Pan Balance 1: Cats and fractions

Usually in elementary mathematics, teaching pan scale balances are used only for display them at the time of equilibrium, to verify that two quantities are equivalent.

In this case, apart of this use, it is also useful imbalance in the balance. 

Many years ago (2003), Wendy Petti of MathCats.com and me, we developed the "MathCats Balance":


" choose from a wide range of objects to place on this scale - from electrons to galaxies! " 
" So how can we balance thin cats with fat cats? You might try multiplying each side by the number shown on the opposite side of the balance. Will 2 x 6 thin cats balance with 5 fat cats? Yes, 12 thin cats do balance with 5 fat cats".

MathCats balance: Android version (youtube)

MathCats Balance App (Google Play)

MathCats Balance (Amazon) 

Many years after this, I developed under the same idea a pan balance of fractions.
Inspired in a old photograph of Maria Montessori and his son Mario:

 ( from Getty Images: www.gettyimages.es/fotos/maria-montessori )

Pan balance to weigh fractions:

This imbalance, when the imbalance ratio under certain conditions is proportional to the ratio of content of the dishes, is also the result of the division. The slope of a straight line. In this case, the result of the division of fractions.

To view it, you can multiply the contents of each dish, until the balance is obtained, the numbers for which has multiplied each dish are in turn the result  fraction of the division.
The program only multiply by prime numbers, because any number can be built with them.

 This is the "Fractions Balance" Android App:

 Fractions Balance (Youtube) promotional video

 Fractions Balance App (Google Play)
Fractions Balance App (Amazon)

Comparing: 
1/2 + 1/3 + 1/6 
with: 1/3 + 1/5 + 1/6 + 1/8 + 1/10 + 1/12
And with: 1/3 + 1/4 + 1/5 + 1/8 + 1/12

 Comparing very near fractions addition (youtube)

I hope it helps on teaching division of fractions.

Place Value Time

After the experience of my old "Time Calculator" (for mathcats.com 2004) I tried to develop an app that explains how this webpage subtract time.

Over the years I have received all kinds of comments on how to make this calculation.
This time, I tried to explain graphically how the original application does.

And I think I've managed to build what should have been the original application.
What was missing was to be able to modify the difference between the two dates.
(and save last configuration) 

I must thank all those who have been making comments over many years

Here you can find the new: "Place Value Time"  (for Android devices) 
( https://play.google.com/store/apps/details?id=com.nummolt.time.place.value  ) 

 Here is the video of the 'Place Value Time' Android app:

 In the video:

• - From countdown to elapsed time.
• - my age.
• - when I should have celebrated my 500,000 hours of life.

The app is available on Google Play, Samsung Galaxy Apps and Amazon.
Place Value Time - Google Play
Place Value Time - Amazon
Is a paid app.(the paid version of the previous 'Time Calculator' (Android)

Explore Ahmes 2/n table RMP and Akhmin n/17 table with 'Proper Fractions'

This summer, by accident I developed the 'Proper Fractions' app. (Pr.Fr. Amazon)

I was developing a new version of the 'Equivalent Fractions' But the development has been stopped for unknown reasons.
This left on my hands a circle, a square and a number line of fractions, all of them interactive.

Surprisingly for me, it proved to be very useful for exploring in unit fractions.
One issue that concerned me since I developed "Old Egyptian Fractions" for Mathcats.
I decided to do some videos to show it. It is a collection in a YouTube Playlist:
https://www.youtube.com/playlist?list=PLo4AMY8jDHYZbDL_FIAAeA6qkY3-X-oDC

The main videos in the list:
"Explore The Ahmes 2/p table of the Rhind Mathematical Papyrus": 

Explore the Akhmim n/17 Table:

Proper Fractions It is a tool that can be useful in solving problems of Egyptian fractions:
Like the MathCats game:
http://www.mathcats.com/explore/oldegyptianfractions/index.html

More on Egyptian Fractions:
References:
Milo Gardner:
http://mathforum.org/kb/message.jspa?messageID=5527254
Egyptian Unit Fractions:
http://www.mathpages.com/home/kmath340/kmath340.htm

I have only to add as a thank you: 
All this would not have been possible without the help and support of: +Milo Gardner (cryptanalyst)

Registered in MathTools (MathForum) http://mathforum.org/mathtools/tool/272979/ 

Exploring Mersenne primes 2^p -1

Just a note:
My app "Touch Integers ℤ (+ - × ÷)" is useful to explore the Mersenne primes:
Here there's a tiny exploration from 2^2 -1 to 2^23 -1.
We found all numbers prime, except when p==11 and p==23
Like everybody knows:

The search is made with:  "Touch Integers ℤ (+ - × ÷)" Android app:

https://play.google.com/store/apps/details?id=com.nummolt.touch.integers

http://www.nummolt.com

///////     UPDATE:  (August 2016)    ////////

Same exploration on Wagstaff Prime Numbers:  

 

 Woodall Primes:

 

     

Euler's Lucky numbers:

 

   

And Goldbach's Conjecture:

 
 

The exploration has been made with:  "Touch Integers ℤ (+ - × ÷)" Android app:

https://play.google.com/store/apps/details?id=com.nummolt.touch.integers

http://www.nummolt.com

Fluidic computers - Logic gates

In an innocent way, and mainly based on my ignorance, in 2001 I developed an online toy about building tiny computing objects with logic blocks:
Is my OBBLOG

Ignoring the equivalences of the Karnaugh table, OBBLOG contains all the possible logic gates with two inputs and one output.

