How to find the Area of a Parallelogram: Geometry

Hey math peeps! In today’s post, we are going to go over how to find the area of a parallelogram. There is an easy formula to remember, A=bh, but we are going to look at why this formula works in the first place and then solve a few examples. Just a quick warning: The following examples do use special triangles and if you are need of a review, check out the posts here for 45 45 90 and 30 60 90 special triangles. Also, don’t forget to watch the video and try the practice problems below. Thanks so much for stopping by and happy calculating!

Area of a Parallelogram Formula:

How to find the Area of a Parallelogram

Why does the Formula for Area of a Parallelogram work?

Did you notice that the formula for area of a parallelogram above, base times height, is the same as the area formula for a rectangle?  Why?

If we cut off the triangle that naturally forms along the dotted line of our parallelogram, rotated it, and placed it on the other side of our parallelogram, it would naturally fit like a puzzle piece and create a rectangle! Check it out below:

How to find the Area of a Parallelogram

Now that we know where this formula comes from, let’s see it in action in the examples below:

Example #1:

How to find the Area of a Parallelogram

Step 1: Write out the formula:

Step 2:  Fill in the formula with values found on our parallelogram, b=12 inches h=4 inches, and multiply them together to get 48 inches squared.

How to find the Area of a Parallelogram

That was a simple example, but lets try a harder one that involves special triangles.

Example #2:

How to find the Area of a Parallelogram

Step 1: Write out the formula:

Step 2:  Label the values found on our parallelogram, b=10 ft and notice that we are going to need to find the value of the height.

Step 3: In order to find the value of the height, we need to remember our special triangles! We are not given the value of the height, but we are given some value of the triangle that is formed by the dotted line.  Let us take a closer look and expand this triangle:

Step 4: We can add in the missing 45º degree value so that our triangle now sums to 180º.

Step 5: Remember 45 45 90 special triangles(If you need a review click the link). Because that is exactly what we are going to need to find the value of the height! Below is our triangle on the left, and on the right is the 45 45 90 triangle ratios we need to know to find the value of the height.

Based on the above ratios, we can figure out that the height value is the same value as the base of the triangle, 2.

Step 6: If we place our triangle back into the original parallelogram, we can plug in our value for the height, h=2, into our formula to find the area:

How to find the Area of a Parallelogram

When you’re ready, check out the practice questions below!

Practice Questions:

Find the area of each parallelogram:

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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Quadratic Equations with Two Imaginary Solutions

Hi everyone and welcome to MathSux! In today’s post we are going to be solving quadratic equations by using the quadratic formula. You may have used the quadratic formula before, but this time we are working with quadratic equations with two imaginary solutions. All this means is that there are negative numbers under the radical that have to be converted into imaginary numbers. If you need a review on imaginary numbers or the quadratic formula before reading this post, check out these links! Thanks so much for stopping by and happy calculating! 🙂

What is the Quadratic Formula?

The Quadratic formula is a formula we use to find the x-values of a quadratic equation. When we find the x-value of a quadratic equation, we are actually finding its x-values on the coordinate plane. Check out the formula below:

where, a, b, and c are coefficients based on the quadratic equation in standard form:

What does it mean to have “Imaginary Roots”?

When we solve for the x-values of a quadratic equation, we are always looking for where the equation “hits” the x-axis. But when we have imaginary numbers as roots, the quadratic equation in question, never actually hit the x-axis. Ever. This creates a sort of “floating” quadratic equation with complex numbers as roots. See what it can look like below:

Quadratic Equations with Two Imaginary Solutions

Ready for an Example?  Let us see how to use the quadratic formula specifically, quadratic equations with two imaginary solutions:

Quadratic Equations with Two Imaginary Solutions
Quadratic Equations with Two Imaginary Solutions
Quadratic Equations with Two Imaginary Solutions

Think you are ready to try practice questions on your own? Check out the ones below!

Practice Questions:

Quadratic Equations with Two Imaginary Solutions

Solutions:

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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Factor By Grouping Examples: Algebra

Hey there math peeps and welcome to MathSux! In today’s post we are going to cover factor by grouping examples, a surprisingly cool and easy factoring method used to factor quadratic equations when “a” is greater than one. It can also be used to factor four term polynomials. We are going to look at an example of each below. If you have any questions, please don’t hesitate to check out the video and try the practice problems at the end of this post. Thanks for stopping by and happy calculating! 🙂

What is Factor by Grouping?

Factor by Grouping is a factoring method that groups common factors of an algebraic expression together.  Many times, we use factoring to find the x-values of a quadratic equation when the coefficient “a” is greater than 1.

