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! π

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! π

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

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! π

What are Perpendicular Lines ?

Lines that intersect to create four 90ΒΊ angles about the two lines.

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! π

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! π

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.

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.

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! π

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! π

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):

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

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

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.

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

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

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! π

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! π

Hi there and welcome to MathSux! Today we are going to break down dilations; what they are, how to find the scale factor, and how to dilate about a point other than the origin. Dilations are a type of transformation that are a bit different when compared to other types of transformations out there (translations, rotations, reflections). Once a shape is dilated, the length, area, and perimeter of the shape change, keep on reading to see how! And if you’re looking for more transformations, check out these posts on reflections and rotations. Thanks so much for stopping by and happy calculating! π

What are Dilations?

Dilations are a type of transformation in geometry where we take a point, line, or shape and make it bigger or smaller, depending on the Scale Factor.

We always multiply the value of the scale factor by the original shape’s length or coordinate point(s) to get the dilated image of the shape. A scale factor greater than one makes a shape bigger, and a scale factor less than one makes a shape smaller. Letβs take a look at how different values of scale factors affect the dilation below:

Scale Factor >1 Bigger

Scale Factor <1 Smaller

Scale Factor=2

In the below diagram the original triangle ABC gets dilated by a scale factor of 2. Notice that the triangle gets bigger, and that each length of the original triangle is multiplied by 2.

Scale Factor=1/2

Here, the original triangle ABC gets dilated by a scale factor of 1/2. Notice that the triangle gets smaller, and that each length of the original triangle is multiplied by 1/2 (or divided by 2).

Properties of Dilations:

There are few things that happen when a shape and/or line undergoes a dilation. Letβs take a look at each property of a dilation below:

1. Angle values remain the same.

2. Parallel and perpendicular lines remain the same.

3. Length, area, and perimeter do not remain the same.

Now that we a bit more familiar with how dilations work, letβs look at some examples on the coordinate plane:

Example #1: Finding the Scale Factor

Step 1: First, letβs look at two corresponding sides of our triangle and measure their length.

Step 2: Now, letβs look at the difference between the two lengths and ask ourselves, how did we go from 3 units to 1 unit?

Remember, we are always multiplying the scale factor by the original length values in order to dilate an image. Therefore, we know we must have multiplied the original length by 1/3 to get the new length of 1.

When it comes to dilations, there are different types of questions we may be faced with. In the last question, the triangle dilated was done so about the origin, but this wonβt always be the case. Letβs see how to dilate a point about a point other than the origin with this next example.

Example #2: Dilating about a Point other than the Origin

Step 1: First, letβs look at our point of dilation, notice it is not at the origin! In this question, we are dilating about point m! To understand where our triangle is in relation to point m, letβs draw a new x and y axes originating from this point in blue below.

Step 2: Now, letβs look at coordinate point K, in relation to our new axes.

Step 3: Letβs use the scale factor of 2 and the transformation rule for dilation, to find the value of its new coordinate point. Remember, in order to perform a dilation, we multiply each coordinate point by the scale factor.

Step 4: Finally, letβs graph the dilated image of coordinate point K. Remember we are graphing the point (6,4) in relation to the x and y-axis that stems from point m.

Check out these dilation questions below!

Practice Questions:

1) Plot the image of Point Z under a dilation about the origin by a scale factor of 2.

2) Triangle DEF is the image of triangle ABC after a dilation about the origin. What is the scale factor of the dilation?

3) Point L is dilated by a scale factor of 2 about point r. Draw the dilated image of point L.

4) Line DE is the dilated image of line AB. What is the scale factor of the dilation?

Solutions:

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

Ahoy and welcome math friends! For the latest installment, here is the Algebra 2 Cheat Sheet & Review made just for you to prepare for finals. On this page, you’ll also find links to the come math friends! For the latest installment, here is the Algebra 2 lesson playlist, the NYS Algebra 2 Common Core Regent’s Playlist, and the library of Geometry blog posts. Hope you find these resources helpful as the end of the school year approaches. Good luck on finals and happy calculating! π

Algebra 2 Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to graphs, it’s on here.

Looking for a more detailed review? Check out the Youtube playlist for Algebra 2 below. It includes every MathSux video related to Algebra 2 and will be sure to help you ace the test!

Algebra 2 Common Core Regents Review:

This playlist is made especially for New York State dwellers as it goes over each and every question of the NYS Common Core Regents. Perfect if you are stuck on that one question! You will surely find the answer here.

Algebra 2 Blog Posts:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Algebra 2 library organized by topic here.

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

Get everything you need to know with this Algebra 2 Cheat Sheet and Review! Download and print the pdf for reviewing Algebra 2 or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Greeting math peeps! As promised here is the Geometry Cheat Sheet and Review made just for you to prepare for finals. On this page, you’ll also find links to the Geometry lesson playlist, the NYS Geometry Common Core Regent’s Playlist, and the library of Geometry blog posts. Hope you find these resources helpful as the end of the school year approaches. Good luck on finals and happy calculating! π

Geometry Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to shapes, it’s on here.

Looking for a more detailed review? Check out the Youtube playlist for Geometry below. It includes every MathSux video related to Geometry and will be sure to help you ace the test!

