CITE THIS NOTEBOOK: Wolfram Cloud as an interactive tool to support secondary school teaching by David Nocar, Jiri Vaško and Tomáš Zdráhal. Wolfram Community DEC 13 2022.
ORIGINAL ARTICLE: D. Nocar, J. Vaško, T. Zdráhal (2022) WOLFRAM CLOUD AS AN INTERACTIVE TOOL TO SUPPORT SECONDARY SCHOOL TEACHING, ICERI2022 Proceedings, pp. 3133-3137. https://dx.doi.org/10.21125/iceri.2022.0783.
Project page: https://www.researchgate.net/project/Computer-Based-Mathematics-Teaching-at-Elementary-School.
FULL ARTICLE: https://wolfr.am/InteractiveTeaching.
ORIGINAL ARTICLE: D. Nocar, J. Vaško, T. Zdráhal (2022) WOLFRAM CLOUD AS AN INTERACTIVE TOOL TO SUPPORT SECONDARY SCHOOL TEACHING, ICERI2022 Proceedings, pp. 3133-3137. https://dx.doi.org/10.21125/iceri.2022.0783.
Project page: https://www.researchgate.net/project/Computer-Based-Mathematics-Teaching-at-Elementary-School.
FULL ARTICLE: https://wolfr.am/InteractiveTeaching.
Article Abstract: Lately, in the Czech Republic, there is a very current demand for the development of students’ digital literacy in the teaching of mathematics, while there are more and more digital tools on the market that can be used for this purpose. And we’re not just talking about the hardware side, but the software as well. There is a lot of software that is free, but the question is how applicable it is in education. In particular, it is about the software being intuitive and running ideally through a web browser. The use of mathematical software in school mathematics is a current global trend, which is supported, for example, by the Wolfram Research company mentioned here. The purpose of the investigation was to verify the fact that step number three of the four-step scheme for solving mathematical problems (see Conrad Wolfram’s book - The Math(s) Fix: And Education Blueprint for the AI Age) can really be successfully implemented, for example, using the Wolfram Cloud.
In this post, we will discuss the Wolfram Cloud tool from Wolfram Research, which is behind the well-known not only mathematical software Wolfram Mathematica. We tested this tool on 3 groups of pupils at a secondary school in the Czech Republic. In total, there were over 40 pupils who took part in the survey. They were 7th and 8th graders. The goal was to show the pupils prepared materials in the Wolfram Cloud environment on the topic of least common multiple, greatest common divisor and Diophantine equations. The pupils had already covered the curriculum, they should have the appropriate mathematical apparatus to solve such problems and they should also already have experience with the Wolfram Cloud platform. Each material contained a basic introduction to the given mathematical apparatus, but also interactive elements of examples, and the emphasis was placed on the element helping pupils to understand the problem and to be able to derive the consequences of changing the parameters entered into the interactive element.
The pupils devoted themselves to the activity as part of the project day one morning, when they had enough time to go through everything and consult with the teacher. An investigation was subsequently carried out with the pupils in the form of an interview (joint discussion) to obtain feedback.
Despite the initial fear of the activities, all the pupils actively participated in exploring the materials (Wolfram Notebooks, technologies and features), and in the end there was a very pleasant atmosphere in the class, when the students themselves came up with what they could use the given tool for and how the visualization of the task helped them. Subsequently, the pupils started editing the materials themselves and testing for other examples that they came up with.
So it can be summarized that most of the pupils were excited about working with this tool, but what is more important, all pupils were actively involved in research with this tool and experienced success there themselves in the form of solving mathematical problems. This approach corresponds both to the development of pupils’ digital literacy and to the constructivist approach known as Inquiry-based teaching.
In this post, we will discuss the Wolfram Cloud tool from Wolfram Research, which is behind the well-known not only mathematical software Wolfram Mathematica. We tested this tool on 3 groups of pupils at a secondary school in the Czech Republic. In total, there were over 40 pupils who took part in the survey. They were 7th and 8th graders. The goal was to show the pupils prepared materials in the Wolfram Cloud environment on the topic of least common multiple, greatest common divisor and Diophantine equations. The pupils had already covered the curriculum, they should have the appropriate mathematical apparatus to solve such problems and they should also already have experience with the Wolfram Cloud platform. Each material contained a basic introduction to the given mathematical apparatus, but also interactive elements of examples, and the emphasis was placed on the element helping pupils to understand the problem and to be able to derive the consequences of changing the parameters entered into the interactive element.
