Pan Balance 2: Fractions Scroll (Fulcrum And Lever)

The Well-sorted Irreducible Fractions

After the development of the MathCats balance and the Balance of Fractions, I realized that the new fulcrum of the MathCats Balance had not been sufficiently exploited.
The fulcrum of the MathCats Scale was passive. Was expressing only the result of the supposed slope of the balance. It was not really interactive.

In the new Android program, Fractions Scroll, the fulcrum of MathCats is already interactive. It responds to the touch, being able to move the fractions with the finger. From left to right (swipe)

This causes the slope indicator corresponding to the chosen fraction (on the right) to be tilted,

Scrolling up and down widens or reduces the range of fractions used in the program: Increases or decreases the maximum denominator and numerator used.

Fractions Scroll Video: https://www.youtube.com/watch?v=bQbdxCfAPEc

Fractions Scroll App: https://play.google.com/store/apps/details?id=com.nummolt.slope.tap

In the paid version: "Fractions Scroll Gravity Lever" this interaction can also be obtained by tilting the device during the fifteen seconds of use of the accelerometers after pressing the corresponding button.
 
In this version, when the accelerometers are running, the bar that indicates the slope corresponding to the chosen fraction always remains horizontal.

Fractions Scroll Gravity Lever Video: https://www.youtube.com/watch?v=F_fH5XMlIRI

Fractions Scroll Gravity Lever App: https://play.google.com/store/apps/details?id=com.nummolt.slope.gravity  

Fractions Scroll Gravity Lever: Also available at Amazon:

https://www.amazon.com/nummolt-apps-Fractions-Scroll-Gravity/dp/B01MZ18BZP/ref=sr_1_1 

Blog about MathCats Balance and Fractions Balance:

http://nummolt.blogspot.com.es/2016/10/pan-balance-cats-and-fractions.html

 

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)

Touch Integers ℤ (+ - × ÷)

Touch Integers is the evolution of the Touch decimals Place value ±. (in the same blog)
Touch decimals could not easily multiply or divide numbers:

I've started my reflections about this 20 years ago:

Is very easy add and subtract graphically. One can regroup the tokens of each order, regroup, carry or borrow tokens, and you can obtain the result in a simulation of abacus.

But not so easy to practice multiplication or division in this visual and interactive way

I looked the inside of the numbers:

Inside the numbers there are the components of the number: The prime factors.

To multiply two integers you must regroup the components of the two numbers.

To divide a integer, you must separate the components.

The program only works with integers. adds, subtract, multiplies and divides (but only exact division) 

Is my latest Android App: 

At Google Play: 
https://play.google.com/store/apps/details?id=com.nummolt.touch.integers
I hope you will find it useful for teaching.

Some animations:

At left: two abacuses (two numbers stacked). 
At right two circles with the prime factors. (two circles with prime numbers stacked)
At right edge: all the prime numbers available to the app. 

To create a number:  Tap on the cells at left. The app shows the number
To add: Drag the tokens from one abacus to the other.
To subtract: Tap the sign key and drag from one abacus to the other.
To multiply: (the numbers must be previously created with the earlier previous steps)
Drag from one prime circle to the other prime circle.
To Divide a number:
Drag the prime numbers outside the prime circle:
Release prime factors to the other prime circle (integer division and multiplication) 
Release prime factors between the prime circles (integer division)
Release prime factors in the list of the right edge: (integer division and erase the prime factor)
Scroll and pick a prime number from the list of the right edge:

And release it in the free zone, or in a prime circle (multiplication) 

Playing with 12*12:

Creating two prime numbers: 
Multiply them. 
Restoring to the original state.
Throwing the prime numbers to the primes list. 

Picking numbers from the big list of prime numbers:
Playing with 2; 3; 5; 7; 11; 13.
1001; 30 and 30,030

(In the current version the top prime number available is 19,874,419)

Multiplication Table with Prime Numbers made with elements from Touch Integers:

 

 

///////////////////////////////////////////////////ACKNOWLEDGEMENTS///////////////////////////////////////////////////////

(Expanded from the Google Play page https://play.google.com/store/apps/details?id=com.nummolt.touch.integers)

Jacobo Bulaewsky: (Arcytech.com (broken)) (12/08/1955 - 08/25/2004)

http://mathforum.org/mathtools/search/index.html?execute=search&database=catalog&keywords=Bulaevsky

There's a software restoration made by Jill Britton.

Brian Sutherland: ( http://www.our-montessori.ca ) Montessori methods adapted to computer. 

(Shockwave Player activities: covering addition, subtraction, multiplication and division) 
Long multiplication: https://www.youtube.com/watch?v=qDMXNjtuqqo
Long division: https://www.youtube.com/watch?v=seJC-1BR_gQ 

Agustín Rayo: (Philosopy professor at MIT) 

And his article about the Prime Numbers, at Scientific American (Spanish version - 02/2010)):

http://www.investigacionyciencia.es/files/3486

"Ladrillos, candados y progresiones.

