Geometry: 45º 45º 90º Special Triangles

Greetings math folks! In this post we are going to go over 45º 45º 90º special triangles and how to find the missing sides when given only one of its lengths. For even more examples, check out the video below and happy calculating! 🙂

Why is it “special”?

 The 45º 45º 90º triangle is special because it is an isosceles triangle, meaning it has two equal sides (marked in blue below).  If we know that the triangle has two equal lengths, we can find the value of the hypotenuse by using the Pythagorean Theorem.  Check it out below!

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two equal legs. This creates a ratio that applies to all 45º 45º 90º triangles!

How do I use this ratio?

Knowing the above ratio, allows us to find any length of a 45º 45º 90º triangle, when given the value of one of its sides.

Let’s try an example:

Now let’s look at an example where we are given the length of the hypotenuse and need to find the values of the other two missing sides.

Now try mastering the art of the 45º 45º 90º special triangle on your own!

Practice Questions: Find the value of the missing sides.

Solutions:

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

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Algebra: Variance and Standard Deviation

Greetings math friends! In this post we’re going to go over variance and standard deviation. We will take this step by step to and explain the significance each have when it comes to a set of data. Get your calculators ready because this step by step although not hard, will take some serious number crunching! Check out the video below to see how to check your work using a calculator and happy calculating! 🙂

What is the Variance?

The variance represents the spread of data or distance each data point is from the mean.  When we have multiple observations in our data, we want to know how far each unit of data is from the mean.  Are all the data points close together or spread far apart?  This is what the variance tells us!

Don’t freak out but here’s the formula for variance, notated as sigma squared:

This translates to:

Let’s try an example:

What is Standard Deviation?

Standard deviation is a unit of measurement that is unique to each data set and is used to measure the spread of data. The formula for standard deviation happens to be very similar to the variance formula!

Below is the formula for standard deviation, notated as sigma:

Since this is the same exact formula as variance with a square root, all we need to do is take the square root of the variance to find standard deviation:

Now try calculating these statistics on your own with the following practice problems!

Practice Questions:

Solutions:

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

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Algebra: Combining Like Terms and the Distributive Property

Greetings math peeps! In today’s post we are going to review some of the basics: combining like terms and the distributive property. It’s so important to master the basics such as these, so you’re prepared and ready to handle the harder stuff that’s just around the corner, trust me they’re coming! And for those who already feel comfortable with these topics, great! Skip ahead and try the practice questions at the bottom of this post and happy calculating! 🙂

When do we combine “like terms?”

Combining like terms allows us to simplify and calculate our answer with terms that have the same variable and same exponent values only. For example, we can combine the following expression:

How do we combine like terms?

We add or subtract the whole number coefficients and keep the variable they have in common.

Why? We could not add these two terms together because their variables do not match! 2 is multiplied by x, while 3 is multiplied by the variable xy.

Why? We could not add these two terms together because their variables and exponents do not match! 2 is multiplied by x, while 3 is multiplied by the variable x^2 . Exponents for each variable must match to be considered like terms.

Distributive Property:

Combining like terms and the distributive property go hand in hand.  The distributive property rule states the following:

There are no like terms to combine in the example above, but let’s see what it would like to use the distributive property and combine like terms at the same time with the following examples:

Example #1:

Example #2:

In some cases, we also have to distribute is the -1 that can sometimes “hide” behind a parenthesis.

Try the following questions on your own on combining like terms and the distributive property and check out the video above for more!

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

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Algebra 2: Imaginary and Complex Numbers

Happy Wednesday and back to school season math friends! This post introduces imaginary and complex numbers when raised to any power exponent and when multiplied together as a binomial. When it comes to all types of learners, we got you between the video, blog post, and practice problems below. Happy calculating! 🙂

What are Imaginary Numbers?

Imaginary numbers happen when there is a negative under a radical and looks something like this:

Why does this work?

In math, we cannot have a negative under a radical because the number under the square root represents a number times itself, which will always give us a positive number.

Example:

But wait, there’s more:

When raised to a power, imaginary numbers can have the following different values:

Knowing these rules, we can evaluate imaginary numbers, that are raised to any value exponent! Take a look below:

-> We use long division, and divide our exponent value 54, by 4.

-> Now take the value of the remainder, which is 2, and replace our original exponent. Then evaluate the new value of the exponent based on our rules.

What are Complex Numbers?

Complex numbers combine imaginary numbers and real numbers within one expression in a+bi form. For example, (3+2i) is a complex number. Let’s evaluate a binomial multiplying two complex numbers together and see what happens:

-> There are several ways to multiply these complex numbers together. To make it easy, I’m going to show the Box method below:

Try mastering imaginary and complex numbers on your own with the questions below!

Practice:

Solutions:

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

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Geometry: How to Construct an Equilateral Triangle?

