Teacher Archives - Page 4 of 5

How to Construct an Equilateral Triangle?: Geometry

Happy Wednesday math peeps! This post introduces constructions by showing us how to construct an equilateral triangle by using a compass and straightedge (or ruler). For anyone new to constructions in geometry, this is the perfect topic for art aficionados since there is more drawing here than there is actual math!

What is an Equilateral Triangle?

Equilateral Triangle: An equilateral triangle is 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+ straightedge). Take a look below:

equilateral triangle
equal sides — Equilateral Triangle: Triangle with all equal sides

Construct an Equilateral Triangle Example:

Check out the GIF below to see how to construct an equilateral triangle step by step using a compass and straightedge with pictures and explanations below!

Solution:

How to Construct an Equilateral Triangle b c

What’s Happening in this GIF?

1. Using a compass, we measure the distance of line segment .

2. With the compass point remaining on point A, we then draw an arc that has the same distance as line segment .

3. With the compass now placed on point B, draw an arc that has the same distance as line segment .

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

5. We have now officially constructed a triangle with all equal side lengths!

Constructions and Related Posts:

Looking to construct more than just an equilateral triangle? Check out these related posts on geometry constructions!

Construct a Perpendicular Bisector

Perpendicular Line through a Point

Angle Bisector

Construct a 45º angle

Altitudes of a Triangle (Acute, Obtuse, Right)

Construct a Square inscribed in a Circle

Best Geometry Tools!

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!

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

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Looking to learn more about triangles? Check out this post on right triangle trigonometry here! And if you’re looking for even more geometry constructions, check out the link here!

Expanding Cubed Binomials: Algebra 2/Trig.

Greetings math friends! This post will go over expanding cubed binomials using two different methods to get the same answer. We’re so used to seeing squared binomials such as, , and expanding them without a second thought. But what happens when our reliable squared binomials are now raised to the third power, such as,? Luckily for us, there is a Rule we can use:

But where did this rule come from? And how can we so blindly trust it? In this post we will prove why the above rule works for expanding cubed binomials using 2 different methods:

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 Method

Step 1: First, focus on the left side of the equation by expanding (a+b)³:

Step 2: Now we are going to create our first box, multiplying (a+b)(a+b). Notice we put each term of (a+b) on either side of the box. Then multiplied each term where they meet.

Step 3: Combine like terms ab and ab, then add each term together to get a²+2ab+b².

Step 4: Multiply (a²+2ab+b²)(a+b) making a bigger box to include each term.

Step 5: Now combine like terms (2a²b and a²b) and (2ab² and ab²), then add each term together and get our answer: a³+3a²b+3ab²+b³.

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

Method #2: The Distribution Method

Screen Shot 2020-08-18 at 10.17.54 PM.png

Let’s expand the cubed binomial using the distribution method step by step below:

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.

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!

Recursive Rule

Welcome to Mathsux! This post is going to show you everything you need to know about how to use a Recursive Formula by looking at three different examples of a recursive rule. Check out the video below for more of an explanation and test your skills with the practice questions at the bottom of this page. Please let me know if you have any questions in the comments section below and happy calculating! 🙂

What is a Recursive Formula or a Recursive Rule?

A Recursive Formula is a type of formula that forms a sequence based on the previous term value. The recursive rules for each formula vary, but we are always given the first term and a formula to find the consecutive terms in the recursive sequence.

Recursive formula can be written as an arithmetic sequence (a sequence where the same number is either added or subtracted to each subsequent term to form a pattern) and recursive formulas can also be written as arithmetic sequences (a sequence where the same number is either multiplied or divided to each subsequent term to form a pattern). We’ll go over an example of each but both types of recursive rules are treated the same exact way!

What does all of this mean? Check out the example below for a clearer picture.

Example #1: Arithmetic Recursive Sequence

Step 1: First, let’s decode what these formulas are saying.

Step 2: The first term, represented by a₁, is and will always be given to us. In this case, our first term has the value a₁=2 and represents the first term of our recursive sequence.

a₁= First Term=2

Step 3: We then plug in the value of our first term, which is a₁=2 into our formula a_n+4 to get 2+4=6. The number 6 now has the value of our second term in the recursive sequence.

a₁= 2 First Term

a₂= (2)+4=6 Second Term

Step 4: Now we are going to continue the pattern, plugging in the value of each previous term to find the next consecutive terms in our recursive sequence.

