Can’t seem to find the mistake in your math? Instead of painfully combing through each line, let the new “Check my work” operation in Maple Learn help! Now in Maple Learn, you can type out a solution to a variety of math problems, and let Maple Learn check your work! Additionally, by signing on to Maple Learn and the Maple Calculator app, you can take a photo of your handwritten math, import it into Maple Learn, and check your work with the click of a button.

Whether you’re solving a system of linear equations or an algebra problem, computing an integral or a partial derivative, “Check my work” can help. Maple Learn will tell you which steps are “Ok” and which steps to double-check. If you get a step wrong, Maple Learn will point out which line has an error, then proceed to check whether the rest of your work followed the right procedure.

Here’s an example of a solution to a system of linear equations written out by hand. All I had to do was snap a picture in the Maple Calculator app, and Maple Learn instantly had my equation set ready to go in the Cloud Expressions menu. Then, I just clicked “Check my work” in the Context Panel.

Maple Learn identified that I was trying to solve a system with 3 equations, checked my steps, and concluded my solution set was correct.

What happens if you make a mistake? Here’s an example of evaluating an improper integral with a u-substitution that involves a limit. This time, I directly typed my steps into Maple Learn and pressed “Check my work” in the Context Panel. Check my work recognized the substitution step and noted what step was incorrect; can’t forget to change the limits of integration! After pointing out where my mistake was, Maple Learn continued to evaluate the rest of my steps while taking my error into account. It confirmed that the rest of the process was correct, even though the answer wasn’t.

After making my change in Maple Learn and checking again, I’ve found the correct value.

Checking your work has never been easier with Maple Learn. Whether you want to type your solution directly in Maple Learn or import math with Maple Calculator, the new “Check my work” feature has you covered. Visit the how-to document for more examples using this new feature and let us know what you think!

What was established in 1788 in Prussia, is derived from the Latin word for “someone who is going to leave”, and can be prepared for using the many capabilities of Maple Learn? Why, it’s the Abitur exam! The Abitur is a qualification obtained by German high school students that serves as both a graduation certificate and a college entrance exam. The exam covers a variety of topics, including, of course, mathematics.

So how can students prepare for this exam? Well, like any exam, writing a previous year’s exam is always helpful. That’s exactly what Tom Rocks Math does in his latest video—although, with him being a math professor at Oxford University, I’d wager a guess that he’s not doing it as practice for taking the exam! Instead, with his video, you can follow along with how he tackles the problems, and see how the content of this particular exam differs from what is taught in other countries around the world.

Oh, but what’s this? On question 1 of the geometry section, Tom comes across a problem that leaves him stumped. It happens to the best of us, even university professors writing high school level exams. So what’s the next step?  Well, you could use the strategy Tom uses, which is to turn to Maple Learn. With this Maple Learn document, you can see how Maple Learn allows you to easily add a visualization of the problem right next to your work, making the problem much easier to wrap your head around. What’s more, you don’t have to worry about any arithmetic errors throwing your whole solution off—Maple Learn can take care of that part for you, so you can focus on understanding the solution! And that’s just what Tom does. In his video, after he leaves his attempt at the problem behind, he turns to this document to go over the full solution, making it easy for the viewer (and any potential test-takers!) to understand where he went wrong and how to better approach problems like that in the future.

So to all you Abitur takers out there—that’s just one problem that can be transformed with the power of Maple Learn. The next time you find yourself getting stuck on a practice problem, why not try your hand at using Maple Learn to solve it? After that, you’ll be able to fly through your next practice exam—and that’ll put you one step ahead of an Oxford math professor, so it’s a win all around!

This is a reminder that we are seeking presentation proposals for the Maple Conference.

Details on how to submit your proposal can be found on the Call for Participation page. Applications are due July 11, 2023.

We would love to hear about your work and experiences with Maple! Presentations about your work with Maple Learn are also welcome.

We’re now coming to the end of Pride Month, but that doesn’t mean it’s time to stop celebrating! In keeping with our celebration of queer mathematicians this month, we wanted to take some time to highlight the works of LGBT+ mathematicians throughout history. While it’s impossible to say how some of these individuals would have identified according to our modern labels, it’s still important to recognize that queer people have always existed, and have made and continue to make valuable contributions to the field of mathematics. It’s challenging to find records of LGBT+ people who lived in times when they would have been persecuted for being themselves, and because of that many contributions made by queer individuals have slipped through the cracks of history. So let’s take the time to highlight the works we can find, acknowledge the ones we can’t, and celebrate what the LGBT+ community has brought to the world of mathematics.

