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!

Happy Springtime to all in the MaplePrimes Community! Though some in our community may not live in the northern hemisphere where flowers are beginning to bloom, many will be celebrating April holidays like Ramadan, Passover, and Easter.

One of my favorite springtime activities is decorating eggs. Today, the practice is typically associated with the Christian holiday of Easter. However, painted eggs have roots in many cultures.

For over 3,000 years, painting eggs has been a custom associated with the holiday of Nowruz, or Persian New Year, during the spring equinox. Furthermore, in the Bronze Age, decorated ostrich eggs were traded as luxury items across the Mediterranean and Northern Africa. Dipped eggs have also played an important role in the Jewish holiday of Passover since the 16th century.

To celebrate this tradition, I would like to invite all of the Maplesoft community to create a decorated egg of their own with the Easter Egg Art Maple Learn document. In this document, an ovoid egg equation is used to define the shape of an egg. 

The ovoid egg equation mimics the shape of a typical hen’s egg. Each bird species lays differently shaped eggs. For example, an ostrich’s egg is more oblong than an owl’s, and an owl’s egg is rounder than a goose’s. Surprisingly, every egg can be described by a single equation with four parameters:

Learn more about this equation and others like it with John May’s Egg Formulas Maple Learn document.

The Easter Egg Art document includes 9 different decorative elements; users can change the color, position, and size of each in order to create their own personal egg! The egg starts out looking like this:



In just a couple of minutes, you can create a unique egg. Have fun exploring this document and share a screenshot of your egg in the comments below!  Here’s one I made:

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!

The recent Maple 2023 release comes with a multitude of new features, including a new Canvas Scripting Gallery full of templates for creating interactive Maple Learn documents.

The Maple Learn Scripting Gallery can be accessed through Maple, by searching “BuildInteractiveContent Maple2023” in the search bar at the top of the application and clicking on the only result that appears. This will bring you to the help page titled “Build and Share Interactive Content”, which can also be found by searching “scripting gallery” in the search bar of a Maple help page window. The link to the Maple Learn Scripting Gallery is found under the “Canvas Scripting” section on this help page and clicking on it will open a Maple workbook full of examples and templates for you to explore.

The interactive content in the Scripting Gallery is organized into five main categories – Graphing, Visualization, Quiz, Add-ons and Options, and Applications Optimized for Maple Learn – each with its own sub-categories, templates, and examples.

One of the example scripts that I find particularly interesting is the “Normal Distribution” script, under the Visualizations category.

All of the code for each of the examples and templates in the gallery is provided, so we can see exactly how the Normal Distribution script creates a Maple Learn canvas. It displays a list of grades, a plot for the grade distribution to later appear on, math groups for the data’s mean and variance, and finally a “Calculate” button that runs a function called UpdateStats.

The initial grades loaded into the document result in the below plot, created using Maple’s DensityPlot and Histogram functions, from the Statistics package. 

The UpdateStats function takes the data provided in the list of grades and uses a helper function, getDist, to generate the new plot to display the data, the distribution, the mean, and the variance. Then, the function uses a Script object to update the Maple Learn canvas with the new plot and information.

The rest of the code is contained in the getDist function, which uses a variety of functions from Maple’s Statistics package. The Normal Distribution script takes advantage of Maple’s ability to easily calculate mean and variance for data sets, and to use that information to create different types of random variable distributions.

Using the “Interactive Visualization” template, provided in the gallery, many more interactive documents can be created, like this Polyhedra Visualization and this Damped Harmonic Oscillator – both from the Scripted Gallery or like my own Linear Regression: Method of Least Squares document.

Another new feature of Maple 2023 is the Quiz Builder, also featured in the Scripting Gallery. Quizzes created using Quiz Builder can be displayed in Maple or launched as Maple Learn quizzes, and the process for creating such a quiz is short.

The QuizBuilder template also provides access to many structured examples, available from a dropdown list:

As an example, check out this Maple Learn quiz on Expected Value: Continuous Practice. Here is what the quiz looks like when generated in Maple:

This quiz, in particular, is “Fill-in the blank” style, but Maple users can also choose “Multiple Choice”, “True/False”, “Multiple Select”, or “Multi-Line Feedback”. It also makes use of all of the featured code regions from the template, providing functionality for checking inputted answers, generating more questions, showing comprehensive solutions, and providing a hint at the press of a button.

