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Geometry Expressions Newsletter
Geometry Expressions Newsletter
August 2013   
Common Core Nuggets
Free apps, created by Larry Ottman using Geometry Expressions.  Each app is directly focused on a specific topic in the Common Core Standards.  Download directly here either as browser apps, as native Android apps (iOS coming soon), or as Lua apps for the TI Nspire. 

If you like the apps, you might like Larry's books:  The Farmer and the Mathematician, The Tortoise and Archimedes and The Farmer and the Mathematician 2.

Geometry Expressions 3.1 SP1 Released
Geometry Expressions 3.1 SP1 has now been released.  It is a free upgrade for any Geometry Expressions 3.0 or 3.1 user.


How to... Create an App with Random Inputs
One of the new input types for apps created with Geometry Expressions 3.1 is random.

Whenever the page is refreshed, an input of type random is given a new, random, value.

To use random inputs in a testing app for the cosine rule:
  1. Draw a triangle and constrain two sides to have lengths a and b, and the angle between them to be theta.
  2. Create a measurement for the length of AC.
  3. Use Draw/Expression to create output measurements for a, b and theta.  (these will be given random values in the app, but having outputs will let the user see their values.)  
  4. In the Variable panel, set the range for each variable.  In the app random values will be generated within these ranges. 
  5. Use File/Export/HTML5/JavaScript App to create your browser app.
  6. Set a, b and theta to have Input type Random. 
  7. Select a Show/Hide button for the output type of the answer. 

 

The resulting app may be viewed here.

A video showing how to create this model is here.

A tns file to load on the TI-Nspire is here.
Problem of the Month
  
This one involves a little physics ...

Equal masses B and C are joined by equal length rods AB and BC of negligible mass. 

A is fixed and C is constrained to lie on an inclined plane.  Gravity acts downwards.

For a given angle of the inclined plane, what is the angle of AB.

Clearly if the plane is horizontal or vertical, the answer is that AB lies vertically.  How about in between?

   

An app which lets you explore this problem is here.



May Problem Solution
Pythagorean generalization   
Given a triangle ABC, parallelograms ADEB and BFGC are constructed. H is the intersection of DE and GF, I is the intersection between HB and AC.  AJ and CK are parallel to HI. J lies on DH and K lies on GH.

 

Show that the area of AJKC is the sum of the areas ADEB and BCGF.

 

Pappus of Alexandria presents this theorem as a generalization of the Pythagorean Theorem.  Show that it is.

   

An app which lets you explore this problem is here.


Proof of pythagorean generalization
The Geometry Expressions Model above shows that the purple parallelogram has an area equal to the sum of the other parallelograms.
pythagorean proof
Setting the triangle to be right and the parallelograms to be square yields a diagram that in itself could constitute a visual proof of the Pythagorean Theorem.

The crucial issue here is to prove that BH, which defines the directions of AK and CJ is in fact perpendicular to AB.  We can then show that the rectangle ACJK is in fact a square.

Easy enough with Geometry Expressions.  By hand proof left to the reader.

 

eBooks
Pythagoras proof
Our Euclid's Elements contains over 120 interactive diagrams.
Available on iTunes

Sample chapters of other iPad eBooks are available here

Interactive math Apps which can be viewed on your iPad are available from Euclid's Muse
11 Geometry Expressions  eBooks are now available as a bundle for the recession-friendly price of $85.95

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