Category Archives: Elementary Mathematics

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 ››

Interactive Array Explorations

Arrays can be enormously helpful tools for helping young learners to visualize multiplication.  Early work with arrays also sets the stage for more advanced mathematics, like binomial multiplication. In this blog post, I present several interactive arrays built with Web Sketchpad as part of the Dynamic Number project. The interactive array model below (and … Continue Reading ››

Fifth Graders Investigate a Mathematical Code

How much tolerance do you have for puzzlement? When faced with a mathematical conundrum, do you embrace the challenge, or do you feel some trepidation at confronting the unknown? For many of us, an unfamiliar mathematical task is sure to raise our heart rate a beat or two. As teachers, it’s easy to translate these fears … Continue Reading ››

Deducing the “Mystery” Fraction

Estimation is an important mathematical skill, yet we rarely ask students to make estimates that relate to fractions. As part of the Dynamic Number project, we created a "mystery" fraction challenge  that presents a green point somewhere between 0 and 1 on the number line. The point's location can be represented as a fraction with … Continue Reading ››

Cross Number Puzzles

We live in a golden age of number puzzles. Sudoku is probably the most famous of all modern-day number puzzles, but there are many Japanese puzzles that are also gaining popularity, such as KenKen and Menseki Meiro. In this post, I'd like to introduce a number puzzle for young learners that predates … Continue Reading ››

Exploring Multiples on a Dynamic Number Grid

What would it take to build a better number grid for young learners? A typical number grid contains 10 columns with the numbers progressing from 1-10, 11-20, 21-30 and so on, from row to row. We decided to upend this tradition and make a dynamic number grid with Web Sketchpad that allows students to choose … Continue Reading ››

Creating Animated Factorization Diagrams

Last year, I had the pleasure of co-organizing a geometry-focused coaching collaborative led by Metamorphosis, a New York-based organization that offers professional content coaching to transform the mindset and practices of teachers and administrators. I had so much fun that I decided to do it again! My workshop partners were Metamorphosis staffers Toni Cameron, Ariel Dlugasch, … Continue Reading ››

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 ››