This is the partial living OBBLOG Truth Table:

(image from the obblog2 G+ community)

For to build a operating toy, I added forks, crossings at different levels and  bit visualizers. (Similar to a light bulb)

However, the toy did not have flip-flops, so you could not make a numbers counter, although small binary calculators 1 to 16 bits.

The toy allows to build little and simple animated toys:

(digital fireplace)

Or more complex constructions:
Binary Tree:

(OBBLOG binary tree)

But from the initial basic logic gates, I discovered that it was possible to build a flip-flop:
(And consequently a counter of binary numbers):

 (The obblog allows a closed loop that continues 

to circulate indefinitely if not obstructed)

Thanks to this, the obblog could become able to build calculators calculating in binary system and be able to show results in decimal system. But being only in 2 dimensions can never bequeath to do what you can get to perform with Minecraft. (Koala steamed) (see: official Minecraft logic gates)

(OBBLOG feature video)

( 4 bits + sign adder with overflow control + hexadecimal digit with sign )

(without flip-flops)

( work process  with obblog )

I posted all this, because yesterday I found similar artifacts along the history of computing:
The article published in Scientific American the December of 1964:

Fluid logic devices, all based in the Coanda effect: 

( from: http://miriam-english.org/files/fluidics/FluidControlDevices.html )

(Fluidic amplifier: https://en.wikipedia.org/wiki/Fluidics)

FLUID CONTROL DEVICES:

FLUID CIRCUIT: performs the operation of dividing 

by 10 in an all-fluid digital computer

And the conference at MIT:  
Referred by:  Fluidic computing at Bowles Fluidics: 
http://strangebits.blogspot.com.es/2007/05/i-had-pleasure-of-participating-in.html

Coding and computation in microfluidics.
http://cba.mit.edu/events/07.05.fluid/

And the conference from Manu Prakash, 
http://cba.mit.edu/events/07.05.fluid/Prakash.pdf

The manu Prakash Thesis: (About "microfluidic bubble logic"), and inside this beautiful Truth table about logic gates and its equivalents in transistors, valves, electrical devices, and fluidics:

(from http://strangebits.blogspot.com.es/2007/05/i-had-pleasure-of-participating-in.html)

Some examples of projected fluidic devices:
(from Prakash thesis)
 Integrated fluidic logic gates with a schematic integrated control circuit 

  
The main patents about fluidics, come from the sixties:
https://www.google.com/patents/US4854176
(from wikipedia)

And schema about Coanda effect from Popular Science (Jun 1967 Pag 118) : 

And the:  Fluidics: Basic components and applications 

By: James W. Joyce  (1977; Unclassified: 1983): 

http://www.dtic.mil/dtic/tr/fulltext/u2/a134046.pdf

http://www.dtic.mil/dtic/tr/fulltext/u2/a134046.pdf

And from the conference:

APPLIED HYDRAULICS AND PNEUMATICS U5MEA23 Prepared by Mr. Jayavelu.S & Mr. Shri Harish Assistant Professor, Mechanical. http://slideplayer.com/slide/5675986/

I would like to see these companions of my program, running.

Anyway, everybody can see them inside the Bowles products:

 

 

 

And inside of the Theranos patented devices: (Lab-on-a-chip)

Fluidic medical devices and uses thereof

 //////////////////////////////ADDED 2019/09/30//////////////////////////////////////

An the works of William Phillips (economist)

https://en.wikipedia.org/wiki/William_Phillips_(economist)

His contribution to Economics and Mathematics: The Phillips Curve:

William Phillips and his: MONIAC (Monetary Income Analogue Computer) Also known as: Phillips Hydraulic Computer:

William Phillips and his MONIAC

The Moniac machine is a dynamic model of a working economy:

Img: From:  Moniac machine is now on loan to the Reserve Bank museum in Wellington

"Phillips ended up at the London School of Economics (LSE) and became very interested in Keynesian theory. It was at this time that he built the Moniac machine. In 1958, Phillips published his seminal work on the relationship between inflation and unemployment. He moved back to Australia  in 1967, before passing away in Auckland in March 1975."