If you ask anyone to name a queer mathematician, chances are—well, chances are they won’t have an answer, because unfortunately the LGBT+ community is largely underrepresented in mathematics. But if they do have an answer, they’ll likely say Alan Turing. Turing (1912-1954) is widely considered the father of theoretical computer science, largely due to his invention of the Turing machine, which is a mathematical model that can implement any computer algorithm. So if you’re looking for an example of his work, look no further than the very device you’re using to read this! He also played a crucial role in decoding the Enigma machine in World War II, which was instrumental in the Allies’ victory. If you want to learn more about cryptography and how the field has evolved since Turing’s vital contributions, check out these Maple MathApps on Vigenère ciphers, password security, and RSA encryption. And as if that wasn’t enough, Turing also made important advances in the field of mathematical biology, and his work on morphogenesis remains a key theory in the field to this day. His mathematical model was confirmed using living vegetation just this year!

In 1952, Turing’s house was burgled, and in the course of the investigation he acknowledged having a relationship with another man. This led to both men being charged with “gross indecency”, and Turing was forced to undergo chemical castration. He was also barred from continuing his work in cryptography with the British government, and denied entry to the United States. He died in 1954, from what was at the time deemed a suicide by cyanide poisoning, although there is also evidence to suggest his death may have been accidental. Either way, it’s clear that Turing was treated unjustly. It’s an undeniable tragedy that a man whose work had such a significant impact on the modern era was treated as a criminal in his own time just because of who he loved.

Antonia J. Jones (1943-2010) was a mathematician and computer scientist. She worked at a variety of universities, including as a Professor of Evolutionary and Neural Computing at Cardiff University, and lived in a farmhouse with her partner Barbara Quinn. Along with her work with computers and number theory, she also wrote the textbook Game Theory: Mathematical Models of Conflict. If you want to learn more about that field, check out this collection of Maple Learn documents on game theory. As a child, Jones contracted polio and lost the use of both of her legs. This created a barrier to her work with computers, as early computers were inaccessible to individuals with physical disabilities. Luckily, as the technology developed and became more accessible, she was able to make more contributions to the field of computing. And that’s especially lucky for banks who like having their money be secure—she then exposed several significant security flaws at HSBC! That just goes to show you the importance of making mathematics accessible to everyone—who knows how many banks’ security flaws aren’t being exposed because the people who could find them are being stopped by barriers to accessibility?

James Stewart (1941-2014) was a gay Canadian mathematician best known for his series of calculus textbooks—yes, those calculus textbooks, the Stewart Calculus series. I’m a 7th edition alumni myself, but I have to admit the 8th edition has the cooler cover. To give you a sense of his work, here’s an example of an optimization problem that could have come straight from the pages of Stewart Calculus. Questions just like this have occupied the evenings of high school and university students for over 25 years. I suspect not all of those students really appreciate that achievement, but nonetheless his works have certainly made an impact! Stewart was also a violinist in the Hamilton Philharmonic Orchestra, and got involved in LGBT+ activism. In the early 1970s, a time where acceptance for LGBT+ people was not particularly widespread (to put it lightly), he brought gay rights activist George Hislop to speak at McMaster University. Stewart is also known for the Integral House, which he commissioned and had built in Toronto. Some may find the interior of the house a little familiar—it was used to film the home of Vulcan ambassador Sarek in Star Trek: Discovery!

Agnes E. Wells (1876-1959) was a professor of mathematics and astronomy at Indiana University. She wrote her dissertation on the relative proper motions and radial velocities of stars, which you can learn more about from this document on the speed of orbiting bodies and this document on linear and angular speed conversions. Wells was also a woman’s rights activist, and served as the chair of the National Woman’s Party. In her activism, she argued that the idea of women “belonging in the home” overlooked unmarried women who needed to earn a living—and women like her who lived with another woman as their partner, although she didn’t mention that part. There is a long-standing prejudice against women in mathematics, and it’s the work of women like Wells that has helped our gradual progress towards eliminating that prejudice. To be a queer woman on top of that only added more barriers to Wells’ career, and by overcoming them, she helped pave the way for all queer women in math.

