Online Learning Has a Cheating Problem, And It’s Growing
As education shifts rapidly to online and hybrid formats, academic dishonesty is becoming more prevalent and harder to detect. A comprehensive review in the scientific magazine Springer Nature Link of self-reported student behavior found that cheating in online exams rose from 29.9% before COVID to 54.7% during the pandemic.
A study published in the Online Learning Journal examined unproctored online exams in a large introductory biology course. Using LMS activity logs to detect whether students were navigating away from the test window, researchers found that up to 70 % of students did so, often on most questions. Appeals to student integrity or honor pledges had little effect, but a simple warning about potential detection combined with a penalty policy reduced this behavior to just 15 %. This study clearly demonstrates that the way an exam is designed—and students’ perception of oversight—has a much greater impact on academic honesty than the technology used itself.
These figures underscore a worrying trend: as reliance on screen-sharing, messaging apps, and generative AI increases, traditional anti-cheating measures struggle to keep pace. While most institutions promote academic honesty, digital environments introduce new vulnerabilities that traditional exams don’t account for. That’s where technology can help educators turn the tide.
The Risks of Academic Dishonesty in Online Assessments
Digital learning offers flexibility, but it also exposes gaps in control and assessment integrity. In particular, math quizzes and STEM evaluations are among the most affected. Why?
Students may copy solutions from peers in real time. During online exams or quizzes, students can easily communicate through messaging apps, video calls, or shared screens. In many cases, this allows them to collaborate during assessments that are meant to be individual. The lack of in-person supervision makes real-time copying far more difficult to detect than in a physical classroom.
Answer-sharing platforms often publish full collections of test questions and answers taken from real courses. This allows students to search for specific problems and find exact solutions, making it easier to cheat, especially in online assessments. These platforms pose a serious threat to academic integrity.
Automatic feedback loops allow repeated guessing until success. Many online quiz systems offer immediate feedback and unlimited attempts without any variation in the questions. While useful for formative learning, this can be exploited during summative assessments, as students may guess repeatedly or try out answers until they find the correct one, rather than understanding the underlying concepts.
Identity verification is difficult in unsupervised settings. Without secure login procedures or proctoring tools, it becomes difficult to ensure that the student completing an online exam is actually the one enrolled in the course. This opens the door to impersonation, paid test-taking services, and other forms of academic fraud that are difficult to trace after the fact.
Randomized Math Quizzes: A Smart Strategy for Academic Honesty
To protect the integrity of online math assessments, many institutions are turning to randomized math quizzes. These assessments use variable-driven content to generate unique versions of the same problem for each learner.
Why Randomization Works
Randomization is effective because it ensures that each student receives a unique version of the same problem. By generating different sets of variables, it becomes nearly impossible for learners to share answers or copy from one another. This approach maintains the same logical structure and learning objectives for all, but with varying inputs that require independent resolution. It also enables educators to automatically regenerate problems, scaling assessments without compromising integrity. Ultimately, randomized math quizzes mirror real-world problem-solving environments, where conditions change and memorization alone is insufficient, encouraging students to apply concepts rather than recall patterns. Randomized quizzes not only deter academic dishonesty: they also promote deeper learning by encouraging individual problem-solving strategies.
Insert random variables into a math question using WirisQuizzes.
How WirisQuizzes Prevents Academic Dishonesty
WirisQuizzes is Wiris’ robust assessment solution designed specifically for STEM education. It enables educators to create rich, dynamic math questions adapted to each student’s context. By introducing random variables, the platform generates virtually unlimited versions of the same problem, discouraging cheating and promoting deeper engagement. Its intelligent grading engine automates correction and delivers immediate feedback, encouraging learning through exploration and repetition.
WirisQuizzes offers built-in random variable generation designed specifically for math and STEM contexts. Unlike generic quiz platforms, WirisQuizzes understands the structure of math and allows educators to define flexible, yet pedagogically consistent, question templates.
With WirisQuizzes, educators can:
Define variable ranges for values like x, a, b, or θ
Create multiple conditions to ensure solvable and meaningful outputs
Randomize both numerical and algebraic parameters
Use automatic validation logic to verify correct answers per version
Preview multiple quiz versions before deploying
This dynamic approach allows hundreds of quiz variations to be generated from a single question model, keeping assessments fair, secure, and scalable.
Real-World Applications: From Schools to Universities
The effectiveness of randomized math quizzes powered by WirisQuizzes has been demonstrated across various educational levels:
Secondary Education
In middle and high schools, teachers use WirisQuizzes to generate personalized homework assignments and quizzes tailored to each student. Even though the problems test the same concepts, every learner receives a unique version with different variables. This significantly reduces opportunities for copying among classmates, while still aligning with the curriculum. As a result, students engage more deeply with the material and take greater ownership of their learning process.
Higher Education
In colleges and universities, professors rely on WirisQuizzes to deliver fair and secure assessments for large cohorts without manually creating dozens or hundreds of different exam versions. By using random variables to generate unique yet equivalent problems, educators can maintain academic integrity even in high-stakes midterms and finals. This is especially valuable in large-enrollment STEM courses like calculus, linear algebra, and statistics.
Online Learning Platforms
EdTech platforms and online learning providers integrate WirisQuizzes into their learning management systems (LMS) to offer secure, scalable assessments to a broad user base. Randomized question generation ensures each learner faces a unique challenge, making answer sharing or mass cheating far more difficult. Combined with immediate feedback and intelligent grading, this approach supports adaptive learning and high levels of academic integrity across thousands of users.
More Than Prevention: Promoting Academic Honesty by Design
Preventing academic dishonesty is only one side of the equation. Tools like WirisQuizzes also encourage academic honesty by fostering independent thinking and personalized learning.
When students know that each problem is unique to them, they are more likely to engage with the material rather than memorize solutions, develop critical thinking and conceptual understanding, and trust that their efforts are being fairly evaluated. This shift in mindset contributes to a culture of integrity.
Make Cheating Obsolete With Randomization
In an era where cheating can happen with a few clicks, prevention must be proactive, not punitive. Randomized math quizzes provide a smart, scalable way to ensure fair assessments and promote academic honesty.
WirisQuizzes empowers educators to build dynamic, variable-driven math assessments that are both secure and student-centered.
