Mathematics

We study a LOT of Math / Maths here at the Sculley Academy! Could it be because it was Paul’s and Karen’s favorite subject in school? Or because they we love to play with numbers? Or because we took God seriously when he said to “go forth and multiply?!” In any case, we all love math here!

Course of Study (tailored to fit each child)

Manipulatives

Miquon Math

QuarterMile Math

“Key to . . .” series

Harold Jacobs’ Elementary Algebra

Harold Jacobs’ Geometry

Paul Foerster’s Algebra & Trigonometry

Calculus: Concepts & Applications

Harold Jacobs’ Mathematics: A Human Endeavor

Other Math Stuff

AP Statistics

We’ll also throw in some calculus, AP courses, etc., as we figure this thing out!

Also, once they hit 4th grade level, they join our very own math club!

Check these out, too:

Other math websites

Math contest sites

Problem of the week sites

Manipulatives:

We have lots of math manipulatives which we encourage use of from early ages–colored counters, pattern blocks, tangrams, geoboards, math puzzles, games, linking cubes, attribute blocks, parquetry blocks, etc. Any time a school supply store has a sale, I’m there!

My opinion of Kindergarten math programs–don’t waste your money! Just my opinion, of course 🙂

Miquon Math:

We begin formal math study with Miquon Math. The curriculum makes use of Cuisenaire rods, which are a very fun manipulative to help with all kinds of math. The six books in this uncluttered series take children from 1st to 3rd grade math skills.

Orange Book [counting, addition, subtraction, multiplication, fractions, equalities and inequalities, number lines and functions, geometric recognition, length, area, and volume, clock arithmetic]
Red Book [odd-even, addition, subtraction, multiplication, fractions, division, equalities and inequalities, factoring]
Blue Book [odd-even, addition, subtraction, multiplication, fractions, division, geometric recognition, grid and arrow games]
Green Book [addition, subtraction, multiplication, division, equalities and inequalities, place value, number lines and functions, factoring, squaring, length, area, and volume, series and progressions, mapping, clock arithmetic]
Yellow Book [addition, subtraction, multiplication, fractions, division, equalities and inequalities, place value, number lines and functions, factoring, simultaneous equations, geometric recognition, length, area, and volume, series and progressions, grid and arrow games, clock arithmetic]
Purple Book [addition, subtraction, multiplication, fractions, division, equalities and inequalities, place value, squaring, graphing equations, grid and arrow games, clock arithmetic, sets, word problems]
QuarterMile Math:

We all enjoy playing Quarter Mile Math on the computer–it’s the most appealing way to do math drills I’ve ever seen! You can choose either horse racing or car racing, and the program covers concepts from K – 9th grade. I’m planning on using this for many years to come. The neatest thing about the program is that you compete against yourself only (it stores your best 5 times for each race), so it doesn’t matter how you do compared to siblings or parents or other kids who have played (though you can check your overall ranking if you really want to)!

Here’s a summary of skills covered:

Introduction Rounding Divisibility, prime numbers
Keyboarding Squares and cubes GCF, LCM, and LCD
Alphabet Powers of 10 Reducing, converting
Whole Numbers Mean, mode, median Fractions
Counting Base 2 and base 16 Computing with fractions
Numbers, number size Math Strategies Decimals
Addition Multiplication and division strategies Naming and rounding decimals
Subtraction Wild strategies Decimal computation
Multiplication Estimation Percents
Division Estimating with whole numbers Converting
Place value Adding multiple numbers Computing with percents
Grouping / regrouping Estimating with fractions Pre-algebra
Doubling / halving Estimating with percents Integer introduction
Twins and neighbors Special percents Integer computing
Combination facts Estimating with decimals Equation introduction
Important 10s facts Estimating with money Equations with whole numbers
Missing number Morse code Equations with integers
Missing operator(s) Fraction Introduction
“Key to . . .” series [Decimals, Fractions, Percents, Measurement, Metric Measurement]:

We started this series in 2002-03, after having used a variety of other curricula for this level of study (poor old P is our experiment again!). I love the “Key to . . .” series, and it is put out by the same folks who publish Miquon Math. This is the course of study we’re doing:

