The Math Education Blog
Below is a problem taken from Dietmar Küchemann’s Algebradabra site. Many problems that mix geometry with algebra invariably shortchange the geometry. For example, the angles of a triangle might be labeled x, 2x, and 3x, and students are asked to find the value of x. Other than knowing that the angles of a triangle sum...
It’s a gripe I’ve shared before, but I’ll repeat it—the typical high school geometry approach to introducing transformations is boring. Fresh from learning the definition of a translation, reflection, rotation, or translation, students are whisked off to the safety of the coordinate plane and asked to explore the numerical effect of reflecting a point over...
Students often have difficulty connecting the Cartesian graph of a function to the actual motion of the variables. Their challenge in visualizing and describing covariation (the way y varies as x varies steadily) relates to their understanding of function families: function families are distinguished by the nature of their covariation. A function whose growth rate remains constant is a...
Several days ago, I was reminded of an interactive pattern block puzzle that I designed during the pandemic in collaboration with Toni Cameron of Reimagined. It provides an engaging opportunity to promote proportional reasoning in the context of geometry. On page 1 of the websketch below (and here), students are given the numerical values of...
When I taught a geometry methods course at City College last fall, I devoted an entire class to investigating area. We focused on problems where triangles were sheared, transforming into new triangles, but maintaining their area. The two Web Sketchpad activities that follow introduce shearing and present a problem with a surprising result that can...
In the February 1954 issue of Mathematics Teacher, Paul C. Clifford describes an optimization problem from his trigonometry class. Of all isosceles triangles ABC with sides AB and BC of length 12, which one has the maximal area? Clifford told his class that an exact solution to the question required calculus. One student, however, proved...
In the fall of 2023, I taught a geometry methods course at City College here in New York. While my goal was to make use of Web Sketchpad throughout the semester, I knew that the most effective use of the software required activities that coupled it with hands-on modeling. Two such opportunities arose with the...
How might we help students connect the unit-circle representation of trigonometric functions with the graphs of these same functions? Below (and here) is a Web Sketchpad model that gives students the tools to construct the graphs of trigonometric functions by using the unit circle as the driving engine. To get started, just follow the bulleted...
With nothing but a compass, students can construct a lovely daisy design consisting of seven interlocking circles, all of the same size. I was delighted to see that the U.S. postal service chose to feature four variations of this design in their floral geometry stamps shown below. I even spotted the daisy design on these...
Eleven years ago, I wrote a post titled What is All the Fuss About Lines? In it, I discussed the difficulties that students encounter when asked to determine the equation of a line. Faced with formulas for calculating slope, the point-slope form of a line, and the slope-intercept form, students lose their common sense as...