Author: Aida Alsina

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AI in Mathematics: The Revolution in Scientific Writing

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.

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Wiris in the Classroom: Learn and Practice with Unlimited Exercises

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.

 

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What is the Birthday Paradox?

birthday paradox

The Fascinating Probability of Sharing a Date

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%

CalcMe representation of the equation

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.

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.

 

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Gamification in Math Learning: Does it Improve Learning?

Teenager playing learning

Gamification in Math Learning: Benefits & Challenges

Struggling to keep students engaged? Gamification, the use of game mechanics in learning, helps increase motivation, comprehension, and retention. The abstraction of concepts and the difficulty in linking them to real-life situations can generate disinterest, frustration and even anxiety in students. In this context, gamification has emerged as an innovative strategy that aims to transform the learning experience by integrating game elements into the educational process.

But what exactly is gamification, and what impact does it have on the teaching of mathematics? According to Deterding et al. (2011), it is defined as the use of game design elements in non-game contexts to enhance the participation, motivation, and engagement of individuals. In the educational setting, this involves not only the incorporation of game mechanics such as challenges, immediate feedback, points, and rewards, but also the creation of more dynamic and interactive experiences that encourage exploration and active learning.

Throughout this article, we will explore the existing evidence on the effectiveness of gamification in math education and see how interactive games like LearningLemur can benefit students’ learning.

Why Gamification Works: Key Benefits for Math Learning

Gamification has proven to be an effective tool for improving the learning experience in various disciplines, and mathematics is no exception. Its impact goes beyond making classes more entertaining; it influences key aspects such as motivation, understanding of concepts, and the development of cognitive and social skills. Below are some of the main benefits supported by research.

Increased Motivation and Engagement

One of the main challenges in teaching mathematics is to maintain students’ interest, especially when they perceive the exercises as repetitive or difficult. Gamification transforms these tasks into interactive and stimulating experiences. According to a study by Hamari, Koivisto, and Sarsa (2014), the implementation of gamified strategies has been shown to significantly increase motivation and academic performance in various disciplines, including mathematics.

Improved Comprehension and Retention of Concepts

 

Games allow students to interact with mathematical concepts in a hands-on, visual way, facilitating deeper understanding. Instead of memorizing formulas without context, students can experiment with problems in dynamic scenarios. A study by Wouters et al. (2013) indicated that gamified environments not only improve knowledge retention but also strengthen the ability to apply what is learned in different situations. 

Cognitive and Social Skills Development

Beyond learning mathematical concepts, gamification promotes essential skills for critical thinking and problem-solving. Solving challenges within a gamified environment requires students to analyze situations, explore different strategies, and adapt their approach based on the results obtained. In addition, many gamified experiences include collaborative dynamics that foster communication and teamwork. It also encourages resilience by allowing students to learn from their mistakes without the fear of conven

tional academic failure.

Mathematics Anxiety Reduction

Math anxiety is a common problem that can affect students’ performance and attitude toward the subject. Presenting problems in a playful and less structured format can reduce the pressure associated with formal learning. According to Sailer et al. (2017), gamification helps to generate a more positive relationship with mathematics, allowing students to face challenges without fear of failure. Immediate feedback and reward systems reinforce individual progress, helping to build confidence and autonomy in learning.

Gamification Challenges: What Educators Need to Know 

While gamification offers multiple benefits in mathematics learning, its effectiveness depends largely on how it is designed and implemented. Some key factors to consider for successful implementation are:

  • Designing appropriate content: Creating gamified activities aligned with curricular objectives requires time and planning on the part of educators.
  • Balancing Play and Learning: It is essential that the fun of the game does not overshadow the educational content, ensuring that students acquire the intended mathematical skills. To avoid this problem, it is key to design activities where progress in the game depends on mastery of the mathematical concepts, ensuring that the mechanics reinforce the content rather than distract from it.
  • Adaptability: Each student learns at their own pace and in a different way. An effective gamified approach must be flexible and allow for customization according to learners’ skill levels and learning styles.