Key to Decimals
Decimal Concepts
Adding, Subtracting, and Multiplying
Dividing
Using Decimals
Key to Fractions
Fraction Concepts
Multiplying and Dividing
Adding and Subtracting
Mixed Numbers
Key to Percents
Percent Concepts
Percents and Fractions
Percents and Decimals
Key to Measurement (optional)
English Units of Length
Measuring Length and Perimeter using English Units
Finding Area and Volume using English Units
English Units for Weight, Capacity, Temperature, and Time
Key to Metric Measurement (optional)
Metric Units of Length
Measuring Length and Perimeter using Metric Units
Finding Area and Volume using Metric Units
Metric Units for Mass, Capacity, Temperature, and Time
Key to Geometry (optional)
Lines and Segments
Circles
Constructions
Perpendiculars
Squares and Rectangles
Angles
Perpendiculars and Parallels, Chords and Tangents, Circles
Triangles, Parallel Lines, Similar Polygons
Harold Jacobs’ Elementary Algebra:

This is a well-written, highly-recommended text from Harold Jacobs. Chapter titles are:

Fundamental Operations Simultaneous Equations Square Roots
Addition Simultaneous Equations Squares and Square Roots
Subtraction Solving by Subtraction Square Roots of Products
Multiplication More on Solving by Addition and Subtraction Square Roots of Quotients
Division Graphing Simultaneous Equations Adding and Subtracting Square Roots
Raising to a Power Inconsistent and Equivalent Equations Multiplying Square Roots
Zero and One Solving by Substitution Dividing Square Roots
Several Operations Mixture Problems Radical Equations
Parentheses Exponents Quadratic Equations
The Distributive Rule Large Numbers Polynomial Equations
Functions and Graphs A Fundamental Property of Exponents Polynomial Functions
An Introduction to Functions Two More Properties of Exponents Solving Polynomial Equations by Graphing
The Coordinate Graph Zero and Negative Exponents Solving Quadratic Equations by Factoring
More on Functions Small Numbers Solving Quadratic Equations by Taking Square Roots
Direct Variation Powers of Products and Quotients Completing the Square
Linear Functions Exponential Functions The Quadratic Formula
Inverse Variation Polynomials The Discriminant
The Integers Monomials Solving Higher-Degree Equations
The Integers Polynomials The Real Numbers
More on the Coordinate Graph Adding and Subtracting Polynomials Rational Numbers
Addition Multiplying Polynomials Irrational Numbers
Subtraction More on Multiplying Polynomials More Irrational Numbers
Multiplication Squaring Binomials Pi
Division Dividing Polynomials The Real Numbers
Several Operations Factoring Fractional Equations
The Rational Numbers Prime and Composite Numbers Ratio and Proportion
The Rational Numbers Monomials and Their Factors Equations Containing Fractions
Absolute Value and Addition Polynomials and Their Factors More on Fractional Equations
More on Operations with Rational Numbers Factoring Second-Degree Polynomials Solving Formulas
Approximations Factoring the Difference of Two Squares More on Solving Formulas
More on Graphing Functions Factoring Trinomial Squares Inequalities
Equations in One Variable More on Factoring Second-Degree Polynomials Inequalities
Equations Factoring Higher-Degree Polynomials Solving Linear Inequalities
Inverse Operations Fractions More on Solving Inequalities
Equivalent Equations Fractions Absolute Value and Inequalities
Equivalent Expressions Algebraic Fractions Number Sequences
More on Solving Equations Adding and Subtracting Fractions Number Sequences
Length and Area More on Addition and Subtraction Arithmetic Sequences
Distance, Rate, and Time Multiplying Fractions Geometric Sequences
Rate Problems More on Multiplication Infinite Geometric Sequences
Equations in Two Variables Dividing Fractions
Equations in Two Variables Complex Fractions
Formulas
Graphing Linear Equations
Intercepts
Slope
The Slope-Intercept Form
Harold Jacobs’ Geometry:

This is a great text for geometry. Chapter titles are:

