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.

Touch Fraction ℚ (1.4.x)

New version of Android App "Touch Fraction ℚ"  (version 1.4.x and up)
https://play.google.com/store/apps/details?id=nummolt.touch.fraction


Evolution of the app over the years: (1993-2015)
In 1993 I developed the first executable (for Windows 3.1) of Touch Fraction ℚ (Racional.exe):

http://www.nummolt.com/rebost/index.html

It was my first attempt of to show an interactive representation of Rational numbers as fibres in (Z * Z {0}) and arrangement in Q.
As the Rational representation in Wikipedia:  Rational Numbers:

 https://en.wikipedia.org/wiki/Rational_number

In 2013 I made the new version of the app: The translation of the original executable to the Android OS.

The interaction became extended to the fraction:

The pie fraction to navigate across the fractions, and the rational representation to navigate across the equivalent fractions. (and select the range of available denominators)

In 2015 after the development of Touch Integers ℤ (+ - × ÷) (https://play.google.com/store/apps/details?id=com.nummolt.touch.integers)

This time, I was able to afford the next step:

Explain fractions and rational numbers as the prime factorization of its members: numerator and denominator.

This is the new version of Touch Fraction ℚ:

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

Interact with the fraction, build fractions adding or removing prime factors in the numerator or in the denominator.

Simplify fractions dragging common prime fractions to the "common" zone, and invert fractions with the "^-1" button.  

Touch Fraction ℚ is a complete tool to understand positive fractions, negative fractions, positive and negative numerators, positive and negative denominators, equivalent fractions, and inverted fraction.

(Video of the app, at: Google Play)

Fractions.... ugh !

I have read many complaints online from math teachers.
Specifically on Linkedin [sci Math primary/secondary education]:
http://www.linkedin.com/groups/Fractions-ugh-69765.S.5941088157259800577

I would like much to some expert tell me if I'm wrong!.
There (at the SciMath group of the Linkedin) I wrote this:

I'm not an expert, and surely I am wrong in thinking that I will develop.
Anyway I'll try, and I'm sure I will not say anything new, but I'd do at least a
little humor:

I think we are talking about the understanding of the fundamentals of elementary
mathematics:
If you have not settled well the fundamentals of elementary mathematics, when it
comes to fractions can not understand anything. (And I'm not oblivious to this
state)
The difficulty of the fractions is that it requires an understanding of the
division.
And the division, requires an understanding of subtraction, and ultimately of
the sum.

Electronic calculators that are available to all people, both children and
adults do not differentiate between operations: all are on the same level
represented by a key with a painted sign. But conceptually, the knowledge
necessary to understand each of the elementary operations should be
superimposed, and with a particular order. (Although sometimes, as a first
track, in some calculators the sum key is larger)

First we must become familiar with numbers, identified and recognized.
And it seems that we are forced to choose the base 10 by a consensus not really
know where it came from.

From this point, propose a specific order to teach children basic
operations:
As is evident, we must begin by the sum.
But the next step, not subtraction.
The next step is multiplication.
That is, from knowledge of the positive integers, to play with them.
And games and experiments can be the sum, or multiplication.
The amount is usually depicted as a line, that is a dimension, although not
mandatory.
Multiplication games, involve working in two dimensions.
Squares and rectangles: Here, as a game that takes place in a plane or a sheet
of paper should work both multiplication as operation.
I think it's important vision of multiplication as an array of big dots in a
plane.
Here you should begin to explain about numbers that can be obtained from a
multiplication: it should be clear that there are some numbers that can not be
the result of a multiplication.
At this point, children know and can recognize all the numbers perfectly, and
can clearly see that there are numbers that can not fully populate a square or
rectangle.
I think at this time of learning the knowledge of the concept of prime number is
as important as knowledge of multiplication tables. (The prime numbers are the
numbers that you never will find as a result in any multiplication table, and at
the same time any number can be obtained by multiplying prime numbers between
them). The prime numbers are the basic bricks of the complete construction of
the numbers building.

And then I think you have to start again:
We must expand the knowledge of numbers.
In our world, we are used to seeing positives and negatives:
Are protons and electrons, positive and negative charges, matter and antimatter,
attraction and repulsion, money in the bank and debt (currently precisely, more
debt than money), in the old photographic culture negative film, and copies
positive, and finally even the worst example: positive and negative
temperatures.

