# 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 ››

# 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 ››

# 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 ››

# Bunny Times Brings Conceptual Understanding to Multiplication

Four years ago, my colleague Scott Steketee and I set out to develop an interactive game to help students develop strategies for thinking about and solving multiplication problems. As we examined the existing apps on the market, we discovered that most focused on the drill aspect of learning one's multiplication facts. We set our goals higher. We … Continue Reading ››

# Zooming Integers: Magnifying the Number Line

In my prior post, I presented a "zooming" number line model that allowed students to estimate the location of a point along a number line and then repeatedly magnify that portion of the number line to obtain ever-finer estimates, accurate to tenths, hundredths, thousandths, and beyond. In a sense I got ahead of myself because I … Continue Reading ››

# Zooming in on Place Value

One of the aspects I like best about Dynamic Geometry software like Web Sketchpad is its ability to illustrate concepts that cannot effectively be represented with static media. Take, for example, a number line that we draw on a white board. Showing the number line labeled with integers is easy. Adding tenth-mark divisions to the … Continue Reading ››

# Dancing Unknowns: You Haven’t Seen Simultaneous Equations Like These!

When it comes to simultaneous equations, I like to push the bounds of conventional pedagogical wisdom. In an earlier post, I offered a puzzle in which elementary-age students solve for four unknowns given eight equations. Now, I'd like to present a puzzle that might sound even more audacious: Solving for ten unknowns. Oh, … Continue Reading ››