The Nature of Deductive Reasoning Inequalities Circles
Drawing Conclusions Postulates of Inequality Circles, Radii, and Chords
Conditional Statements The Exterior Angle Theorem Tangents
Equivalent Statements Triangle Side and Angle Inequalities Central Angles and Arcs
Valid and Invalid Deductions The Triangle Inequality Theorem Inscribed Angles
Arguments with Two Premises Parallel Lines Secant Angles
Undefined Terms and Definitions Parallel Lines Tangent Segments
More on Definitions Perpendicular Lines Chord and Secant Segments
Postulates about the Undefined Terms The Parallel Postulate The Inverse of a Point
Direct Proof: Arguments with Several Premises Some Consequences of the Parallel Postulate Inverses of Lines and Circles
Indirect Proof The Angles of a Triangle The Concurrence Theorems
Some Theorem Proofs Two More Ways to Prove Triangles Congruent Concyclic Points
Fundamental Ideas: Lines and Angles Quadrilaterals Cyclic Quadrilaterals
The Distance Between Two Points Quadrilaterals Incircles
Betweenness of Points Parallelograms Ceva’s Theorem
Rays and Angles Quadrilaterals That Are Parallelograms The Centroid of a Triangle
Angle Measurement Kites and Rhombuses Some Triangle Constructions
Complementary and Supplementary Angles Rectangles and Squares Regular Polygons and the Circle
Betweenness of Rays Trapezoids Polygons
Some Consequences of the Ruler and Protractor Postulates The Midsegment Theorem Regular Polygons
Some Basic Postulates and Theorems Area The Perimeter of a Regular Polygon
Postulates of Equality Polygonal Regions and Area The Area of a Regular Polygon
Two Bisection Theorems Squares and Rectangles Limits
Some Angle Relationship Theorems Parallelograms and Triangles The Circumference and Area of a Circle
Theorems about Right Angles Trapezoids Sectors and Arcs
Some Original Proofs The Pythagorean Theorem Geometric Solids
Congruent Triangles Heron’s Theorem Lines and Planes in Space
Triangles Similarity Rectangular Solids
Congruent Triangles Ratio and Proportion Prisms
Some Congruence Postulates More on Proportion The Volume of a Prism
Proving Triangles Congruent The Side-Splitter Theorem Pyramids
More Congruence Proofs Similar Triangles Cylinders and Cones
The Isosceles Triangle Theorem The A. A. Similarity Theorem Spheres
Overlapping Triangles Proportional Line Segments Similar Solids
Some Straightedge and Compass Constructions The Angle Bisector Theorem Euler’s Theorem
Transformations Perimeters and Areas of Similar Triangles Non-Euclidean Geometries
The Reflection of a Point The Right Triangle Geometry on a Sphere
More on Reflections Proportions in a Right Triangle The Saccheri Quadrilateral
Line Symmetry The Pythagorean Theorem Revisited The Geometries of Lobachevsky and Riemann
Translations Isosceles and 30 – 60 Right Triangles The Triangle Angle Sum Theorem Revisited
Rotations The Tangent Ratio
Point Symmetry The Sine and Cosine Ratios
Paul Foerster’s Algebra & Trigonometry:

This is the text that Jacob’s recommends following his algebra and geometry texts, and covers 2nd year algebra. Here are the chapter titles:

Preliminary Information Rational Algebraic Functions Probability, Data Analysis, and Functions of a Random Variable
Sets of Numbers Introduction to Rational Algebraic Functions Introduction to Probability
The Field Axioms Rational Function Graphs–Discontinuities and Asymptotes Words Associated with Probability
Variables and Expressions Special Products and Factoring Two Counting Principles
Polynomials More Factoring and Graphing Probabilities of Various Permutations
Equations Long Division of Polynomials Probabilities of Various Combinations
Inequalities Factoring Higher-Degree Polynomials–The Factor Theorem Properties of Probability
Properties Provable from the Axioms Products and Quotients of Rational Expressions Functions of a Random Variable
Functions and Relations Sums and Differences of Rational Expressions Mathematical Expectation
Graphs of Equations with Two Variables Graphs of Rational Algebraic Functions Again Statistics and Data Analysis
Graphs of Functions Fractional Equations and Extraneous Solutions Trigonometric and Circular Functions
Functions in the Real World Variation Functions Introduction to Periodic Functions
Graphs of Functions and Relations Irrational Algebraic Functions Measurement of Arcs and Rotation
Linear Functions Introduction to Irrational Algebraic Functions Definitions of Trigonometric and Circular Functions
Introduction to Linear Functions Graphs of Irrational Functions Approximate Values of Trigonometric and Circular Functions
Properties of Linear Function Graphs Radicals and Simple Radical Form Graphs of Trigonometric and Circular Functions
Other Forms of the Linear Function Equation Radical Equations General Sinusoidal Graphs
Equations of Linear Functions from their Graphs
linear function tool