When discussing fundamental mathematical constants, pi often takes the spotlight. However, another number, equally vital yet less celebrated, is Euler’s number e. Known simply as e, this constant appears in countless mathematical contexts and real-world applications. In this article, we will explore the origin, properties, and applications of e, positioning it as an indispensable tool in the worlds of mathematics, science, and finance.
What Is Euler’s Number e?
Euler’s number e is an irrational and transcendental number, approximately equal to 2.71828. It serves as the base of natural logarithms and arises naturally in various growth processes and limit calculations.
The Origin of e: A Practical Example
Consider the following real-life scenario: Imagine a bank offering a 100% annual interest rate on a principal investment of 1€. If interest is compounded once at the end of the year, you would have 2€. However, if the bank compounds interest semiannually at 50% every six months, the calculation changes:
After six months: 1€ × 1.5 = 1.5€
After another six months: 1.5€ × 1.5 = 2.25€
Now, let’s divide the year into three periods of four months each, applying an interest rate of 33.33% every four months:
After four months: 1€ × 1.33 = 1.3333€
After eight months: 1.3333€ × 1.3333 ≈ 1.7777€
After twelve months: 1.7777€ × 1.3333 ≈ 2.3703€
So, if you receive infinite payments, will you have infinite money? Unfortunately, the answer is no. As the frequency increases -from quarterly to monthly, to daily, and beyond- the final amount approaches a limit. The formula to calculate this is:
Image created with Wiris’ MathType
As n approaches infinity, the value converges to e. This simple yet powerful example demonstrates the natural emergence of e in exponential growth, explaining its foundational role in continuous compounding interest models.
Fundamental Properties of e Number
Irrationality: e cannot be expressed as a simple fraction.
Transcendence: e is not a root of any non-zero polynomial equation with rational coefficients.
Infinite Series: e can be represented as the sum of the infinite series:
Image created with Wiris’ MathType
Natural Logarithm Base: The function uses e as its base, integral to calculus and natural growth models.
Euler’s Identity: Considered one of the most beautiful equations in mathematics:
eiπ + 1 = 0
This equation elegantly connects five fundamental mathematical constants: 0, 1, e, i, and π.
Applications Across Disciplines
Euler’s number e is not only a fundamental constant in pure mathematics but also plays a pivotal role in a range of practical applications across various scientific and technological fields. From modeling growth processes to risk management, e is indispensable in understanding and predicting complex phenomena.
In Mathematics
Calculus: Euler’s number is central to calculus, particularly in the context of exponential growth and decay. One of the most important properties of e is that the derivative of the function ex is ex itself. This unique property makes it extremely useful in solving differential equations that describe dynamic systems, such as population growth, radioactive decay, and heat transfer. It simplifies the analysis of these systems because the function is self-replicating under differentiation.
Complex numbers: Euler’s formula, eix = cos(x) + isin(x), is one of the most profound equations in mathematics, linking exponential functions with trigonometric functions. This relationship is crucial in fields like electrical engineering and signal processing, where it simplifies the analysis of oscillating systems and waveforms. It allows for a more intuitive understanding of rotations and oscillations in the complex plane.
In Physics
Radioactive decay: The process of radioactive decay follows an exponential law, which can be modeled using e. The amount of a radioactive substance remaining after a certain period is given by the equation N(t) = N0eλt, where N(t) is the amount of substance at time t, N(t) is the initial amount, and λ is the decay constant. This model is used to predict the behavior of radioactive materials over time, which has applications in medicine (e.g., radiology), archaeology (e.g., carbon dating), and nuclear physics.
Thermodynamics: In thermodynamics, the Boltzmann factor (see image below) is essential for determining the probability of a system being in a particular energy state, where 𝐸 is the energy, 𝑘 is the Boltzmann constant, and 𝑇 is the temperature. This exponential relationship is vital in understanding systems in equilibrium, such as the distribution of particles in gases or the behavior of molecules in biological processes.
In Economics and Finance
Continuous compounding: One of the most common uses of e is in the calculation of continuously compounded interest. If interest is compounded continuously rather than at discrete intervals, the formula for the accumulated value of an investment becomes A = Pert, where 𝑃 is the initial investment, 𝑟 is the interest rate, and 𝑡 is the time in years. This formula highlights how the frequency of compounding approaches a limit, with e representing the limit as compounding becomes continuous. It is fundamental in understanding the long-term growth of investments and savings.
Risk models: Financial models such as the Black-Scholes model for option pricing rely heavily on e to account for the stochastic nature of asset prices. The formula for option pricing involves exponential functions to model how the price of an option evolves over time. e also plays a key role in various risk models, including Value at Risk (VaR) calculations, where it helps quantify the likelihood of extreme losses in investment portfolios.
In Technology
Algorithms: Euler’s number is used in algorithmic analysis, particularly when analyzing the performance of algorithms that exhibit exponential growth. For instance, certain random processes and algorithms, such as the analysis of QuickSort (a widely used sorting algorithm), can involve e. Specifically, in probabilistic algorithms and complexity theory, e appears in the expected running time of algorithms, as they often follow exponential distributions or involve recursive calculations that converge to e.
Machine learning and data science: In machine learning, e is frequently involved in algorithms that model growth processes or in optimization algorithms that use exponential decay to adjust parameters over time. For instance, in gradient descent, which is used to find the minimum of a function, learning rates may decay exponentially based on e to optimize performance and prevent overshooting.
In Biology and Medicine
Population growth: One of the most well-known real-world applications of e is in modeling population growth. When populations grow in ideal conditions (without external limitations), the number of individuals follows an exponential growth curve, which is modeled by the equation, P(t) = P0ert where 𝑃(𝑡) is the population at time 𝑡, P0is the initial population, and 𝑟 is the growth rate. This principle is not only relevant to ecology but also helps in understanding the spread of diseases in epidemiology.
Pharmacokinetics: The exponential decay model is also used in pharmacokinetics, which studies how drugs are absorbed, distributed, metabolized, and excreted by the body. The concentration of a drug in the bloodstream over time often follows an exponential decay, and the parameter governing the decay is related to e. This helps in determining dosing schedules for medications to ensure therapeutic levels are maintained without causing toxicity.
Visualizing Euler’s Number e with Wiris’ Digital Tools
As digital solutions advance, tools like MathType and CalcMe empower users to work effortlessly with e and other complex mathematical expressions. Whether you’re a researcher, educator, or professional, integrating these Wiris’ technologies into your workflow ensures precision and efficiency.