After familiar with the expansion of natural numbers to integers, you can begin
to explain a new operation: subtraction.
And the rest can use all the analogies offered by nature to experience:
But there is one that is best suited: Matter and antimatter.

Everybody knows that matter and antimatter destroy each other.
And simultaneously, you can create equal amounts matter and antimatter at the
same time from nothing (they can not humans, but somewhere in the universe
happens: To be clear I'm doing a simplification that would make hair tip to a
physics expert)
Imagine games where positive numbers are eliminated with the negative, or the
creation of equal amounts of positive and negative elements from nowhere, with
no any change in the effective amount of all elements above board (the sum of
all them).
Even I have examples in software made by me some years ago.

And I think this is the time to speak of zero. Zero is both "nothing" and also
an undetermined but equally positive and negative elements or matter and
antimatter, pennies in the pocket and debt.

I think the point where we are, the films they have seen children, and games
that will surely develop or imagine, no child will miss in a reflection that use
matter and antimatter as imaginary tool for learning.

At the same time, we could start with the mechanics of operations as it has
always has been done: Subtraction with borrowing and the usual mechanics to
solve these operations.
But with the confidence that the result will always be within the set of the
known numbers. Because we are familiar with a result of both matter and
antimatter, money or money we owe.

We know to add and multiply, and we know subtract.
All the way I'm trying to go, we must never forget that mathematics is a fun
game, and a place to experiment:
Although not essential not think it be reflected by any curriculum, we must not
forget that we have not tried the multiplication of positive and negative
numbers, separated, or mixed. As representable integer multiplication above a
plane, the results are evident. And I guess as a fun game, yet I believe I can
assure you that with correct results guaranteed, although I can understand the
discomfort that may feel a teacher, helpless in front such unusual
situations.

And when we got here, we are ready to face the division.
The result of a division is a sharing out, it is also a proportion.
We're not prepared to afford the mechanism of decimal division.
We would have to broaden our knowledge of numbers:
At this point, we only know the positive and negative integers: natural
numbers.
Before starting to split, one must know the integer division: the sharing
out.

At this point, the division is the inverse operation of multiplication.
When we have the result of a multiplication in one plane (a rectangular matrix)
we can see directly the result of two divisions simultaneously. A square
displays the result of two splits. A square, the same sharing out twice.

Experimenting with splits that to make it easier, at first by simplicity should
be positive numbers, you can reach to see a problem with the distributions of
numbers that are not the result of a multiplication already met the first heard
of multiplication: the prime numbers.
Therefore, we must expand for the third time knowledge of numbers. Must
integrate rational numbers, the fractions.
Each fraction is the representation of a new kind of number: A number that needs
two numbers, one above and one below (or where you want to put). And are both as
a ratio and the description of a sharing out. But in the educational process
should be clear that they are a new type of number: which allows to get the
result of the inverse operation of multiplication for numbers that we could not
arrive until now.

I'll stop my thoughts here.
The explanation about fractions extremely lengthen this already too long
letter.
And I will not do. From now just do a sketch, but I think I managed to
substantiate the basis necessary to prepare the fractions with a little more
comfort than detected in this dialogue so interesting:

At this point, with the basis described, we would have the ability to understand
a little better the fractions, equivalent fractions, the role of prime numbers
in simplifying fractions, and the ease with which we have faced the positive
numbers and negative, integrate them into reflections on fractions, if only to
play but to see also that mathematics is a solid and coherent construction that
which is far from being threatened by the antics that can devise a kid.
And then face the complex world of operations with fractions.
And I think we should stick with the same order we have followed so far:
The sum of rational numbers, multiplication, subtraction and division.
And once experienced this, or at the same time addressing the division with
decimals.
The decimal number as a result of a division is somewhat flawed alternative way
of expressing a rational number also called fraction.

Greetings, and thanks for the opportunity to participate in this dialogue.
This dialogue has been very useful, because it gave me the necessity of
writing, and writing it, I think I have clarified my thoughts a bit. Thank You.
And thanks for the patience if you arrived here.

Adding unit fractions +

 New Android app:  Adding Unit Fractions +

4/5 

The app proposes 21 challenges to overcome.
Obtaining the proper fractions listed at the top of the application, 

adding two or three unit fractions.