Variation Functions with Non-integer Exponents Equations of Sinusoids from their Graphs
Linear Functions as Mathematical Models Functions of More Than One Independent Variable Sinusoidal Functions as Mathematical Models
Systems of Linear Equations and Inequalities Quadratic Relations and Systems Inverse Circular Functions
Introduction to Linear Systems Introduction to Quadratic Relations Evaluation of Inverse Relations
Solution of Systems in Linear Equations Circles Inverse Circular Relations as Mathematical Models
Second-Order Determinants Ellipses Properties of Trigonometric and Circular Functions
f(x) Terminology, and Systems as Models Hyperbolas Three Properties of Trigonometric Functions
Linear Equations with Three or More Variables Parabolas Trigonometric Identities
Solution of Second-Order Systems by Augmented Matrices Equations from Geometric Definitions Properties Involving Functions of More Than One Argument
Solution of Higher-Order Systems by Augmented Matrices Quadratic Relations –xy-Term Multiple-Argument Properties
Higher-Order Determinants Systems of Quadratics Half-Argument Properties
Systems of Linear Inequalities Higher-Degree Functions and Complex Numbers Sum and Product Properties
Linear Programming Introduction to Higher-Degree Functions Linear Combination of Cosine and Sine with Equal Arguments
Quadratic Functions and Complex Numbers Complex Number Review Simplification of Trigonometric Expressions
Introduction to Quadratic Functions Quadratic Equations from their Solutions–Complex Number Factors Trigonometric Equations
Graphs of Quadratic Functions Graphs of Higher-Degree Functions–Synthetic Substitution Triangle Problems
x-Intercepts and the Quadratic Formula Descartes’ Rule of Signs and the Upper Bound Theorem Right Triangle Problems
Imaginary and Complex Numbers Higher-Degree Functions as Mathematical Models Oblique Triangles–Law of Cosines
Evaluating Quadratic Functions Sequences and Series Area of a Triangle
Equations of Quadratic Functions from their Graphs Introduction to Sequences Oblique Triangles–Law of Sines
Quadratic and Linear Functions as Mathematical Models Arithmetic and Geometric Sequences The Ambiguous Case
Exponential and Logarithmic Functions Arithmetic and Geometric Means General Solution of Triangles
Introduction to Exponential Functions Introduction to Series Vectors
Exponentiation for Positive Integer Exponents Arithmetic and Geometric Series Vectors–Resolution into Components
Properties of Exponentiation Convergent Geometric Series Real-World Triangle Problems
Exponentiation for Rational Exponents Sequences and Series as Mathematical Models
Powers and Radicals Without Calculators Factorials
Scientific Notation Introduction to Binomial Series
Exponential Equations Solved by Brute Force The Binomial Formula
Exponential Equations Solved by Logarithms
Logarithms with Other Bases
Properties of Logarithms
Proofs of Properties of Logarithms
Inverses of Functions–The Logarithmic Function
The Add-Multiply Property of Exponential Functions
Exponential and Other Functions as Mathematical Models

Harold Jacobs’ Mathematics: A Human Endeavor:

Funnily enough, the subtitle on this book is “A Book for Those Who Think They Don’t Like the Subject”. Well, this isn’t true of anyone in our family, and we LOVE this book! Here’s what Martin Gardner (a famous mathematician whose books I’ve enjoyed since childhood) says about the book: “There are four reasons [for this book’s success]: The author’s choice of exciting topics, with emphasis on their recreational aspects; the author’s clear, friendly style; his inclusion of amusing cartoons and comic strips along with other art; above all, his enthusiasm for mathematics.” I love it! P did some of this in 2001-02. Here are the chapter titles:

