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Measuring the World

Mathematics for Elementary and Middle School Teachers

by Susan Addington and David Dennis

Table of Contents


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 3-dimensional objects

B. Adding and subtracting volumes

1.4. The size of a set

A. Discrete sets

B. Counting without numbers

One-to-one 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


Adding the opposite

B. Length models for subtraction

C. Properties (and non-properties) 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


Combining like units


Learning the 1-digit 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 1-digit 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 missing-factor multiplication

G. Properties of multiplication and division, order of operations

H. Canceling in fractions

I. The syntax of units

J. Further calculator and non-calculator techniques

4.3. Ratios

A. Definitions and examples of ratios

Part/whole ratios

Part/part ratios

B. Representations of ratios


Parallel number lines


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

Self-explanatory 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


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


6.5. Geometry in three dimensions

A. Types of 3-dimensional 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 3-dimensional geometry

Conic sections

Symmetry of 3-dimensional 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, non-uniform 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

3-dimensional 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. Non-integer 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


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


B. Counting and diagrams

Organized list

Venn diagram

Grid table


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


8.7. Margins of error: What’s the chance your measurement is correct?

Qualitative treatment of standard deviation

8.8 Qualitative treatment of correlation



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

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Last updated January 11, 2010

Copyright 2009-10 David Dennis and Susan Addington. All rights reserved.