Blog

How to Construct the Altitudes of a Triangle ⊿

Hi everyone and welcome back to another week of MathSux! For this week’s math lesson, I bring to you four constructions in one in How to Construct the Altitudes of a Triangle. In the video below, we will:

  • Define what an altitude is
  • Find the altitude of an acute, obtuse, and right triangles
  • Discover and define the orthocenter

I hope you find the below video helpful and interesting! Let me know in the comments below and happy calculating!

How to Construct the Altitudes of a Triangle :

An Altitude is a perpendicular line drawn from the vertex of a triangle to the opposite side, creating a 90º angle.

How to Construct the Altitudes of a Triangle

In the video above, we will look at how to find the altitude of an acute obtuse, and right triangle.

How to Find the Orthocenter of Triangle with a Compass:

The Orthocenter is a point where all three altitudes meet within a triangle.

How to Construct the Altitudes of a Triangle

In order to find the orthocenter using a compass, all we need to do is find the altitude of each vertex. The point at which they meet is the orthocenter. Check out the video above to see how this works step by step.

More Constructions?

If you’re looking for more constructions out there, be sure to check out the MathSux Constructions Playlist on YouTube for even more tutorials! I hope you find this video helpful and hope it motivates you to learn even more about constructions (or you know get through your homework/pass that test). Whatever reason you have to learn about constructions I hope this helps! Stay positive and happy calculating!

Still got questions? No problem! Don’t hesitate to comment with any questions below.

Facebook ~ Twitter ~ TikTok ~ Youtube

Check out more Geometry posts and Constructions step by step here!

Derivatives Practice Questions

Hi everyone and welcome to MathSux! In this weeks post, we will venture into Calculus for the first time! I won’t get too much into the nitty gritty explanation of what derivative are here, but instead will provide a nice overview of Derivatives Practice Questions. This post includes everything you need to know about finding the derivatives of a function including the Power Rule, Product Rule, Quotient Rule, and the Chain Rule. Below you will see examples, a Derivative Rules Cheat Sheet, and of course practice questions! I hope these quick examples help in the classroom or for that test coming up! Let me know if it helps and you want more Calculus lessons. Happy Calculating!

What is a Derivative?

We use the derivative to find the rate of change of a function with respect to a variable. You can find out more about what a derivative is and its proper notation here at mathisfun.com. Read on below for a derivative rules cheat sheet, examples, and practice problems!

Derivative Rules Cheat Sheet:

Power Rule:

The power rule is used for finding the derivative of functions that contain variables with real exponents.  Note that the derivative of any lone constant number is zero.

derivatives of functions with exponents

Product Rule:

The product rule is used to find the derivative of two functions that are being multiplied together.

derivatives practice questions

Quotient Rule:

Applying the quotient rule, will find the derivative of any two functions set up as a ratio.  Be sure to notice any numbers or variables in the denominator that can be brought to the numerator (if that’s the case, can use the more friendly power rule).

derivatives of trigonometric functions

Chain Rule:

The chain rule allows us to find the derivative of nested functions. This is great for trigonometric functions and entire functions that are raised to an exponent.

calculus chain rule examples

Ready for some practice questions!? Check out the ones below to test your knowledge of derivatives!

Derivatives Practice Questions:

Find the derivatives for each function below.

Derivatives Practice Questions

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

*Also, if you want to check out Rate of Change basics click this link here!

Facebook ~ Twitter ~ TikTok ~ Youtube

Holiday Discount at the Math Shop!

Holiday Discount

Seasons greetings math friends! Tis’ the season of giving, celebrating, and of course, glitter! And in honor of the holiday season, I’m here to take a break from providing math lessons this week, to instead, provide a holiday discount at the Math Shop! Did you know we have an exclusive math shop, just for us MathSux nerds? Check out the one-of-a-kind MathSux merchandise designs located in the Math Shop, found in the link below. Don’t forget to use the discount code, ‘CHEERSMATH‘ for a whopping 20% off! Also, check out my my top 3 favorite picks below for a quick preview of what you’ll find in the store. Does anyone else have nerdy math gear that they use in their lives and in their classroom? Let me know in the comments below!