Now, there is a fair amount of debate as to whether or not our next mathematician really was LGBT+, but there is sufficient possibility that it’s worth giving Sir Isaac Newton a mention. Newton (1642-1727) is most known for his formulation of the laws of gravity, his invention of calculus (contended as it is), his work on optics and colour, the binomial theorem, his law of temperature change… I could keep going; the list goes on and on. It’s unquestionable that he had a significant impact on the field of mathematics, and on several other fields of study to boot. While we can’t know how Newton may have identified with any of our modern labels, we do know that he never married, nor “had any commerce with women”^[a], leading some to believe he may have been asexual. He also had a close relationship with mathematician Nicolas Fatio de Duillier, which some believe may have been romantic in nature. In the end, we can never say for sure, but it’s worth acknowledging the possibility. After all, now that more and more members of the LGBT+ community are feeling safe enough to tell the world who they are, we’re getting a better sense of just how many people throughout history were forced to hide. Maybe Newton was one of them. Or maybe he wasn’t, but maybe there’s a dozen other mathematicians who were and hid it so well we’ll never find out. In the end, what matters more is that queer mathematicians can see themselves in someone like Newton, and we don’t need historical certainty for that.

So there you have it! Of course, this is by no means a comprehensive list, and it’s important to recognize who’s missing from it—for example, this list doesn’t include any people of colour, or any transgender people. Sadly, because of the historical prejudices and modern biases against these groups, they often face greater barriers to entry into the field of mathematics, and their contributions are frequently buried. It’s up to us in the math community to recognize these contributions and, by doing so, ensure that everyone feels like they can be included in the study of mathematics.

Some texts distinguish between unary and binary negation signs, using short dashes for unary negation and a longer dash for binary subtraction. How important is this distinction to users of Maple?

Some earlier versions of Maple used to have short dashes for negation (in some places). Maple 2023 has apparently abandoned the short dash for unary negation, and all such signs are now a long dash.

How about math books? Do all texts make this short-long distinction? The typesetters for my 2001 Advanced Engineering Math book also opted for all long dashes and that book was set from the LaTeX exported from Maple 20+ years ago. But I also have texts in my library that use a short dash for unary negation, on the grounds that -a, the additive inverse of "a" is a complete symbol unto itself, the short dash being part of the symbol for that additive inverse.

Apparently, this issue bugs me. Am I making a tempest in a teapot?

We are happy to announce another Maple Conference this year, to be held October 26 and 27, 2023!

It will be a free virtual event again this year, and it will be an excellent opportunity to meet other members of the Maple community and get the latest news about our products. More importantly, it's a chance for you to share the work work you've been doing with Maple and Maple Learn. There are two ways to do this.

First, we have just opened the Call for Participation. We are inviting submissions of presentation proposals on a range of topics related to Maple, including Maple in education, algorithms and software, and applications. We also encourage submission of proposals related to Maple Learn. 

You can find more information about the themes of the conference and how to submit a presentation proposal at the Call for Participation page. Applications are due July 11, 2023.

Presenters will have the option to submit papers and articles to a special Maple Conference issue of the Maple Transactions journal after the conference.

The second way in which to share your work is through our Maple Art Gallery and Creative Works Showcase. Details on how to submit your work, due September 14, 2023, are given on the Web page.

Registration for attending the conference will open later this month. Watch for further announcements in the coming weeks.

I encourage all of you here in the Maple Primes community to consider joining us for this event, whether as a presenter or attendee!

Paulina Chin
Contributed Program Chair

Probability distributions can be used to predict many things in life: how likely you are to wait more than 15 minutes at a bus stop, the probability that a certain number of credit card transactions are fraudulent, how likely it is for your favorite sports team to win at least three games in a row, and many more. 

Different situations call for different probability distributions. For instance, probability distributions can be divided into two main categories – those defined by discrete random variables and those defined by continuous random variables. Discrete probability distributions describe random variables that can only take on countable numbers of values, while continuous probability distributions are for random variables that take values from continuums, such as the real number line.

Maple Learn’s Probability Distributions section provides introductions, examples, and simulations for a variety of discrete and continuous probability distributions and how they can be used in real life. 

One of the distributions highlighted in Maple Learn’s Example Gallery is the binomial distribution. The binomial distribution is a discrete probability distribution that models the number of n Bernoulli trials that will end in a success.