The pupils devoted themselves to the activity as part of the project day one morning, when they had enough time to go through everything and consult with the teacher. An investigation was subsequently carried out with the pupils in the form of an interview (joint discussion) to obtain feedback.
Despite the initial fear of the activities, all the pupils actively participated in exploring the materials (Wolfram Notebooks, technologies and features), and in the end there was a very pleasant atmosphere in the class, when the students themselves came up with what they could use the given tool for and how the visualization of the task helped them. Subsequently, the pupils started editing the materials themselves and testing for other examples that they came up with.
So it can be summarized that most of the pupils were excited about working with this tool, but what is more important, all pupils were actively involved in research with this tool and experienced success there themselves in the form of solving mathematical problems. This approach corresponds both to the development of pupils’ digital literacy and to the constructivist approach known as Inquiry-based teaching.
Introduction
Introduction
The least common multiple (LCM) of one or more integers is the smallest natural number that is divisible by all of those numbers.
One way to find the least common multiple of two numbers is to make a list of the prime divisors of each number. Then multiply each prime number (factor) by the maximum number of times it occurs in any number. If the same factor occurs multiple times in both numbers, you multiply that factor by the largest number of occurrences.
For three or more numbers, proceed similarly, multiply the greatest common power found by the prime factors of the given numbers.
The greatest common divisor (GCD) of two or more integers (at least one of which is not zero) is the largest natural number that divides the numbers without a remainder.
One way to find the least common multiple of two numbers is to make a list of the prime divisors of each number. Then multiply each prime number (factor) by the maximum number of times it occurs in any number. If the same factor occurs multiple times in both numbers, you multiply that factor by the largest number of occurrences.
For three or more numbers, proceed similarly, multiply the greatest common power found by the prime factors of the given numbers.
The greatest common divisor (GCD) of two or more integers (at least one of which is not zero) is the largest natural number that divides the numbers without a remainder.
In order to be able to determine GCD or LCM, we need a known decomposition into prime numbers. And also what a prime number even is.
A prime number is a natural number greater than 1 that is divisible only by 1 and itself. Other numbers, i.e. those that have more than two divisors, are called composite numbers. A composite number can therefore be written as a product of prime numbers. Now we will show how to factor a number into prime numbers. And also how to find out if a given number is prime.
To find out whether a given number is a prime number, we can use the PrimeQ[number] command. We will be returned a True value (if it is a prime number), or a False value (if it is not a prime number and therefore a composite number).
A prime number is a natural number greater than 1 that is divisible only by 1 and itself. Other numbers, i.e. those that have more than two divisors, are called composite numbers. A composite number can therefore be written as a product of prime numbers. Now we will show how to factor a number into prime numbers. And also how to find out if a given number is prime.
To find out whether a given number is a prime number, we can use the PrimeQ[number] command. We will be returned a True value (if it is a prime number), or a False value (if it is not a prime number and therefore a composite number).
PrimeQ[8]
Out[]=
False
PrimeQ[6]
Out[]=
False
So from the examples above, we can see that 5 is a prime number because the result of the statement was True. On the contrary, we can say that the number 6 is not a prime number and is a composite number. So when it is a composite number, we can find its divisor. To find the divisors, we use the Divisors[number] command. The result of this function will be a list of numbers that divide the number we entered. For a prime number, this function returns us only a list of two elements, with the number one and the given prime number. Let's try it right away.