"At LSE [1] Phillips was interested in the circular flow of money model. With his knowledge of hydro-mechanics, Phillips realised that he could build a machine that would take the static circular flow model out of the textbook and into a dynamic 3 dimensional setting, by using water to represent the flow of money. Using a diverse range of materials including bits and pieces from obsolete Lancaster bombers, the first Moniac was created in his landlady’s garage. It was built at a cost of £400 – over $32,000 in today’s money."

"The machine was unveiled in a seminar at LSE in 1949, and Phillips explained how his machine could be used to demonstrate the complex interrelationships between macroeconomic variables such as consumption, taxes, government spending, investment, savings, interest rates and exchange rates. A second machine was built to represent the rest of the world and introduce trade flows into the domestic economy."

Making Money Flow: The MONIAC YouTube: 

reservebankofnz: (Reserve Bank museum in Wellington)

 
From: LSE’s James Meade Archives:

Here is a schematic drawing of MONIAC, courtesy of LSE’s James Meade Archives, with more detail explaining the machinations  (From: http://nautil.us/blog/the-rube-goldberg-machine-that-mastered-keynesian-economics )

A clearer, more modern schematic of the device

 From:

Nicholas Barr, in A.W.H. Phillips: Collected Works in Contemporary Perspective by Robert Leeson 

More:

 

A Brief History of Liquid Computers

By: Andrew Adamatzky

 

 

///////////////////////////////////////////////////////////////////////////////////////////////////

More serious and basic information: Introduction to fluid Mechanics: (Dead Link)

https://vscht.cz/uchi/ped/hydroteplo/materialy/introduction.fluid.mech.pdf  Y. Nakayama (1999).

(Link at Archive.org:  https://web.archive.org/web/20180814192332/https://vscht.cz/uchi/ped/hydroteplo/materialy/introduction.fluid.mech.pdf )

Follow: Reflections about all this at: (Dead Link) https://plus.google.com/communities/102467630739607363236 

(Link at Archive.org: https://web.archive.org/web/20190330044111/https://plus.google.com/communities/102467630739607363236 )

[1] LSE: London School of Economics.

Dancing numbers: (Math Circus)

Dancing numbers: Numbers Circus: Modulo choreography.
( And:  Circus for to tame numbers, too )

After the post dedicated to dancing sort algorithms,  I think it's time to show a program made and published in 2004, which it could be useful in teaching elementary mathematics.


Is the "Dancing numbers" 

Originally made for MathCats.com , it never has been published. Finally I posted it on my page so do not be missed for the moment.

The program allows you to manipulate a lot of integers (up to 2000) with some simple rules: decide the number of columns that are to line up neatly from lowest to highest.
(Also possible to arrange smaller amounts of numbers)

The program allows for reflection on the divisibility of numbers and remainders:

It is also advisable to use the mode 9 columns, that shows the rule of divisibility for 9.
(The sum of all members of any number equals the heading number of the column)

And all the rules explained in https://en.wikipedia.org/wiki/Modular_arithmetic
(related to addition, subtraction, multiplication and division (sometimes), and congruence).

Here is a sample of how the program works:

(Dancing numbers: youtube)

You can use the application here: http://www.nummolt.com/obbl/dancingnumbers/modular_xtd_01.html

I hope its use will be useful to teach elementary mathematics.

Noting mathematics education in the world, from the statistics page of my Android apps

This is just a small observation, based on my own stats:
As the numbers that I show are not too high, observation may not have enough value.

Typically, the math apps downloads have a proportion of users fairly well distributed among all countries. However, there is always a predominance of the US downloads:

Example graph of a usual app of active users statistics:

(humble numbers related to a calculator)

In my  Google Play page, there is one of the oldest applications that I've done, devoted to linear algebra and the  Gauss-Jordan elimination as a method of solving two equations with two unknowns.

For the application to be a little more attractive, I called it "Adding apples and oranges", referring to an equation are getting really different variables to give a result.

(Apples and Oranges free app)

The proportion of current statistics of this application are quite different from other applications. There is a predominance of US downloads:

(humble numbers related to a Apples and oranges app)

While this difference so marked, it makes me think:

It could be that the education authorities of the US had greater knowledge of applied mathematics than education authorities of other countries?

Knowledge of the intellectual basis of the information revolution which we live has influenced the educational guidance of the US?

Or is the result of many parents with deep and old computer knowledge?

What is it that justifies this big difference?

I've searched Common Core related items with the elimination of Gauss-Jordan:

Here are some examples about the related items:

CCSS.MATH.CONTENT.8.EE.C.8 - A, B, C.
CCSS.MATH.CONTENT.HSA.REI.C.6
CCSS.MATH.CONTENT.HSA.REI.D.12

http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf

http://www.corestandards.org/Math/Content/EE/

I can not draw a conclusion. Just leave the data in case anyone is able to solve these questions.

(Apples and oranges, 1 decimal version (paid version))