On the NCTM discussion site myNCTM, there's currently an extended discussion about "Division and multiplication of fractions." As the discussion has continued, I've grown concerned with what I see as a fundamental problem with the way we often introduce multiplication as repeated addition: "Multiplying 4 by 5 means we're combining five groups of four items. … Continue Reading ››
We created the Web Sketchpad game below (and here) as part of our Dynamic Number project. It challenges elementary-age students to uncover the value of a secret number by collecting and analyzing clues that narrow its range of possible values. The game familiarizes students with inequality signs, introduces the use of x to represent … Continue Reading ››
In his 1947 book, One, Two, Three...Infinity, physicist George Gamow poses a pirate treasure problem that has since become a classic. Below is my reworded statement of the puzzle.
Among a pirate's belongings you find the following note: The island where I buried my treasure contains a single palm tree. Find the tree. From the palm tree, … Continue Reading ››
Below are two maps of the United Sates, with the smaller map a 50 percent scaled copy of the original. The edges of the two maps are parallel. Imagine that the maps are printed out, with one resting on top of the other. Believe it or not, you can stick a pin straight through both maps … Continue Reading ››
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 ››
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 ››
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 ››
The picture below on the left shows a right triangle JML with altitude KM. This altitude is defined as the geometric mean of JK and KL. Using similar triangles, you can prove that JK × KL = KM². Another way to think about this construction is shown below on the right. Start with segments JK and … Continue Reading ››
Twitter is a great place to find geometry problems. The July 22, 2017 post of xylem presented the image below with two squares, ABCD and BFGE, sharing a vertex. Given that AE = 5, what is the length of DG? My first thought was that surely the problem was … Continue Reading ››
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 numerator between … Continue Reading ››
My first thought was that surely the problem was …