Check out the Maple Learn Scripting Gallery for yourself and see what kinds of interactive content you can make for Maple and Maple Learn!

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!

Hello everyone! Alex, Sarah, and I decided to create this collection of financial literacy documents as we noticed a lack of resources for this strand in mathematics. With many curricula around the world implementing financial literacy concepts, we thought it might be useful not just for Ontario, but for many jurisdictions around the world. 

There are 4 documents in the Simple Interest collection; Introduction, Equation Generator, Mental Calculations, and Reflection. The Introduction is designed for intermediate and advanced level students as it introduces students to the concept of interest and how to calculate it. Students get to fill in the table by filling in the calculations on the right. This provides enough scaffolding so students of various grades can participate in this activity. 

The Equation Generator document uses sliders to help students investigate linear equations in the form of y=mx+b. It also relates the simple interest equation (I=Prt) to the linear equation by asking students to compare interest rates. The idea behind this document is to bridge concepts outlined in the 2021 grade 9 destreamed math curriculum; in particular, the financial literacy, and linear relations strands. The document provides some reflection questions for students to think about the relationship between the variables. 

The third document in the collection is the mental calculations document which presents a series of questions in increasing difficulty designed to help students compare interest rates. Students are intended to choose which scenario they think is more appropriate without using a calculator. There are hints provided on the right side if students wish for a hint, as well as explanations further to the right of the hints and answers below the main questions. Through our analysis of the curricula around the world, we noticed that many jurisdictions focus on mental math as a skill that their students should develop. Students may not always have access to a calculator and it is important for them to know how to make financially sound decisions or analyze advertisements that they may see around their neighbourhood. 

Lastly, the last document is the reflection page where students are able to analyze their findings. In particular, “interest” may carry a negative connotation for students such that we want them to think of the potential benefits of interest as well. The reflection questions are designed to help students consolidate their learnings and can be further expanded on by the teacher. Such possibilities can include scenario-based questions. 

May you find these documents helpful! 

Happy Valentine’s Day to everyone in the MaplePrimes community. Valentine’s Day is a time to celebrate all things love and romance. To celebrate, we at Maplesoft wanted to share our hearts with you.

Today the heart shape represents love, affection, and a major organ. Though the heart’s full meaning today is unique to the modern era, the shape itself is much older.

 In ancient Greece, the Cyrenese people used the heart-shaped seed of a plant called silphium as a form of contraception. The seed became so widely used that it is featured on Cyrenese currency. This is the first case of the heart shape being connected to love and passion, but the form did not yet have an association with the human heart.

French poet Thibault de Blaison was the first to use a pear-shaped human heart to symbolize love in his thirteenth-century romance “Roman de la Poire”. Later, during the renaissance period, artists began to paint the Sacred Heart of Jesus in a spade-like shape. Depictions of the heart continued to develop and by the Victorian Era, the heart we know and love today had taken shape and started to appear on Valentine’s Day cards.

The simplicity and symmetry of the heart shape, which likely led to its widespread popularity, also makes the form convenient to define mathematically.

To find the equation for your heart, use the Valentine Hearts Maple Learn document. Choose one of four ways to define your heart, then move the sliders and change the color to make a unique equation for your heart. 

Once you’re done, take a screenshot and share it with your Valentine. Who says math isn’t romantic?

When introduced to geometry, one of the first things we learn is the definition of the word “polygon”. A polygon is a closed 2-dimensional shape with at least 3 straight sides and angles. A regular polygon is a polygon with congruent sides and equal angles. A regular polygon with n sides has Schläfli symbol {n}. I’m interested in mathematical history, so when I learned that the idea of higher-dimensional spaces was invented in the middle of the nineteenth century I decided to research more about Ludwig Schläfli and the notation he came up with to describe his ideas.

In general, the Schläfli symbol is a notation of the form {p, q, r, ...} for regular polytopes. Polytopes are geometric objects with flat sides. This week, I will be focusing on 3-dimensional polytopes, also called polyhedra.

Similar to regular polygons, regular polyhedra are 3-dimensional shapes whose faces are all the same regular polygon. A regular polyhedron’s Schläfli symbol is of the form {p, q}, where p is the number of edges each face has and q is the number of faces that meet at each of the polyhedron’s vertices.

