Category Archives: Elementary Mathematics

Make Your Own Fractions

In my very first Sine of the Times blog post from January 2012, I wrote about the paucity of fractions that young learners typically encounter in their math classes. While they might construct visual representations of 1/2, 2/3, and 8/12, it's unlikely they'll create models of 7/31, 36/19, or 5/101. That's a shame because without … Continue Reading ››

Stars, Polygons, and Multiples

I've always found my collaborations with teachers to be a great inspiration for curriculum development, and that was especially true of my work with Wendy Lovetro, an elementary-school teacher in Brooklyn, NY. Wendy coordinated an after-school math club at her school, and I used the setting as an opportunity to develop and field test Sketchpad activities for the … Continue Reading ››

Raz’s Magic Multiplying Machine

Here is a question you don't hear very often: What does it feel like to experience multiplication in our bodies? It's a strange question because our typical exposure to multiplication is numerical. I give you two numbers—say, 3 and 5—and you tell me their product, 15. But multiplication need need not be so static and concrete. Back … Continue Reading ››

Developing Arithmetic Fluency Through “Drill and Thrill”

In a previous post, I described the game Chisla, an app that succeeds in making basic arithmetic both challenging and tense.  Given a numerical target, you have just 15 seconds to pick numbers from a list of possible addends that will sum to that value. A target of 10 is no big deal, but when the target escalates … Continue Reading ››

Adding and Subtracting on a Dynamic Number Chart

Below is a collection of four interactive number charts that we first introduced in our NSF-funded Dynamic Number project. Start by asking students to press the four directional arrows and to explore what they do. The right arrow, for example, moves the shaded square to the right, and wraps the square to the next row up when you … Continue Reading ››

Estimating Angle Measurement

Angles are a thorny concept to teach because of the fundamentally different ways in which they can be used and understood. In the article What's Your Angle on Angles?, the authors divide the concept of angle into three main groups: angle-as-figure, angle-as-wedge, and angle-as-turn. In the Web Sketchpad game below, we focus on angle-as-turn. Given an angle, students enter an estimate … Continue Reading ››

An Equivalent Fractions Game

In my recent posts, I've introduced interactive models for comparing fractions and multiplying fractions. To continue the fraction theme, below is a Web Sketchpad model in which the need for equivalent fractions arises naturally through the rules of a game. The model displays two arrays. Dragging the four points changes the arrays' dimensions. The goal is to drag blue squares into the blue array … Continue Reading ››

A Compendium of Elementary Web Sketchpad Activities

Below is a complete list of all the interactive elementary-themed activities we've offered on our Sine of the Times blog. Most of these activities grew out of the work that we did during our NSF-funded Dynamic Number project. The list includes links to our original blog posts as well as links that take you directly … Continue Reading ››

Comparing and Identifying Fractions Visually

In my previous post, I presented an interactive Web Sketchpad model for visualizing and solving fraction multiplication problems. This week, I'd like to back up a step and focus on the more fundamental skill of visualizing and reasoning about the size of fractions. The fraction game below (and here) presents two random fractions at a time and … Continue Reading ››

Understanding Fraction Multiplication

Last week, Scott and I attended a fraction symposium at NYU, and it made me realize how long it's been since I've written about our Sketchpad work with fractions. Below is a Web Sketchpad model for displaying and solving fraction multiplication problems. Representing fraction multiplication with an area model is a common approach, but it's challenging to sort … Continue Reading ››