When should we use Factor by Grouping?

1) If the first coefficient in a quadratic equation, a, is greater than 1:

Factor By Grouping examples

2) When there is a polynomial with 4 terms:

Factor by Grouping Examples:

Ready for an Example?  Let us look at how to factor a quadratic equation when a is greater than one.

Factor By Grouping examples
Factor By Grouping examples

Now, let’s take a look at another type of example, that can be solved with the help of factor by grouping!

Notice that this question is actually easier to solve than the last! The polynomial above, is already split into 4 terms, therefore, we can jump ahead, skipping the product/sum steps we did in the previous example!

Factor By Grouping examples

Ready to try practice questions on your own? Check them out below to master Factor by Grouping!

Practice Questions:

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 to learn about different Factoring methods? Check out the posts below!

GCF, Product/Sum, Difference of Two Squares, Quadratic Formula

Quadratic Formula, Product/Sum, Completing the Square, Graph

Completing the Square

Free High School Math Resources

Free High School Math Resources

Greetings math friends! Today’s post is for the New York state teachers out there in need of a lesson boost. In this post, we’ll go over what MathSux has to offer for free high school math resources including videos, lessons, practice questions, etc. Remember everything you see here is 100% free and designed to make your life (and your students’ life) easier.

Signing up with MathSux will get you access to FREE:

1) Math Videos

2) Math Lessons

3) Practice Questions

4) NYS Regents Review

Everything designed here aligns with the NYS Common Core Standards for Algebra, Geometry, Algebra 2/Trig. and Statistics.

I am a NYS math teacher that creates free math videos, lessons, and practice questions every week, right here, for you! On the YouTube channel, you’ll also find NYS Common Core Regents questions reviewed one question at a time.

Featured on Google Classrooms around the world, MathSux.org is a great resource especially now, in the time of COVID and zooming and schooling from home. I hope you stick around and find these resources helpful.

And if you’re looking to get the latest MathSux.org videos and emails straight to your inbox, don’t forget to sign up on the right hand-side of the website. Thanks so much for stopping by and happy calculating! 🙂

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Difference between Bar Graphs and Histograms

Hi everyone and welcome to MathSux! Today we will be looking at the difference between Bar Graphs and Histograms. Bar Graphs are something we are already familiar with but now we have histograms, a type of graph that looks somewhat similar. How and why are we using histograms? All questions will be answered in this post! Don’t forget to check your skills with the video and practice questions below. Thanks for stopping by and happy calculating! 🙂

What is the difference between Bar Graphs and Histograms?

Both a bar graph and a histogram allow us to easily visualize a data set that would otherwise just look like a bunch of numbers.  By using a bar graph or histogram, we can more easily look at data, analyze it, and even come to conclusions and make decisions which why they are so useful!  Nothing like a good graph to make cloudy data seem crystal clear!

Below we will look at the same exact data represented in bar graph form and in histogram form.  At a quick glance, what similarities and differences do you notice? Read on for more!

Bar Graph: Simple and to the point, bar graphs measure out the number of times something occurs. Best used for small groups of data, also, note that each bar on the graph has spaces between.

Ex:

Difference between Bar Graphs and Histograms

Histogram:  Numerical data gets grouped together in something called a frequency table.  The data is then graphed based on the number of times something occurs within that interval. Best used for larger groups of data, also note that each interval is represented with bars without spaces in between them.

Ex: *The example below shows the same exact data shown in the bar graph above, but now in the format of a histogram.

Difference between Bar Graphs and Histograms

Now, let’s check out an Example to see how to make a histogram step by step:

Step 1: First, we need to create something called a Frequency Table where we will create intervals of all the test scores, then tally each score given within the data set.

65, 65, 70, 72, 80, 95, 99, 92, 83, 78

Difference between Bar Graphs and Histograms

Step 2: Now we can start creating our graph with the frequency on the y-axis and the test score intervals on the x-axis. Noticed that we also gave the histogram a title.

Difference between Bar Graphs and Histograms

When you’re ready, check out the questions below to practice your histogram skills!

Practice Questions:

1) The science club is measuring the length of different local plants in inches and gather the following data. Create a histogram that represents their findings.

10, 12, 7, 24, 36, 8, 14, 24, 18, 30

2) A class survey is taken to find out how much students earn every week from small jobs and chores. Create a histogram for the following data that was collected:

$0, $0, $0, $15, $20, $50, $47, $25, $30, $52

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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Looking for more? Check out this post on Box and Whisker Plots and the MathSux Statistics page here!