Strategies for Effective Gamification in Mathematics Learning

To maximize the potential of gamification in math, educators and developers must carefully structure their approach. Here are some key strategies to ensure effective implementation:

Setting Clear Learning Objectives

Gamification must have a clear educational purpose to be effective. Before incorporating game elements, it is essential to define precise learning outcomes. Every challenge, reward, or interactive element should directly contribute to developing students’ understanding of mathematical concepts. When games are designed to align with curriculum goals, students remain engaged while making measurable academic progress.

Incorporating Interactive Games for Engagement

Engaging students through interactive math learning is one of the most effective ways to enhance learning. Digital platforms and apps provide adaptive learning experiences that offer students challenges suited to their individual skill levels. Features such as leaderboards, badges, and achievements create a sense of accomplishment, motivating students to persist in solving mathematical problems while reinforcing key concepts.

Providing Instant Feedback and Progress Tracking

Timely feedback is crucial in the learning process. One of the major advantages of gamification in math learning is the ability to provide instant feedback. When students receive immediate results on their performance, they can quickly identify mistakes, adjust their approach, and strengthen their understanding. Many digital platforms include progress-tracking tools, allowing both students and educators to monitor learning trends, address weaknesses, and celebrate achievements.

Encouraging Collaboration and Healthy Competition

Gamified learning is not just about individual progress—it can also foster teamwork and social interaction. Incorporating cooperative challenges and team-based problem-solving enhances mathematics learning by encouraging peer-to-peer discussions and shared strategies. At the same time, friendly competition through scoreboards and ranking systems can inspire students to push themselves further, increasing motivation and engagement in a healthy, constructive way.

Tools and Resources for Gamifying Mathematics Learning

With advancements in technology, numerous tools are available that support gamification in math. These resources offer diverse ways to create a more dynamic learning experience, helping students grasp mathematical concepts more effectively:

  • Digital Learning Platforms: Websites and apps that integrate math practice with games to teach mathematical concepts through fun and challenging activities.
  • Classroom Gamification Kits: Both physical and digital resources are designed to help teachers implement game-based learning strategies in their lessons.
  • Virtual and Augmented Reality: Immersive experiences that provide students with visual and hands-on interactions to better understand abstract mathematical ideas.
  • Customizable Gamification Plugins: LMS-integrated tools that allow educators to personalize their teaching by adding game-based elements to their curriculum.

Try Gamification in Mathematics Learning with Learning Lemur

The benefits of gamification in mathematics learning are clear: increased motivation, improved comprehension, and enhanced engagement. If you’re ready to experience the power of interactive games in education, explore Learning Lemur. Our platform allows educators and students to immerse themselves in engaging, game-based mathematics learning experiences.

Create a free account today and discover how gamification can revolutionize the way you learn mathematics!

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What is the Collatz Conjecture?

What is the Collatz Conjecture?

Mathematical conjectures are riddles that challenge the human mind: seemingly simple problems that, despite having been verified in millions of cases, still lack a proof. They are seemingly intuitive questions that hide unexpected complexity. 

The fascinating thing about conjectures is that, in many cases, they can be empirically verified i.e., verified by an enormous number of numerical examples. However, this verification does not satisfy mathematicians, who seek a theoretical proof, a solid proof that validates the statement in its entirety. Without such a proof, the conjectures remain open challenges. 

The Collatz conjecture is a perfect example of this type of mathematical mystery. It was proposed in 1937 by Lothar Collatz and its statement is as follows: 

Choosing any positive integer we will apply the following steps: 

➔ If the number is even, we will divide it by 2. 

➔ If the number is odd, multiply it by 3 and then add 1. 

➔ Repeat the process with the new number obtained. 

The conjecture says that, no matter what number you start with, you will always get to number 1. Once you get to 1, the process repeats indefinitely: 1 →4 →2 →1

Formally, this can be written as the function f=N →N defined as

Two illustrative examples of the Collatz Conjecture

Let’s look at a couple of simple examples.

