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! 🙂

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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! 🙂

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

How to Solve for Exponents

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.

How to Solve for Exponents
How to Solve for Exponents

Example #2: Solving Exponential Equations with the Same Base

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.

How to Solve for Exponents

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).

How to Solve for Exponents

Step 2: Now we can set the exponent values equal to each other and solve for x now that both sides have matching bases.

How to Solve for Exponents
How to Solve for Exponents

Example #3: Solving Exponential Equations with the Different Bases

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.

How to Solve for Exponents

Step 1: Take log of both sides and solve for x.

How to Solve for Exponents
How to Solve for Exponents

When you’re ready check out the practice questions below to own your new exponent skills!

Practice Questions:

Solve each exponential equation for x.

Solutions:

Looking to review logs? Check out this post here!

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How to Tell if Something is a Function?

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

How to Tell if Something is a Function?

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! 🙂

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Why “MathSux”?

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!

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And for more “Just for Fun” math posts and videos, click the link here.

Back to School Review: Algebra | Geometry | Algebra 2/Trig.

Back to School Review

Hey math friends and welcome back to MathSux! Back to school season is upon us which means most students (and teachers) will need to review a bit before diving into a completely new subject.  In order to alleviate some of the back to school whoas, I bring to you, this back to school review! Check out the videos below to get the math juices flowing whether you’re new to Algebra, Geometry, or Algebra 2/Trig! I hope you find these videos helpful and wish everyone the best of luck in their first days at school! Happy calculating! 🙂

How to Prepare for Algebra: 

Calling all incoming algebra students, Combining Like Terms is a great place to start! You most likely have combined like terms before, but there’s nothing like sharpening your skills before getting the intense Algebra questions that are coming your way. Check out the video below and try the practice questions here!

Practice Problems: https://mathsux.org/2020/09/30/algebra-combining-like-terms-and-distributive-property/

How to Prepare for Geometry: 

Geometry students, you have the world of shapes ahead of you! It’s an exciting time to review basic Area, Perimeter, Circumference, and Pythagorean Theorem rules before moving ahead with this subject. Review the Pythagorean Theorem below from Khan Academy and check out the last page of the review sheet here to review area and perimeter.

How to Prepare for Algebra 2:

Relieve the fond memories of algebra by reviewing all the different ways to Factor and Solve Quadratic Equations! This is a great way to prepare for rational expressions and the harder algebra 2 problems that are right around the corner.  Check out the video below and related practice questions here to reinforce these hopefully not yet forgotten algebra skills!

Practice Problems: https://mathsux.org/2020/06/09/algebra-4-ways-to-factor-trinomials/
Practice Problems: https://mathsux.org/2016/07/06/algebra-2-factor-by-grouping/

Hope you find this quick review helpful before diving in for the real deal! Besides brushing up on these math topics, what type of new school year routines do like to practice in your classroom or at home? Let me know in the comments and happy calculating! 🙂

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Olympics Statistics: Top 10 Medals by Country

Olympics Statistics

Greetings and welcome back to MathSux! This week, in honor of the Tokyo Olympics, I will be breaking down some Olympic Statistics. We will look at the top 10 countries that hold the most medals and then look at the top 10 medals earned by country in relation to each country’s total population. Let’s take a look and see what we find! Also, please note that all data used for this analysis was found on the website, here. Anyone else watching the Olympics? Try downloading the data with the link above and see what type of conclusions you can find! Happy Calculating! 🙂

Top 10 Countries: Total Olympic Medals

Below shows the top 10 total medals earned by country from the beginning of the Olympics in 1896 to present day July 2021. As we can see in the graph below, the United States is way ahead of the game with thousands more Olympic medals when compared to any other country in the entire world! I always knew the U.S. did well in the Olympics, but did not realize it was to this magnitude!

Top 10 Countries: Total Olympic Medals Based on Population

Below is a different kind of graph. This percentage rate represents total medals earned over time from 1896 to July 2021 divided by the country’s total population. In this case, we can see that Lichtenstein has earned way more medals based on their small population size when compared to any other country in the world! This is amazing and unexpected!