For instance, to deepen the understanding of e’s behavior, one can visualize the limit expression
using graphing tools such as CalcMe, a Computer Assisted System (CAS) for algebraic manipulation that allows complex mathematical calculations to be carried over in a self-contained, fast and intuitive way. As increases, the graph approaches the value of number e, offering an intuitive graphical demonstration of this mathematical limit.
Additionally, MathType, the world’s leading equation editor that allows you to write math notation as easily as you write text, enables seamless insertion of complex mathematical expressions into documents and web content, ensuring clarity and precision.
Graph showing the function (1+1/x)xin red, approaching the value of e in blue.
Perfect for students, teachers, editors, and technical writers, MathType streamlines the creation of high-quality technical and scientific content across various platforms.
For seamless handling of mathematical expressions and visualizations, consider incorporating both MathType and CalcMe into your digital toolkit.
Creating engaging and varied math quizzes can be a surprisingly time-consuming task. Teachers often face the dreaded blank page, struggling to come up with fresh problems that match different levels, topics, and student needs. But that’s about to change. With AI for math making its way into classrooms, quiz creation just got a powerful upgrade.
Meet LearningLemur: Your AI Question Generator
LearningLemur is an intuitive math platform integrated with Google Classroom that provides customizable quizzes, automatic grading, and personalized feedback, saving educators time and enhancing student learning. Its extensive library of customizable quizzes and exercises fosters engagement and mastery, while detailed analytics help educators track progress and identify areas for improvement. From simplifying assignment creation to delivering instant feedback, LearningLemur transforms classrooms into dynamic and interactive learning spaces, making math more accessible and engaging for all students.
Now, Wiris has introduced a new feature within LearningLemur: a math question generator that understands your needs and generates ready-made quizzes in seconds. It’s designed to address one of the most common pain points for educators: how to start creating math assessments efficiently without sacrificing quality.
Here’s what LearningLemur’s new AI-powered tool offers:
Describe the type of questions you want.
Receive an instant set of customized math problems.
Edit, refine, and assign them directly to your students.
Forget about starting from scratch. This tool helps you work smarter, not harder, all within a platform that is intuitive and easy to navigate, even for teachers with limited tech experience.
How does it work?
Using this tool is simple and straightforward. Teachers input the number of questions, select the question type (e.g., open answer, multiple choice) and finally provide a prompt (e.g., “Make fractions sum problems for students of grade 12. Ask them to be simplified.”).
The AI handles the rest, offering draft content that aligns with your instructions. And because it’s powered by the trusted Wiris correction engine, you maintain full confidence in the mathematical accuracy.
Why This AI Feature Matters for Educators
The LearningLemur AI question generator isn’t just about saving time. It’s designed to help teachers focus on what really matters: pedagogy, clarity, and curriculum alignment. While the AI handles the generation, teachers can shape the output by ensuring pedagogical alignment with curricular goals, adjusting the complexity and style of each question, and refining the language and clarity for the specific group of learners. This tool does not replace your expertise but accelerates your workflow, allowing you to dedicate more time to effective teaching.
Tips for Writing Effective Prompts
Crafting the right prompt is crucial to getting the most out of the LearningLemur AI for math tool. Think of it as briefing a colleague or delegating to a student teacher. Be clear, be specific, and include the following:
Specify the exact math topic you want to assess in your students. For example, indicate whether you want exercises on fractions or polynomials.
Define the educational level or age of your students to tailor the difficulty. Questions should be accessible yet challenging, matching the knowledge and skills of your learners.
Add any constraints. If you have specific requirements, such as simplifying answers or using decimals only, be sure to include them. This ensures the generated questions meet your pedagogical or exam criteria.
Optionally, use word problems to add context and generate relatable scenarios. Adding context or real-life situations makes problems more engaging and easier to understand for students. For instance, framing a problem in the context of shopping, travel or science helps connect math to everyday life and increases motivation.
Example Prompts
“Create 6 word problems on solving linear equations for Grade 9. Include at least one with negative coefficients and require students to simplify x.”
“Create 10 questions on simplifying algebraic expressions for 11-year-old students. Include at least one question involving the distributive property and another with factoring.”
“Create 5 questions for first-year university math students on calculating the rank of a matrix using elementary row operations.”
Use Cases in the Real Classroom
The LearningLemur math question generator adapts across levels and contexts:
For daily practice, generate variations to reinforce a concept.
For formative assessment, tailor difficulty and format.
For homework, instantly provide differentiated sets by level.
For exam prep, create question banks by topic or skill.
And since you can edit anything, it integrates seamlessly into any teaching style.
Diverse Learning Needs: Easily Tailor Content to Students’ Needs
From the teachers’ perspective, LearningLemur offers great flexibility, as educators with limited time or resources in creating math problems can now generate high-quality content more quickly. These advantages are particularly evident for teachers managing large classes, as they can personalize content while reducing preparation time.
Additionally, LearningLemur enables teachers to address the diverse needs of their students by creating customized quizzes and exercises tailored to different skill levels and learning requirements. Teachers can also assign targeted homework tasks, allowing students to strengthen specific competencies where they may need additional support.
Empowering Teachers Through Intelligent Automation
AI for math isn’t about replacing educators: it’s about empowering them. LearningLemur is a clear example of how AI question generators can reduce friction in the creative process while leaving full pedagogical control in teachers’ hands.
Whether you’re battling writer’s block or trying to generate multiple versions of a quiz quickly, this tool gives you a solid head start.
Technology is no longer a guest in the classroom: it’s become a core part of the educational experience. And in the world of mathematics, it’s proving to be a powerful ally. From interactive whiteboards to adaptive AI systems, technology in math education is reshaping the way students learn and how teachers teach.
But what are the tools driving this transformation? And how can enterprises and institutions make the most of them while navigating the inevitable challenges?
In this article, we break down the evolution of digital learning tools, explore their practical impact, and look ahead to where educational technology trends are taking us.
How Technology Has Evolved Math Education
There was a time when learning math meant pencil, paper, and a lot of chalk. Today, that equation includes dynamic software, mobile apps, and real-time collaboration platforms. Over the years, technology has evolved from a classroom supplement to a strategic cornerstone of math education.