Constructing a Perpendicular Bisector

Hi everyone and welcome to MathSux! In this post we are going to be constructing a perpendicular bisector, a line that cuts a line segment in half and creates four 90º angles. It’s a super fast and super simple construction! If you’re looking for more constructions, don’t forget to check more out here. Thanks so much for stopping by and happy calculating! 🙂

What is the Perpendicular Bisector of a line ?

  • Cut’s our line AB in half at its midpoint, creating two equal halves.
  • This will also create four 90º angles about the line.
Constructing a Perpendicular Bisector

What is happening in this GIF?

Step 1: First, we are going to measure out a little more than halfway across the line AB by using a compass.

Step 2: Next we are going to place the compass on point A and swing above and below line AB to make a half circle.

Step 3: Keeping the same distance on our compass, we are then going to place the point of the compass onto Point B and repeat the same step we did on point A, drawing a semi circle.

Step 4: Notice the intersections above and below line AB!? Now, we want to connect these two points by drawing a line with a ruler or straight edge.

Step 5: Yay! We now have a perpendicular bisector! This cuts line AB right at its midpoint, dividing line AB into two equal halves.  It also creates four 90º angles.

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!

NYS Algebra Regents Review

Hi everyone and welcome to MathSux! In honor of school state tests being back on, I bring to you a playlist that goes over each and every question from the NYS Algebra Regents from January 2020. The entire playlist is 100% free of charge and fear not because there are more Regents review questions and playlists on the way! If you’re looking specifically for NYS Algebra Common Core Regents Review questions check out the links listed at the end of this post. Good luck on the Algebra Regents and happy calculating! 🙂

Algebra Common Core Regents Jan.2020 Playlist:

More NYS Algebra Regents Review:

1) Rate of Change

2) Completing the Square

3) Piecewise Functions

4) Recursive Formulas

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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If you’re looking for a review on all of Algebra, don’t forget to check out the Algebra playlist on Youtube here:

How to Solve Inequalities with 2 Variables: Algebra

Hi everyone and happy Wednesday! Today we are going to look at how to solve inequalities with 2 variables. You may hear this in your class as “Simultaneous Inequalities” or “Systems of Inequalities,” all of these mean the same exact thing! The key to answering these types of questions, is to know how to graph inequalities and to know that the solution is always found where the two shaded regions overlap each other on the graph. We’re going to go over an example one step at time, then there will be practice questions at the end of this post that you can try on your own. Happy calculating! 🙂

How to Solve Inequalities with 2 Variables:

Just to review, when graphing linear inequalities, remember, we always want to treat the inequality as an equation of a line in  form….with a few exceptions:

1)Depending on what type of inequality sign we are graphing, we will use either a dotted line and an open circle (< and >) or a solid line and a closed circle (> or <) and  to correctly represent the solution.

2) Shading is another important feature of graphing inequalities.  Depending on the inequality sign we will need to either shade above the x-axis ( > or > ) or below the x-axis ( < or < ) to correctly represent the solution.

3) Solution: To find the solution of a system of inequalities, we are always going to look for where the shaded regions of both inequalities overlap.

How to Solve Inequalities with 2 Variables

Now that we know the rules, of graphing simultaneous inequalities, let’s take a look at an Example!

How to Solve Inequalities with 2 Variables

Step 1: First, let’s take our first inequality, and get it into y=mx+b form. To do this, we need to move .5x to the other side of the inequality by subtracting it from both sides. Once we do that, we can identify the slope and the y-intercept.

Step 2: Before graphing, let’s now identify what type of inequality we have here.  Since we are working with a < sign, we will need to use a dotted line and open circles when graphing.

Step 3: Now that we have identified all the information we need to, let’s graph the first inequality below:

Step 4: Now it is time for us to shade our graph, since this is an inequality, we need to show all of our potential solutions with shading.  Since we have a less than sign, <, we will be shading below the x-axis.  Notice all the negative y-values below are included to the left of our line.  This is where we will shade.

Step 5: Next, let’s start graphing our second inequality! We do this by taking the second equation, and getting it into y=mx+b form. To do this, we need to move 2x to the other side of the inequality by adding it to both sides. Then we can simplify the inequality even further by dividing out a 2.

Step 6: Before graphing, let’s now identify what type of inequality we have here.  Since we are working with a > sign, we will need to use a solid line and closed circles when creating our graph.

Step 7: Now that we have identified all the information we need to, let’s graph the second inequality below:

Step 8: Now it is time for us to shade our graph.  Since we have a greater than or equal to sign, >, we will be shading above the x-axis.  Notice all the positive y-values above are included to the left of our line.  This is where we will shade.