Let’s take the number n=6 and apply the steps of the Collatz conjecture:

  1. 6 is even so we divide by 2: 6/2=3
  2. 3 is odd so we multiply by 3 and add 1: 3·3+1=10
  3. 10 is even so we divide by 2: 10/2=5
  4. 5 is odd so we multiply by 3 and add 1: 3·5+1=16
  5. 16 is even so we divide by 2: 16/2=8
  6. 8 is even so we divide by 2: 8/2=4
  7. 4 is even so we divide by 2: 4/2=2
  8. 2 is even so we divide by 2: 2/2=1

Once we reach 1, the process is repeated: 1 →4 →2 →1.

So we observe that starting with n=6 the conjecture is satisfied.

Let us now take the number n=21 and apply the steps of the Collatz conjecture:

  1. 21 is odd so we multiply by 3 and add 1: 21·3+1=64
  2. 64 is even so we divide by 2: 64/2=32
  3. 32 is even so we divide by 2: 32/2=16
  4. 16 is even so we divide by 2: 16/2=8
  5. 8 is even so we divide by 2: 8/2=4
  6. 4 is even so we divide by 2: 4/2=2
  7. 2 is even so we divide by 2: 2/2=1

Again, we arrive at number 1.

Both examples have reached the number 1, but in a different number of steps. In fact, although 21 is a larger number, it reached 1 in fewer steps than 6. This conjecture has been tested for an incredible number of numbers, up to more than 2ˆ60 = 1.152.921.504.606.846.976 cases, without finding any counterexample. However, it still remains a mystery whether there is a number that does not satisfy the conjecture.

Graphical representations of the Collatz conjecture

The directed graph of orbits is a visual representation that facilitates the understanding of the behavior of numbers under the rules of the Collatz conjecture. In this graph, each number is represented as a node, and the connections between them show the steps followed by the conjecture process. When following the sequence of a number, the nodes are connected by arrows that indicate how the number is transformed at each step. Although even numbers are generally omitted from the representation for simplicity, the graph illustrates how the numbers “orbit” around certain cycles, such as 1 →4 →2 →1.

In the image above, we can see an example of the directed graph of orbits, where we have highlighted in black the numbers 3 and 21. The number 3 refers to the sequence of 6, since in this graph the even numbers are omitted to simplify the visualization.

We note that 21 quickly reaches 1, since it only passes through even numbers on its way, as we saw in the previous example. On the other hand, the number 6 first becomes 3, then pass through the number 5, which will finally take it to 1 after several passes through even numbers. 

It is also interesting to note that the number 9, although relatively small, follows a longer sequence: it goes through the numbers 7, 11, 17, 17, 13 and 5, and finally reaches 1, after a total of 19 steps. 

Below is a graph in which the X-axis shows the different initial integer values, while the Y-axis represents the number of iterations required for each number to reach the value 1.

In summary, the Collatz conjecture is an apparently simple problem that, to this day, still has no formal proof. Although it may seem a statement of little relevance or practical utility, this conjecture has applications in various fields, such as number theory, cryptography, algorithm analysis and artificial intelligence. In these fields, the study of complex sequences and their behavior under specific rules can offer valuable insights for solving larger problems and understanding mathematical patterns.

Have you been fascinated by the simplicity and mystery of the Collatz conjecture? Now it’s your turn: take the number 27 and start the sequence. How many steps will you need to get to 1?

If this challenge got you, don’t keep it to yourself! Share it with others who are curious about mathematical riddles and find out together who can solve the challenge the fastest.

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MathType Math Equation Writer Joins the Skolon Educational App Store

A New Chapter for MathType: Now on Skolon

Wiris is thrilled to announce that MathType, our renowned math equation writer for creating and editing math equations, is now available on Skolon’s educational App Store. This partnership marks a significant milestone in our mission to simplify mathematical education for students and teachers worldwide. With features that enhance math content formatting, MathType improves how math expressions are presented and understood in digital learning environments.

