Annotated Site Map

An incomplete guide to Henri Picciotto's Math Education Page

Other maps: by grade level | comprehensive | directories

Click the triangles!

K-8

Selected curriculum materials for Middle School

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

Before working in a high school, I taught elementary school for many years. This 1998 article summarizes some of my ideas on K-8 math education: Early Mathematics

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

Algebra
I 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 article on 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.
Intermediate
Seeking Depth in Algebra 2
An outline of some units, and links to some lessons.
Algebra 1 and Algebra 2 materials, 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, plus a video on pedagogy and links to related materials.
Exponential Functions
A "real world" introduction, and a visual lesson about the graphs.
(Also: all exponential graphs are similar.)
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.
Complex Numbers
Games and worksheets, for Algebra 2 and/or Precalculus.
Matrices
An effective introduction, based on geometric transformations and complex numbers.
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.
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.
Rate of Change
A directory of representations and activities to teach this essential concept, going belong the endless repetition of "rise over run".
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.
Constant Sums, Constant Products
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
Polyomino Lessons
Three pentomino puzzle books
Five supertangram puzzle books

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

Intermediate

The Pythagorean theorem home page: applets, work on grid paper, activities on the geoboard.

Notes and links on Transformational Geometry: some materials dircted to students, and some for teachers and curriculum developers.
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
Symmetry
Links to a wealth of K-12 materials
The Two-Color Theorem
A micro-unit that lays some groundwork for graph theory terminology in Discrete Math classes.
Electives After Algebra 2:
Precalculus
Basics, Going Deeper, And More!
Infinity
Infinite sets, proof, chaos, fractals.
Space
Abstract algebra, symmetry, transformational geometry, dimension.
Mathematics Overview
A one-year review: some highlights of secondary school math.

Teaching

Straightforward techniques for Reaching the Full Range of students in a heterogeneous class — in other words, any class.

Against Hyper-Acceleration: too many kids are accelerated in a counter-productive way.

The Assessment Trap: the wrong approach to assessment and the primacy of grades undermines student learning.

Taking Notes vs. Doing Math: Students cannot do both at the same time.

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

Big-Picture Planning: mapping out a whole program, a course, a unit.

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

Videos and links to articles about teaching, including a five-minute rant against time pressure.

Manipulatives

Brief video introduction, and links (virtual manipulatives, activities): Manipulatives

Curricular and enrichment links: Geometric Puzzles.

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

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")

Activities
Physical, in-the-body lessons, spanning many topics in secondary school math:
Kinesthetic Activities

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

Angles (from Geometry Labs)
Perimeter Patterns (in "Early Mathematics".)
...extended for teachers in Pattern Block Trains
Big Dodecagon! — beautiful designs!
Wallpaper Catalog — hands-on symmetry

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

Sample activities using the CircleTrig geoboard (from Geometry Labs):
Geometry: Angles
Trigonometry: The Ten-Centimeter Circle
Books

Working with Pentominoes, a book of lessons available from Didax

Five books of geometric puzzles (free!), using superTangrams

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, many suitable as writing prompts: Applets Directory

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

Using the computer: Math: Visual and Interactive! (2013)

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