El fabuooso mundo de los números primos".

Agustín Rayo (2010): http://www.investigacionyciencia.es/files/3486

Ulrich Kortenkamp: (Professor für Didaktik der Mathematik. Universität Potsdam. 

Author of "Place Value Chart" and "Cinderella") http://kortenkamps.net/index.php/Hauptseite 

Place Value Chart: Web page: Stellenwerttafel: http://kortenkamps.net/index.php/Stellenwerttafel

Place Value Chart: (iOS) https://itunes.apple.com/de/app/stellenwerttafel/id568750442 

Cinderella: http://cinderella.de/tiki-index.php

Ulrich Kortenkamp: Place Value Chart

Christian Urff: http://www.lernsoftware-mathematik.de/?p=1331

Math Tablet: https://play.google.com/store/apps/details?id=air.com.urff.test.TeileGanzes

Wendy Petti (Teacher and author of MathCats): http://www.mathcats.com 

Nearly 20 years of support and patience with me. And specially for the vision about the position of the places in "Touch decimals Place value ±" which allowed continuing the work well.

Our first work as a team: OBBL Architecture blocks: http://www.mathcats.com/explore/obbl/obbl.html
And "Place Value Party Cake":

MathCats: "Place value party": http://www.mathcats.com/explore/age/placevalueparty.html

Jeff LeMieux: (Builder, teacher along 35 years and software developer) 
Scripts Web Page: http://syzygy.virtualave.net/

For his work and the assistance in the development of Touch Decimals: Option without negative numbers.

Joan Jareño (From: CREAMAT team) http://srvcnpbs.xtec.cat/creamat/joomla/ 

Author of "Calaix +ie": http://xtec.cat/~jjareno/    http://calaix2.blogspot.com.es/

And History of numbers: Calculus: http://xtec.cat/~jjareno/calculus/

For their help in the last steps in the development of "Touch Integers".
* * *

Added later:
Playing with the app "Touch Integers ℤ (+ - × ÷)":
Exploring Mersenne primes: 2^p -1 (Not all are primes) 
nummolt.blogspot.com/2016/06/exploring-mersenne-primes-2p-1.html
https://www.youtube.com/watch?v=sOnkpDihIS4



  

Touch decimals: Part 1: References

New Touch decimals: free Android app: Part I

Touch decimals,Place value ±

Introduction:

This is the application with a longer history among which I developed in www.nummolt.com.

The first version, made after two failed attempts was published in 1997:

I named it nummòlt, and in fact was the fundational application of nummolt.com.

Nummòlt was a application developped in C ++, and it was my first OOP program:

(Old nummòlt download page)

It was one of many attempts in the nineties to represent numbers graphically to kids.

I will like to remember some of the contemporaries who tried to represent numbers graphically:

Jacobo Bulaevsky: and www.arcytech.org page:

(Rebuilt by Jill Britton and Suzanne Alejandre of MathForum)

http://britton.disted.camosun.bc.ca/baseten/instructions.html

http://mathforum.org/varnelle/howto.html

http://britton.disted.camosun.bc.ca/patterns/patterns_d.html

(Jacobo Bulaevsky died in 8/13/2004)

(Base Ten Instructions)

Brian Sutherland and his: our-montessori.com:

http://web.archive.org/web/20060113011625/www.our-montessori.com/home.html

A rebuild of the Original:

http://www.our-montessori.ca/

http://ethereal.utm.utoronto.ca/our-montessori/math/stampgame.php(needs Shockwave player ')

http://ethereal.utm.utoronto.ca/our-montessori/math/largebeadframe.php(needs Shockwave player ')

(Stamp game)

In collaboration with Wendy Petti of MathCats, in 2003 we did the "Place Value Party"

http://www.mathcats.com/explore/age/placevalueparty.html

With the "place value" shaped as floors of a virtual birthday cake:

(MathCats.com Place Value Party)

In recent years, with the advent of affordable graphic tablets, there are new approaches:

Mainly: Ulrich  Kortenkamp (The creator of Cinderella). The Place Value Chart

https://itunes.apple.com/us/app/place-value-chart/id568750442

(Kortenkamp: Place Value Chart: iTunes)

(Here I must recommend the book: "Early Mathematics Learning: Selected papers of the POEM 2002 Conference": https://books.google.es/books?id=Lui3BAAAQBAJ ) 

And also:
Christian Urff: Rechentablett:

https://itunes.apple.com/de/app/rechentablett/id574825573

(Urff: Rechentablett: iTunes)

After the appearance of all this, I saw the possibility of bringing my previous programs under the way to represent numbers used by Urff and Kortenkamp.

This is my free: "Touch decimals, Place value ±" for Android.

Place value structure seen before, adding tokens with sign, and with the ability to create the neutral element of the sum: pairs of tokens positive - negative that do not alter the final outcome of any addition or subtraction operation.

https://play.google.com/store/apps/details?id=com.nummolt.math.value.place

(Nummolt's Touch decimals,Place value ±)

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