# Annotated Site Map

An incomplete guide to Henri Picciotto's Math Education Page Click the triangles!

## ▶▼K-8

Selected curriculum materials for Middle School

Before working in a high school, I taught elementary school for many years. This article summarizes my ideas on K-8 math education: Early Mathematics

Fractions: two visual representations, and a great problem to explore.

Number Pyramids, and In Addition: puzzles and problems whose only prerequisite is knowing how to add whole numbers

A naked-eye astronomy unit: Look! Up in the Sky!

## ▶▼Algebra

I have designed some algebra manipulatives (the Lab Gear) and co-authored, with Anita Wah, an algebra textbook (Algebra: Themes, Tools, Concepts -- ATTC). Start on these pages:
The Lab Gear
ATTC
A New Algebra: Tools, Themes, Concepts
This article (published in the Journal of Mathematical Behavior, v 12, #1, March 1993) summarizes much of the pedagogical and curricular thinking behind ATTC.
Algebra for All?
A short presentation of how the Lab Gear and ATTC help make the algebra course both more accessible and more mathematically rich.
ATTC turned out to be somewhat controversial. Read the debates!
My response to a critic.

### ▶▼Basic

Equations vs. Identities
Make These Designs
A powerful activity about y=mx+b, using any electronic grapher.
Left or Right?
Do these lines meet to the left or the right of the y-axis?
A lesson to connect graphical and algebraic approaches to systems of equations.
Comparing Cell Phone Plans
A "real world" application of linear functions and equations, in a fully spelled-out lesson.
Constant Sums, Constant Products

### ▶▼Intermediate

Seeking Depth in Algebra 2
An outline of some units, and links to some lessons.
Iterating Functions
A rich topic, which can serve as an introduction to sequences and series, and is prerequisite to the study of dynamical systems and chaos.
Letters and Postcards
An introduction to linear programming.
A series of worksheets I developed, the core of an Algebra 2 unit, plus some interesting follow-up lessons for teachers. Includes two non-standard proofs of the quadratic formula.
Sequences and Series
A dozen preview / introductory lessons on arithmetic and geometric sequences and series, from ATTC.
Exponential Functions
A "real world" introduction, and a visual lesson about the graphs.
(Also: all exponential graphs are similar.)

Complex Numbers
Games and worksheets, for Algebra 2 and/or Precalculus.
Doctor Dimension
A geometric/visual context to discuss rate of change and piecewise functions.

### ▶▼All Levels

Electronic Graphing
Lessons on linear functions, equations and inequalities, quadratics, polynomials, exponential functions, logarithms, square roots, absolute value.
Function Diagrams
A parallel axes representation, which provides a way to look at many familiar ideas from a fresh perspective. Includes many animations and worksheets, and ranges from middle school to calculus.
Functions
The Common Core State Standards include functions starting in 8th grade, and throughout high school. I have developed chunks of curriculum at all those levels.

### ▶▼Enrichment

Introduction to Abstract Algebra
A playful approach to group theory for a pre-college audience.

## ▶▼Geometry

### ▶▼Basic

Three sample lessons:Angles
An extension: Soccer Angles

An applet about the area of a circle.

Background, links, activities, and articles on:
Geometric Puzzles in the Classroom
A book of related activities: Polyomino Lessons
Four books of geometric puzzles, using plastic superTangrams:
SuperTangram Activities, Books 1 and 2
SuperTangrams for Beginners, Books 1 and 2

Tiling: links to tessellation activities, using manipulatives or grid paper.

### ▶▼Intermediate

Notes and links on Transformational Geometry
Isometries applets

Information about a book I co-authored with George W. Hart, about polyhedra and many other topics in two, three and four dimensions: Zome Geometry (off site)

Teaching Proof in high school
Proofs "without words":
Sum of angles in a triangle
Pythagorean theorem
Completing the square
Parallelogram area

Construction (compass and straightedge basics, interactive geometry software puzzles, concurrence theorems)

The geometry of graphs:
y = mx + b. (Similar triangles and the graphs of linear functions)
Geometry of the Parabola: 2D (Geometric definition of the parabola, the reflection property, evidence that all parabolas are similar)
Geometry of the Parabola: 3D (Proof that it is indeed a conic section)
The Geometry of Conic Sections: 2D | 3D
Unexpectedly related problems: minimizing certain sums of distances, and the locus of points from where a triangle...
looks isosceles!

### ▶▼Enrichment

The Two-Color Theorem
A micro-unit that lays some groundwork for graph theory terminology in Discrete Math classes.

## ▶▼Electives

Alternate electives after Algebra 2:
Infinity
Infinite sets, proof, chaos, fractals.
Space
Abstract algebra, symmetry, transformational geometry, dimension.
Mathematics Overview
A one-year review of the highlights of secondary school math.

## ▶▼Teaching

I analyzed the Common Core State Standards for high school math: CCSS-M, A Closer Look

For a Tool-Rich Pedagogy: how manipulatives and technology can transform the classroom

Against Hyper-Acceleration

The Assessment Trap

Reaching the Full Range of students in a heterogeneous class — in other words, any class

## ▶▼Manipulatives

Brief video introduction, and links: Manipulatives

Introduction to Base 10 blocks and the Lab Gear, followed by a general discussion of the uses and limitations of manipulatives: Manipulatives (in "Early Mathematics")

Debate on the appropriateness of using the geoboard to introduce the Pythagorean Theorem: an exchange with Don Chakerian.

### ▶▼Activities

Links to geoboard activities, most of which can also be carried out on dot paper.

Sample activities using the geoboard and the Lab Gear: Area (in "A New Algebra".)

Angles (from Geometry Labs)
Perimeter Patterns (in "Early Mathematics".)
...extended for teachers in Pattern Block Trains
Sample activities using the CircleTrig geoboard (from Geometry Labs):
Geometry: Angles
Trigonometry: The Ten-Centimeter Circle
Physical, in-the-body lessons, spanning many topics in secondary school math:
Kinesthetic Activities

### ▶▼Books

Information about a book of activities that use a broad range of manipulatives: Geometry Labs (free!)

Four books of geometric puzzles, using plastic superTangrams (free!):
SuperTangram Activities, Books 1 and 2
SuperTangrams for Beginners, Books 1 and 2

Information about a book (co-authored with George W. Hart) using the Zome System -- the ultimate 3D manipulative: Zome Geometry

## ▶▼Technology

Visual illustrations of various concepts, usually suitable as writing prompts: Applet Directory

Using the computer: Math: Visual and Interactive!

Electronic Graphing: many activities to deepen students' understanding of fundamental concepts (middle school to precalculus.)

The Turtle in the Age of the Mouse: why I teach programming (1997).

## ▶▼For Teachers

Workshops, math program review, coaching
Math Education Consulting
Professional development for math teachers:
Summer workshops
Math lessons for math teachers
Teachers' Mathematics 