Mathematical Ways of Thinking Symmetry and Regular Figures An Introduction to Statistics
The Path of a Billiard Ball Symmetry Organizing Data: Frequency Distributions
More Billiard-Ball Mathematics Regular Polygons The Breaking of Ciphers and Codes: An Application of Statistics
Inductive Reasoning: Finding and Extending Patterns Mathematical Mosaics Measures of Central Tendency
The Limitations of Inductive Reasoning Regular Polyhedra: The Platonic Solids Measures of Variability
Deductive Reasoning: Mathematical Proof Semiregular Polyhedra Displaying Data: Statistical Graphs
Number Tricks and Deductive Reasoning Pyramids and Prisms Collecting Data: Sampling
Number Sequences Mathematical Curves Topics in Topology
Arithmetic Sequences The Circle and the Ellipse The Mathematics of Distortion
Geometric Sequences The Parabola The Seven Bridges of Konigsberg: An Introduction to Networks
The Binary Sequence The Hyperbola Euler Paths
The Sequence of Squares The Sine Curve Trees
The Sequence of Cubes Spirals The Moebius Strip and Other Surfaces
The Fibonacci Sequence The Cycloid
Functions and Their Graphs Methods of Counting
The Idea of a Function The Fundamental Counting Principle
Descartes and the Coordinate Graph Permutations
Functions with Line Graphs More on Permutations
Functions with Parabolic Graphs Combinations
More Functions with Curved Graphs The Mathematics of Chance
Interpolation and Extrapolation: Guessing Between and Beyond Probability: The Measure of Chance
Large Numbers and Logarithms Dice Games and Probability
Large Numbers Probabilities of Successive Events
Scientific Notation Binomial Probability
An Introduction to Logarithms Pascal’s Triangle
Decimal Logarithms The Birthday Problem: Complementary Events
Logarithms and Scientific Notation
Exponential Functions
AP Statistics

We use the text, Introduction to the Practice of Statistics, and the video series, Against All Odds.

P uses a TI-89 calculator

ResamplingStats (for Microsoft Excel).

Other Math Stuff:

Zome Geometry makes use of ZomeTool, which is one of the best geometry manipulatives around. Although the book claims the curriculum is for grades 9 to 12, we think it’s much too fun to save until high school. With the struts and 62-hole connectors, you can build prisms, dodecahedrons, all the platonic solids, hypercubes, etc, many of which are wonderful 3-D bubble makers, all the while learning geometry!
Geometry Labs makes use of protractors, pattern blocks, tangrams, polyominoes, geoboards, etc, for a wonderful hands-on study of geometry.

There’s a really neat-looking web site, AAA Math, (thanks, Michelle!) that is a great tool for working on any area that needs extra work. 🙂 You can select topics by grade or general category (which I *love*, since you can tailor it very specifically to where a child is at). General topics are:

Addition Algebra
Comparing Counting
Decimals Division
Equations Estimation
Exponents Fractions
Geometry Graphs
Measurement Mental Math
Money Multiplication
Naming Numbers Patterns
Percent Place Value
Practical Math Properties
Ratios Statistics
Subtraction
Other math websites:

Armada — Classic Battleship Game
Cool Math
Pentominoes:
Pentominoes Page
Pentominoes Puzzle Solver
Math Forum
Set Game
Exploring Data
Names of Big Numbers — fun to know!
Gelfand Correspondence Program (out of Rutgers University, but have to be at least 13 years old to apply)
Mathematical Association of America bookstore
The Magic of Math
Georgia Council of Teachers of Mathematics
Math resources (comprehensive)
Harold Reiter’s home page]
PBS Teacher Search
Math curriculum of Westminster Schools and course sequencing
Math Contest Sites:

Mathematical Olympiads (4th grade and up) *****
American Mathematics Competitions (6th grade and up) *****
MathCounts (6th grade and up) *****
Mandelbrot Competition (highschool level)
Math League Contests (4th grade and up)
List of math competitions on the web
American Region Math League Competition (highschool level)
Woodward Academy’s Junior Varsity Math Tournament

GACS Dwight Love Math Tournament

UGA Math Tournament

GA Tech High School Math Competitions

GCTM Math Tournaments

Georgia math competitions

Long list of contests

Problem of the week sites:

MathCounts problem of the week
Math Forum problems of the week
Ole Miss problems of the week
Quote from a math professor:

There are three things that I would say are vital for a GOOD high school mathematics experience.