Holiday Discount:

Math Shop: https://mathsux.creator-spring.com/

Discount: ‘CHEERSMATH

Top 3 Gifts (for the Math Nerd in your Life):

The Math Shop has one-of-a-kind math themed designs on t-shirts, stickers, and posters in a variety of colors for all the math nerds in your life. These items are perfect for math fans, teachers, and students to keep studying the subject light and positive! This can be perfect for those frustrated learning moments, for example, quotes like “Keep Calm and Calculate on” will carry any math learner through to the more enlightened side of math. Although this was hard to do, check out my personal top 3 favorites from the Math Shop:

Holiday Discount

1) Peace, Love, Pi T-Shirt in Pink:

2) MathSux Stickers

3) Keep Calm and Calculate on Stickers

Hoping you’re all having a wonderful holiday season and a Happy New Year! Last but not least, of course I also want to wish you happy calculating! Be back with more math lessons in 2022!

Facebook ~ Twitter ~ TikTok ~ Youtube

NYS Regents Review – Algebra June 2021

Greetings math friends! Today we are going to break down the NYS Regents, specifically the Algebra NYS Common Core Regents from June 2021, one question at a time. The following video playlist goes over each and every question one step at a time. I’ve been working on this playlist slowly adding new questions and videos every week and now that it is complete, it is time to celebrate (and/or study)! Please enjoy this review along with the study aids and related links that will also help you ace the NYS Regents. Happy calculating!

NYS Regents

NYS Regents – Algebra June 2021 Playlist

Study Resources:

Looking to ace your upcoming NYS Regents!? Don;t forget to check out the resources below, including an Algebra Cheat Sheet, and important topics and videos to review. Good luck and happy calculating!😅

Algebra Cheat Sheet

Combing Like Terms and the Distributive Property

How to Graph an Equation of a Line

Piecewise Functions

NYS Regents – Algebra 2020

How to Study Math?

How is one supposed to study math!? Well, there is usually only one way, and that is to practice, practice, practice! But don’t get too stressed, because you can also make practice fun (or at least more pleasant).

Add some background music to your study session and make a nice cup of tea before diving in for the brain marathon. Another idea is to study only 1 hour at a time and to be sure to take breaks. Can’t seem to get that one question? Take a break and walk a way, or even better find a new study spot! It’s been scientifically proven that studying in different places can boost your memory of the very information you’re trying to understand.

What study habits do you have that have worked for you in the past? Let me know in the comments and good luck on your upcoming test!

Facebook ~ Twitter ~ TikTok ~ Youtube

** Be sure to check out the full list of Algebra lessons and old Regents questions review!**

Legs of a Right Triangle (when an altitude is drawn)

Greetings math peeps and welcome to another week of MathSux! In todays post we are going to explore how to find the legs of a right triangle when an altitude is drawn from the vertex to its hypotenuse. We are going to take this step by step on how to solve a problem like this. Although, I go over the long way to solve this problem, there is also short cut many people use called the “geometric means” which is also briefly mentioned in this post (under the Tip! section). Whichever method you choose, do what makes most sense to you! Happy calculating!

Similar Right Triangles (with Altitude drawn):

When two triangles have equal angles and proportionate sides, they are similar.  This means they can be different in size (smaller or larger) but if they have the same angles and the sides are in proportion, they are similar! Triangles can be proven similar by AA, SAS, or SSS. For more on similar triangles, check out this post here.

There is a special type of scenario that happens with similar right triangles. When an altitude is drawn from the vertex of a right triangle, it forms two smaller triangles, which creates three right triangles that are similar to the original triangle, based on Angle Angle (AA). Check out the example to see how it works!

In triangle ABC, an altitude is drawn from angle A to its hypotenuse BC. Notice that this creates three 90º right triangles in total (ABC, ABD, and ADC). 

Drawing an altitude created three total right triangles, broken out below:

If you take a closer look, at the triangles above you’ll notice that these two new triangles (ABD and ADC) share an angle with the original bigger triangle ABC and a 90º angle. This makes each of the new triangles similar to the original triangle by AA. We can therefore say that:

How are the New Right Triangles Similar?

1) Triangle ABC ~ ADC

2) Triangle ABC ~ ADB

Ready for an Example?  Lets check out the one below!

Legs of a Right Triangle

Step 1: To find the length of the missing legs of a right triangle, first, let’s separate each right triangle to see what we’re working with, along with the values of each length that was provided.

Legs of a Right Triangle

Step 2: We want to find the length of side AC, so let’s use the triangles whose sides include AC.  This leads us to use triangles ABC and ADC.

Legs of a Right Triangle

Step 3:  Now, let’s set up our proportion to find our missing side.  To make it easier for ourselves let’s first flip and rotate triangle ADC to line up with triangle ABC.

Legs of a Right Triangle
Legs of a Right Triangle

Think you’re ready to try some practice questions on your own? Check out the ones below!