What changed? The tools got smarter, more intuitive, and better aligned with pedagogical goals. Technology in math education now means:
Receiving instant feedback and adaptive assessments.
Engaging in problem-solving through interactive scenarios.
Collaborating beyond borders with cloud-based platforms.
Visualizing abstract concepts through simulations.
Digital tools for math teaching are not only enhancing understanding but also making math more approachable, flexible, and inclusive.
Key Technologies Enhancing Math Learning
From immersive 3D platforms to algebra systems that respond in real time, the toolbox for math educators is broader and more innovative than ever. Let’s look at some of the most impactful digital learning tools reshaping classrooms today.
AI and adaptive learning
Artificial intelligence is reshaping math education by enabling more personalized and adaptive learning experiences. By analyzing individual performance, adaptive systems deliver tailored exercises, offer real-time hints, and adjust difficulty levels on the fly. Many platforms now include AI-driven features to support this responsiveness.
A clear example is MathType handwriting, a tool that uses artificial intelligence to convert handwritten expressions into clean, digital equations ready to be used in your documents. These innovations represent a step forward in simplifying the transition from analog input to digital output and supporting more intuitive interaction with mathematical content in digital learning environments.
Graphing calculators and CAS tools
Graphing calculators and Computer Algebra Systems (CAS) allow students to explore equations, functions, and transformations in real time. These solutions shift the focus from mechanical calculation to analytical insight. A prime example is CalcMe, Wiris’ JavaScript-based CAS, which enables users to perform complex calculations and visualize mathematical concepts interactively across various devices.
Interactive software
Visual and interactive platforms offer higher engagement. Programs like WirisQuizzes and Learning Lemur enable learners to build, manipulate, and visualize mathematical models.
With MathType, creating complex equations becomes intuitive, helping students and teachers focus on the math, not the formatting.
LMS integration
Platforms such as Moodle and Canvas have become exponentially more powerful with integrated digital math tools. Embedding the previously mentioned platforms within these systems turns traditional coursework into interactive, dynamic experiences that meet learners where they are.
Online collaborative platforms
Math isn’t always a solo activity. Digital platforms make it easy for students to co-create solutions, share ideas, and solve real-world problems together, whether in the classroom or worldwide. Working in teams, sharing solutions, and learning from peers isn’t just good practice; it’s real-world preparation.
Challenges on the Digital Road
Of course, no transition comes without friction. Schools and universities face several challenges in adopting digital tools for math teaching:
Access inequality: Not every student has reliable internet or a personal device.
Teacher training gaps: Tools are only as effective as those using them.
Overreliance risk: Tech should enhance, not replace, mathematical reasoning.
Privacy concerns: Data protection must be part of the planning.
Maintenance and cost: Edtech isn’t a one-time purchase; it’s an evolving ecosystem.
These issues are real, but not insurmountable. They call for thoughtful planning, continuous support, and strategic investment.
Smart Implementation: Making the Most of Digital Tools
What does it take to ensure technology truly enhances math education? How can enterprises and institutions deploy digital learning tools effectively while avoiding common pitfalls?
Start with clear goals. Tech without purpose is just noise. Define your outcomes first, then choose solutions that help achieve them.
Empower teachers and invest in training. Teachers need to feel confident, not just compliant, when using digital tools for math teaching.
Ensure access, make inclusion a priority. Equitable access to devices and software should be foundational, not optional.
Blend old and new. Combine traditional instruction with digital methods. Balance is key: let tech support, not replace, core learning.
Safeguard data. Choose platforms with robust privacy protocols. Make sure every tool complies with education data standards.
Encourage exploration. Let students use technology to tinker, test, and create. Math becomes more meaningful when they take ownership of their learning.
Looking Ahead: What’s Next in EdTech for Math?
The pace of innovation is only accelerating. These educational technology trends are already reshaping what’s possible:
Smarter AI: Learning systems that adapt in real time based on student engagement.
Blockchain records: Tamper-proof credentials that simplify academic verification.
Game-based curricula: Turning math practice into immersive, story-driven experiences.
Immersive AR/VR: Math will move off the page and into the student’s physical space, making abstract ideas concrete.
Global classrooms: Seamless, multilingual collaboration across borders.
Modular microlearning: Bite-sized lessons for modern attention spans.
Accessibility: Digital tools are increasingly designed to support learners with disabilities or diverse learning needs, including visual impairments, dyslexia, and motor challenges.
How Wiris Is Leading the Change
At Wiris, we believe technology in math education should empower without overwhelming. That’s why we’ve built solutions that integrate seamlessly into classrooms while amplifying learning outcomes.
WirisQuizzes: create auto-graded, math-rich assessments inside your LMS.
MathType: author complex math expressions with ease, across any device.
Learning Lemur: generate math content and adapt the learning process to individual student needs.
CalcMe: perform complex calculations and visualize mathematical concepts interactively across various devices.
Whether you’re teaching derivatives or designing STEM assessments, Wiris helps educators turn digital potential into real progress.
Final Thoughts
The classroom is changing, students are changing, technology is changing, and math education must evolve to meet the moment.
By thoughtfully and strategically integrating digital learning tools, schools can unlock deeper understanding, greater engagement, and lifelong skills. At Wiris, we’re proud to help educators worldwide navigate this journey.
Ready to bring your math instruction into the future? Discover how Wiris can help you lead the way with cutting-edge educational technology.
Step-by-step Solution to an Oxford Access Exam Problem.
The Oxford University access exams are famous for their difficulty and for challenging students with mathematical problems of high complexity. In this article, we will focus on a 2020 exam problem that involves calculating the area between two curves. Through a clear and detailed analysis, we will break down each step necessary to approach and solve this mathematical challenge.
If you are interested, feel free to also visit our other article, “How to solve an equation of degree 16”, where we explore another challenging Oxford exam problem.
The problem statement is as follows:
A line is tangent to the parabola y=x²at the point (a,a²), where a>0.
The area of the region bounded by the parabola, the tangent line, and the x-axis equals
This is a classic calculus problem that, as we will see, requires the use of both derivatives and integrals. Don’t worry; we’ll break it down into three clear and simple steps. Let’s get to it!
How to Find the Equation of the Tangent Line?
Before starting with the calculations, let’s visualize the problem. Below, you’ll see a graph illustrating the parabola y=x² and the tangent line at the point (a, a²), where a>0. In this graph, we can see how the tangent line touches the curve of the parabola at a single point without crossing it.