What Does This Mean for Educators and Students?

Through Skolon’s platform, educators can seamlessly integrate MathType’s equation editor into their teaching resources. With its intuitive interface and powerful features, MathType makes it easier than ever to create professional-quality math equations and notations. For students, this tool provides an accessible way to engage with complex mathematics, fostering a deeper understanding of the subject. Additionally, its formatting tools ensure that mathematical expressions remain clear and well-structured in different digital contexts.

Why Choose MathType Math Equation Writer on Skolon?

By joining forces with Skolon, we ensure that educators and students gain easy access to MathType’s math equation writer’s capabilities. Skolon’s platform simplifies the process of discovering, purchasing, and deploying educational tools, making it an ideal partner for delivering MathType’s benefits to a wider audience. Now, schools can quickly adopt MathType as part of their digital learning ecosystem, streamlining both teaching and learning experiences. With enhanced formatting and compatibility features, MathType further improves the clarity of mathematical content across multiple formats.

About Skolon’s Educational App Store

Skolon is a trusted name in the education technology space in Northern Europe, offering a comprehensive App Store that caters to the diverse needs of schools and educators. Their platform enables users to access and manage a wide range of digital tools in one convenient location, and we’re proud to see MathType’s math equation writer among their trusted offerings. The ability to generate clear and structured mathematical expressions ensures that digital math content remains visually precise and easily shareable.

Get Started with MathType Today

We invite you to explore MathType on Skolon and experience firsthand how its math equation writer can enhance your educational efforts. Whether you’re a teacher aiming to enrich your curriculum or a student looking for a better way to approach mathematics, MathType on Skolon is here to support you. With advanced formatting options, MathType ensures mathematical clarity across digital platforms.

Discover MathType on Skolon Now

Looking Ahead

At Wiris, we’re committed to empowering the education sector through innovative tools like MathType. Partnering with Skolon brings us closer to our goal of making math accessible, engaging, and enjoyable for everyone. By leveraging MathType’s math equation writer and its enhanced presentation features, we continue to enhance digital learning worldwide.

Stay tuned for more updates as we continue to expand our reach and improve learning experiences globally.

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How to solve an equation of degree 16

Step-by-step solution of an Oxford access exam problem

Every year, thousands of students face the challenge of the admission exams to enter the prestigious University of Oxford, a process that tests not only their knowledge, but also their ability to solve under pressure. In particular, the math exam is known for its complexity, posing problems of all kinds, from solving equations to questions of logic, advanced algebra, calculus and number theory. In this article, we will explore one of the problems presented in October 2023. 

The problem poses the following question: how many real solutions does the following equation have?

Do you dare to solve the equation before reading the complete solution in the blog? 

At first glance, the statement seems simple: an equation with a single variable, where all the numbers involved are integers between 1 and 4. However, the real difficulty lies in correctly undoing the squared parentheses, making sure to consider all possible cases. 

A key detail in this type of equations is that, when squaring a number, the same result is obtained for both its positive and negative values. For example: 

This occurs because squaring a number eliminates the negative sign. Therefore, when solving a quadratic equation we must take into account the two possible options: the positive and the negative number that can generate that result. 

One last observation: if we were to fully develop the parentheses, we would obtain an equation of the form xˆ16+…, which indicates that the equation is of degree 16 and, at most, could have 16 real solutions for x. Therefore, we cannot rule out any of the options presented to us as possible answers.

Since for equations of degree 16 there is no systematic formula like Ruffini’s method for polynomials of degree 3, we will solve this equation by working the parentheses progressively, from the outermost to the innermost.

 

Specific example of how to solve an equation of degree 16

Let’s get on it! 

Let’s start with the outermost parenthesis. If we define t as ((x²-1)²-2)²-3 , then we get t²=4. This gives us two possible values for t: t=2 or t=-2. 