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating!

Facebook ~ Twitter ~ TikTok ~ Youtube

Check out more posts on Similar Triangles here!

Sum of Infinite Geometric Sequence

Hey math friends and happy Wednesday! Today we are going to take a look at how to find the sum of infinite geometric sequence (aka series) in summation notation. This may sound complicated, but lucky for us there is an already existing formula that is ready and easy for us to use! So, let’s get to it! Also, don’t forget to check out the video and practice problems below for even more. Happy calculating! 🙂

What does it mean to find the “Sum of Infinite Geometric Sequence” (Series)?

We already know what a geometric sequence is: a sequence of numbers that forms a pattern when the same number is multiplied or divided to each term.

Example:

But when what happens if we wanted to sum all the terms of our geometric sequence together?

Example:

We can also write our infinite geometric series using Summation Notation:

How would we calculate that?  That’s where our Infinite Geometric Series formula will come in handy! But before we find a solution, lets take a closer look at what geometric series qualify for this formula.

Divergence Vs. Convergence:

There are two types of Infinite Geometric Series:

Type 1: Infinite Geometric Series that diverges to infinity

Type 2: Infinite Geometric Series that converges to a numeric value (-1 < r < 1)

Check out the differences in the example below:

Infinite Geometric Series Formula:

Note that the below infinite geometric series formula can only be used if the common ratio, r, is less than 1 and greater than -1. If the common ratio, r, is not between -1 and 1, then the sum of the geometric sequence diverges to infinity (and the formula cannot be used).

Now that we have a formula to work with and know when to use it (when -1 < r < 1), let’s take another look at our question and apply our infinite geometric series formula to find a solution:

Sum of Infinite Geometric Sequence

Step 1: First, let’s identify the common ratio to make sure that its between -1 and 1.

Sum of Infinite Geometric Sequence

Step 2: Now that we know we can use our formula, let’s write out each part and identify what numbers we are going to plug in.

Sum of Infinite Geometric Sequence

Step 3: Now let’s fill in our formula and solve with the given values.

Sum of Infinite Geometric Sequence

Practice Questions:

State if each series converges or diverges, then if applicable find the solution.

Sum of Infinite Geometric Sequence

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

*Also, if you want to check out Finite Geometric Series click this link here!

Facebook ~ Twitter ~ TikTok ~ Youtube

Mobius Bands

Greeting math friends and welcome to another fabulous week of MathSux! Today we are going to look at how to make a Mobius Strip otherwise known as “mobius bands.” Whatever you call them, they are a one of a kind shape that leads to all sorts of questions and math exploration. So let’s get to it! Let’s start off by explaining what a mobius strip is:

What are Mobius Bands?

A mobius strip (or band) is a two-dimensional shape that only has one surface.  Invented by German mathematician August Ferdinand Mobius and also independently by Johann Benedict Listing in the 1800s, it is considered a mathematical phenomenon! We are going to create our own mobius strip today to understand why this shape is so fascinating!

*Note: You may also see different spelling of the mobius band: including Möbius or Moebius

How to Make a Mobius Strip:

Mobius Bands

Step 1: First, let’s gather out materials. For this activity we will need a piece of paper, scissors, tape, and a pencil.

Mobius Bands

Step 2: Let’s cut out a rectangle. The size of the one shown above is 11 X 1 inches. Once you have your rectangle, take one end of the rectangle and give it a half twist.

Mobius Bands

Step 3: Now take the half twisted piece and attach it to the other end of the rectangle with a piece of tape. Behold the beauty of our mobius strip!

How to make a mobius strip

Step 4: Now we are going to take a pen or pencil and draw a line going all around the mobius strip without lifting the pen. See if you can move he mobius strip along, while the pencil remains where it is.  Notice anything special happen? We created a line around the entire shape without lifting our pencil!

Real-World Mobius:

Have you been pondering where we can find Mobius bands in the real world? I thought so! Take a look at the list below:

  • Printer ink cartridges
  • Serpentine Belt in a car
  • VHS tapes (if anyone remembers or knows what those are)
  • Can you think of anymore? Let me know in the comments below!