 

Happy Wednesday math peeps! This post introduces constructions by showing how to construct an equilateral triangle by using a compass and a ruler. For anyone new to constructions, this is the perfect topic for art aficionados since there is more drawing than there is actual math.  Screen Shot 2020-08-25 at 4.09.58 PM.pngEquilateral Triangle: A triangle with three equal sides.  Not an easy one to forget, the equilateral triangle is super easy to construct given the right tools (compass+ ruler). Take a look below:Screen Shot 2020-08-25 at 3.56.17 PM.png

Solution:

Construction-GIF-v2

What’s Happening in this GIF? 

1. Using a compass, measure out the distance of line segment  Screen Shot 2020-08-25 at 4.19.02 PM.

 2. With the compass on point A, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

 3. With the compass on point B, draw an arc that has the same distance as Screen Shot 2020-08-25 at 4.19.02 PM.

4. Notice where the arcs intersect? Using a ruler, connect points A and B to the new point of intersection. This will create two new equal sides of our triangle!

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

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Algebra 2: Expanding Cubed Binomials

Greetings math friends! This post will go over how to expand and simplify cubed binomials 2 different ways. We’re so used to seeing squared binomials such as, Screen Shot 2020-08-19 at 11.29.14 AM.png, and expanding them without a second thought.  But what happens when our reliable squared binomials are now raised to the third power, such as,Screen Shot 2020-08-19 at 11.29.48 AM?  Luckily for us, there is a Rule we can use:

Screen Shot 2020-08-18 at 10.12.33 PM

But where did this rule come from?  And how can we so blindly trust it? Which is why we are going to prove the above rule here and now using 2 different methods:Screen Shot 2020-08-19 at 11.31.13 AM

Why bother? Proving this rule will allow us to expand and simplify any cubic binomial given to us in the future! And since we are proving it 2 different ways, you can choose the method that best works for you.

Method #1: The Box MethodScreen Shot 2020-08-18 at 10.14.37 PMScreen Shot 2020-08-18 at 10.14.55 PM.pngScreen Shot 2020-08-18 at 10.15.06 PMScreen Shot 2020-08-18 at 10.15.39 PM.pngScreen Shot 2020-08-18 at 10.15.50 PM

Screen Shot 2020-08-19 at 2.24.54 PMScreen Shot 2020-08-19 at 2.53.43 PM

Screen Shot 2020-08-19 at 2.29.22 PM.pngScreen Shot 2020-08-18 at 10.17.19 PM.png

Screen Shot 2020-08-19 at 2.27.56 PMScreen Shot 2020-08-19 at 2.54.36 PM.png

Screen Shot 2020-08-18 at 10.21.05 PM.png

Method #2: The Distribution MethodScreen Shot 2020-08-18 at 10.17.54 PM.pngScreen Shot 2020-08-18 at 10.19.49 PMScreen Shot 2020-08-19 at 2.42.11 PM

Screen Shot 2020-08-18 at 10.21.05 PM.png

Now that we’ve gone over 2 different methods of cubic binomial expansion, try the following practice questions on your own using your favorite method!

Practice Questions: Expand and simplify the following.

Screen Shot 2020-08-18 at 10.21.56 PM

Solutions:

Screen Shot 2020-08-18 at 10.22.19 PM.png

Still, got questions?  No problem! Check out the video above or comment below! Happy calculating! 🙂

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**Bonus: Test your skills with this Regents question on Binomial Cubic Expansion!

 

Algebra: How to use Recursive Formulas?

Welcome to Mathsux! This post is going to show you everything you need to know about Recursive Formulas by looking at three different examples. Check out the video below for more of an explanation and test your skills with the practice questions at the bottom of this page.  Happy calculating! 🙂

What is a Recursive Formula?

A Recursive Formula is a type of formula that forms a sequence based on the previous term value.  What does that mean?  Check out the example below for a clearer picture:

Example #1:

Screen Shot 2020-08-11 at 8.12.21 AM.pngScreen Shot 2020-08-11 at 8.12.33 AMScreen Shot 2020-08-11 at 9.18.18 AM

Screen Shot 2020-08-11 at 8.13.07 AMScreen Shot 2020-08-11 at 8.13.36 AM.pngScreen Shot 2020-08-11 at 9.21.01 AM.pngScreen Shot 2020-08-11 at 8.14.34 AM.pngScreen Shot 2020-08-11 at 8.14.49 AMExample #2:

Screen Shot 2020-08-11 at 8.15.19 AM.png

Screen Shot 2020-08-11 at 8.15.38 AMScreen Shot 2020-08-11 at 9.22.10 AMScreen Shot 2020-08-11 at 8.52.36 AMScreen Shot 2020-08-11 at 8.52.52 AM.pngScreen Shot 2020-08-11 at 9.23.54 AM.png