The pattern can be more easily seen below. Notice we are able to find the value of all 5 terms of the recursive sequence for the solution only given the first term and recursive formula at the beginning of our question.

Step 5: We found the recursive sequence we were looking for: 2, 6, 10, 14, 18. Since the question was originally only asking for the value of the fifth term we know our solution only needs to be the value of the fifth term which is 18.

Example #2: Geometric Recursive Sequence

Step 1: First, let’s decode what these formulas are saying.

Step 2: The first term, represented by a₁, is and will always be given to us. In this case, our first term has the value a₁=1 and represents the first term of our recursive sequence.

a₁= First Term=1

Step 3: We then plug in the value of our first term, which is a₁=1 into our formula 2^a_n+1 to get 2¹+1=3. The number 3 now has the value of our second term in the recursive sequence.

a₁= 1 First Term

a₂= 2⁽¹⁾+1=3 Second Term

Step 4: Now we are going to continue the pattern, plugging in the value of each previous term to find the next term in our recursive sequence.

The pattern can be more easily seen below. Notice we are able to find the value of all 3 terms of the recursive sequence for the solution only given the first term and recursive formula at the beginning of our question.

***Note this was written in a different notation but is solved in the exact same way! This recursive formula is a geometric sequence.

Step 5: We found the recursive sequence we were looking for: 1,3,9. Since the question was originally only asking for the value of the third term we know our solution only needs to be the value of the third term which is 9.

Example #3:

Step 1: First, let’s decode what these formulas are saying.

Step 2: The first term, represented by a₁, is and will always be given to us. In this case, our first term has the value a₁=4 and represents the first term of our recursive sequence.

a₁= First Term=4

Step 3: We then plug in the value of our first term, which is a₁=4 into our formula 3a_n-1-2 to get 3(2)-1=5. The number 5 now has the value of our second term in the recursive sequence.

a₁= 4 First Term

a₂= 3(2)-1=5 Second Term

Step 4: Now we are going to continue the pattern, plugging in the value of each previous term to find the next term in our recursive sequence.

Step 5: We found the recursive sequence we were looking for: 4,10,28. Since the question was originally only asking for the value of the third term we know our solution only needs to be the value of the third term which is 28.

Think you are ready to solve a recursive equation on your own?! Try finding the specific term in each given recursive function below:

Practice Questions:

Solutions:

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 arithmetic sequence or geometric sequence posts next!

Arithmetic Sequence

Geometric Sequence

Finite Arithmetic Series

Finite Geometric Series

Infinite Geometric Series

Golden Ratio in the Real World

Fibonacci Sequence

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!

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***Bonus! Want to test yourself with a similar NYS Regents question on Recursive Formulas? Click here! And if you want to learn about even more sequences, check out the link here!

Intersecting Secants Theorem: Geometry

Ahoy! Today we’re going to cover the Intersecting Secants Theorem! If you forgot what a secant is in the first place, don’t worry because all it is a line that goes through a circle. Not so scary right? I was never scared of lines that go through circles before, no reason to start now.

If you have any questions about anything here, don’t hesitate to comment below and check out my video for more of an explanation. Stay positive math peeps and happy calculating! 🙂

Wait, what are Secants?

Intersecting Secants Theorem: When secants intersect an amazing thing happens! Their line segments are in proportion, meaning we can use something called the Intersecting Secants Theorem to find missing line segments. Check it out below:

Let’s now see how we can apply the intersecting Secants Theorem to find missing length.

Screen Shot 2020-07-14 at 10.45.29 PM.png

Step 1: First, let’s write our formula for Intersecting Secants.

Step 2: Now fill in our formulas with the given values and simplify.

Step 3: All we have to do now is solve for x! I use the product.sum method here, but choose the factoring method that best works for you!

Step 4: Since we have to reject one of our answers, that leaves us with our one and only solution x=2.

Screen Shot 2020-07-14 at 10.14.41 PM.png

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

Practice Questions: Find the value of the missing line segments x.

Solutions:

Screen Shot 2020-07-20 at 9.30.55 AM.png

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

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To review a similar NYS Regents question check out this post here.

Circle Theorems & Formulas

Central Angle Theorem

Inscribed Angles & Intercepted Arcs

Area of a Sector

Circle Theorems

Absolute Value Equations: Algebra

Happy Wednesday math friends! Today, we’re going to go over how to solve absolute value equations. Solving for absolute value equations supplies us with the magic of two potential answers since absolute value is measured by the distance from zero. And if this sounds confusing, fear not, because everything is explained below!

