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".
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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 5. Collaboration

The 07/04/2015 after some days from the first post of "Touch decimals":
https://play.google.com/store/apps/details?id=com.nummolt.math.value.place 
I received this animation from JeffL (member of MathTools & author of: 
"Jeff's Interactive K12 Math Scripts Page") :

http://www.nummolt.com/LeMieux.gif 

This led me to develop an alternative program with its version.
And I added it to my original program.
With translation between the two systems.
(One with 'Tokens' signed and  'Places' unsigned, and the other with color
'Tokens'  unsigned and 'Places' signed).
I think the program has been enriched by the two visions.

The pre-release videos:

Borrowing in subtraction

Borrowing in subtraction and the color of Tokens

Transition between the two methods

The YouTube Video about the posted version (July 15 2015): Touch decimals v.1.1.0 r.4:

Touch decimals promotional video

I should also like to thank Wendy Petti for comments made in the early development of this program.

From 2015/09/1 There's a online version of "Touch decimals Place value"  JavaScript version:

www.nummolt.com/touchdecimals/index.html 

Touch decimals: Part 4: Borrowing in subtraction. Carry in Addition.

Touch decimals Place value ±

Borrowing in subtraction:

Carry in addition:

MERGED 2016/11/10:

And from a dialog on Twitter: https://twitter.com/macajo/status/793790918788837376

The following Video:  Subtraction: 534-486:

 

Touch decimals Part 3: Basic interaction

Touch decimals,Place value ±

Basic interactions:

Video:

Touch decimals: Part 2: Tokens and Places

New: Touch decimals: free Android app: Part II

Touch decimals,Placevalue ±

Main components of Touch Decimals:

The main board with two charts of identic number of inner rectangles.

Each chart with:

A number with dot that represents each one of the two numbers.

A button to change the sign of all the inner components of each chart.

A row of a variable number of rectangles named: “Places”

A variable number of big dots (with numbers inside), named: “Tokens”.

In the “settings” button or option button: the way to refresh the charts.

PLACES:

A place is a rectangle, and a rectangle has 4 borders:

The top and bottom border may limit another equivalent place, or an abyss where can fall tokens, and become lost forever.

A token is free to cross any of those boundaries under its responsibility.

The left border is impenetrable:

Only it is authorized by the passage of a token if it gets the support of 10 tokens of the same type.

When the 10 tokens have crossed the left border, become one token

The right border:

Right border can be crossed freely

If a token cross the right boundary, it becomes 10 tokens of the same type in the next place.

The maximum reccomended number of tokens into each place is: nine.

TOKENS:

A token can be either positive or negative

A positive token is born with the tap of a finger on the screen

A pair of positive-negative tokens is born with a tap of two fingers on the screen

A token disappears if it crosses the top or bottom border.

Positive and negative tokens have the same behavior:

A token can cross at will the intermediate horizontal border

When a token crosses the border on the right, it becomes 10 tokens.

A token need 9 companions from the old place to cross the border to the left.

Positive and negative tokens may be in the same place. 

But if two tokens of different signs meet, they cancel each other.

Touch decimals,Place value ±

This is how the tokens live in their places in "Touch decimals, Place value ±"

The behavior of places and tokens of this app describes the behavior of the numbers in the decimal system.

Development:

All in this text: charts (twins), places and tokens also describes the internal structure of the app and his internal java classes:

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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