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
About
- 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.
- A discussion about geoboards.
Basic
- Equations vs. Identities
- Three downloadable lessons from ATTC.
- 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?
- Add Till It's Plaid!
- 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
- Making Sense in Algebra 2
- An outline of some units, and links to some lessons.
- Parabolas and Quadratics
- 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.
Advanced
- 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
- An untraditional approach to several traditional topics, including the quadratic formula.
Enrichment
- Introduction to Abstract Algebra
- A playful approach to group theory for a pre-college audience.
Geometry
Basic
- My book Geometry Labs (Free download)
- 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
- Books of related activities (all free downloads):
- Tangram Puzzles
- 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)
Advanced
- 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
About
Brief video introduction, and links (virtual manipulatives, activities): Manipulatives
Links and curricular materials about algebra manipulatives: The Lab Gear
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.
- Links to pattern block activities:
- 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
Activities that use a broad range of manipulatives: Geometry Labs (free download)
Algebra Lab Gear books: free download | available from Didax
Three books of pentomino puzzles (free download!)
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)
Programming: The Turtle in the Age of the Mouse (1997) | Boxer: A Teacher's Experience (2022).
For Teachers
- Workshops, math program review, coaching
- Math Education Consulting
- Professional development for math teachers:
- Summer workshops
- Math lessons for math teachers
- Teachers' Mathematics
