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Presentations and Workshops

Handouts and links for some of my talks.
I can present any of them to your school or district.
(Math Education Consulting)

See my résumé for a full list of presentations.

Follow me on Twitter: @hpicciotto

Transformational Proof

in High School Geometry
with Lew Douglas

Level: Teachers' mathematics, relevant to grades 8-10

Description: A Deep Dive Into Transformational Proof in High School Geometry

In this mini-session, we will provide a detailed framework for transformational proof, including a set of clearly-specified assumptions. We will use these assumptions to prove basic transformational theorems. With these in hand, you can prove triangle congruence and similarity conditions (formerly taken as postulates) and proceed traditionally, or prove the customary theorems without using congruent or similar triangles. It is also possible to combine transformational and traditional proofs. This session is for you as a teacher-learner. We will not focus on activities for students. That said, we will include interactive components and whole-group discussion.

big P iconOn this site:
Transformational Geometry. (Scroll down to Transformational Proof.)

Computing Transformations

Using Complex Numbers and Matrices

Level: Grades 10-12

I will assume familiarity with the basics of transformational geometry, and present topics for possible use in grades 10-12. An introduction to the mathematics underlying computer graphics: a visual approach to complex numbers in Algebra 2, including review and extension of trigonometry; application of complex numbers to the computation of geometric transformations; and finally 2 by 2 and 3 by 3 matrices for these computations, including how complex numbers help us find the matrix for rotations.

big P iconOn this site:
Complex Numbers in Algebra 2 (PDF)
Complex Number Arithmetic Games
Computing Transformations (PDFs): GeoGebra | TI-89
Related materials:
Transformational Geometry
Seeking Depth in Algebra 2

The Lab Gear

Manipulatives for Algebra

Level: Grades 6-11

Given technology, speed and accuracy in algebraic manipulation no longer constitute legitimate priorities. However a grasp of the fundamental structures of algebra (the meaning of variables, operations, functions, equations) remains crucial. Intelligent use of manipulatives can help. The Lab Gear provides a hands-on approach where the inner logic of the model replaces the memorization of seemingly arbitrary rules. This powerful learning tool facilitates communication about abstract ideas and helps improve the discourse in the algebra class.

big P iconOn this site:
The Lab Gear

the Full Range

Level: Grades 7-12

As everyone knows, students learn math at different rates. What should we do about it? I propose a two-prong strategy based on alliance with the strongest students, and support for the weakest. On the one hand, relatively easy-to-implement ways to insure constant forward motion and eternal review. On the other hand, a tool-based pedagogy (using manipulatives and technology) that supports multiple representations, and increases both access and challenge.

Slides: Online | Keynote
Video (75 minutes) | Webinar recording (45 minutes)

big P iconOn this site:
Heterogeneous Classes
For a Tool-Rich Pedagogy
Visual and Interactive!
Electronic Graphing
Common Core: A Closer Look
big P iconBlog Posts:
Extending Exposure
Lagging Homework
Separating Related Topics
Pruning the Curriculum

Function Diagrams

Level: Grades 7-12

An encyclopedic introduction to function diagrams and their pedagogical applications to arithmetic, basic algebra, dynamical systems, and calculus. Much of this illustrated with the help of GeoGebra.

big P iconOn this site:
Function diagrams overview
Function diagram applets
Electronic tools for function diagrams
Function diagram PDFs
The Geometry of y=mx+b
Iterating Linear Functions
Presentation slides: PDF | Keynote

Pattern Blocks

for middle and high school

Level: Grades 6-10

Pattern blocks are ubiquitous in elementary schools, but they're not commonly seen in middle school or high school. Yet, they do offer plenty of interesting curricular opportunities. (And yes, they're fun!) I present an encyclopedic tour of the puzzles, activities, lessons, and connections they suggest about area, perimeter, angle measurement, symmetry, tiling, "π" for regular polygons, and rate of change.

big P iconOn this site:
Pattern Blocks
Geometry Labs, Labs 1.1, 5.6, 7.2-7.4, 11.8
Wallpaper starters
Pattern Block Trains
Presentation slides (PDF | Keynote)

Three Paths to the Quadratic Formula

Level: Grades 8-11

A sequence of lessons on parabolas, quadratic functions, and quadratic equations. The unit works well with Algebra 2 students, and includes activities with manipulatives, graphing, and symbol manipulation. These approaches lead to three distinct proofs of the quadratic formula, including a new one.

Bibliography: For the hands-on approach to quadratics and completing the square, see Lab Gear Activities for Algebra 1, by Henri Picciotto, Creative Publications. (It is currently unavailable, but a new edition is in the works. It will be published by Didax.)

big P icon On this site:

An introduction to the Lab Gear.

Two graphical approaches
Parabolas and Quadratics
Constant Sums, Constant Products

Common Core
a closer look

Level: Grades 9-12

The Common Core State Standards introduce significant and generally positive changes to the high school math curriculum, but they do not mandate a specific sequence in grades 9-11. This deliberate omission may allow educators to escape the tyranny of tradition, and re-sequence the high school curriculum in a way that is consistent with students' mathematical maturity and brain development, on the one hand, and with the new possibilities offered by advances in pedagogy and by new technologies, on the other. Unfortunately, the large number of standards, and the sequences suggested in the CCSS Appendix undermine these possibilities.

big P iconOn this site:
Presentation slides
In-depth analysis of the high school standards

Making Sense
in Algebra 2

Level: High School

Units from a course I developed with my colleagues in the Math Department at the Urban School of San Francisco.

This presentation is based on "Seeking Depth in Algebra 2" (see below.)

big P icon On this site: Seeking Depth in Algebra 2

Abstract Algebra

Level: Grades K-12

Even though Abstract Algebra is a college-level course, it is possible to have a lot of fun with this topic at any age by using an informal approach. I have taught these lessons in one form or another to students in Kindergarten through 12th grade, and to teachers, since 1971. Taken together, they are a good introduction to the power and beauty of mathematical structure. The approach is playful and founded on student experience, discussion, and reflection. The key concept is that of a group, with a special emphasis on the identity and inverse elements, which are essential understandings throughout K-12 mathematics.

big P iconOn this site:
Abstract Algebra

The Geometry of Conic Sections

Level: Grades 9-12

Most high school curricula seem to forget that the conic sections are geometric objects! I will explain in several ways that contrary to popular belief, all parabolas have exactly the same shape. I will use interactive software (both 2D and 3D) to construct the conics, prove their reflection properties, and show that they are indeed the result of slicing a cone. Finally, I will explore a question about soccer that unexpectedly leads to a hyperbola.

big P iconOn this site:
Geometry of the Parabola
Geometry of the Conic Sections
Soccer Angles

An alternate elective
after Algebra 2

Level: Grades 11-12

An advanced geometry elective I have taught biennially since 1992. Three components: symmetry (introduction to abstract algebra, recognizing symmetry groups around a point, along a line, and in the plane, art projects, tiling); transformations (complex numbers review, matrices, isometries); dimension (polyhedra, Platonic and Archimedean solids, duality, Euler's and Descartes' theorems, the fourth dimension.) Using Cabri 2 and 3D software, building with the Zome system, reading Abbott's Flatland.

big P icon On this site: Space

High School Math

Level: Grades 9-12

High school math classes look very much the same from year to year and from school to school. Yet, other models are possible! In addition, technological advances mean that speed and accuracy are no longer legitimate priorities. We can no longer divorce skills from understanding, nor can we consider obsolete skills to be foundational. What we need is an eclectic mix of approaches that prioritize student learning and habits of mind.

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Presentation slides
Urban School Math Department
Nothing Works

the Dots

Level: Grades 7-10

Accessible hands-on activities on the geoboard (or dot paper) lead to many ideas in arithmetic, geometry, and algebra: equivalent fractions, slope, the Pythagorean theorem, and simplifying radicals. This session is suitable for middle school and high school math teachers who are looking for Common Core-compatible approaches and content which will work with a wide range of students.

big P iconOn this site:
Geoboard Activities (includes presentation slides)
Dot papers
Geometry Labs (especially labs 8.5, 8.6, 9.2, 9.3, 9.4, 10.1, 10.2)
Presentation slides
Geoboard diagonals
The Pythagorean Geoboard

Strengthening Mathematics Departments

with Laura Hawkins

Audience: This session will be of particular interest to department chairs and anyone involved in school change.