This distribution is used in many real-life scenarios, including the fraudulent credit card transactions scenario mentioned earlier. All the information needed to apply this distribution is the number of trials, n, and the probability of success, p. A common usage of the binomial distribution is to find the probability that, for a recently produced batch of products, the number that are defective crosses a certain threshold; if the probability of having too many defective products is high enough, a company may decide to test each product individually rather than spot-checking, or they may decide to toss the entire batch altogether.

An interesting feature of the binomial distribution is that it can be approximated using a different type of distribution. If the number of trials, n, is large enough and the probability of success, p, is small enough, a Poisson Approximation to the Binomial Distribution can be applied to avoid potentially complex calculations. 

To see some binomial distribution calculations in action and how accurate the probabilities supplied by the distribution are, try out the Binomial Distribution Simulation document and see how the Law of Large Numbers relates to your results. 

You can also try your hand at some Binomial Distribution Example Problems to see some realistic examples and calculations.

Visit the Binomial Distribution: Overview document for a more in-depth explanation of the distribution. The aforementioned Probability Distributions section also contains overviews for the geometric distribution, Poisson distribution, exponential distribution, and several others you may find interesting!

2-dimensional motion and displacement are some of the first topics that high school students learn in their physics class. In my physics classes, I loved solving 2-dimensional displacement problems because they require the use of so many different math concepts: trigonometry, coordinate conversions, and vector operations are all necessary to solve these problems. Though displacement problems can seem complicated, they are easy to visualize.
For example, below is a visualization of the displacement of someone who walked 10m in the direction 30^o North of East, then walked 15m in the direction 45^o South of East:

From just looking at the diagram, most people could identify that the final position is some angle Southeast of the initial position and perhaps estimate the distance between these two positions. However, finding an exact solution requires various computations, which are all outlined in the Directional Displacement Example Problem document on Maple Learn.

Solving a problem like this is a great way to practice solving triangles, adding vectors, computing vector norms, and converting points to and from polar form. If you want to practice these math skills, try out Maple Learn’s Directional Displacement Quiz; this document randomly generates displacement questions for you to solve. Have fun practicing!

In March of 2023, two high school students, Calcea Johnson, and Ne’Kiya Jackson, presented a new proof of the Pythagorean Theorem at the American Mathematical Society’s Annual Spring Southeastern Sectional Meeting. These two young women are challenging the conventions of math as we know it.
The Pythagorean Theorem states that in a right angle triangle, the sum of the squares of the legs is equal to the square of the hypotenuse: 

The theorem has been around for over two thousand years and has been proven hundreds of times with many different methods. So what makes the Johnson-Jackson proof special? The proof is one of the first to use trigonometry.
For years, mathematicians have been convinced that a trigonometric proof of the Pythagorean Theorem is impossible because much of trigonometry is based upon the Pythagorean Theorem itself (an example of circular reasoning).
That said, some results in trigonometry are independent of the Pythagorean Theorem, namely the law of sines, and the sine and cosine ratios; the latter is a result that 12-year-old Einstein used in his trigonometric proof of the theorem.
Though all the details of the Johnson-Jackson proof have not been made public, there was enough information for me to recreate the proof in Maple Learn. The idea of the proof is to construct a right angle triangle with an infinite series of congruent right angle triangles (the first of which has side lengths a, b, and c). Then, using the sine ratio, solve for the hypotenuse lengths of each small congruent triangle. To explore this construction see Johnson and Jackson’s Triangle Construction on Maple Learn. 

Next, find the side lengths of the large triangle (A and B) by evaluating an infinite sum (composed of the hypotenuse lengths of the small congruent triangles). Finally, apply the law of sines to the isosceles triangle made from the first 2 congruent triangles. After simplifying this expression, the Pythagorean relationship (c² = a² + b²) emerges.
 

To see more details of the proof, check out Johnson and Jackson’s Proof of Pythagorean Theorem on Maple Learn.
This new proof of the Pythagorean Theorem shows that discoveries in math are still happening and that young people can play a big role in these discoveries!

The most recent shift in education has seen countries adopting a more student-centered approach to learning. This approach involves enabling students to make sense of new knowledge by building on their existing knowledge. Many countries have embraced this approach in their educational systems. Teachers are no longer the sage on the stage, and gone are lectures and one-way learning. This new era of learning lends itself to the social constructivist framework of teaching and learning. 

