American Journal of Interdisciplinary Research and Development 

ISSN Online: 2771-8948 

Website: www.ajird.journalspark.org 

Volume 27, April - 2024 

110 | P a g e  
 

ON THE USE OF THE GRAPHING CALCULATOR IN SOME 

HIGH SCHOOL MATH TOPICS 
Alexander Dryakhlov 

A Teacher of Academic Lyceum “International House 

Tashkent” TIIAME National Research University 

 

Abstract 

This paper investigates Desmos Graphing Calculator's effectiveness as a teaching tool for 

high school math (academic lyceum). It showcases Desmos' applications for both senior 

and junior students. For seniors, it demonstrates how Desmos tackles continuous random 

distributions, solving for probabilities and moments (mean, median, etc.) of a given 

probability density function. For juniors, it exemplifies how Desmos aids in real-world 

data modeling with sine functions, fitting a curve to Tashkent's monthly temperature data. 

While initial syntax learning is required, Desmos' benefits outweigh the effort, promoting 

visualization, calculation speed, and ultimately, enhanced student understanding and 

engagement. 

 

Keywords: Desmos Graphing Calculator, high school mathematics, continuous random 

distributions, sine functions, data modeling. 
 

 

Introduction 

Desmos Graphing Calculator can be an invaluable tool in teaching many topics of high 

school mathematics. This tool is available as an app for a phone from the Google Play Store, 

App Store as well as a web interface (desmos.com).  

In this paper we consider the application of the tool in teaching some topics of mathematics 

at an academic lyceum to junior and senior students. There is a certain learning curve 

involved in studying the syntax of the app but it’s worth it. 

 

Year 2 (Seniors) 

The topic is Continuous Random Distributions. Calculating the probabilities and some 

characteristics of a distribution requires integration. Although the integration was quite 

extensively covered by the syllabus, at times it’s beneficial to use the technology to speed up 

the process. In the following example improper integration is used which is not studied in 

the Calculus I course. 

Let 𝑋 be a random variable with probability density function (pdf) given by 

𝑓(𝑥) = 2𝑒−2𝑥, 𝑥 ≥ 0.  

1. Verify it’s a valid probability density function (pdf) and find 

2. Its mean 

3. Median 

4. Variance and standard deviation. 

http://desmos.com/


 

American Journal of Interdisciplinary Research and Development 

ISSN Online: 2771-8948 

Website: www.ajird.journalspark.org 

Volume 27, April - 2024 

111 | P a g e  
 

Using Desmos Calculator  

● We define function 𝑓(𝑥). 

● Then we calculate ∫ 𝑓(𝑥)𝑑𝑥
∞

0
 to make sure its value is 1 (thus it’s a valid pdf).  

● We evaluate the first moment 𝑚1 (the expectation, mean) as ∫ 𝑥𝑓(𝑥)𝑑𝑥
∞

0
. 

● And the second moment 𝑚2 = ∫ 𝑥2𝑓(𝑥)𝑑𝑥
∞

0
. 

● The variance is found by the formula 𝑉𝑎𝑟(𝑋) = 𝐸(𝑋2) − (𝐸(𝑋))2 = 𝑚2 − 𝑚1
2. 

● The median of the distribution is found as the intersection of the graph of the 

Cumulative Distribution Function (cdf) 𝐹(𝑥) = ∫ 𝑓(𝑡)𝑑𝑡
𝑥

0
 and the horizontal line 𝑦 = 0.5. 

Both graphs are shown below. The median value is 0.347 (or, 
𝑙𝑛2

2
 exactly). 

● The working example1 is saved in desmos.com. 

 

● The shaded area below the graph 𝑦 = 𝑓(𝑥) which is equal to 
1

2
 for 0 ≤ 𝑥 ≤

0.347(median) is shown below 

 
1 https://www.desmos.com/calculator/sfxqg53buw 

http://desmos.com/


 

American Journal of Interdisciplinary Research and Development 

ISSN Online: 2771-8948 

Website: www.ajird.journalspark.org 

Volume 27, April - 2024 

112 | P a g e  
 

 
Year 1 (Juniors) 

The topic is Modeling with Sine and Cosine Functions. The following data taken from this 

source2 is to be modeled by the General Sine Function 𝑇 = 𝑎 𝑠𝑖𝑛(𝑏(𝑡 − 𝑐)) + 𝑑, where 𝑡 is 

time in months and 𝑇 is the temperature in degrees Celsius. This data represents the mean 

monthly maximum temperature for Tashkent, Uzbekistan. 

Jan Feb Mar Apr May  Jun Jul Aug Sep Oct Nov Dec 

6.2 8.2 15.1 20.5 26.7 32.1 34.6 33.7 28.6 21 12.9 7.1 

 

The task is done in Desmos Graphing Calculator.  

● First, the data is entered as a table assuming Jan=1, Feb=2, etc.  

● The amplitude 𝑎 and the average 𝑑 are calculated using the maximum and the minimum 

of the temperature over all 12 months. 

● The coefficient 𝑏 is found via the period of 12 months (assuming the pattern will continue 

next years) as 
2𝜋

12
. 

● The most problematic is to find the appropriate value of coefficient 𝑐 which signifies the 

horizontal shift of the sine curve. But the calculator allows us to use parameters in the 

equation. Each such parameter introduces a slider for changing the value of the parameter. 

So, by moving the slider for parameter 𝑐, we move the curve horizontally to match the scatter 

plot the best we can. 

● The resulting equation for the data happens to be 𝑇 = 14.2 𝑠𝑖𝑛(
𝜋

6
(𝑡 − 4.1)) + 20.4 

 
2 https://en.climate-data.org/asia/uzbekistan/tashkent/tashkent-485/#climate-table 



 

American Journal of Interdisciplinary Research and Development 

ISSN Online: 2771-8948 

Website: www.ajird.journalspark.org 

Volume 27, April - 2024 

113 | P a g e  
 

● The working example3 is saved in desmos.com. 

●  

 
 

References: 

1. https://www.desmos.com/calculator/sfxqg53buw; 

2. https://en.climate-data.org/asia/uzbekistan/tashkent/tashkent-485/#climate-table; 

3.   https://www.desmos.com/calculator/gqvkhrokze. 

 

 
3 https://www.desmos.com/calculator/gqvkhrokze 

http://desmos.com/