement is correct? Qualitative treatment of standard deviation 8.8 Qualitative treatment of correlation
Appendices A1. Review of measurement concepts Concepts and techniques from the text, collected in one place. A2. Tables with a spreadsheet program A. What do spreadsheets do? B. Entering data C. Formulas D. Formatting E. Graphs A3. Dynamic geometry software A. Geometric constructions B. Transformations C. Graphing A4. Calculators Techniques from the text, collected in one place A5. Latin and Greek roots for math terms A6. Selected answers and hints Contents of Measuring the World 

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Last updated January 11, 2010 Copyright 200910 David Dennis and Susan Addington. All rights reserved. 