Substituting t by its original value in each case, we obtain the following results: 

1. If t=2 then: ((x²-1)²-2)²-3 =2 ⇒ ((x²-1)²-2)²=5 and again, substituting r= (x²-1)²-2 we obtain that r²=5 . Let’s look at the two new cases that arise:

1.1 r=√5: substituting r for its original value we obtain (x²-1)²-2 =√5 ⇒ (x²-1)²=2+√5 and again, one last time, we substitute s=x²-1, we obtain that s²=2+√5 and observe the two results again:

When trying to calculate the square root to obtain the value of x, we would be taking the root of a negative number. As a result, the two values obtained will be imaginary numbers.

1.2 r=-√5 substituting r for its original value we obtain (x²-1)²-2=-√5 ⇒ (x²-1)²=2-√5 . Following the same reasoning as in the previous section, given that √4=2, it follows that √5>2 . This implies that 2-√5<0, and when trying to calculate its square root, we would obtain an imaginary number. Therefore, when we continue developing to obtain the value of x, this would also be an imaginary number.

2. If t=-2 then: ((x²-1)²-2)²-3 =-2 ⇒ ((x²-1)²-2)²=1 and again, substituting r= (x²-1)²-2 we obtain that r²=1 . Let’s look at the two cases:

2.1 r=1: substituting r for its original value we obtain (x²-1)²-2 =1 ⇒ (x²-1)²=3 and again, one last time, we substitute s=x²-1, we obtain that s²=3 and observe the two results again:

2.1.1 s=√3 and substituting s for its value, we finally obtain: x²-1=√3 ⇒ x²=1+√3 and therefore

2.1.2 s=-√3 substituting s for its original value we obtain x²-1=-√3 ⇒ x²=1-√3 . This subtraction is negative, since √2≈1,41 and therefore, √3>1,41 with which we deduce that 1-√3<0 , and when trying to calculate its square root, we would obtain an imaginary number.

2.2 r=-1 substituting r for its original value we obtain (x²-1)²-2 = -1 ⇒ (x²-1)² = 1 and again, one last time, we substitute s=x²-1 and we obtain that s²=1 and observe the two results again:

2.2.1 s=1 and substituting s for its value, we finally obtain: x²-1 = 1 ⇒ x²=2 and therefore

2.2.1.1 x=√2 which is a real solution.

2.2.1.2 x=-√2 which is a real solution.

2.2.2 s=-1 and substituting s for its value, we finally obtain: x²-1 = -1 ⇒ x²=0 and therefore the only possible solution is x=0 .

Let’s review all the possible real values we have obtained:

 

This gives us a total of 7 real solutions for x, so the correct answer is option (c).

Did you get your answer right?

Drawing this equation with our graphing tool is a simple and effective way to check the number of real solutions. Looking at the graph, you can clearly see how the curve interacts with the x-axis, confirming the previous result. Below, we show you the graph to see for yourself:

Done with the WirisQuizzes Assessment tool

 

Will you dare to take on more challenges in the future?

If you found this content useful or inspiring, please share it with your friends and fellow number lovers!

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Instant Feedback in Open-Ended Math Questions: Key Takeaways from BETT 2025

Bett congress 2025 Wiris Tech in Action

Recently, we had the exciting opportunity to showcase our approach at BETT 2025: Enhancing Open-Ended Math Questions with Instant Feedback to teachers, publishers and educators alike at the conference in London. Feedback plays a crucial role in guiding learning, and at the event, we focused on how instant feedback in open-ended math questions can dramatically enhance students’ mathematical understanding, all the while assisting teachers and making their day to day more efficient.

Unlike traditional multiple-choice problems, open-ended mathematics questions require deeper thinking as they are questions that a student can freely answer without a predefined format, making personalized feedback more vital. In this article, we will explore the significance of instant feedback in open-ended math questions and discuss how different types of feedback—corrective feedback, confirmatory feedback, and suggestive feedback—can help students improve.