Now, to proceed, we first need to find the equation of the tangent line at that specific point. Since the tangent touches the parabola at point (a, a²), we need to determine both its slope and its equation.
This is where the derivatives come into play. The slope of the tangent at any point on the parabola is given by the derivative of the function y=x². We start by calculating the derivative:
This means that the slope of the tangent at any point on the parabola is 2x. To find the slope at the tangent point (a, a²), we substitute in the derivative:
Therefore, the slope of the tangent line at point (a, a²) is 2a.
Now that we have the slope, we can write the equation of the tangent line using the formula for the line in its form:
y – y0 = m (x – x0)
Where m is the slope of the line, and (x0, y0) is the point through which the line passes. In our case, the point of tangency is (a, a²), and the slope is 2a, so we substitute these values in the formula for the line:
y – a²= 2a (x-a)
y – a²= 2ax – 2a²
y = 2ax – a²
So, the equation of the tangent line at the point (a, a²) is:
y = 2ax – a²
Find the Intersection Points
Now that we have the equation of the tangent, we need to find the points of intersection between this tangent and the x-axis, that is, when y=0. This will allow us to determine the limits of the region whose area we want to calculate.
The area to be calculated is marked in orange in the graph below.
Substituting y=0 in the tangent equation and solving for x we obtain:
0= 2ax – a²
2ax = a²
x = a/2
So the tangent cuts the x-axis at point
Calculate the Area Between the Parabola, the Tangent and the X-axis
Once we have the equation of the tangent and we know the points of intersection, we can proceed to calculate the area of the region bounded by the parabola, the tangent and the x-axis. For this, we use the definite integral, which allows us to find the area between two curves.
The area we want to calculate is obtained by subtracting two regions. In the first graph, you can see the total region we are interested in. However, part of this area is specifically delimited in the second graph, which we must subtract to be left with only the part we are looking for.
Let us calculate both integrals:
a) Integral of graph 1:
b) Integral of graph 2:
Now that we have solved all the integrals, we subtract the results to obtain the total area:
In conclusion, the area of the region bounded by the parabola y = x², the tangent y = 2ax – a²and the x-axis, is:
Therefore, option c is the correct answer
And so we come to the end of this fascinating problem! If you enjoyed this analysis or found it useful to better understand the solving process, feel free to share it with other math enthusiasts!
Now we challenge you: do you dare to calculate the area formed by the curve y = x3 and its tangent at the point (1,1)? Share your solution or your ideas in the comments. We’d love to hear your approach!
From Textbooks to a Personalized Digital Math Learning Platform
When La Salle San Ildefonso set out to replace printed math textbooks with digital tools, they needed more than just online worksheets—they needed a complete math learning platform. With WirisQuizzes integrated into Moodle, their teachers built a collaborative system that supports independent learning, personalized feedback, and scalable assessment. What started as one teacher’s initiative has become a model adopted across multiple La Salle campuses, reshaping how students engage with math.
The Challenge
When La Salle San Ildefonso first decided to phase out physical math textbooks, they faced a critical question: How do we offer high-quality math learning in a wholly digital format without sacrificing feedback, rigor, or accessibility?
As teacher Javier Melchior explained, early digital tools offered little beyond binary right-or-wrong grading
“The only feedback we could give was whether the answer was right or wrong. That wasn’t enough”, Javier explained. ”What students needed was guidance – not just the solution, but the how behind the solution”.
This insight became a turning point for La Salle’s approach to digital education. It wasn’t just about digitizing content but about ensuring that students could learn actively and reflectively within a new system.
At the same time, the shift to digital demanded a scalable infrastructure for:
Creating large volumes of exercises quickly.
Providing real-time, personalized feedback.
Supporting teachers across varying levels of technical proficiency.
Giving students the tools to practice independently and meaningfully.
Other platforms like Google Classroom, while helpful for document sharing, lacked the sophistication for in-depth math evaluation and formative feedback.
The Solution
That’s when WirisQuizzes came into focus—not just as a plugin but as the core engine behind La Salle’s custom-built math learning platform.
Led by Javier and math teacher Guillermo Pérez González, the department began building a shared bank of exercises directly within Moodle using WirisQuizzes. These weren’t just questions—they were intelligent, dynamic exercises that could offer step-by-step feedback, track partial understanding, and regenerate with different data for unlimited practice.
“We discovered the true power of WirisQuizzes when we realized we could break down a problem into parts,” said Javier. “It’s not just about getting the final answer. If a student does the first 20% right, the platform can recognize and score that.”
WirisQuizzes became more than an assignment tool for students—it was their personal study assistant. “Now they ask us, ‘Is there a quiz for this topic?” Guillermo shared. “It’s become a natural part of how they study and prepare.”
The Implementation Process
La Salle’s implementation of WirisQuizzes was a grassroots initiative that grew into an institutional innovation. Here’s how it evolved.
Assessment
The shift began nearly a decade ago when La Salle removed printed textbooks for some math levels. With a digital-first model on the horizon, Javier started exploring alternatives. “I found WirisQuizzes online by chance,” he said. “And after testing its capabilities, I pitched it to my colleagues. From there, it just took off.”
Customization
Using Moodle’s system-level question bank, the teachers created hundreds of questions organized by topic: polynomials, functions, geometry, probability, and more. Every question included custom feedback. “Almost all our questions show the solution process,” said Javier. “This makes it easier for students to learn independently—even their parents can follow along.”
Importantly, this work was collaborative. Teachers built and reused each other’s content. “We never assigned categories formally,” said Guillermo. “If someone had free time, they’d jump in and add to the bank. It’s a shared resource for everyone.”
Deployment
WirisQuizzes was deployed through Moodle across the school. Teachers could:
Select questions from a shared bank.
Build quizzes quickly.
Offer time-limited assessments or practice exercises.
Use quizzes for classwork, homework, or even during substitute hours.
The school also exported question banks to other La Salle campuses in Tenerife, Madrid, and Gran Canaria. Adoption varied depending on each site’s IT support and teacher training, but the system was designed for easy export/import.
Training
While many core users were math or STEM teachers with programming experience, others found the “programming” aspect intimidating. Guillermo acknowledged this gap. “Some teachers hear ‘programming’ and think of C++,” he joked. “But it’s not like that. They’d see how simple it is if they just took one step.”