Social Constructivism. Students adopt new knowledge through interacting with others to share past experiences and make sense of the learned concepts together. Perhaps the most well-known applications of social constructivist classrooms are Thinking Classrooms popularized by Peter Liljedahl in 2021 (the same age as Maple Learn!). In a Thinking Classroom, groups of students collaborate to discuss potential solutions to solve open-ended problems. Ideas are recorded on vertical surfaces so that all students, including those from different groups, have access to one another’s ideas. The teacher is hands-off in this type of classroom, with students asking each other questions if stuck or unsure. This approach facilitates the exchange of ideas and encourages collaboration among students. Sadly, this innovative idea was brought into the classroom at a peculiar time, at the height of the pandemic when less socialization was happening. 

Nevertheless, teachers were intrigued by this idea, and like any good idea, it spread like wildfire. For the first time, many teachers have reported that they observed their students engage in active thinking, rather than just mechanically plugging and chugging numbers into formulae, as was traditionally done in math education. This shift in approach has led to a deeper understanding of mathematical concepts and improved problem-solving skills among students. At the same time, students were more uncomfortable than ever before because they were not accustomed to the feeling of “not knowing.” The strongest students were often the most uncomfortable as they were conditioned to view mathematics as having only one correct answer. This discomfort is a natural part of the learning process, as it indicates that students are grappling with the new concepts and expanding their understanding. This new approach, which emphasized exploration and problem-solving over rote memorization, challenged their existing beliefs and required them to think in new ways. Over time, as they become more familiar with this approach, students develop greater confidence in their mathematical skills and improve their abilities to think critically and creatively.

Social Constructivism in Maple Learn. As a secondary math teacher, I’ve been using Maple Learn to support my students’ learning. I’ve mainly created projects and collections of financial literacy documents that are not only informative but also exploratory for students to engage with at their own pace. Here is where I see the potential of Maple Learn - not only to support teachers in the classroom but also to act in place of the teacher for asynchronous class work by being the guide on the side. 

The project-based ideas such as “Designing a roller coaster or slide,” “Exploring the rule of 72,” and open-ended questions such as “Designing a cake” and “Moving sofa” can lend themselves to creative discussions using mathematics. This is because Maple Learn offers its users the chance to visualize dynamic representations. Users can relate the algebraic, graphical, text-based, and/or geometrical representations of the same math concept. The convenience of having everything on one page encourages students to take away what they deem are the most important pieces of information as opposed to the teacher telling them what the major takeaways are. Due to their diverse backgrounds and unique mathematical identities, different students tend to focus on different aspects of a given concept. However, it is precisely these differences that can lead to a deeper understanding of the topic at hand. By sharing their perspectives and insights with one another, students can gain a more complete and nuanced understanding of mathematical concepts, and develop a broader range of problem-solving strategies. 

Source: Double angle identity. Illustration provides geometrical and algebraic representations side by side.  

In addition, the different functions that Maple Learn offers, allow students with varying mathematical backgrounds to have an equitable chance at learning. Some students may be better at manipulating equations, while others might be more visual. Maple Learn provides students with a blank canvas to explore mathematical concepts on their own, without the stress of mental calculations, the need to access different functions on a calculator, or the necessity to search for explanatory videos online. Maple Learn can also have embedded hyperlinks which can be important concepts or documents. These links can provide an easier learning platform for students to construct their own knowledge. An example can be found here. By removing these barriers, students are free to delve into the material and develop a deeper understanding of the underlying principles. This approach can further foster creativity, curiosity, and a passion for learning among students, while also equipping them with the tools they need to succeed in their future academic pursuits. 

Arguably the most difficult aspect of social constructivism to implement using Maple Learn is the “social” aspect of it all which requires a bit of creativity. The goal is not to eliminate the use of teachers, but rather have teachers present the material in a different light. The teacher still decides what students learn in the classroom (or maybe that’s already decided by  the government agency) but how they learn the material is up to the teacher. After interacting with Maple Learn and coming up with interesting solutions, students can trade their responses with their peers to evaluate one another’s responses, approaches, ideas, and solutions to a problem. Students definitely learn more from each other and I believe as teachers, we should capitalize on this aspect. With many jurisdictions around the world adopting a student-centered approach to learning, it is time we advance our teaching styles. Even with the recent advances in AI, we still need to teach our kids how to think, and to think deeply. Tech can definitely help in this regard. 