 

 

Why Instant Feedback Matters in Open-Ended Math Questions

The Power of Feedback in the Learning Process

Feedback is integral to the learning process, helping students refine their thinking and improve their understanding. This is especially true for open-ended mathematics questions, where there is often no one right answer. In such cases, instant feedback in open-ended math questions can make a significant difference in guiding students toward the correct solution.

During BETT 2025, we discussed how instant feedback in math questions not only helps students correct mistakes quickly and builds their confidence but how it also assists teachers in the class environment making the teaching and learning process much more efficient.

 

Types of Feedback for Open-Ended Math Questions

Different types of feedback are required depending on the student’s needs and the nature of the question. The three main types of feedback we focused on at BETT 2025 are suggestive feedback, confirmatory feedback, and corrective feedback.

Suggestive Feedback in Math Questions

Suggestive feedback in math questions guides students to discover the solution themselves. Instead of providing the correct answer outright, suggestive feedback encourages independent problem-solving. For example, if a student suggests “2 and 8” as the two numbers whose sum is 10 and product is maximum, suggestive feedback might be: “What happens if you try numbers closer together?”

 

Confirmatory Feedback in Math Questions

Confirmatory feedback in math questions is designed to reinforce correct answers. When students solve problems correctly, feedback like “That’s correct, well done!” helps to affirm their understanding and boosts their confidence.

In the case of open-ended math questions, confirmatory feedback is invaluable for ensuring students stay motivated, especially when they are tackling complex problems.

This type of feedback fosters critical thinking and encourages exploration, making it an essential tool for developing problem-solving skills in open-ended math questions.

 

Corrective Feedback in Math Questions

Corrective feedback in math questions identifies mistakes and provides the correct answer. For example, if a student incorrectly solves x^2=4 as x=4, the corrective feedback would be “The correct solution is x= +-2”.

While effective for addressing misunderstandings, corrective feedback should be used in moderation, as over-reliance on it can hinder the development of problem-solving skills.

 

 

The Teaching Cycle: A Framework for Feedback Application

At BETT 2025, Mrs. Brook, a fictional high school math teacher, shared her teaching cycle, which integrates corrective, suggestive, and confirmatory feedback seamlessly. Her cycle is designed to maximize learning outcomes by aligning feedback strategies with the different stages of teaching.

The Four Stages of Mrs. Brook’s Teaching Cycle

  • Delivering the Content

Mrs. Brook begins her teaching by introducing and explaining the content to her students. This step ensures all students have a baseline understanding of the topic.

  • End-of-Class Knowledge Validation

Following the lesson, Mrs. Brook conducts an end-of-class knowledge check. This is where corrective feedback plays a crucial role. For example, if a student misunderstands a concept or calculation, Mrs. Brook provides immediate corrective feedback to clarify misconceptions.

 

  • Recommending Practice at Home

Mrs. Brook encourages her students to practice independently. At this stage, she often employs suggestive feedback to guide students without directly giving them answers, helping them to think critically and explore solutions. Additionally, confirmatory feedback is used to reinforce correct solutions and build student confidence. When students solve problems accurately, Mrs. Brook provides positive reinforcement such as “Great job!” to motivate them and affirm their understanding.

 

  • Assessing Students’ Knowledge Levels

After students have had the opportunity to practice, Mrs. Brook assesses their progress to identify areas of improvement. During this phase, she focuses on encouraging students to reflect on their learning journey and strive for continued growth. She fosters a positive environment by recognizing their efforts and offering motivation to help them stay engaged and confident in their abilities.

This cyclical model of teaching and feedback showcases how structured feedback enhances learning and ensures students’ needs are addressed at every stage​Tech.

 

 

Wrapping up: Embracing the Future of Math Education with Wiris

At WIRIS, we’re committed to supporting educators with the tools they need to provide effective feedback. Our Learning Lemur and WirisQuizzes products ensure that instant feedback in open-ended math questions is accessible and customizable, helping students succeed in their math learning journey.

As technology continues to evolve, the integration of instant feedback into open-ended math questions will only become more critical. We look forward to the continued evolution of math education, driven by personalized learning and real-time feedback.

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