Each proposed proper faction has a variable number of solutions.
And different levels of difficulty
You can not repeat unit fractions with the same value.
In the app you'll find a button to delete all the solutions found in the current problem, and to start from scratch.
The littlest unit fraction used in this app is 1/28.
The program is designed to show the usefulness of the subtraction of fractions in solving such problems.

Some hints: 

In the Rhindt Mathematical Papyrus (RMP) in 1650 BC the scribe Ahmes
copied the now-lost test from the reign of the king Amenemamhat III .

The first part of the papyrus is taken up by the 2/n table.

The fractions 2/n for odd n ranging from 3 to 101 are expressed as sums of unit fractions. 

In this app you can build some of the Ahmes  decompositions ( 2/3 , 2/5,2/7, 2/9 ) and 

the discarded ones by him also.

The app allow to decompose also: 

3/4, 3/5, 4/5, 5/6, 3/7, 4/7, 5/7, 6/7, 3/8, 5/8, 7/8, 4/9, 5/9, 7/9, 8/9, 3/10, 7/10, 9/10.

You can use the knowledge acquired solving the 2/X decompositions to solve the rest of the problems

.....   

At first glance we can try the most elementary mechanisms: Subtraction of an essay:

2/3:

2/3 - 1/2 = 4/6 – 3/6 = 1/6; 2/3 = 1/2 + 1/6.

2/3 - 1/4 = 8/12 - 3/12 = 5/12; 2/3 = 1/4 + 5/12 = 1/4 + 4/12 + 1/12 = 1/4 + 1/3 + 1/12.

2/5 – 1/3 = 6/15 – 5/15 = 1/15; 2/5 = 1/3 + 1/15.

2/5 – 1/4 = 8/20 – 5/20 = 3/20; 2/5 = 1/4 + 3/20 = 1/4 + 2/20 + 1/20 = 1/4 + 1/10 + 1/20.

2/7:

2/7 – 1/4 = 8/28 – 7/28 = 1/28; 2/7 = 1/4 + 1/28.

2/7 – 1/6 = 12/42 – 7/42 = 5/42; 2/7 = 1/6 + 5/42 = 1/6 + 3/42 + 2/42 = 1/6 + 1/14 + 1/21

2/9:

2/9 – 1/6 = 4/18 – 3/18 = 1/18; 2/9 = 1/6 + 1/18.

To solve the basic problems of the 2/n table, Milo Gardner 

in the Wolfram's Math World suggests this basic rule first published in 2002:

2/(p*q) = (2/A)*(A/(p*q))  where A=(p+1)

Rule applied: 

2/(3*1) = (2/(3+1))((3+1)/(3*1))= (1/2)*((3/3)+(1/3))=(3/6)+(1/6) = 1/2+1/6.

2/(5*1) = (2/(5+1))((5+1)/(5*1))= (1/3)*((5/5)+(1/5))=(5/15)+(1/15) = 1/3+1/15.

2/(7*1) = (2/(7+1))((7+1)/(7*1))= (1/4)*((7/7)+(1/7))=(7/28)+(1/28) = 1/4+1/28.

2/(9*1) = (2/(9+1))((9+1)/(9*1))= (1/5)*((9/9)+(1/9))=(9/45)+(1/45) = 1/5+1/45

2/(3*3) = (2/(3+1))((3+1)/(3*3))= (1/2)*((3/9)+(1/9))=(3/18)+(1/18) = 1/6+1/18

From the basic 2/n table, we can afford easily the solution of the other problems:

3/4 = 2/4 + 1/4 = 1/2 + 1/6 + 1/4

3/5 = 2/5 + 1/5 = 1/3 + 1/15 + 1/5.

4/5 = 2/5 + 2/5 = 2/3 + 2/15 = 1/2 + 1/6 + 2/15 = 1/2 + 5/30 + 4/30 = 1/2 + 6/30 + 3/30 = 1/2 + 1/5 + 1/10.

5/6 = 4/6 + 1/6 = 2/3 + 1/6 = 1/2 + 1/6 + 1/6 = 1/2 + 1/3.

3/7 = 2/7 + 1/7 = 1/4 + 1/28 + 1/7.

4/7 = 2/7 + 2/7 = 2/4 + 2/28 = 1/2 + 1/14.

….........