In summary, by emphasizing collaboration, critical thinking, and exploration, social constructivism encourages students to build their understanding of new concepts through interaction with others. Often seen as Thinking Classrooms, Maple Learn can supplement social constructivist classrooms by offering a blank canvas for students to explore mathematical concepts on their own, free from the limitations of traditional calculations and rote memorization. Together, these approaches can empower students to become active learners and critical thinkers, setting them on a path towards success in the classroom and beyond. Here are some “How-To” videos to help you get started with creating your own documents in Maple Learn. You can also browse the example gallery with thousands of existing examples here. Happy creating!

A geometric transformation is a way of manipulating the size, position, or orientation of a geometric object. For example, a triangle can be transformed by a 180^o rotation: 

Learning about geometric transformations is a great way for students, teachers and anyone interested in math to get comfortable using x-y coordinates in the cartesian plane, and mapping functions from R² to R². Understanding geometric transformations is also an essential step to understanding higher-level concepts like the Transformations of Functions and Transformation Matrices.
Check out the Geometric Transformations collection on Maple Learn to learn about this topic. Start out by playing with the Geometric Transformations Exploration document to build intuition about how objects are affected by each of the four transformation types: Dilation, Reflection, Rotation, and Translation. Once you are confident in your skills, try using the Single Geometric Transformation Quiz to test your knowledge.
For those looking to expand their understanding of geometric transformations, the Combined Transformations Exploration document will let you explore how multiple transformations and the order of said transformations affect the final form of an object. For example, the blue polygon can be transformed into 2 different pink polygons depending on whether the reflection or rotation is performed first:

Once you have the hang of combined transformations, try answering questions on the Combined Geometric Transformations Quiz. 

How Can Maple Learn Help Address Math Anxiety in Classrooms?

Math anxiety is referred to as negative behaviours such as uneasiness and general avoidance when asked to solve math problems. For teachers and teacher candidates, this can be due to various reasons such as previous negative experiences in math classes, learning styles that conflict with their math teacher, lack of self-confidence, low self-esteem, and stereotype issues related to the belief that math is for men only. Although it is commonly believed that math anxiety only exists in students, research has shown that math anxiety is present among elementary teacher candidates and elementary teachers, particularly women. Furthermore, research has shown that female teachers who suffer from math anxiety have a tendency to pass down their math anxious behaviours to students, particularly affecting more girls than boys. Since the majority of the elementary teaching staff are women, it is possible that a cyclic pattern will arise where teachers will pass down math anxiety to students, and these students will grow up dealing with math anxiety.

As a current PhD candidate, I have taught elementary teacher candidates basic math knowledge. It was clear to me from the first day, math anxiety was very present within the students I had. Many of these teacher candidates had candidly revealed that they have not taken any math classes since Grade 11, which is the final grade in Ontario where math is mandatory. With Maple Learn, because manyof the documents are created by educators, these documents can function as learning materials which a teacher can use for extra practice and guidance. 

One strategy to combat math anxiety in general is developing greater self-efficacy and confidence in their math skills. For example, using the Converting and Decimals to Fractions document, teachers and teacher candidates can use this as a tool to support their understanding and can help double-check their work. Unlike students, when learning about math concepts and skills in class, in addition to using online resources they also can ask teachers for help. Whereas for adult learning, it is possible that some may feel shy or embarrassed to seek help from others. On Maple Learn, there are multiple quizzes where a teacher can use as practice to further their understanding. In addition to the solution, these features also provide hints and a “check your work” button so that it can guide the teacher in solving such problems if stuck on a question. One of the cool features of these solutions is that they don’t just reveal the answer, but also include steps to solve the question whenever a teacher gets stuck.

Furthermore, additional visualizations could be a useful tool for visual learners and serve as another method to understand and solve such math problems rather than solely relying on algebra. 

The documents provided in the example gallery provide multiple different methods on understanding and solving math problems. For example, when multiplying fractions, one can either simplify before multiplying the fractions together or they can first multiply the fractions, then simplify.

The more practice one does, the better they become at solving math problems, and if interested, Maple Learn has many quizzes that one can use to improve their math skills. For more fractions documents, check out this page here!

