Weekly Math Problem! #21

Integration. So, here we are again with another integration problem. If you've done last week's problem, then you've already done some of the work needed to do this week's problem. 🤓 Wait, you haven't seen last week's problem?! Pause....check out WMP! #20. 

Last week I mentioned that partial fraction decomposition is an important technique that allows you to "simplify" some complicated rational expressions by breaking them down into smaller parts. WMP! #21 is such a problem that requires the technique of partial fraction decomposition.

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️ How to integrate rational expressions

        ✔️ Partial fraction decomposition

WMP! #21 wants us to...

Happy solving!

Check back on Friday, September 25th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

Again, if you take a look at WMP! #20, you will see that it has the same rational expression as this week's problem. So, when you're given a rational expression, similar to that one, to integrate, you can see what needs to be done to carry out the integration. Here's how I carried it out:

That's all I've got! (I actually really like this problem.)  

 

▪️ How did you feel about this week's problem?? 

▪️ Did you carry out the integration differently than I did??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

See you right back here next week for WMP! #22, the last problem of September.

🤓

Cheers!

The Younge Lady

Weekly Math Problem! #15

Quadrilaterals. Listen, geometry is not necessarily a favorite topic in mathematics for a lot of people. So, now you're thinking, "What about you, Lori? How do you feel about geometry?"  Well, it's not my favorite either 😅; I'm not in love with geometry. 🤷🏿 I will say, I am very visual, so I love the fact images, and the need to draw pictures, are a crucial part of geometry. I have lots of respect for this topic. Geometry is everywhere.

This week's problem was taken from the last administered NYS Regents High School Examination in Geometry before quarantine--January 2020, Part III, question 32. To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Plotting points on the coordinate plane

    ✔️  Slope, distance, midpoint formulas

    ✔️  Properties and theorems for quadrilaterals

    

WMP! #15 says...

Happy solving! 

Check back on Friday, August 14th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

In high school I learned the two-column proof method for geometry proofs. The proofs I remember doing the most were triangle proofs. I rarely, if ever, used the paragraph-style for writing geometric proofs. So, for this problem I'll complete the necessary computations and accompany them with statements that support those computations. 

There are various way to prove that a quadrilateral is a rhombus. This is because there are various ways to prove that a quadrilateral is a parallelogram, of which a rhombus is a special case. I will prove that quadrilateral NATS is a rhombus with the following method:

            I.   Prove that quadrilateral NATS is a parallelogram

            II.  Prove that the parallelogram is a rhombus.

 

But first, here is an image of quadrilateral NATS (the rhombus in question):

**This plot was created using GeoGebra's Geometry App.

I. Prove that quadrilateral NATS is a parallelogram. I will do this by showing that the diagonals of NATS bisect each other. The midpoint formula was used.

Diagonal NT and diagonal AS share a midpoint. NM = TM and AM = SM. Therefore, diagonals NT and AS bisect each other. Therefore, quadrilateral NATS is a parallelogram. ✔️

II. Prove that parallelogram NATS is a rhombus. I will do this by showing that the diagonals are perpendicular. 

The slopes of diagonal NT and diagonal AS are negative reciprocals of one another. Therefore, diagonal NT and diagonal AS are perpendicular. Therefore, parallelogram NATS is a rhombus. ∎

It's been quite a while since I've done one of these. This has me thinking 🤔 back to when I took my high school math Regents. I remember doing logic proofs. Maybe I'll explore that topic in a future WMP!

◾️ How do you feel about geometry?? 

◾️ What method did you use to complete the proof?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Without further ado, on to WMP! #16 

Cheers!

The Younge Lady

Weekly Math Problem! #14

Matrix Operations. Ahh...matrices. Just when you thought basic math operations couldn't get more complicated...enter numbers in block form. 😅🤦🏿‍♀️ In my years of doing mathematics, I see that every area of mathematics has practical applications. The truth is, most people won't learn math at the levels where such applications exist. Also for the people who do go on learn math at those levels where these applications exist, accompanying software is also learned. Doing that type of math by hand just isn't efficient. Linear Algebra has some cool applications but, before you can learn the math needed for such applications, you have to start with the basics.  

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  How to perform basic matrix operations

    

Here is WMP! #14...

Happy solving! 

Check back on Friday, August 7th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

In colored fashion, here's how the matrix calculations go:

Matrix multiplication is definitely tricky but, with a keen eye, a pattern can be seen. Yeah...don't have much to say this week; it's been a hectic one. In an upcoming WMP, I'll do a problem from the math journey I took this summer. 👀 Wondering what it is? You should stick around to find out. 😃

◾️ Have you ever done linear algebra or learned about matrix theory before?? 

