A Classic Algebra/Geometry Inequality Proof Without Words

In case one forgets that math can be beautiful...

Diagram is far from perfect. Hopefully you can make sense of it.


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Explore Inscribed Square in Rt Triangle - CCSS Activity

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A Crazy Ages Problem - Is there a place for this in CCSS

A variation on a classic for your algebra group!

J is as old as K was when J was half the age he'll be in 10 years. If K is y yrs old, express J's current age in terms of y.

Want answer and detailed solution sent to your inbox? By now you should know what you have to do. Join the other teachers who have figured it out...

Reflections...
So you're probably thinking this question is either too hard or irrelevant in the Common Core. I beg to differ!

Before having students attempt to solve this algebraically I would encourage my students to experiment with numerical values. For example suggest a value for y, say y=40 and have them guess some values for x, J's current age. I'm sure you'd agree that this is still a challenging problem but someone will probably guess the correct value, x=30. In fact just verifying that 30 is correct is formidable enough!

Is making a table still a good idea for organizing information? I'll let you decide...

         Present         10 yrs from now

J           x                          x+10

K           y                          y+10

So is there still a value to these "un-real" types of puzzle problems? Do you see the benefits that I do?

f(x)=(25--x^2)(16-x^2) Algebra 2 CCSS Challenge

f(x)=(25--x^2)(16-x^2)
P=product of x-int, Q=y-int of graph of f
|P| - |Q| =?
Generalize!

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Back to School Challenge SAT/CCSS Problem

Posted on Twitter this morning...

(100+99+98+...+51)-(1+2+3+...+50)=?

NO calculator/Noformulas!
Mental math - 30 seconds!

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Free CCSS/SAT/Challenge Math Problems/Solns sent to your inbox?

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An Algebra Puzzle to Start The Year! Grades 6-10

As I posted on Twitter today (@dmarain)...

If I were to give you $50 we'd have the same amount. If you were to give me $50 I'd have 9 times as much as you. How much do we each have?

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Other comments welcome.

Reflections...

1. This was not intended to be highly challenging. It might engage students early in the year and I designed it to be accessible to most.

2. The language is open to interpretation by design! We want students to feel some disequilibrium. But we don't have to resolve ambiguities. I let my students do that among themselves.

3. Before jumping into an algebraic solution, I would allow my students to experiment with numbers - call it "plug in" or Guess-Test-Revise. After all on standardized tests this is what many will do in spite of all the algebra we teach!

Never ASS-U-ME in Geometry: A Triangle Problem to Get Them Thinking!

Not quite back to school for most but the problem above might prove interesting to review some geometric/deductive reasoning.

For new geometry students, replace 'a' by a value, say 40, and ask them to fill in all the missing angles. Most should deduce that angle 5 = 50, but my educated guess is that many will assume b = 40, so
angle 5= angle 6 = 50 and angle 3 = angle 4 = 40. From there to angle 1 = angle 2 = 50, so
angle 2 + angle 3 = 90. QED!  Not quite...

Well, the '90' is correct but the reasoning is another story! So this is all about justifying, checking validity of mathematical arguments, sorta' like some of the Eight Mathematical Practices of the Common Core!

In fact, you might ask them to redraw the diagram, keeping the given conditions but making it clear that b does not have to be 40 and that Angles 3&4 also do not have to be 40!

My Rant on MathShare 8 yrs ago

Haven't looked at the Yahoo group, MathShare I moderated several years ago. Phased it out when I started this blog.

Best way to describe what appears below? How about, " The more things change, the more they stay the same." Sad, but this could've been written today, except I'm retired...

From MathShare

Here's a novel thought...
I'd like to start a thread to explore what seems obvious to all of my teachers and probably to you and all of your colleagues as well. Not one of the teachers in my department teaches skills in a vacuum as in 'kill and drill.' They demand that students take notes and practice with many exercises until the concepts/skills are set in place and then come back to it in another context later on. They ask many questions in class to explore a topic, deepen conceptual understanding and assess: 'Show me why the absolute value of (x-1) is not x+1! How can you demonstrate this is not always true. Turn to your partner and convince each other!"
Technology enhances all of this but technology, standards, standardized tests and a plethora of new reports from various curriculum groups and governmental agencies WILL NEVER LEAD TO IMPROVEMENT of learning in the classroom and you all know why!
It's time to stop the political rhetoric and address what is really going on in the classroom. Effective teachers have always been effective. These are teachers who ravenously explore new instructional strategies, read everything they can get their hands on and then decide what will work best for their students. They are not mired in the the past nor are they easily swayed by buzz words or glitz. They are open to change but they will never abandon FIRST PRINCIPLES of learning. They will always be here after all the experts are long gone. ISN'T IT TIME WE CELEBRATE THEM AND USE THEM AS THE TRUE MODEL OF EDUCATION. THEY ARE THE GOLD STANDARD! Their students certainly know the truth: "She was the hardest and most demanding teacher I ever had. But boy did I learn!"

C'mon – don't be afraid to share 'self-evident' truths!! After all, someone has to tell the emperor he is naked! Ok, my rant is done!
Dave Marain