*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 ››

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 ››

In my previous post, I wrote about cross number puzzles—puzzles that mix arithmetic and logic to introduce students to place value, commutativity, and the addition and subtraction algorithms.
Now, I'd like to present a variant of cross number puzzles that adds some algebra to the mix. Below (and here on its own page) are … Continue Reading ››

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 ››

In the interactive Web Sketchpad model below (and here on its own page), *ABCD* is an arbitrary quadrilateral whose midpoints form quadrilateral *EFGH*. Drag any vertex of *ABCD*. What do you notice about *EFGH*?
The midpoint quadrilateral theorem, attributed to the French mathematician Pierre Varignon, is relatively new in the canon of geometry theorems, dating to 1731. Mathematics … Continue Reading ››

In my last post, I presented a lovely geometry problem from Japan that was ideally suited to a dynamic geometry approach. Below is a new problem whose construction is nearly identical to the original one. The text says, "Five isosceles triangles have their bases on one line, and there are 10 rhombi. One length of the rhombus … Continue Reading ››

Here is a wonderful geometry problem from Japan: The five triangles below are all isosceles. The quadrilaterals are all rhombi. The shaded quadrilateral is a square. What is the area of the square?
I wondered at first whether the English translation of the problem was correct because with so many side … Continue Reading ››

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 ››

If there were an award for 'Mathematical Theorem Most Amenable to a Visual Proof,' the Pythagorean Theorem would surely win. The title of this post is a nod to the Sketchpad activity module *Pythagoras Plugged In *by Dan Bennett. Dan's book contains 18 visual, interactive proofs of the Pythagorean Theorem. And there are more: *The Pythagorean … **Continue Reading ›› *

Several years ago, I wrote a blog post about the value that students derive from writing mathematics with Sketchpad. The post included an example of a simple Logo iteration, easily implemented in Sketchpad, that produces some very complex and interesting shapes depending on the values of several input parameters. In the … Continue Reading ››