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, … Continue Reading ››

# Category Archives: Algebra

# Slope Games Aplenty

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 … Continue Reading ››

# Dynagraphs of Linear Functions

This post provides three interactive examples of dynagraphs–a powerful representation of functions that emphasizes the behavior and relationship of a function’s independent and dependent variables.

# A Triple Number Line Model for Visualizing Solutions to Equations

In Algebra 1, I was the king of solving for *x*. Algebraic manipulation was fun and satisfying, and I was good at it. But my confidence was shaken when I encountered a test question of the variety 4*x* + 5 = 4*x* – 3. After subtracting 4*x* from both sides, I was … Continue Reading ››

# The Polar-Cartesian Connection

The Web Sketchpad model below (and here) shows the function *f(θ)* = 1 – cos 2*θ* in both Cartesian and polar form. For each graph, the independent variable appears as a red bar that corresponds to a particular value of *x* (for Cartesian) or *θ* (for polar). The red bar has … Continue Reading ››

# Introducing Web Sketchpad at the 2021 NCTM Annual Meeting

NCTM’s Virtual 2021 Annual Meeting ran from April 21 through May 1, and in Session 299 Daniel Scher, Karen Hollebrands, and I presented an on-demand video workshop to introduce Web Sketchpad (WSP). Even if you weren't able to attend the conference, you can still take … Continue Reading ››

# The Perimeter Problem

At a recent meeting of mathematics content coaches (many from the organization Reimagined) we investigated the following problem: What is the perimeter of the polygon below?

It appears at first that there isn't enough information to solve the problem. Indeed, the lengths of only three of the polygon's … Continue Reading ››

# Adding and Subtracting Integers

A recent post on the my NCTM discussion group asked about tools to help students visualize and understand addition and subtraction of integers.

I always found this confusing for some of my Algebra 1 students, mainly because they had been told to memorize some rules about whether to add or subtract the two integers and … Continue Reading ››