Also, if you have any questions about anything here, don’t hesitate to comment. Happy calculating! 🙂

Absolute Value measures the “absolute value” or absolute distance from zero. For example, the absolute value of 4 is 4 and the absolute value of -4 is also 4. Take a look at the number line below for a clearer picture:

Absolute Value

Now let’s see how we can apply our knowledge of absolute value equations when there is a missing variable! Absolute Value Equations example

Absolute Value Equations Now let’s look at a slightly different example:

Absolute Value Equations example Screen Shot 2020-07-08 at 2.07.59 PM

Absolute Value Equations

Screen Shot 2020-07-08 at 2.08.26 PM.png

Absolute Value Equations

Practice Questions: Given the following right triangles, find the missing lengths and side angles rounding to the nearest whole number.

Absolute Value Equations examples

Solutions:

Absolute Value Equations solutions

Still got questions? No problem! Check out the video the same examples outlined above. Happy calculating! 🙂

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Also, if you’re looking for a review on combining like terms and the distributive property, check out this post here.

How to Solve Log Equations: Algebra 2/Trig.

Welcome to Mathsux! Today, we’re going to go over how to solve log equations, yay! But before we get into finding x, though, we need to go over what log equations are and why we use them in the first place…..just in case you were curious!

Also, if you have any questions about anything here, don’t hesitate to comment below or shoot me an email. Happy calculating! 🙂

What are Log Equations?

Logarithms are the inverses of exponential functions. This means that when graphed, they are symmetrical along the line y=x. Check it out below!

When on the same set of axis, notice how the functions are symmetrical over the line y=x:

We use logarithms to find the unknown values of exponents, such as the x value in the equation,. This is a simple example, where we know the value of x is equal to 2,(). But what if it were to get more complicated? That’s where logs come in!

How to Solve Log Equations?

Logarithms follow a swooping pattern that allows us to write it in exponential form, let’s take a look at some Examples below:

But wait there’s more! Logs have a set of Rules that makes solving log equations a breeze!

We can use these rules to help us algebraically solve logarithmic equations, let’s look at an example that applies the Product Rule.

Example:

Screen Shot 2020-06-24 at 9.36.08 PM.png

Screen Shot 2020-06-24 at 9.46.07 PM.png

Try the following practice questions on your own!

Practice Questions:

Screen Shot 2020-06-24 at 9.39.16 PM.png

Solutions:

Still got questions? No problem! Check out the video that goes over the same example outlined above. And for more info. on logarithms check out this post that goes over a NYS Regent’s question here. Subscribe below to get the latest FREE math videos, lessons, and practice questions from MathSux. Thanks for stopping by and happy calculating! 🙂

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****Check out this Bonus Video on How to Change Log Bases****

How to Graph Equation of a Line, y=mx+b: Algebra

Hi everyone, welcome back to Mathsux! This week we’ll be reviewing how to graph an equation of a line in y=mx+b form. And if you have not checked out the video below, please do! Happy calculating! 🙂

An equation of a line can be represented by the following formula:

y=mx+b

Y-Intercept: This is represented by b, the stand-alone number in y=mx+b. This represents where the line hits the y-axis. This is always the first point you want to start with when graphing at coordinate point (0,b).

Slope: This is represented by m, the number next to x in y=mx+b. Slope tells us how much we go up or down the y-axis and left or right on the x- axis in fraction form:

Now let’s check out an Example!

Graph the following:

-> First, let’s identify the slope and y-intercept of our line.

-> To start, let’s graph the first point on our graph, the y-intercept at point (0,1):

-> Now for the slope. We are going to go up one and over to the right one for each point, since our slope is 1/1.

-> Connect all of our coordinate points and label our graph.

Try the following practice questions on your own!

Practice Questions:

Want more Mathsux? Don’t forget to check out our Youtube channel and more below! And if you have any questions, please don’t hesitate to comment below. Happy Calculating! 🙂

Need to brush up on slope? Click here to see how to find the rate of change.

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Perpendicular & Parallel Lines Through a Given Point: Geometry

Happy Wednesday math friends! Today we’re going to go over the difference between perpendicular and parallel lines, then we’ll use our knowledge of the equation of a line (y=mx+b) to see how to find perpendicular and parallel lines through a given point. This is a common question that comes up on the NYS Geometry Regents and is something we should prepare for, so let’s go!