Remember that all data for the above graphs were made from the following website, here. Are you surprised by the above graphs and conclusions? Try downloading the data on your own and see what you can conclude using your own Olympics Statistics skills! Happy calculating! 🙂

Looking to apply more math to the real world? Check out how to find volume of the Hudson Yards Vessal in NYC here

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How to Construct a Perpendicular Line through a Point on the Line

Greetings math peeps and welcome to another week of MathSux! In this post, we will learn how to construct a perpendicular line through a point on the line step by step. In the past, we learned how to construct a perpendicular bisector right down the middle of the line, but in this case we will learn how to create a perpendicular line through a given point on the line (which is not always in the middle). Following along with the GIF or check out the vide below. Thanks for stopping by and happy calculating! 🙂

What are Perpendicular Lines ?

  • Lines that intersect to create four 90º angles about the two lines.
How to Construct a Perpendicular Line through a Point on the Line

What is happening in this GIF?

Step 1: First, we are going to gather materials, for this construction we will need a compass, straight edge, and markers.

Step 2: Notice that we need to make a perpendicular line going through point B that is given on our line.

Step 3: Open up our compass to any distance (something preferably short though to fit around our point and on the line).

Step 4: Place the compass end-point on Point B, and draw a semi-circle around our point, making sure to intersect the given line.

Step 5: Open up the compass (any size) and take the point of the compass to the intersection of our semi-circle and given line.  Then swing our compass above the line.

Step 6: Keeping that same length of the compass, go to the other side of our point, where the given line and semi-circle connect.  Swing the compass above the line so it intersects with the arc we made in the previous step.

Step 7: Mark the point of intersection created by these two intersecting arcs we just made and draw a perpendicular line going through Point B!

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

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Want to see how to construct a square inscribed in a circle? Or maybe you want to construct an equilateral triangle? Click on each link to view each construction!

Origami and Volume of a Box and Square Base Pyramid

Greetings and happy summer math peeps! In honor of the warm weather and lack of school, I thought we’d have a bit of fun with origami and volume! In this post, we will find the volume of a box and the volume of a square base pyramid. We will also be creating each shape by using origami and following along with the video below. For anyone who wants to follow along with paper folding tutorial, please note that we will need one piece of printer paper that is 8.5″ x 11″and one piece of square origami paper that is 8″ x 8″. If you’re interested in more math and art projects check out this link here. Stay cool and happy calculating! 🙂

Volume of Box (or Rectangular Prism):

To get the volume of our origami box (video tutorial above), we are going to multiply the length times the width times the height. All the values and units of measurement were found by measuring the box we made in inches in the video above with 8.5 x 11 inch computer paper.

Origami and Volume
Origami and Volume

Volume of Square Base Pyramid:

Below is a diagram of the square base pyramid we created via paper folding (watch video tutorial above to follow along!). Please note that if you used a different sized paper (other than 8 X 8 inches), you will get a different value for measurements and for volume.

Origami and Volume
Origami and Volume

For step by step instruction, don’t forget to check out the video above to see how to paper fold a box and square base pyramid. I hope this post made math suck just a little bit less and finding volume a bit more fun. Still got questions or want to learn more about Math+ Art? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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For more Math + Art, check out this post on Perspective Drawing here.

The Original Spirograph: Math + Art

Happy Summer everyone! Now that school is out, I thought we could have a bit of fun with Math and Art! In this post, we will go over how to make a the original spirograph (by hand) step by step using a compass and straight edge. Follow along with the video below or check out the tutorial in pictures in this post. Hope everyone is off to a great summer. Happy calculating! 🙂

What is a Spirograph?

The childhood toy we all know and love was invented by Denys Fisher, a British Engineer in the 1960’s.But the method of creating Spirograph patterns was invented way earlier by engineers and mathematicians in the 1800’s.

The Original Spirograph (by hand):

The Original Spirograph

Step 1: Gather materials, for this drawing, we will need a compass and straight edge.

The Original Spirograph

Step 2: Using our compass, we are going to open it to 7 cm and draw a circle.

The Original Spirograph

Step 3: Next, we are going to open the compass to 1cm, making marks all around the circle, keeping that same distance on the compass.

The Original Spirograph

Step 4: Draw a line connecting two points together (any two points some distance apart will do).

The Original Spirograph

Step 5: Now, we are going to move the straight edge forward by one point each and connect the two points with another line.

The Original Spirograph

Step 6: Continue this pattern of moving the ruler forward by one point and connecting them together all the way around.

Step 7: We have completed our Spirograph drawing! Try different sized circles, points around the circle, colors, and points of connections to create different types of patterns and have fun! 🙂

Still got questions or want to learn more about Math+ Art? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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For more Math + Art, check out this post on Perspective Drawing here.