Below are two regular polyhedra: a cube (also known as a hexahedron) and a great stellated dodecahedron. The cube is one of five Platonic solids, and the great stellated dodecahedron is one of four Kepler-Poinsot polyhedra – all of these can be represented by Schläfli symbols. The cube has Schläfli symbol {4, 3}, since squares have 4 equal sides, and each vertex of a cube is created by the vertices of 3 squares meeting.

Can you figure out the Schläfli symbol for the great stellated dodecahedron?

The great stellated dodecahedron has the Schläfli symbol {5/2, 3}. This is because great stellated dodecahedrons are regular star polygons. As a result, the first number in their Schläfli symbol is an irreducible fraction whose numerator represents a number of sides and whose denominator corresponds to a turning number. The particular fraction 5/2 corresponds to a pentagram – a regular star polygon with 5 points – and great stellated dodecahedrons are composed of 12 of these pentagrams, where 3 pentagrams meet at each vertex of the shape.

One notable example of a regular polytope in pop culture is the tesseract, which has the Schläfli symbol {4, 3, 3}. This is an extension of the cube’s Schläfli symbol, {4, 3}, and the last number indicates that there are three cubes folded together around every edge. Below are two representations of a tesseract: one that uses a Schlegel diagram (left) and one from the 2012 movie Avengers (right).

Try out our Regular Polyhedra Visualization Using Schlafli Symbol Notation! In this document, you can test out your own Schläfli symbols for regular polyhedra. If they are valid Schläfli symbols, you’ll be provided with a 3-D visualization of the shape. If they are invalid, you can check out the logic for finding the specifications for regular polyhedra and this document, which provides all the 3-D regular polyhedra for you to try out.

Happy Lunar New Year to everyone in the MaplePrimes community, as we enter the Year of the Rabbit. The rabbit symbolizes longevity, positivity, auspiciousness, wittiness, cautiousness, cleverness, deftness and self-protection!

To celebrate, one of our Maple Learn content developers, Laura Layton, made a Lunar New Year Color by Number:

In this puzzle, your goal is to simplify the modulo equations in each square, and then fill in the square with the color that corresponds to the answer.

I hope you have fun solving this puzzle and revealing the hidden images and I wish everyone good health and happiness in the coming year!

Last week, one of our Maple Learn developers, Valerie McKay-Crites, published a Maple Learn document, based on the very popular Maple application by Highschool Teacher, Jason Schattman called "Just Move It Over There, Dear!".

In the Maple application, Schattman explains the math behind moving a rectangular sofa down a hallway with a 90-degree turn. In the 3D Moving Sofa Problem Estimate, Valerie uses Schattman’s math to determine the largest rectangular sofa that can be taken down a flight of stairs and down a hallway with a 90-degree turn. Both applications reminded me of how interesting the Moving Sofa Problem is, which inspired me to write a blog post about it!

If you’ve ever been tasked with moving a rectangular sofa around a 90-degree turn, you might wonder:

What is the largest sofa that can make the move?

Following these steps as outlined in Schattman’s "Just Move It Over There, Dear!", will guarantee that the sofa will make the turn:

1. Measure the width of the hallway (h)
2. Measure the length (L) and width (w) of the sofa.
3. If L + 2w is comfortably less than triple the width of the hall, you'll make it!

When we work out the math exactly, we see that if the sofa's length plus double its width is less than , the sofa will make the turn!

This problem is easy if we only consider rectangular sofas, however, the problem becomes significantly more complex if we consider sofas of different shapes and areas. In mathematics, this problem is known as the Moving Sofa Problem, and it is unsolved. If we look at a hallway with a 90-degree turn and legs of width 1 m (i.e. h = 1 above), the largest known sofa that can make the turn is Gerver’s Sofa which has an area of 2.2195 m², this area is known as the Sofa Constant. Gerver’s Sofa, created in 1992, was constructed with 18 curve sections:

Check out this GIF of the sofa moving through the turn. It provides some insight into why Gerver’s sofa is such an interesting shape:

What is fascinating is that no mathematician has yet to prove that Gerver’s sofa is the sofa with the largest area capable of making the 90-degree turn.

The Moving Sofa Problem, is a great example of how math is embedded in our everyday lives. So, don’t stop being curious about the math around you as it can be fascinating and sometimes unproven!

