I added a third dynamic proof of the Pythagorean theorem on my Web site. It is better-looking, but more difficult to justify than the previous two. You can see all three, starting here. For a brief discussion of how you might use these in the classroom, see this post. --Henri
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Geometric Series
... geometric in more than one way! Start with a right triangle ∆ABC, with hypotenuse AB = 1, and AC = b.Draw a parallel to AB through C.Draw a circle centered at C, with radius b.Label the intersection of the line and the circle D.Drop a perpendicular from D to line AC, intersecting it at… Continue reading Geometric Series
Pythagorean Proofs
I just added a second dynamic geometry proof of the Pythagorean theorem on my Web site.Both are proofs "without words", which in reality means that you should use them to generate discussion. Indeed, many words are needed for students to fully grasp what they see, but the words should not come exclusively from the teacher.… Continue reading Pythagorean Proofs
Interactive Whiteboards
Some years ago, I wrote about interactive whiteboards (IWBs), in response to a passionate anti-IWB opinion piece I stumbled upon. The author of that piece objected to IWBs on multiple grounds, some of them legitimate. But I disagreed with his main point, which was to counterpose IWBs to student-centered pedagogy. To me, those are not… Continue reading Interactive Whiteboards
Make These Designs
Among the activities I have developed, "Make These Designs" is among the most popular. Students enjoy it, and teachers appreciate how it can be used as an engaging introduction to, or interesting review of, an important topic: graphs of linear functions, and the parameters m and b in y=mx+b.When electronic graphing first came onto the… Continue reading Make These Designs
Proportional Relationships
One good thing about the Common Core middle school standards is the emphasis on proportional relationships, and the fact that they are approached in a multidimensional way. In addition to "set up a proportion and solve it", which is probably the most common way to teach this, the standards propose multiple representations and a variety… Continue reading Proportional Relationships
Patterns
A correspondent writes:We emphasize the idea that students should approach problems in multiple ways. This has caused me to wonder about patterns. For example:students might conclude that 3^0=1 because of the pattern 3^4=81, 3^3=27, 3^2=9, 3^1=1orwhen the second difference is constant, students will conclude that the function is quadraticorwhen a function is concave up, the… Continue reading Patterns
Triangle Similarity Update
As you may remember from my previous posts on this subject, I have been thinking a lot about the Common Core approach to secondary-school geometry, specifically the logical switcheroo that makes triangle congruence and similarity a consequence of assumptions about geometric transformations, rather than the other way around. To support this change, I have started… Continue reading Triangle Similarity Update
Fathom: Free for Now!
See below for a letter from Bill Finzer, the leader of the Fathom team, dated June 15, 2014. Fathom is outstanding software for statistics, but also for basic algebra. If you are not familiar with it, I strongly recommend you check it out. It is free for the duration of this school year.Once you have… Continue reading Fathom: Free for Now!