To address this, La Salle and Wiris discussed tailored training sessions that start simple, focusing on quiz-building basics before moving into more advanced logic and scripting.
The Results
La Salle’s math learning platform, powered by WirisQuizzes, delivered transformative results across several dimensions.
Improved Student Autonomy & Motivation
Students began voluntarily requesting quizzes to practice new topics. “It’s like they have a personal tutor,” said Javier. In one memorable example, he logged in on a weekend to find nine students had already completed a practice quiz and scored a perfect 10. “They were competing healthily. It became fun.”
Collaborative Efficiency Among Teachers
The math department created a bank of over 600–700 questions, reusable across grade levels and subjects. This significantly reduced prep time while increasing question quality and consistency.
Parental Engagement and Transparency
Because quizzes included feedback, parents could follow along, even if they weren’t strong in math. “Some told us they were thrilled to see their child working independently,” Guillermo shared.
Formative Assessment at Scale
Teachers used quizzes to diagnose student progress week by week. “In Bachillerato, I might run three quizzes weekly,” said Guillermo. “And they all count toward the grade. It keeps everyone working consistently.”
“WirisQuizzes isn’t just a quiz engine,” Javier added. “It’s a full feedback system. It’s the backbone of our math learning platform.”
The Future of WirisQuizzes at La Salle
La Salle continues to expand the use of WirisQuizzes beyond math:
They’ve developed questions for economics, technology, and even geography.
Some teachers are exploring how AI-based feedback could provide personalized remediation paths.
A growing number of educators across La Salle’s network are joining upcoming training sessions to learn how to build questions and customize feedback.
They’ve also expressed interest in exploring remote proctoring, integrating with other LMS platforms like Google Classroom, and co-developing content with other schools.
Rounding Off
La Salle’s transformation is a shining example of how a school can build its own collaborative, intelligent, and scalable math learning platform. By integrating WirisQuizzes into every level of instruction—from the classroom to home study—teachers empowered students with autonomy, parents with transparency, and the school with data-driven teaching strategies.
What started as one teacher’s experiment has become a system-wide initiative that is changing how La Salle teaches and how students learn.
👉 Want to build your own feedback-driven math platform?
The Towers of Hanoi is one of the most iconic puzzles in the field of mathematics and computer science. With its simple structure and transparent rules, this problem has captivated mathematicians and amateurs alike, making it an intriguing challenge since its creation.
According to the legend, the Towers of Hanoi were conceived by a group of monks in an ancient temple in India. In the center of this temple, three golden pillars stood, upon which rested 64 disks of various sizes, arranged in the first pillar in an orderly fashion: the largest disk at the base, with progressively smaller disks stacked on top, until reaching the smallest disk, placed at the top.
The monks’ objective was to move all the disks from the first pillar to the third, using the central pillar as an auxiliary. To accomplish this task, they had to follow a strict set of rules:
Only one disk can be moved at a time.
A disc can never be placed on top of a smaller disc.
Legend says that when all the disks were correctly transferred to the last pillar, the world would come to an end. However, how close is the end of the world, really? How long would it take to complete this challenge?
This challenge was officially introduced by the French mathematician Édouard Lucas in 1883 as part of his research in number theory and quickly became a popular problem.
Next, we will analyze the solution of this problem from a mathematical perspective, breaking down its key principles and explore how it transforms into an interesting programming challenge.
Recursion as a Solution to the Puzzle
The challenge of the Towers of Hanoi is not only in moving the disks from one pillar to another following the strict rules but also in calculating how many moves are needed to solve the problem of moving n disks from the origin pillar to the destination pillar.
The most efficient way to approach this problem is through recursion, a fundamental technique in programming. Recursion allows the problem to be divided into smaller subproblems, where the solution of each subproblem leads to the solution of the original problem.
In the case of the Towers of Hanoi, recursion applies naturally, and we can divide the problem of moving n disks into three steps:
Move the first n – 1 disks from the origin pillar to the auxiliary pillar, using the destination pillar as the auxiliary.
Move the largest disc (the disc n) from the origin pillar to the destination pillar.
Move the n – 1 disks from the auxiliary pillar to the destination pillar, using the origin pillar as an auxiliary.
We can repeat this pattern recursively, reducing the number of discs with every step until we reach a base case where only one disk remains, which we just directly move.
In mathematical terms, the minimum number of moves to solve the problem with n disks equals the number of moves needed to solve it for n – 1 disks, plus one to move the largest disk, plus the number of moves needed to solve it for n – 1 disks again. This formula yields the result:
TH( n ) = TH( n – 1 ) + 1 + TH( n – 1 ) = 2TH( n – 1 ) + 1
Where TH( n ) represents the number of moves needed to solve the problem with n disks.
From the above formula, we will substitute each TH( i ) by its corresponding expression until we reach TH( 1 ) . Then, we will replace TH( 1 ) by 1, since, with only one disc, it is possible to move it directly to the destination pillar, which justifies that the minimum number of moves in this case is 1.
Let’s see the expression developed:
We first substitute TH( n – 1 ), and we obtain:
TH( n ) = 2TH( n – 1 ) + 1
TH( n ) = 2( 2TH( n – 2 ) + 1 ) + 1 = 2²TH( n – 2 ) + 3
Now we substitute TH( n – 2 ):
TH( n ) = 2²TH( n – 2 ) + 3
TH( n ) = 2²( 2TH( n – 3 ) + 1 ) + 3 = 2³TH( n – 3 ) + 7
If we continue substituting in this way, we will arrive at a general formula:
TH( n ) = 2kTH( n – k ) + (2k – 1)
Where k is the number of steps backward in the recursion.
For k = n – 1 we arrive at TH( 1 ), the base case:
TH( n ) = 2²TH( n – 2 ) + 3
We know that TH( 1 ) = 1, then we obtain:
TH( n ) = 2n – 1 · 1 + ( 2n – 1 – 1 ) = 2n – 1
We have shown that the number of moves needed to solve the Towers of Hanoi with n disks is:
TH( n ) = 2n– 1
Solving the Puzzle Using Mathematical Induction
Another way to approach the solution to the Towers of Hanoi puzzle is to use mathematical induction, a technique used to prove that a statement is true for all natural numbers, based on two fundamental steps: the base case and the inductive step.