***Note this was written in a different notation but is solved in the exact same way!

Screen Shot 2020-08-11 at 8.53.24 AM.pngScreen Shot 2020-08-11 at 8.53.35 AM

Example #3:Screen Shot 2020-08-11 at 8.54.05 AM.pngScreen Shot 2020-08-11 at 8.54.19 AMScreen Shot 2020-08-11 at 9.24.42 AMScreen Shot 2020-08-11 at 8.54.49 AMScreen Shot 2020-08-11 at 9.25.41 AM.pngScreen Shot 2020-08-11 at 8.56.22 AMScreen Shot 2020-08-11 at 8.56.36 AM.pngPractice Questions:

Screen Shot 2020-08-11 at 10.04.21 AM

Solutions:

Screen Shot 2020-08-11 at 9.02.18 AM.png

Still got questions? No problem! Check out the video above for more or try the NYS Regents question below, and please don’t hesitate to comment with any questions. Happy calculating! 🙂

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***Bonus! Want to test yourself with a similar NYS Regents question on Recursive Formulas?  Click here!

Geometry: Reflections

Greetings and welcome to Mathsux! Today we are going to go over reflections, one of the many types of transformations that come up in geometry.  And thankfully, it is one of the easiest transformation types to master, especially if you’re more of a visual learner/artistic type person. So let’s get to it!

What are Reflections?

Reflections on a coordinate plane are exactly what you think! When a point, a line segment, or a shape is reflected over a line it creates a mirror image.  Think the wings of a butterfly, a page being folded in half, or anywhere else where there is perfect symmetry.

Check out the Example below:

Screen Shot 2020-08-04 at 5.19.40 PM

Screen Shot 2020-08-04 at 4.57.07 PMScreen Shot 2020-08-04 at 4.57.34 PM.pngScreen Shot 2020-08-04 at 4.57.55 PMScreen Shot 2020-08-04 at 4.58.10 PM.pngScreen Shot 2020-08-04 at 4.59.19 PMScreen Shot 2020-08-04 at 4.59.36 PM.pngScreen Shot 2020-08-05 at 9.16.37 AM.png
Screen Shot 2020-08-04 at 5.00.19 PM.pngScreen Shot 2020-08-04 at 5.00.43 PMScreen Shot 2020-08-04 at 5.01.02 PM.png

Practice Questions:

Screen Shot 2020-08-04 at 5.13.52 PM

Solutions:

Screen Shot 2020-08-04 at 5.15.37 PM.png

Still got questions?  No problem! Check out the video above or comment below! Happy calculating! 🙂

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Algebra: Piecewise Function Review

Greetings, today’s post is for those in need of a piecewise function review!  This will cover how to graph each part of that oh so intimidating piecewise function.  There’s x’s, there are commas, there are inequalities, oh my! We’ll figure out what’s going on here and graph each part of the piecewise-function one step at a time.  Then check yourself with the practice questions at the end of this post. Happy calculating! 🙂

 

Wait, what are Piece-Wise Functions? Exactly what they sound like! A function that has multiple pieces or parts of a function.  Notice our function below has different pieces/parts to it.  There are different lines within, each with their own domain.Screen Shot 2020-07-21 at 10.01.59 AM

Now let’s look again at how to solve our example, solving step by step:

Screen Shot 2020-07-21 at 10.02.29 AM.pngScreen Shot 2020-07-21 at 10.02.41 AMScreen Shot 2020-07-21 at 10.03.06 AM.png

Translation: We are going to graph the line f(x)=x+1 for the domain where x > 0

To make sure all our x-values are greater than or equal to zero, we create a table plugging in x-values greater than or equal to zero into the first part of our function, x+1.  Then plot the coordinate points x and y on our graph.

Screen Shot 2020-07-21 at 10.04.33 AM

Screen Shot 2020-07-21 at 10.05.00 AM.png

 

Screen Shot 2020-07-21 at 10.06.46 AM

Translation: We are going to graph the line  f(x)=x-3 for the domain where x < 0

To make sure all our x-values are less than zero, let’s create a table plugging in negative x-values values leading up to zero into the second part of our function, x-3.  Then plot the coordinate points x and y on our graph.

Screen Shot 2020-07-21 at 10.07.33 AM.png

Screen Shot 2020-07-21 at 10.07.57 AM

Ready to try the practice problems below on your own!?

Practice Questions: Graph each piecewise function:

Screen Shot 2020-07-21 at 10.08.32 AM.png

 

 

 

 

 

 

 

 

Solutions:

Screen Shot 2020-07-21 at 10.09.20 AM

Screen Shot 2020-07-21 at 10.09.58 AM.png

Still got questions?  No problem! Check out the video above or comment below for any questions. Happy calculating! 🙂

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***Bonus! Want to test yourself with a similar NYS Regents question on piecewise functions?  Click here.