If you are curious to learn more about the moving sofa problem check out this video by Numberphile, featuring Dan Romik from UC Davis: https://www.youtube.com/watch?v=rXfKWIZQIo4&t=1s

With the winter solstice speeding towards us, we thought we’d create some winter themed documents. Now that they’re here, it’s time to show you all! You’ll see two new puzzle documents in this post, along with three informative documents, so keep reading.

Let’s start with the tromino tree!

First, what’s a tromino? A tromino is a shape made from three equal sized squares, connected to the next along one full edge. In this puzzle, your goal is to take the trominos, and try to fill the Christmas tree shape.

There’s a smaller and larger tree shape, for different difficulties. Try and see how many ways you can fill the trees!

Next, we’ll look at our merry modulo color by numbers.

In this puzzle, your goal is to solve the modulo problems in each square, and then fill in the square with the color that corresponds to the answer. Have fun solving the puzzle and seeing what the image is in the end!

Snowballs are a quintessential part of any winter season, and we’ve got two documents featuring them.

The first document uses a snowball rolling down a hill to illustrate a problem using differential equations. Disclaimer: The model is not intended to be realistic and is simplified for ease of illustration. This document features a unique visualization you shouldn’t miss!

Our second document featuring snowballs talks about finding the area of a 2-dimensional snowman! Using the formula for the area of a circle and a scale factor, the document walks through finding the area in a clear manner, with a cute snowman illustration to match!

The final document in this mini-series looks at Koch snowflakes, a type of fractal. This document walks you through the steps to create an iteration of the Koch snowflake and contains an interactive diagram to check your drawings with!

I hope you’ve enjoyed taking a look at our winter documents! Please let us know if there’s any other documents you’d like to see featured or created.

Welcome back to another Maple Learn blog post! Today we’re going to talk about the gift-wrapping algorithm, used to find the convex hull of a set of points. If you’re not sure what that means yet, don’t worry! We’re going to go through it with four Maple Learn documents; two which are background information on the topic, one that is a visualization for the gift-wrapping algorithm, and another that goes through the steps. Each will be under their own heading, so feel free to skip ahead to your skill level!

Before we can get into the gift-wrapping algorithm we need to define a few terms. Let’s start by defining polygons and simple polygons.

Polygon: A closed shape created by joining a series of line segments.

Simple polygon: A polygon without holes and that does not intersect itself.

So, what are convex and concave polygons? Well, there are three criteria that define a convex polygon. A polygon that is not convex is called concave. The criteria are…

1. Any line segment connecting any two points within the polygon stays within the polygon.
2. Any line intersects a polygon’s boundary at most twice.
3. All interior angles are less than 180 degrees or pi radians.

Because the criteria are equivalent, if any one is missing, the shape is concave. AKA, all three criteria must be present for a shape to be convex. Most “regular shapes”, such as trapezoids, are convex polygons!

A shape that satisfies convex criteria but not the criteria for being a polygon is called a convex set.

As mentioned at the start of this post, the gift-wrapping algorithm is used to find the convex hull of a set of points. Now that we know what convex polygons and convex sets are, we can define the convex hull!

Convex hull: The convex set of a shape or several shapes that fully contains the object and has the smallest possible area.

Why was the convex polygon important? Well, the convex hull of a set of points is always a convex polygon. Some of the points in the set are the vertices of said polygon, and are called extreme points. You can find the convex hull of either concave or convex polygons.

This document amazed me when I tried it for the first time. Here, you can generate a set of points with the “Generate Another” button, and then press the “Visualize” button. The document then calculates the perimeter of the convex hull of the set of points! The set can be further customized below the buttons, by changing the number of points. The other option below it allows you to slow down or speed up the visualization. Pretty cool, huh? It’s like it’s thinking!

Try the document out a few times, or watch the gif below to get a quick idea of it.

This final document walks you through the steps of how to use the gift wrapping algorithm. It is a simple loop of 4 steps, with one set-up step. Unlike the other documents in this post, I won’t be delving too far into the math behind the steps. I want to encourage you to check this one out yourself, as it’s really quite a fun problem to solve once you have some time!

I hope you check out the documents in this post. Please let us know below if there’s any other documents you’d like to see featured!