If you need any further explanation, don’t hesitate to check out the Youtube video below that goes into detail on how to find perpendicular and parallel lines through a given point one step at a time. Happy calculating! 🙂

Perpendicular Lines:

Perpendicular & Parallel Lines Through a Given Point

Perpendicular Lines: Lines that intersect to create a 90-degree angle and can look something like the graph below. Their slopes are negative reciprocals of each other which means they are flipped and negated. See below for example!

Example: Find an equation of a line that passes through the point (1,3) and is perpendicular to line y=2x+1 .

Parallel Lines:

Parallel lines are lines that go in the same direction and have the same slope (but have different y-intercepts). Check out the example below!

Example: Find an equation of a line that goes through the point (-5,1) and is parallel to line y=4x+2.

Try the following practice questions on your own!

Practice Questions:

1) Find an equation of a line that passes through the point (2,5) and is perpendicular to line y=2x+1.

2) Find an equation of a line that goes through the point (-2,4) and is perpendicular to line

3) Find an equation of a line that goes through the point (1,6) and is parallel to line y=3x+2.

4) Find an equation of a line that goes through the point (-2,-2) and is parallel to line y=2x+1.

Solutions:

Need more of an explanation? Check out the video that goes over these types of questions up on Youtube (video at top of post) and let me know if you have still any questions.

Happy Calculating! 🙂

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Looking for more on Perpendicular and parallel lines? Check out this Regents question on perpendicular lines here!

4 Ways to Factor Trinomials: Algebra

Greeting math peeps and welcome to MathSux! In this post, we are going to go over 4 ways to Factor Trinomials and get the same answer, including, (1) Quadratic Formula (2) Product/Sum, (3) Completing the Square, and (4) Graphing on a Calculator. If you’re looking for more don’t forget to check out the video and practice questions below. Happy Calculating! 🙂

Also, if need a review on Factor by Grouping or Difference of Two Squares (DOTS) check out the hyperlinks here!

*If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

We will take this step by step, showing 4 ways to factor trinomials, getting the same answer each and every time! Let’s get to it!

4 Ways to Factor Trinomials

(1) Quadratic Formula:

____________________________________________________________________

(2) Product/Sum:

4 Ways to Factor Trinomials ____________________________________________________________________

(3) Completing the Square:

4 Ways to Factor Trinomials ____________________________________________________________________

(4) Graph:

Choose the factoring method that works best for you and try the practice problems on your own below!

Practice Questions:

Solutions:

Want a review of all the different factoring methods out there? Check out the ones left out here (DOTS and GCF) and happy calculating! 🙂

For even more ways to factor quadratic equations, check out How to factor by Grouping here! 🙂

Looking for more on Quadratic Equations and functions? Check out the following Related posts!

Factoring Review

Factor by Grouping

Completing the Square

The Discriminant

Is it a Function?

Imaginary and Complex Numbers

Quadratic Equations with 2 Imaginary Solutions

Focus and Directrix of a Parabola

Also, if you want more Mathsux? Don’t forget to check out our Youtube channel and more below! If you have any questions, please don’t hesitate to comment below. Happy Calculating! 🙂

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Median of a Trapezoid Theorem: Geometry

Hi everyone and welcome to Math Sux! In this post, we are going to look at how to use and applythe median of a trapezoid theorem. Thankfully, it is not a scary formula, and one we can easily master with a dose of algebra. The only hard part remaining, is remembering this thing! Take a look below to see a step by step tutorial on how to use the median of a trapezoid theorem and check out the practice questions at the end of this post to truly master the topic. Happy calculating! 🙂

*If you haven’t done so, check out the video that goes over this exact problem, also please don’t forget to subscribe!

Step 1: Let’s apply the Median of a Trapezoid Theorem to this question! A little rusty? No problem, check out the Theorem below.

Median of a Trapezoid Theorem: The median of a trapezoid is equal to the sum of both bases.Step 2: Now that we found the value of x , we can plug it back into the equation for median, to find the value of median

Want more practice? Your wish is my command! Check out the practice problems below:

Practice Questions:

1.is the median of trapezoid ABCDEF, find the value of the median, given the following:2. is the median of trapezoid ACTIVE, find the value of the median, given the following:3.is the median of trapezoid DRAGON, find the value of the median, given the following: