Presentations and Workshops

Level: Grades 6-12

Description:
People learn at different rates: all classes are heterogeneous. I share manageable ways to differentiate by time, not content, in order to both challenge and support every student: constant forward motion, eternal review, and extended exposure. I also share some other ideas from my new book. (See the link below.)

On this site:

All the presentation links

The slides: PDF | Keynote

Make These Designs

The Art of Teaching

About Teaching (many more relevant links)

Elsewhere:

A book:
There Is No One Way to Teach Math: Actionable Ideas
by Henri Picciotto and Robin Pemantle
Until January 31, 2025 20% off with code SMA24

Level: Preservice math teachers

Description:
I share activities I developed for middle and high school classroom use. Tessellation puzzles provide a rich environment for basic geometric facts and for an introduction to the isometries of the plane, while making connections to art. Pentominoes, supertangrams, and rep-tiles illustrate the relationships between similar figures. I discuss ways to use those puzzles to help students discover or confirm different parts of the high school geometry curriculum, and in some cases go further.

On this site:

Geometric Puzzles for Prospective Math Teachers

The Place and Purpose of Puzzles in Math Curriculum

Tiling

Geometry Labs

Transformational Geometry

Transformational Geometry for Teachers

Pentominoes home page

Supertangrams home page

Rep-tiles

Geometric Puzzles

Level: Grades 3-5

Description:
I present how to use well-chosen rectangles to compare, add, subtract, multiply, and divide fractions. Also: how to use pie slices to see equivalence and make connections with angles, time, money, decimals, and percent. These representations are intended to complement, not replace approaches you already use. We end with a high-ceiling Egyptian fractions challenge.

On this site:

Fractions

Level: Math Teachers' Circle

Description:
We'll explore puzzles which are appealing to both children and adults: tangrams, pentominoes, supertangrams, and rep-tiles. All of them involve the use of specific tiles to cover various figures. In most cases, we'll use hands-on materials. Although most of the puzzles we'll tackle can be solved, we'll prove some are impossible. Solvable puzzles will range in difficulty from easy to, ahem, funstrating.

These geometric puzzles touch on symmetry, congruence, similarity, convexity, the Pythagorean theorem, and tiling. They can be used in the classroom, K-12. I may reference those uses, but we will not dwell on them: our main focus will be on solving the puzzles and thinking about them.

On this site:

Links

Slides

Geometric Puzzles

Level: Leaders of Math Circles and Math Teachers' Circles

Description:
Tangram and pentomino puzzles which are fun and interesting for all ages. The activities will touch on angles, convexity, area, congruence, and similarity. Some impossibility proofs.

On this site:

Links

Geometric Puzzles

On YouTube:

Recording

Level: General Interest

Description:
A hands-on workshop.
Tiling the plane (tessellation) enhances the introduction of geometry in middle and high school. It provides opportunities for students to do creative work they take pride in. It connects with art (e.g. Escher) and culture (e.g. Islamic design). And it provides a rich context for basic geometry (sum of the angles in a polygon, exterior angle theorem, parallels and transversals, regular polygons) and transformational geometry (rigid motions and basic theorems about them).

On this site:

Links

Tilings to Analyze

Level: General Interest

Description:
Symmetric designs are found in virtually every culture, and are interesting to students across the grades. We will explore line and rotational symmetry with activities on free online platforms. We will discuss rosette symmetry (finite figures), frieze symmetry (infinite designs that extend in one dimension), and wallpaper symmetry (two-dimensional designs and tessellations). We will analyze examples from all over the world. Bonus: links to online symmetry projects, as well as hands-on activities using mirrors, pattern blocks, tangrams, pentominoes, templates, and special papers for when you're back in the classroom.

On this site:

"Handout" with links

Symmetry

Level: Grades 7-10

Description:
I present five free virtual manipulatives I created on a foundation of Google Drawings, Google Slides, and GeoGebra: 11x11 Geoboard, Algebra Lab Gear, Tiling, Grid Paper, and Pentominoes. For each one, I share a problem-rich, low-threshold / high-ceiling lesson — suitable for middle school or high school. (The topics are appropriate for students in pre-algebra, basic algebra, and basic geometry.)

On this site:

Geoboard

Grid Paper

Lab Gear

Triangles and Quadrilaterals

Pentominoes

About virtual manipulatives: Part 1 | Part 2

Level: Grades 7-10

Description:
Tangrams and pentominoes are geometric puzzles which are fun and interesting for all ages. Can students use those manipulatives in an age of remote instruction? Yes! In this workshop, I will share virtual tools I created to make that possible. I will propose problems suitable for ages 9-99, and we will work as a team to generate dozens of solutions. The activities will touch on symmetry, area, congruence, and similarity.

On YouTube:

Session recording

On this site:

"Handout" with links

Slides: PDF | Keynote

Virtual Manipulatives

Geometric Puzzles

Level: Grades 7-12

Description:
Transformational geometry can help bring in new approaches to teaching, broaden and deepen the understanding of proof, and make connections with several other parts of mathematics. Unfortunately, bringing a transformational approach to schools is hampered by lack of clarity and the weight of tradition. I suggest some possibly useful guidelines to facilitate this transition, including what topics should be prioritized in teacher training.

On this site:

Blog post (detailed expansion of the talk)

Slides: PDF | Keynote

Transformational Geometry

Level: General Interest

Description:
We will explore a wide range of problems and puzzles on geoboards. (Dot paper works also.) Some provide a hands-on approach to secondary school topics such as slope, the Pythagorean theorem, and simplifying radicals. Some are currently unsolved and are fun to think about. Some are "teacher-level" problems, and fall somewhere in between. Participants will walk away with many ready-to-use activities, suitable for middle school, Algebra 1, Geometry, and their own mathematical recreation.

On this site:

Using the Geoboard

Level: Grades 7-11

Description:
Lew Douglas, a leading Bay Area math educator, passed away in April 2019. Lew delighted in math and developed lessons to allow others to share in that wonder. Come celebrate his life and contributions as we share some of our favorite classroom activities designed or inspired by Lew Douglas.

Kim Seashore will share what she learned from him when she was a young teacher of geometry. Henri Picciotto will share Lew's introduction to the golden ratio (based on the geometry and algebra of the pentagram); and an activity about symmetric polygons, which can be foundational in a transformational approach to geometry. Bring your own memories and lessons from Lew to share!

On this site:

Pentagrams and Spirals (by Lew Douglas, edited by Henri Picciotto)

Symmetric Polygons (by Lew Douglas and Henri Picciotto)

Transformational Geometry (several items co-authored by Lew Douglas)

The Radian Dance (by Kim Seashore, inspired by Lew Douglas)

Kinesthetic Activities (that whole approach inspired in part by Lew Douglas)

Level: Grades 6-11

Description:
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.

On this site:

The Lab Gear | Q and A

Lab Gear videos and animated slides

Level: Grades 7-12

Description:
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 (64 minutes)

Webinar recording (45 minutes)
Webinar recording (40 minutes)

On this site:

Extending Exposure

For a Tool-Rich Pedagogy

Heterogeneous Classes

Group Work

Manipulatives

Visual and Interactive!

Electronic Graphing

These links and more, annotated: About Teaching

Curriculum items mentioned in the talk:

Pattern Blocks

Geometric Puzzles

Lab Gear

Geoboard

Make These Designs

Geometric Construction

Function Diagrams

The Ten-Centimeter Circle

Algebra: Themes, Tools, Concepts

Geometry Labs

Blog Post:

Rich Activities

Level: Grades 7-12

Description:
Teaching math in longer periods presents some challenges, but on balance it is a tremendous gift to both teachers and students. For teachers, it is a chance to develop professionally and expand their pedagogical repertoire. For students, it is a chance to experience a more hands-on and more varied program, which can result in better understanding and retention.