Geometry Common Core Regents Review:

This playlist is made especially for New York State dwellers as it goes over each and every question of the NYS Common Core Regents. Perfect if you are stuck on that one question! You will surely find the answer here.

Geometry Blog Posts:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Geometry library organized by topic here.

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

Get everything you need to know with this Geometry Cheat Sheet and Review! Download and print the pdf for reviewing Geometry or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

It’s that time of year again, summer is coming, the vacation vibes are calling, but so, unfortunately, are the test cramming and non-stop class reviewing that is coming our way. Nothing like going over topics mentioned at the beginning of the school year to bring us down. How is one supposed to remember everything? Fear not, because I have made a special cheat sheet and review for Algebra, (with Geometry and Algebra 2/Trig. soon to be on the way). I hope you’re staying safe, cool, and calm as the end of the year approaches. Good luck on finals and tests and happy calculating! π

Algebra Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to parabolas, it’s on here.

Looking for a more detailed review? Check out the Youtube playlist for Algebra below. It includes every MathSux video related to Algebra and will be sure to help you ace the test!

Algebra Common Core Regents Review:

This playlist is made especially for New York State dwellers as it goes over each and every question of the NYS Common Core Regents. Perfect if you are stuck on that one question! You will surely find the answer here.

Algebra Blog Posts:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Algebra library organized by topic here.

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

Get everything you need to know with this Algebra Cheat Sheet and Review! Download and print the pdf for reviewing Algebra or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Happy Wednesday math peeps! Today we are going to go over the Finite Arithmetic Series Formula; What it is, how to use it, and even do a little derivation. Before going any further though, please make sure you know how arithmetic sequences work here. If you have any questions, please don’t hesitate to comment below and to check out the video and practice questions below. Happy calculating! π

What does it mean to find the βSum of the Finite Arithmetic Sequenceβ?

We already know what an arithmeticsequence is: a sequence of numbers that forms a pattern when the same number is added or subtracted to each term.

Example:

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

Example:

More specifically, what if we wanted to find the sum of the first 20 terms of the above arithmetic sequence? How would we calculate that? Thatβs where our Finite Arithmetic Series formula comes in handy!

Finite Arithmetic Series Formula:

Looking at the above formula, I have to wonder, what happens if we are not given the value of the last term of the sequence for “a sub n”? What would we do? Do not worry, because there is another way to use this formula if we expand and simplify it, check it out below:

*Bonus* Arithmetic Series Formula:

Plug in the arithmetic sequence formula for “a sub n,” then combine like terms.

Letβs take a closer look at what each part of our bonus formula represents below:

Now that we have two formulas to work with, letβs take another look at our question now applying our finite arithmetic series formula:

Step 1: First letβs write out our formula and identify what each part represents/what numbers need to be filled in. Since we are not given the value of the last term, βa sub nβ we can use the second bonus formula we previously derived.

Step 2: Now letβs fill in our formula and calculate.

Practice Questions:

1) Find the sum of the first 15 terms of the following arithmetic sequence:

4, 8, 12, 16, β¦.

2) Find the sum of the first 24 terms of the following sequence:

2, 7, 12, 17, β¦.

3) Find the sum of the first 32 terms of the following arithmetic sequence:

100, 97, 94, 91, β¦.

4) Find the sum of the first 50 terms of the following arithmetic sequence:

5, 7, 9, 11, 13, β¦.

Solutions:

1) 480

2) 1,428

3) 1,712

4) 2,700

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

Happy Wednesday math peeps! TIn today’s post, we are going to go over Proving Similar Triangles, by going over:

1) What it means when two triangles are similar?

2) How to prove two triangles similar?

3) How to find missing side lengths given triangles are similar?

For even more practice, don’t forget to check out the video and practice problems below. Happy calculating! π

What are Similar Triangles?

When two triangles have congruent angles and proportionate sides, they are similar. This means they can be different in size (smaller or larger) but as long as they have the same angles and the sides are in proportion, they are similar! We use the “~” to denote similarity.

In the Example below, triangle ABC is similar to triangle DEF:

How can we Prove Triangles Similar?

There are three ways to prove similarity between two triangles, letβs take a look at each method below:

Angle-Angle(AA): When two different sized triangles have two angles that are congruent, the triangles are similar. Notice in the example below, if we have the value of two angles in a triangle, we can always find the third missing value which will also be equal.

Side-Side-Side (SSS): When two different sized triangles have three corresponding sides in proportion to each other, the triangles are similar.

Side-Angle- Aside (SAS): When two different sized triangles have two corresponding sides in proportion to each other and a pair of congruent angles between each proportional side, the triangles are similar.

Letβs look at how to apply the above rules with the following Example:

Step 1: Since, we know the triangles ABC and DEF are similar, we know that their corresponding sides must be in proportion! Therefore, we can set up a proportion and find the missing value of length EF by cross multiplying and solving for x.

Practice Questions:

1) Are the following triangles similar? If so, how? Explain.

2) Are the following triangles similar? If so, how? Explain.

3) Given triangle ABC is similar to triangle DEF, find the side of missing length AB.

4) Given triangle ABC is similar to triangle PQR, find the side of missing length AC.

Solutions:

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