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 – 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!😅
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!
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!
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.
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.
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.
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!
Hey math friends and happy Wednesday! Today we are going to take a look at the infinite geometric series formula in summation notation. First, we’ll review what a geometric sequence (or geometric progression) is, then look at the difference between divergence and convergence based on the common ratio, and then finally look at the formula. The unique thing about infinite geometric sequences is that the formula doe snot always apply. We only know if the formula applies to our series based on the value of the common ratio.
This may sound a bit complicated, with a few new concepts, but fear not as divergence and convergence are easy to identify! If you have any questions, don’t forget to check out the video and practice problems below. 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 by each successive term.
The number is multiplied or divided to each term is called the common ratio. We can see below that each number in the sequence is being multiplied by 1/2. Therefore, we can say that the common ratio of the geometric sequence below is 1/2.
Example:
But then, what happens if we wanted to sum all the terms of our geometric sequence together? The sequence of terms is now considered to be an infinite geometric series since we are summing all of the terms of our series together, infinitely. Take a look below:
Example:
Summation Notation:
We can also write our infinite geometric series using something called Summation Notation shown below. This is a different format, also known as sigma notation, if you need a refresher on how to use and read summation notation, please check out the link in this paragraph. When writing in summation notation, we are actually writing the sum of the geometric series in a different way. Check it out below:
Now that we know what each part of this summation notation looks like, let’s actually identify what each part of this equation means in relation to our geometric sequence:
a1=The first term. In this case, we can see that the first term of our geometric sequence is the number 5 in the example below. Therefore, we can say a1=5.
r= The common ratio is the number that is multiplied or divided by each consecutive term within the sequence. In the example below, each number is multiplied by 1/2, therefore we can say, r=1/2.
n= The total number of terms we are trying to sum together. In the example mentioned below, we are trying to sum an infinite number of terms, so in this case, n starts at 1 and then continues infinitely to increase by 1 for each subsequent term.
If we were to expand our inifite series summation it would look something like this:
But how would we calculate this? This is an infinite series represented in summation notation, therefore, it goes on infinitely, how does this work?
That’s where our Infinite Geometric Series formula comes in handy! But before we jump into finding a solution and looking at our formula, let us take a closer look at what types of geometric series qualify for using the infinite series formula in the first place.
Divergence Vs. Convergence: When do we use the Infinite Series formula?
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:
Now the moment you’ve all been waiting for… the Inifite Geometric Series Formula! Remember, the use of this formula is purely based on the common ratio of our geometric sequence.
The 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 the 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:
Step 1: First, let’s identify the common ratio to make sure that it is between numbers -1 and 1. If the common ratio is a fraction between these two numbers, then we can use our infinite series formula.
Step 2: Now that we know we can use our infinite series formula, let’s write out each part of the formula and identify what numbers we are going to plug in.
Step 3: Last but not least, let’s fill in our formula with the correct numbers and solve with the given values.
Think you are ready to try inifite geometric series examples on your own? Check out the following infinite geometric series practice questions below to determine if a series converges or diverges.
Practice Questions:
State if each given series converges or diverges, then if applicable find the solution.
Solutions:
Related Posts:
Looking to learn more about sequences? You’ve come to the right place! Check out these sequence resources and posts below. Personally, I recommend looking at the finite series posts next including: finite geometric sequence or finite arithmetic series!
Still, got questions? No problem! Don’t hesitate to comment below or reach out via email. And if you would like to see more MathSux content, please help support us by following ad subscribing to one of our platforms. Thanks so much for stopping by and happy calculating!
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:
Step 1: First, let’s gather out materials. For this activity we will need a piece of paper, scissors, tape, and a pencil.
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.
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!
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! 🙂
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:
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?
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:
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.
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?
An Angle Bisector is a line that evenly cuts an angle into two equal halves, creating two equal angles. For example, if we have a 70-degree acute angle and we create an angle bisector this would create two equal angles of 35 degrees each, dividing 70 by 2. Check out how to do this construction step by step with pictures and explanations below.
Angle Bisector Example:
Step 1: First, we start by placing the point of our compass on the point of the angle, which in this case is 70 degrees.
Step 2: Next, we are going to draw an arc that intersects both lines that stem from the angle we want to bisect.
Step 3: Now, take the point of our compass to where the lines and arc intersect, and draw an arc towards the center of the angle.
Step 4: Keeping that same distance on our compass, we are going to 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 or ruler, connect it to the center of the original angle.
Step 6: We have officially bisected our angle into two equal 35-degree halves, creating an angle bisector!
*Please note that the above example bisects a 70º angle, but this construction method will work for an angle of any size acute or obtuse!🙂
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! 🙂
Constructions and Related Posts:
Looking to construct more than just an angle bisector? Check out these related posts and step-by-step tutorials on geometry constructions below!
Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only $7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!
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! And if you’re looking for even more geometry constructions, check out the link here!
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.
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:
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.
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:
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.
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:
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! 🙂
Greetings math peeps! In today’s post, we are going to look at how to solve for exponents. At this point, we are familiar with solving for unknown variables within an equation, but never before did we have to solve for an exponent! Understanding how to do this and what method(s) to use will take our algebra skills to a whole new level. Also, don’t forget to check out the video and practice questions in this post for even more examples. Happy calculating! 🙂
What is an Exponential Equation?
An exponential equation, is an equation where the exponent is an unknown variable and takes the following form:
Sometimes we will be asked to solve for the unknown variable in an exponential equation. There are two main ways of solving for this type of equation and we’ll go over each type below.
(Method 1) Same Base
(Method 2) Different Base
Let’s dive right in and look at three different examples applying one of the two methods above.
Example #1: Solving Exponential Equations with the Same Base
Whenever an equation has the same base, like in the example below (both bases are 5), we can set the exponents equal and solve for the missing variable.
Step 1: Since both sides of the equation have the same base (a base of 5), we can set the exponent values equal to each other and use basic algebra to solve for x.
Example #2: Solving Exponential Equations with the SameBase
We can set exponents equal to each other whenever an equation can be written to have the same base on either side. Like in the example below both bases can be written as 5, we can again, set the exponents equal and solve for the missing variable.
Step 1: Re-write the right part of the equation so the bases on either side match. In this case we need to re-write 25 as 5^2. Make sure to keep the part of the exponent, 2x, that was already attached to the 25 so it becomes, 5^2(2x).
Step 2: Now we can set the exponent values equal to each other and solve for x now that both sides have matching bases.
Example #3: Solving Exponential Equations with the DifferentBases
When an exponential equation doesn’t have the same base and cannot be written to have the same base, we must use logs to solve for the unknown variable! So, make sure you are comfortable with the log rules listed here in order to solve these types of questions. We are mainly going to be working with the power rule of logs.
Step 1: Take log of both sides and solve for x.
When you’re ready check out the practice questions below to own your new exponent skills!
Greetings math peeps and welcome to MathSux! In todays post we will be exploring how to tell something is a function or not. We’ll start off by defining what a function is, go over its notation, and then look at several different examples of how to recognize a function in different formats including function maps, tables, and graphs! Functions are a concept that is seen throughout algebra and mathematics so understanding it well is key to learning more! Also, don’t forget to check out the video and practice problems at the end of this post to practice your skills! Happy calculating! 🙂
What is a Function?
A function works like a machine with inputs and outputs. When we input a number into a function, a new number pops right back out. We can say that a function is like a machine because it transforms one number into a completely different number once it enters this so called “machine.”
Let’s look at an example in action, where f(x)=2x+1 is the function and we want to see what happens when we plug inout a 3 into our function.
Notice we plugged in 3 into the variable x and solved, to get the output 7.
Function Notation:
Functions have their own notation, when we have f(x)=2x+1, this can also be written as g(x), h(x), or any other letter you can think of!
In the previous input/output example, the input value 3 is plugged into the function f(x)=2x+1 for the missing variable, x to get the output 7.
Another way to write what we did in function notation is to say that we found the value for f(3)=6 for the function f(x).
How to Tell if Something is a Function?
A main and important rule of functions is that there is always one unique input, but there can be different or repeating outputs. Let’s look at some examples below, at how to identify a function.
Example #1:Function Maps
Example #2: Tables
Example #3: Graphs
In order to know if a function is a function when looking at graph, we perform something called a Vertical Line Test. All we must do is draw a vertical line, if the line hits the graph one time, the graph is a function! If the vertical line his more than that, the graph is not a function.
Try the following practice questions to test your skills below!
Practice Questions:
Identify if the following are functions or not and explain your reasoning.
Solutions:
Looking to learn more about Algebra? Check out the algebra lessons page here. Thanks for stopping by and happy calculating! 🙂
Hi Everyone and welcome to MathSux! Today, I wanted to answer a question I get a lot which is why name your Blog and YouTube channel, “MathSux”? Clearly, I love math, but with the name “MathSux” I wanted to show that it can also be hard and even I can think that it suck sometimes. When we don’t understand something it can be frustrating whether its related to math or really anything! The point is we’ve all gotten frustrated when learning something new at some time, but that’s ok, and that’s exactly what MathSux stands for! 🙂
Check out the video below to hear why I chose the name “MathSux” while doodling math art . I hope you enjoy it and happy calculating! 🙂
Why is it called “MathSux”?
*New lessons will be coming your way starting next Wednesday. Also be on the lookout for Regents review questions up on YouTube tomorrow and Friday! 🙂
If you are a teacher or student, have you ever thought math sucked at some time in your life? Let me know in the comments below!