Worksheet:

I also made this FREE worksheet to go along with the above video and lesson for anyone interested. Let me know if this is helpful! 🙂

If you’re looking for more fun math projects, check out my “Just for Fun” page here. And if you want to get to see the latest MathSux content, don’t forget to follow along with us and subscribe via the links below. Thanks so much for stopping by and happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

Math Jewelry Review: Hanusa Design

Math Jewelry Review Interview

Hi Everyone and welcome to MathSux! I was sent three pieces of math jewelry from the jewelry brand, Hanusa Design. Math jewelry?! What is that?! Each piece you see here was inspired by mathematical art and created using 3D printing. I’m wearing the mobius necklace above and below you’ll also see mini pi and golden ratio earrings. For the full un-boxing and math jewelry review check out the video below and if you’re interested and want to learn more about Hanusa designs and the 3-d printing process, keep reading for the full interview I had with founder, Chirs Hanusa himself in this blog post.

Math Jewelry Review Interview

Math Jewelry Review:

Hanusa Design Interview with Founder, Chris Hanusa:

1.What made you start Hanusa Design? What led you to making jewelry as someone interested in math?

My adventure into 3D printing started in 2015 when I was updating a course in Mathematica I was teaching at Queens College. I was intrigued by 3D printing and I noticed that it was possible to use Mathematica as 3D design software, so I included a 3D design project as part of the class. As my students and I explored 3D printing, I recognized the universal appeal of the beauty and precision of mathematical concepts, and turned these ideas into jewelry. In turn, I founded Hanusa Design in 2017.

2. The jewelry is made through a 3D printing process. Can you explain the process from start to end? Is there a difference between the use of metal vs. nylon?

The design process starts with a mathematical concept that I’ve seen in my research, in mathematical texts, or as “found math” in the real world. I use Mathematica to do the 3D design, using three-dimensional coordinates, parametric functions, and aesthetic choices that turn the idea into a 3D model. The model is then exported directly from Mathematica to an STL file, which is basically a way to represent the boundary of the 3D object as a collection of triangles. The STL files are then sent to a 3D printer.

Once there, the same STL file can be used to create a nylon or metal piece of jewelry. The colorful nylon pieces are created using a SLS (selective laser sintering) process, where a thin layer of nylon powder is spread out and precisely fused to the previous layer using a laser. The excess powder is removed and then I hand dye the models using fabric dye. In contrast, the metal pieces are created using a lost-wax casting process. First, the models are 3D printed in high-resolution wax, then a plaster mold is created around the wax, and then the wax is replaced by molten metal.

3.I saw on your website that you are a mathematician and mathematical artist. Do you teach mathematical art at a university? If so, what types of topics do you cover? What is your favorite form of mathematical art?

I do teach two different courses that involve mathematical art. I teach a class called Mathematical Design that explores art that is created with functions, parametric functions, and polar functions using Desmos. This year I hope to give my Mathematical Design students the opportunity to use the Queens College Makerspace to take their digital art and bring it into reality using a laser cutter, a sewing machine, or a pen plotter. My other class is called Mathematical Computing. In this class I teach my students the computational software Mathematica, including how to use the software to do 3D modeling.  By the end of the semester, the students have designed and 3D printed a mathematical sculpture.

I suppose my favorite type of mathematical art is the visualization of complex mathematical concepts. It’s hard to understand certain concepts, like constructions in the fourth (or higher) dimension. Any picture or sculpture that helps clarify these difficult ideas is important, and it’s even better when it’s created with an eye to the aesthetic. I highly recommend any work by Henry Segerman.

4.I saw on your website that Hanusa Design has been featured in both New York Fashion Week (NYFW) and MoMath.  In what capacity?  Are you being featured in this week’s NYFW Fall 2021?

A wide variety of Hanusa Design jewelry has been available in the gift shop at the National Museum of Mathematics since 2018, including my dangling cubes earrings and interlocking octahedron necklace. I was asked to participate in a New York Fashion Week-adjacent show in Spring 2019 and enjoyed the experience. I am looking forward to eventually participate in New York City Jewelry Week.

5.Where can we find Hanusa Design, in stores or online? 

A variety of museums, galleries, and stores stock Hanusa Design Jewelry. As I mentioned before, it is available at the National Museum of Mathematics in New York, NY. It can also be found at the Exploratorium in San Francisco, CA, the Queens Museum in Queens, NY, Gallery North in Setauket, NY, Because Science in Vienna, VA, and in the Wolfram Store in Champaign, IL.

Hanusa Design Discount:

Hanusa has been kind enough to give MathSux readers an exclusive 10% off discount with the code ‘MATHROCKS’ now through December 31st 2021. I know its a bit early but this would be the perfect gift for the holiday season which is coming around the corner! Check out the full collection on their website here for even more designs and colors!

MathSux Giveaway:

math jewelry review

I’m going to be giving away a pair of pi earrings to one lucky MathSux reader! All you have to do is watch the YouTube video above, subscribe to MathSux, and comment below.