Several studies, such as “Seeing and feeling volumes: The influence of shape on volume perception”, have shown that people have a tendency to overestimate the volume of common objects, such as glasses and containers, that are tall and thin and underestimate those that are short and wide; this phenomenon is called “elongation bias”. 

Sue Palmberg, an instructor at Edwin O. Smith High School, created and shared with us a lab activity for students to design a glass in Maple and use volumes of revolution to determine the amount of liquid it can hold. This lab was then turned into this Maple Learn document: Piecewise Volumes of Revolution Activity.

Use this document to create your own glass or goblet shape and determine its volume. Simply create a piecewise function that will define the outside shape of your glass between your chosen bounds and another piecewise function to define the hollowed-out part of your creation. The document will graph the volumes of revolution that represent your glass and calculate the relevant volume integral for you.

Here is my own goblet-shaped creation: 

I used this piecewise function to define it:

After creating the outline of my goblet, I constructed a function for the hollow part of the goblet – the part that can actually hold liquid.

Using Context Panel operations and the volume integral provided by the document, I know that the volume of the hollow part of my goblet is approximately 63.5, so my goblet would hold around 63.5 units³ of liquid when full.

Create your own goblets of varying shapes and see if their volumes surprise you; elongation bias can be tricky! For some extra help, check out the Piecewise Functions and Plots and Solids of Revolution - Volume Derivation documents!

In an age where our lives are increasingly integrated online, cybersecurity is more important than ever. Cybersecurity is the practice of protecting online information, systems, and networks from malicious parties. Whenever you access your email, check your online banking, or make a post on Facebook, you are relying on cybersecurity systems to keep your personal information safe. 

Requiring that users enter their password is a common security practice, but it is nowhere near hacker-proof. A common password-hacking strategy is the brute-force attack. This is when a hacker uses an automated program to guess random passwords until the right one is found. The dictionary attack is a similar hacking strategy, where guesses come from a list like the 10,000 Most Common Passwords.

The easiest way to prevent this kind of breach is to use strong passwords. First, to protect against dictionary attacks, never use a common password like “1234” or “password”. Second, to protect against brute-force attacks, consider how the length and characters used affect the guessability. Hackers often start by guessing short passwords using limited types of characters, so the longer and more special characters used, the better.

Using the Strong Password Exploration Maple Learn document, you can explore how susceptible your passwords may be to a brute-force attack. For example, a 6-character password using only lowercase letters and numbers could take as little as 2 seconds to hack.

Whereas an 8-character password using uppercase letters, lowercase letters, and 10 possible special characters could take more than 60 hours to crack.

These hacking times are only estimations, but they do provide insight into the relative strength of different passwords. To learn more about password possibilities, check out the Passwords Collection on Maple Learn

The areas of statistics and probability are my favorite in mathematics. This is because I like to be able to draw conclusions from data and predict the future with past trends. Probability is also fascinating to me since it allows us to make more educated decisions about real-life events. Since we are supposed to get a big snow storm in Waterloo, I thought I would write a blog post discussing conditional probability using the Probability Tree Generator, created by Miles Simmons.

If the probability of snowfall on any given day during a Waterloo winter is 0.75, the probability that the schools are closed given that it has snowed is 0.6, and the probability that the schools are closed given that it has hasn’t snowed is 0.1, then we get the following probability tree, created by Miles’s learn document:

From this information we can come to some interesting conclusions:

What is the probability that the schools are closed on a given day?

From the Law of total probability, we get:

Thus, during a very snowy Waterloo winter, we could expect a 0.475 chance of schools being closed on any given day. 

One of the features of this document is that the node probabilities are calculated. You can see this by comparing the second last step to the number at the end of probability trees' nodes.

What is the probability that it has snowed given that the schools are closed?

From Bayes’ Theorem, we get:

Thus, during a very snowy Waterloo winter, we expect there to be a probability of 0.947 that it has snowed if the schools are closed. 

We can also add more events to the tree. For example, if the students are happy or sad given that the schools are open:

Even though we would all love schools to be closed 47.5% of the winter days in Waterloo, these numbers were just for fun. So, the next time you are hoping for a snow day, make sure to wear your pajamas inside out and sleep with a spoon under your pillow that night!

To explore more probability tree fun, be sure to check out Miles’s Probability Tree Generator, where you can create your own probability trees with automatically calculated node probabilities and export your tree to a blank Maple Learn document. Finally, if you are interested in seeing more of our probability collection, you can find it here!