In this case, the goal is to show that to move disks from the first pillar to the third pillar, 2n– 1 moves are required. Let’s break this process down with a demonstration by induction.
Base case:
Suppose we have only one disk. In this scenario, it is clear that only one move is needed to move the disk from the source abutment to the target abutment. That is, for n = 1, the number of moves required is 21– 1 = 1, which is true.
Inductive step:
Now, suppose that the statement is true for n = k, i.e., that to move k disks requires 2k– 1 moves. What we must prove is that the statement is also true for n = k + 1.
To move k + 1 disks, we first need to move the upper disks from the origin pillar to the auxiliary pillar, using the destination pillar as an auxiliary. According to our inductive hypothesis, this will take 2k– 1 moves.
Next, we move the larger disk (the disk k + 1) from the origin pillar to the destination pillar, which requires just 1 move.
Finally, we move the k disks from the auxiliary pillar to the destination pillar, using the origin pillar as an auxiliary, which will also take 2k– 1 moves, according to our inductive hypothesis.
Therefore, the total number of moves required to move k + 1 disks is:
(2k– 1) + 1 + (2k– 1) = 2k+1– 1
Thus, we have shown that if the statement is true for n = k, it is also true for n = k + 1.
As we have observed, it is true for 1. By mathematical induction, we can conclude that for n disks, the minimum number of necessary moves is 2n– 1.
Going back to the legend, if the monks were extremely fast and could move a disk in just one second, we could calculate how long it would take to complete the challenge with 64 disks. According to the formula, the total number of moves would be 264– 1, approximately 18.4 quintillion moves. With each of these taking just 1 second, this gives us 18.4 quintillion seconds. Converting these into years, we get that the total time would be approximately 581.4 billion years. So, if the legend turned out to be true, the end of the world would still be a long way away!
We note that the function describing the number of moves needed to solve the Towers of Hanoi is exponential, meaning that it grows rapidly as we increase the number of disks. In the following graph, we can visually see how this function spikes, clearly illustrating the exponential growth.
This visual support allows us to confirm, once again, that completing the challenge with 64 discs would take an extraordinary amount of time.
If you liked this mathematical challenge and you were surprised by its solution, don’t hesitate to share this article! And if you dare, test your skills by solving the Towers of Hanoi puzzle: will you manage to move the disks in the exact number of steps? Let us know your experience!
Artificial Intelligence (AI) is a field of computer science that seeks to develop systems able to perform tasks that traditionally require human intelligence, such as logical reasoning, problem solving, and learning. Based on advanced algorithms and machine learning models, AI can analyze large volumes of data, identify complex patterns and adapt autonomously to new contexts.
In mathematics, these capabilities are transforming the way knowledge is generated, verified, and communicated. From automating demonstrations to generating detailed explanations, AI is facilitating the work of researchers, teachers, and students. In addition, its ability to process natural language is improving the writing of mathematical papers, making texts more accessible, accurate and structured.
In this article, we will explore how AI in math is revolutionizing mathematical writing and how Wiris’ tools are part of this advancement.
The Role of AI in math and the Creation and Editing of Mathematical Content
Writing scientific papers in mathematics has always been a challenge due to the complexity of their notations and symbols. Traditionally, mathematicians have relied on handwriting or the use of complex typesetting languages. However, with the advent of AI, the process of creating and editing scientific documents has evolved exponentially.
Artificial intelligence has brought multiple benefits to the writing of mathematical papers, facilitating both the writing and the editing and publication of academic papers. Below are some of the main ways in which AI is impacting this area.
Handwriting recognition
One of the most significant advances has been the development of systems capable of interpreting and digitizing handwritten notes. Traditionally, mathematicians wrote their equations and proofs on paper, which was time-consuming if they needed to transcribe them into a digital format. With AI-based handwriting recognition tools, such as those built into MathType, it is now possible to directly handwrite equations into our touch devices and automatically convert them into editable digital text.
This not only saves time but also reduces transcription errors and allows you to work more efficiently. In addition, this type of technology is especially useful for students and teachers, as it facilitates the conversion of notes into organized documents without the need for manual typing.
Formula automation
In the creation of mathematical articles, writing formulas accurately is essential. Traditionally, this required specialized tools that, while powerful, could be complex and demanded a deep understanding of their syntax and structure, such as LaTex.
Advancements in technology have enabled the development of tools like MathType, which simplify equation writing without the need for coding. Instead of memorizing commands and code structures, users can simply enter their formulas through an intuitive graphical interface. This democratizes access to mathematical writing, allowing more people, regardless of their technical expertise, to create high-quality mathematical documents.
Another key aspect is the automatic correction of errors in equations. AI in math can identify inconsistencies in formula writing and suggest real-time corrections, preventing mistakes that could compromise the validity of an academic paper.
Optimization of editing
Another benefit that AI has brought is the optimization of the editing of mathematical documents. In the past, mathematicians and scientists had to manually check their documents for errors in equations, notation and text structure. Today, there are AI algorithms embedded in editing programs that can analyze the consistency of mathematical expressions, suggest improvements in writing and detect inconsistencies.
AI-based writing assistants, such as those integrated into advanced text processors, can help improve the clarity and readability of mathematical articles. In addition, they provide stylistic recommendations to give the paper a logical and easy-to-follow structure, which is crucial in academic paper writing.
The integration of AI into mathematical writing is not just an evolution but a revolution that is transforming the way we create, edit, and share scientific knowledge. Tools like Wiris, with their powerful capabilities, are making mathematical writing more accessible, efficient, and error-free. As these technologies continue to advance, they will further bridge the gap between human intuition and machine accuracy, making the work of researchers, teachers, and students alike easier. If you found this exploration interesting, share it with your peers and join the conversation about the future of AI in math. If you’d like to find out more about our products, please feel free to get in touch with our Sales team at sales@wiris.com for more information.
Learning mathematics requires more than just understanding theories: the key to mastering complex concepts and honing skills lies in continuous practice. However, traditional methods often limit the learning process: once the student has solved a problem, the challenge disappears, and exams often repeat the same questions, thus losing their value as an assessment tool. Imagine having the possibility to generate unlimited exercises with random variables, adapted to the needs of each student. With Wiris, this is now possible.