How to Solve Inequalities with 2 Variables

Where is the solution?!

Step 9: The solution is found where the two shaded regions overlap. In this case, we can see that the two shaded regions overlap in the purple section of this graph.

How to Solve Inequalities with 2 Variables

Step 10: Check!  Now we can finally check our work.  To do that, we can choose any point within our overlapping purple shaded region, if the coordinate point we choose holds true when plugged into both of our inequalities then our graph is correct!

Let’s take the point (-4,-1) and plug it into both original inequalities where x=-4 and y=-1.

Practice Questions:

Solutions:

How to Solve Inequalities with 2 Variables
How to Solve Inequalities with 2 Variables

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 to review graphing linear inequalities Check out this post on here!

Inscribed Angles & Intercepted Arcs: Geometry

Ahoy math friends and welcome to MathSux! In this post, we are going to go over inscribed angles and intercepted arcs. We’ll break down the main basic rule for inscribed angles and the three theorems associated with this rule. If you are looking for more circle theorems, check out these posts on the Intersecting Secants Theorem and Central Angles Theorem. Also, don’t forget to check out the video and practice questions to truly master the topic below. Happy calculating! 🙂

Inscribed Angles:

When two chords come together to touch the outline of a circle, they create something called an inscribed angle. An inscribed angle is equal to half the value of the arc length.

Inscribed Angles & Intercepted Arcs

Inscribed Angle Theorems:

There are three inscribed angle theorems to know based on the rule stated above, check them out below!

Theorem #1: (Intercepted Arcs) In a circle when inscribed angles intercept the same arc, the angles are congruent.

Inscribed Angles & Intercepted Arcs

Theorem #2: In a circle when an angle is inscribed by a semicircle, it forms a  90º angle.

Theorem #3: When a quadrilateral is inscribed in a circle, opposite angles are supplementary (add to 180º). (The proof below shows angles A and C as supplementary, but this proof would also work for opposite angles B and D).

Inscribed Angles & Intercepted Arcs

Let’s look at how to apply these rules with an Example:

a) Step 1: To find the value of angle CDB we need to look at our given information. We know that angle CAB=85º, notice that this follows theorem number 3, “When a quadrilateral is inscribed in a circle, opposite angles are supplementary.” Therefore, we must subtract 110º from 180º to find the value of angle CDB.

b) Step 2: For finding angle ABD, we’re going to use the same theorem we used in part a, opposite supplementary angles of an inscribed quadrilateral are supplementary.

c) Step 3: Next, to find the value of arc ABD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to half the value of its arc. Then use some basic algebra to solve for arc ABD.

d) Step 4: To find arc ACD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to  the value of its arc, then use algebra to solve similar to part c.

If this looks confusing, check out the video above! And when you are ready master this topic with the practice questions below!

Practice Questions:

Solutions:

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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Summation Notation: Algebra 2

Hi everyone and welcome to MathSux! In this post we are going to go over summation notation (aka sigma notation). The summing of a series isn’t hard as long as you know how to read the notation! We will go over an example and breakdown what each part of this notation represents step by step. When you are ready, please don’t forget to check out the practice questions at the end of this post to truly master the topic. Thanks for stopping by and happy calculating! 🙂

What is Summation Notation?

Summation notation lets us write a series in an easy and short-handed way.  Before we go any further we also need to define a series!

Series: The sum of adding each term within an infinite sequence. This can include arithmetic or geometric sequences we are already familiar with. For example, let’s say we have the arithmetic sequence: 2,4,6,8, ….. now with a series we are adding all of these terms together: 2+4+6+8+……

Now back to summations. Summations allow us to quickly understand that the sequence being added together is done so on an infinite or finite basis by giving us a range of values for which the unknown variable can be evaluated and summed together.  Summation notation is represented with the capital Greek letter sigma, Σ, with numbers below and above as limits for calculation and the series that must be evaluated to the right.

If this sounds confusing, don’t worry, it might sound more confusing than it actually is! Take a look at the breakdown for sigma notation below:

Summation Notation

Wait, what does the above summation say?

Translation: It tells us to evaluate the expression, n+1 by plugging in 1 for n, 2 for n, and 3 for n and then wants us to sum all three solutions together.

Take a look below to see how to solve this step by step:

Summation Notation

Check out the video above to see more examples step by step! When you’re ready to try them on your own, check out the practice problems 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! 🙂

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Looking for something similar to sigma notation? Check out this post on geometric sequences here!