Divisors[5]
Out[]=
{1,5}
Divisors[6]
Out[]=
{1,2,3,6}
Divisors[2048]
Out[]=
{1,2,4,8,16,32,64,128,256,512,1024,2048}
Prime factorization
Prime factorization
Factorization is done by successive division (without a remainder) by prime numbers (from the lowest to the highest) until we are left with the number 1. For the prime factorization of a composite number, it is enough to know the prime factors up to the value of the square root of the composite number. Different forms of notation can be chosen on paper, but in the end we get numbers, which when we multiply them, we get the original number. For example, by decomposing the number 6, we get the prime numbers 2 and 3. We check by multiplying these prime numbers and it must be true that we get the original number. Which can be clearly seen, because 2*3 = 6. In Wolfram Mathematica, we use the command FactorInteger[number] for decomposition. The result of the command is a list in the form {{base, power},{base, power},...}. So if the prime number 2 appears for example three times in the decomposition, the list will look like {{2,3}}. So it will be the number 2^3 = 2 * 2 * 2 = 8. Now we will look at specific examples and we will also check back that this is really the correct decomposition.
FactorInteger[48]
Out[]=
{{2,4},{3,1}}
2^4*3^1
Out[]=
48
2*2*2*2*3
Out[]=
48
We see that the number 48 can be decomposed into the product of prime numbers 2*2*2*2*3. For such a low number, this is a fairly simple matter. Therefore, we will now try to find the prime factorization of a much higher number.
FactorInteger[108993654]
Out[]=
{{2,1},{3,3},{7,1},{11,2},{2383,1}}
2*3^3*7*11^2*2383
Out[]=
108993654
2*3*3*3*7*11*11*2383
Out[]=
108993654
Such a decomposition would be very difficult to find if we had to do it only by the paper and pencil method. Also, some of you may have a question, what are any prime factorizations of higher numbers good for at all. And precisely the complexity of decompositions of higher numbers is the basis of encryption in computer science, where it relies on the fact that finding the decomposition of an extremely high number is still an unsolvable problem even for the most powerful computers. So rather an unsolvable problem in real time. Therefore, you can use a secure connection to websites or mobile applications with complete peace of mind.
Least common multiple
Least common multiple
We already have the definition of LCM in the introduction. Now we will see how we can determine the LCM using Wolfram Mathematica software. To find the LCM of two or more numbers, we use the command LCM[number, number, ...]. The result will be a number that can be divided by the original numbers without a remainder and is the smallest number. For example, we find the LCM for the numbers 10 and 20. This is a very simple example and it should be clear to everyone what the result will be.
LCM[10,20]
Out[]=
20
We got the result 20 as expected. The smallest number that is divisible by 10 and 20 is of course the number 20. We wouldn't even need any prime factorization for that. Now let's look at a more complex example. We will try to find the LCM for numbers 10238 and 23426.
LCM[10238,23426]
Out[]=
119917694
We can see that this is already a fairly high number. If we wanted to solve it on paper, we would put a lot of work into calculating it. But with the help of Wolfram Mathematica, we have the result in a fraction of a second.
We also find LCM for numbers 45, 36, 35, 68.
We also find LCM for numbers 45, 36, 35, 68.
LCM[45,36,35,68]
Out[]=
21420
We see that finding the LCM even for multiple numbers is a matter of time. It takes us the longest to write the command.
Greatest common divisor
Greatest common divisor
The greatest common divisor of two or more numbers is the number that divides all the entered numbers without a remainder and is also the greatest divisor. To find the GCD in the Wolfram Mathematica program, we will use the function GCD[number, number, ...]. To begin with, we find the GCD for the numbers 10 and 20. No one should have a problem doing this in their head, as this is a really simple example. Let's get to him.
GCD[10,20]
Out[]=
10
It probably won't surprise anyone that the result is the number 10, because it is actually the largest number that divides both the number 10 and the number 20. This was a really simple example, let's look at a more complex one. We find the GCD for the numbers 251160 and 3947944.
GCD[251160,3947944]
Out[]=
728
If we wanted to calculate this GCD by hand, we have to spend a lot of effort to find it. These are complex prime number decompositions that we would have to perform. But it's not unrealistic. Just time consuming. Meanwhile, Wolfram Mathematica managed it immediately after evaluating the command. Undoubtedly, it can be said that the use of such software is beneficial for more complex examples. We will now try to look at more complex word problems, which we will solve using the Wolfram Mathematica software.