In addition, it has an automatic correction functionality which means that while the exercise is being generated, the answer is calculated simultaneously. This allows instant feedback, making it easier to learn and correct mistakes immediately.
For teachers, this tool offers a key advantage: the possibility of generating an infinite number of exercises, with each one being unique. In this way, teachers can create varied and personalized learning experiences, ensuring that students face new challenges every time they practice.
For their part, students enjoy constant practice which allows them to reinforce their skills without the worry of repeating the same exercises. In addition, this ability to generate unlimited exercises gives them excellent preparation for exams.
Practical example: Solving a system of linear equations
To illustrate how this tool works and how it can be used in the classroom, we will focus on solving systems of linear equations. We chose these types of problems because they represent a common mathematical challenge faced by all students and offer the possibility to explore different problem-solving methods. In addition, systems of equations can have various characteristics, such as being compatible, incompatible, or indeterminate, allowing students to work with a wide range of situations.
Let us imagine that we have the following system of linear equations with three unknowns:
We are going to solve it using Gauss’s Method. The first step is to write the system in the form of an augmented matrix:
Now, we will apply elementary operations to reduce the matrix to its row echelon form.
Therefore, we obtain:
From equation 3 of the system, we find the variable z:
Unlimited generation of systems of linear equations with random variables
Now that we have solved this system, for further practice, we would like to be able to generate new systems of equations of the same form but with random values.
These systems would take the following form:
Below, we will show you a code example to implement this functionality and generate given random systems of equations. With Wiris, it is totally possible!
r() := random(-4,4)
sol = [r() with i in 1..3]
[a,b,c] = sol
repeat
A = [[r() with i in 1..3] with j in 1..3]
until determinant (A) != 0
b = A * sol
ec = {}
for i in 1..3 do
ecaux = A.i * [x,y,z]
ec = append(ec, ecaux = b.i)
end
{ec1, ec2, ec3} = ec
r() := random(-4,4): This function generates a random number between -4 and 4.
sol = [r() with i in 1..3]: Here, three random values are generated that will represent the solutions of the system of equations.
[a,b,c] = sol: The generated values are assigned to the unknowns of the system.
repeat…until determinant(A) != 0: This block ensures that the coefficient matrix A is invertible (i.e., its determinant is not zero), which guarantees the system has a unique solution.
b = A * sol: The result vector b is calculated by multiplying the coefficient matrix A by the solutions sol
for i in 1..3 do: In this cycle, the three equations are generated, using each of the rows of the matrix A and the solutions sol.
{ec1, ec2, ec3} = ec: Finally, the three generated equations are stored in the variables ec1, ec2, and ec3, ready to be used in new exercises.
Using WirisQuizzes and implementing this code as a question, we can obtain the following results:
Reloading another system with the symbol “=” generates a new set of equations, as shown below:
For more details, you can consult the complete product documentation here.
If you have found this article useful, we would love you to share it with other colleagues, teachers or students. We would also be delighted to hear your opinion and any suggestions you may have.
The birthday paradox is a mathematical concept that invites us to challenge our intuition. A paradox, in essence, is a statement or result that, although seemingly contradictory or contrary to common logic, proves to be true when analyzed closely. In the case of the birthday paradox, the scenario is simple but puzzling: how many people must be in a room for the probability of at least two sharing the same birthday to be greater than 50%?
The result will be surprisingly low, but do you dare to answer before you continue reading to reveal the answer?
The most intuitive way to approach the paradox would be to think that it takes at least 183 people, half the days of the year, for the probability of two people sharing the same birthday to be greater than 50%. However, the reality is very different.
To understand how this probability is calculated, let us recall that the basic formula of any probability consists of dividing the favorable cases by the total cases.
In this case, rather than calculating the probability of a coincidence directly, it is easier to first calculate the probability that no two people share a birthday. After determining this probability, subtracting it from 1 (that is, calculating the complement) gives the probability that at least two people share a birthday.
To calculate the probability that no two people share a birthday, we will proceed constructively, evaluating scenario by scenario.
If there is only one person in a room, there is no chance that they will share their birthday with someone else since no other people are present. Therefore, the probability that their birthday is unique is 100%, i.e., 365/365.
Now, if a second person comes in, the probability that their birthday will be different from the first person’s is 364/365, since there are only 364 days remaining in which they can be born to avoid a coincidence.
If a third person is added, the probability that their birthday does not match anyone else’s is 363/365 since there are now two busy days.
Therefore, for n people, all the above conditions must be met simultaneously: that the second person does not share a birthday with the first, that the third person does not share a birthday with the previous two, and so on. This implies that the probability that none of the n people share the same birthday is the product of the individual probabilities:
And finally, we obtain:
Analyzing probabilities for different values of n
Let us examine the results for different values of n:
For n=10 we obtain that P (at least one coincidence) = 11,61%
This calculation has been created using CalcMe
For n=15 we obtain that P (at least one coincidence) = 25,03%
For n=23 we obtain that P (at least one coincidence) = 50,05%
For n=50 we obtain that P (at least one coincidence) = 96,53%
For n=60 we obtain that P (at least one coincidence) = 99,22%
As we can see from the examples above, to achieve a probability of more than 50%, only 23 people in the same room are needed. This result may seem surprising, but it makes sense if we consider that with 23 people, 253 different pairs can be formed, and each of these pairs represents a chance for two people to share the same birthday. Even more surprising is that to have a probability greater than 99%, only 60 people are needed, demonstrating how the number of possible combinations grows rapidly with every new person added.
This is the graph of the probability distribution:
Graph showing the probability of two people sharing a birthday, with the number of people on the X-axis and the probability on the Y-axis, expressed as a percentage.
In contrast, if instead of calculating the probability of coincidence between any pair of people we consider the probability that someone in a room of n people (excluding you) specifically shares your birthday, the calculation is different. This probability is given by:
For n = 22, this probability is about 0.059 (5.9%), which is quite low. In fact, there would need to be at least 253 people in the room for this probability to exceed 50%.
The birthday paradox is a fascinating reminder of how our intuition can fail when confronted with mathematical logic.
Now that you know the birthday paradox and its key concepts, can you calculate the probability that exactly k people within a group of n people share their birthday? We’d love to hear your results in the blog comments! If this article surprised you, share it with your friends so that they can learn more about this mathematical paradox. What’s more, if you’re interested in trying these calculations for yourself, have a look at CalcMe.