First, students MUST develop good algebra skills. Algebra is the toolbox of mathematics. A carpenter who didn’t know how to use a hammer or saw wouldn’t be much of a carpenter. If a student is getting no better than C’s in algebra then I would suggest redoing the course
and getting a better grasp.

Second, a good geometry course is important. I don’t care what Saxon says, he didn’t do enough geometry. If every student were to go all the way through Saxon’s advanced mathematics book, then, maybe that’s enough. But it still wasn’t a concentrated study of geometry.
Geometry is some of the more useful mathematics for the non- mathematician. It has applications in all kinds of areas. Also, the logic of proofs and geometric constructions are very valuable in developing sound logical thinking skills.

Third, every student should take a mathematics course in every semester of every year they are in school. This is especially true for any child who will go on to college. The old adage “If you don’t use it you’ll lose it” is especially true in mathematics. We constantly see students who run into trouble because they didn’t take a mathematics course since their sophomore or junior year in high school and then
didn’t take one here until their junior or senior year.

Now, regarding your specific questions. “Then what do we do for 11th and 12th grade?”

Here’s what I wrote on the subject a few years ago.

Number Theory – Number Theory can be a very challenging graduate level course. Yet, it can also be taught on a high school, or even junior high school, level. Properties of prime numbers and their uses can be fascinating. The uses of number theory for drivers’ license numbers, credit card security and book ISBN numbers are all interesting. And they can be taught by the average home school parent. There are a number of good number theory books available. Just look for ones that do not demand anything beyond high school mathematics.

Probability and Statistics – For anyone to be able to listen intelligently to public debate a basic understanding of probability and statistics is essential. When politicians start throwing statistics around it is extremely helpful to have a deeper understanding of these ideas than what most people have. There is a great book, though a little old now, called How to Lie with Statistics. That book alone would make for a great course in statistics. Studying probability will help give quantitative reasons, rather than just moral and ethical ones, for avoiding lotteries and other forms of gambling.

Solid Geometry – Most college mathematics faculty have probably never had a course in solid geometry. Yet it can be taught at a level that anyone who has gotten through high school geometry can grasp. It is a fascinating topic. Since you probably never studied it yourself it can give you an opportunity to learn something new, even if you had a lot of mathematics when you were in school. I do not know how many publishers have books available in solid geometry but a good library should have a couple.

Logic – Does the conclusion of an argument follow from the hypotheses?
Is the reasoning used in a debate sound? A course in logic can help answer those questions. Puzzle books are a good source of problems in logic. By puzzle books, I mean the kind with things like “Five men are in a room. One has a brown hat. The man with the umbrella is an accountant.” In the end you have to state each man’s name, occupation and what he is wearing. These sorts of problems, in addition to be educational, are just plain fun.

History of Mathematics – This is an absolutely fascinating area of study. Where did all of the great stuff in mathematics originate? Who did it? Who were these people? Was Gauss just the greatest mathematician of all time or was he also an interesting person? This is another topic that many university faculty never study but one that would benefit anyone.

Analytic Geometry – Studying analytic geometry can help you know why a satellite dish is shaped like a paraboloid. Or why an elliptical pool
table would make for much less interesting games.

Survey of Mathematics – This is a course that is actually often used as the general education mathematics course in colleges. Typically such a course will cover a little of probability and statistics, set theory and logic, algebra, geometry and number theory. This also goes by the name of Mathematics for the Liberal Arts or other names.

Trigonometry – Trigonometry is basically a merging of algebra and geometry. It includes all kinds of interesting applications of mathematics to real life.

Pre-Calculus – Some high school publishers call this “Advanced Mathematics.” It is basically an Algebra 3/Trigonometry course. This would be especially useful for students who are likely to go into technical fields in college. It will help extend and solidify the algebra skills that are so essential for success in collegiate mathematics.

Each of these topics can be studied at a level anywhere from early high
school or junior high up through graduate school.

“He wants to pursue a career in computer animation–what kinds of math will help him most, and where would we look for them?”

A solid algebra and geometry background