On this site:

Teaching in the Long Period

Math in the Long Period

Tool-Rich Pedagogy

Manipulatives

Level: Grades 9-12

Description:
Geometric puzzles are accessible to all students, and provide a popular change of pace from the daily routine. They offer opportunities for hands-on explorations and challenging problems about area, perimeter, congruence, similarity and scaling, symmetry, and the square root of two. In this workshop, you will make tangrams by tearing, discover pentominoes and supertangrams in order to use them in puzzles of increasing difficulty, and use the Pythagorean theorem to get insight into rep-tiles.

On this site:

Geometric Puzzles

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.

On this site:

Transformational Geometry. (Scroll down to Transformational Proof.)

Level: Grades 8-10

Description:
Algebra manipulatives provide an environment where students can make sense of two ways to solve quadratic equations: factoring and completing the square. Graphing technology allows students to link those approaches to quadratic functions. Using these tools and connecting these concepts makes the algebra come to life for all students.

On this site:

The Lab Gear

Parabolas and Quadratics

Constant Sums, Constant Products

Five Representations | Applet

Geometry-Graphing Connection

Completing the Square | Other applets

Common Core: A Closer Look

Level: Teachers' Mathematics, relevant to grades 7-12

Description:
Many concepts depend on distance: the triangle inequality, the definition of a circle, the value of π, the properties of the perpendicular bisector, the geometry of the parabola, etc. In taxicab geometry, you can only move horizontally and vertically in the Cartesian plane, so distance is different from the usual "shortest path" definition. We will explore the implications of taxicab distance. There are no prerequisites, other than curiosity and a willingness to experiment on graph paper.

On this site:

Geometry Labs

Taxicab Geometry

Level: Grades 10-12

Description:
I 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.

On this site:

Complex Numbers in Algebra 2 (PDF)

Complex Number Arithmetic Games

Computing Transformations (PDFs): GeoGebra | TI-89

Matrices overview

Related materials:

Transformational Geometry

Seeking Depth in Algebra 2

Precalculus

Level: Grades 7-12

Description:
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.

On 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

Level: Grades 6-10

Description:
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.

On this site:

Pattern Blocks

Geometry Labs, Labs 1.1, 5.6, 7.2-7.4, 11.8

Wallpaper starters

Angles

Pattern Block Trains

Presentation slides (PDF | Keynote)

Level: Grades 8-11

Description:
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.)

On this site:

An introduction to the Lab Gear.

Two graphical approaches

Parabolas and Quadratics

Constant Sums, Constant Products

Level: Grades 9-12

Description:
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.

On this site:

Presentation slides

In-depth analysis of the high school standards

Level: High School

Description:
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.)

On this site: Seeking Depth in Algebra 2

Level: Grades K-12

Description:
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.

On this site:

Abstract Algebra

Level: Grades 9-12

Description:
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.

On this site:

Geometry of the Parabola

Geometry of the Conic Sections

Soccer Angles

Level: Grades 11-12

Description:
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.

On this site: Space

Level: Grades 9-12

Description:
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.

On this site:

Presentation slides

Urban School Math Department

Nothing Works

Level: Grades 7-10

Description:
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.

On 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

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.

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.

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.

Level: Grades 9-12

Description:
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.

On this site: About Teaching

Level: Grades 11-12

Description:
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.

On this site: Infinity

Level: High School Minicourse

Description:
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.

Bibliography:

- Hands-on approach to quadratics and completing the square: Algebra Lab Gear two books by Henri Picciotto, from Didax.

- Geometric approach to complex numbers: Algebra II/Trigonometry: A Guided Inquiry, by Stein, Crabill, and Chakerian (out of print)

On this site: Seeking Depth in Algebra 2

Using Manipulatives to Teach about Angles and to Introduce Trigonometry

Level: Grades 8-11 Workshop

Description:
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.

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

Minicourse

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.

Bibliography:
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.