Solving word problems using Wolfram Mathematica
Solving word problems using Wolfram Mathematica
1) The teacher wants to divide the fruit among the students so that everyone has the same number of different types. He has 8528 pieces of cherries, 1640 pieces of apples and 3936 pieces of plums at his disposal. To how many children can the fruit be divided if we know that there will be no fruit left? How many individual pieces of fruit does each student get?
GCD1=GCD[8528,1640,3936]
Out[]=
328
Cherries=8528/GCD1
Out[]=
26
Apples=1640/GCD1
Out[]=
5
Plums=3936/GCD1
Out[]=
12
He can distribute the fruit among up to 328 pupils and each of them will receive 26 cherries, 5 apples and 12 plums.
2) The rectangular tiles have dimensions of 6715 mm and 2690 mm, please determine the smallest side length of the square that can be paved with these tiles. How many tiles will be needed?
LCM1=LCM[6715,2690]
Out[]=
3612670
number=LCM1*LCM1/(6715*2690)
Out[]=
722534
The smallest side length of the square to be tiled is 3612670 mm and 722534 tiles will be needed.
3) From the five bars 6636 mm, 2844 mm, 3792 mm, 6004 mm and 1580 mm long, we want to cut the same length pieces so that there are no scraps and there are as few as possible. How many cm will these pieces measure and how many will there be?
GCD2=GCD[6636,2844,3792,6004,1580]
Out[]=
316
pieces=6636/GCD2+2844/GCD2+3792/GCD2+6004/GCD2+1580/GCD2
Out[]=
66
The pieces will measure 31.6 cm and there will be 66 of them.
Interactive panel on LCM and GCD
Interactive panel on LCM and GCD
An interactive element, where pupils could find the GCD and LCM of two numbers by selecting a slider. They also saw the prime
number decomposition of the given numbers.
number decomposition of the given numbers.
In[]:=
togExp[f1_,f2_,f3_,num1_,num2_]:=Framed[Text[Style[Grid[{{Framed[f1[num1,num2],Background->Nest[Lighter,Blue,3]]},{Framed[f2[num1,num2],Background->Nest[Lighter,Green,2]]},{Framed[f3[num1,num2],Background->Nest[Lighter,Yellow,2]]}},Alignment->Center,ItemSize->43.5],Black,14.7,Bold,FontTracking->"Narrow"]],Background->Lighter[RGBColor[1,1/2,0,0.8]],FrameMargins->2]
In[]:=
togExp[f1_,f2_,num1_,num2_]:=Framed[Text[Style[Grid[{{Framed[f1[num1,num2],Background->Nest[Lighter,Blue,3]]},{Framed[f2[num1,num2],Background->Nest[Lighter,Green,2]]}},Alignment->Center,ItemSize->43.5],Black,14.7,Bold,FontTracking->"Narrow"]],Background->Lighter[RGBColor[1,1/2,0,0.8]],FrameMargins->2]
In[]:=
lcmExp[num1_,num2_]:=Module[{fac1=fac[num1],fac2=fac[num2],faclcm=fac[LCM[num1,num2]],lcm=LCM[num1,num2]},Grid[{{"LCM[",num1,",",num2,"] = LCM[",fac1,",",fac2,"] =",faclcm,"=",lcm}}]]
In[]:=
fac[num_]:=If[Length[FactorInteger[num]]==1,Extract[Superscript@@@FactorInteger[num],1],CenterDot@@Superscript@@@FactorInteger[num]]
In[]:=
gcdExp[num1_,num2_]:=Module[{fac1=fac[num1],fac2=fac[num2],facgcd=fac[GCD[num1,num2]],gcd=GCD[num1,num2]},Grid[{{"GCD[",num1,",",num2,"] = GCD[",fac1,",",fac2,"] =",facgcd,"=",gcd}}]]
In[]:=
Manipulate[Quiet[togExp[lcmExp,gcdExp,num1,num2]],{{num1,126,"First number"},0,200,1,Appearance->"Labeled"},{{num2,98,"Second number"},0,200,1,Appearance->"Labeled"},FrameMargins->0,SaveDefinitions->True]
Out[]=