What do you guys think of Hanusa Design? Would you wear mathematical jewelry? What about the 3D printing process? Don’t forget to check out the video above for the full math jewelry review. Let me know what you guys think and happy calculating! 🙂

If you’re looking for more mathematical reviews, check out my review on the NumWorks calculator here.

Facebook ~ Twitter ~ TikTok ~ Youtube

Angle Bisector Definition & Example

Hi everyone and welcome to another fabulous week of MathSux! I bring to you the first construction of the back to school season! In this post, we are going to go over the angle bisector definition and example. First, we will define what an angle bisector is, then we’ll take our handy dandy compass and straight edge to construct an angle bisector that will bisect an angle for any size! Check out the video and GIF below for more and happy calculating! 🙂

What is an Angle Bisector?

A line that evenly cuts an angle into two equal halves, creating two equal angles.

Angle Bisector Example:

Angle Bisector Definition & Example

Step 1: Place the point of your compass on the point of the angle.

Step 2: Draw an arc that intersects both lines that stem form the angle you want to bisect.

Step 3: Take the point of your compass to where the lines and arc intersect, then draw an arc towards the center of the angle.

Step 4: Now keeping the same distance on your compass, take the point of your compass and place it on the other point where both the line and arc intersect, and draw another arc towards the center of the angle.

Step 5: Notice we made an intersection!? Where these two arcs intersect, mark a point and using a straight edge, connect it to the center of the original angle. 

Step 6: We have officially bisected our angle into two equal 35º halves.

*Please note that the above example bisects a 70º angle, but this construction method will work for an angle of any size!🙂

What do you think of the above angle bisector definition & example? Do you use a different method for construction? Let me know in the comments below! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

Looking for more constructions? Check out how to construct a square inscribed in a circle and an equilateral triangle by clicking on their respective links!

AnAngle Bisector Definition & Example

How to Find Expected Value

Greeting math friends! Today, we are going to dive into statistics by learning how to find the expected value of a discrete random variable. To do this we will need to know all potential numeric outcomes of a “gamble,” as well as be able to repeat the gamble as many time as we want under the same conditions, without knowing what the outcome will be. But I’m getting ahead of myself, all of this will be explained below with two different examples step by step! Don’t forget to check out the video and practice questions at the end of this post to check your understanding. Happy calculating! 🙂

What is Expected Value?

Expected Value is the weighted average of all possible outcomes of one “game” or “gamble” based on the respective probabilities of each potential outcome.

Expected Value Formula: Don’t freak out because below is the expected value formula.

How to Find Expected Value

In essence, we are multiplying each outcome value by the probability of the outcome occurring, and then adding all possibilities together!  Since we are summing all outcome values times their own probabilities, we can re-write the formula in summation notation:

How to Find Expected Value

Does the above formula look insane to you?  Don’t worry because we will go over two examples below that will hopefully clear things up! Let check them out:

Example #1: Expected Value of Flipping a Coin

Step 1:  First let’s write out all the possible outcomes and related probabilities for flipping a fair coin and playing this game.  Making the below table, maps out our Probability Distribution of playing this game.

How to Find Expected Value

Step 2: Now that, we have written out all numeric outcomes and the probability of each occurring, we can fill in our formula and find the Expected Value of playing this game:

How to Find Expected Value

Ready for another?  Let’s see what happens in the next example when rolling a die.

Example #2: Expected Value of Rolling a Die

Step 1:  First let’s write out all the possible outcomes and related probabilities for rolling a die. In this question, we are assuming that each side of the die takes on its numerical value, meaning rolling a 5 or a 6 is worth more than rolling a 1 or 2.  Making the below table, maps out our Probability Distribution of rolling the die.

How to Find Expected Value

Step 2: Now that, we have written out all numeric outcomes and the probability of each occurring, we can fill in our formula and find the Expected Value of playing this game:

How to Find Expected Value

Check out the practice problems below to master your expected value skills!

Practice Questions:

(1) An unfair coin where the probability of getting heads is .4 and the probability of getting tails is .6 is flipped.  In a game where you win $10 on heads, and lose $10 on tails, what is the expected value of playing this game?

(2) An unfair coin where the probability of getting heads is .4 and the probability of getting tails is .6 is flipped.  In a game where you win $30 on heads, and lose $50 on tails, what is the expected value of playing this game?

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

Facebook ~ Twitter ~ TikTok ~ Youtube

Looking for something similar to Expected Value? Check out the statistics page here!