Description: How do we build a culture of teacher collaboration? How do we spread effective approaches across the department? How do we incorporate new ideas into our program? How do we respond to administrative directives, as well as to the needs of our students? What should we ask of our administrators? We will share our tentative answers, and would love to hear yours. Join us in a conversation about what it takes to strengthen a math department.

Teacher Collaboration
A key to improving
math instruction

Audience: This session will be of particular interest to department chairs and anyone involved in school change.

Description: Teachers value autonomy and specialization, yet the advantages of collaboration and flexibility are many. So are the complications. Hear the rationale for one department's move to intensive mentoring and the development of a collaborative ethic. I will assess decades of experience in this practice, and reflect upon its impact on teachers, curriculum, pedagogy, and learning.

big P icon On this site:
Presentation slides.
Teacher Collaboration (an article from Independent School, co-authored with Jonathan Howland).
Escape from the Textbook! sharing and collaboration network.

Nothing Works!
The Art of Teaching

Level: Grades 9-12

Teaching high school math is a complex endeavor, where apparently contradictory approaches can complement each other: there is no one way that works with all teachers and all students. I will present my mix of techniques for organizing curriculum, sequencing concepts, designing rich activities, working with (somewhat) heterogeneous classes, leading effective class discussions, using cooperative learning groups, assigning homework, assessing student understanding, and other day-to-day concerns.

big P icon On this site: About Teaching

An alternate elective
after Algebra 2

Level: Grades 11-12

Syllabus and highlights of an alternate math elective after Algebra 2, which I have been teaching biennially since 1991: paradoxes involving infinity, proof by contradiction, Cantor's discoveries, mathematical induction, chaos, fractals; connections to literature, philosophy, science, and computer programming. Readily available materials on these subjects tend to be written for either the general public or college students. My presentation will focus on how to make this content accessible in high school.

big P icon On this site: Infinity

Level: High School Minicourse

Units from a course I developed with my colleagues in the Math Department at the Urban School of San Francisco. Our approach is to cover fewer topics in greater depth and to use a variety of learning tools, both manipulative and electronic.

This presentation was initially created by Naoko Akiyama and Scott Nelson as a one-hour presentation. I joined them to expand it to a three-hour minicourse. Get the slides.

Bibliography: Much of the material is unpublished, but see
Hands-on approach to quadratics and completing the square: Lab Gear Activities for Algebra 1, by Henri Picciotto, Creative Publications (It is currently unavailable from the publisher. Contact me if you need it.)
Geometric approach to complex numbers: Algebra II/Trigonometry: A Guided Inquiry, by Stein, Crabill, and Chakerian (out of print)

big P icon On this site: Seeking Depth in Algebra 2

Geometry Labs

Using Manipulatives to Teach about Angles and to Introduce Trigonometry

Level: Grades 8-11 Workshop

What is an angle? Interior and exterior angles in a polygon. Inscribed and central angles. Soccer angles. Trig ratios.

Puzzles and problems using pattern blocks and circular geoboards, plus the students' own bodies.

These new approaches to old topics provide both access and challenge and work well with heterogeneous classes. The labs enhance discourse and deepen understanding.

(I also offer an all-day version of this, covering more topics.)

Bibliography: See my book Geometry Labs.

big P icon On this site: See Angles and The Ten-Centimeter Circle


Description: Make regular polygons, pyramids, prisms, and antiprisms. Explore the relationships between the dodecahedron, the icosahedron, the rhombic triacontahedron, and more... Identify the components of icosahedral symmetry.

A hands-on lab with an amazing manipulative, making connections with many traditional geometry and trigonometry topics.

The handouts are excerpted from Zome Geometry (Key Curriculum Press) a book I co-authored with George W. Hart, using the Zome System

On the Web:
For a taste of what is possible with the Zome System, check out George Hart's page about Zome Polyhedra, and his Advanced Zome Constructions.