◾️ If so, how long ago and what was your experience like?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Next up...WMP! #15 😏

Cheers!

The Younge Lady

Weekly Math Problem! #12

Solving A STATISTICS Word Problem. One of my favorite sections of math is stats! 😍 I really like it because it was the first time I really noticed math that I learned in school all around me--in advertisements, commercials, in education, in finance, in the medical field, pharmaceuticals, etc. This is one area in mathematics that I want to be so much more comfortable with and better at. I have a few books that I've purchased over the years from the time I first started learning and understanding stats in undergrad until now. Two of those titles include The Humongous Book of Statistics Problems by W. M. Kelley and R. A. Donnelly Jr., and Statistics in a Nutshell, Second Edition by Sarah Boslaugh. Of course there are also electronic textbooks and stats PDF files in my possession. **Does anyone else scour the internet for PDF files? 👀 Asking for a friend. 😅** Also, because I tutor statistics, I make sure to have resources on hand to refer to, so that the people I work with get the best help they can get from me. I have also purchased and read Numbers Rule Your World by Kaiser Fung. I need to read this book again. I enjoyed it the first time I read it and definitely wouldn't mind reading it again.

Even though I really like statistics, word problems can definitely be annoying. If you're like me, you need to read word problems at least three times before feeling comfortable enough to start solving. 

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Working with normally distributed data

    ✔️  Working with "small" sample sizes

    ✔️  How to conduct a t-test

    ✔️  Hypothesis testing 

    

Here is WMP! #12...

Happy solving! 

Check back on Friday, July 24th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

I'm working with some students this summer and after their review of pertinent information concerning methods to solve specific types of word problems, they freely expressed how much word problems gave them a problem. I totally get it. Word problems definitely make me sigh quite a few times while reading them. So, I decided to color-code this week's word problem with the information and formula used to come to the appropriate conclusion.

**The critical value above came from this t-table.

GeoGebra Classic  has a cool probability tool that shows you the graph of some discrete and continuous probability distributions. It also generates some statistical results of a few different types of statistical tests that are typical when learning statistics. (I encourage you to check it out.) Here are the graph and results of the t-distribution with information from the word problem:

◾️ How do you feel about statistics?? 

◾️ Were there any differences between how you solved the word problem and what I presented above?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

See you next week for...WMP! #13 👩🏿‍🏫

Cheers!

The Younge Lady

Weekly Math Problem! #11

Conic Sections. Graphing is cool. I like pictures a lot; I'm very visual. Conic sections are cool. I just thought I'd throw this problem in the mix, so I could do some more graphing.  

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Completing the square method

    ✔️  Factoring polynomials

    ✔️  Other algebra techniques 🤷🏿

    

I present to you WMP! #11...

Happy solving! 

Check back on Friday, July 17th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

I previously used the completing the square method in WMP! #7 which required me to solve a quadratic equation. Here's how the completing the square method looks for the curve from this week's problem:

L👀ks like we have a...🥁drum roll please🥁...HYPERBOLA! 🎉 Now that we know we have a HYPERBOLA, we can find the center, vertices, and asymptotes. I color coordinated the important information from the standard form with the components associated with a HYPERBOLA.

Here is the graph of the HYPERBOLA with the center, vertices, and asymptotes identified:

*This graph was generated on Geogebra.org.*

😄 I had fun with this one. 😄

◾️ Have you ever studied conic sections?? 

◾️ If so, what was your experience like?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

See you next week for...WMP! #12 📚

Cheers!

The Younge Lady

Weekly Math Problem! #10

Solving An Exponential Equation. Exponential equations are very different from polynomial equations (linear, quadratic, etc.) not just in the way they look, but also in behavior and methods needed to solve them. Exponential equations sometimes require a different type of "manipulation" via logarithmic properties, as opposed to what is needed to solve polynomial equation types, which do not require that. 

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  Logarithmic properties

    ✔️  How to solve polynomial equations

    

I present to you WMP! #10...

Happy solving! 

Check back on Friday, July 10th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

Listen here...I looked at this problem an automatically assumed that all I needed to do was, somehow, get the exponential term in one spot--factoring it out--then I'll be good. Nah fam, not at all. 😕 When I tell you that wasn't it....that was not it. I needed another technique and wasn't quite seeing it. I was stumped for a bit. What I needed wasn't apparent to me because I was not used to using this technique to solve a logarithmic equation of this form. Factoring is what I needed to do...just not in the way I originally thought. Here's what I did:

I had to solve this thing by factoring a trinomial. It took me too long to see it. Once I saw it, it was solved in minutes. Even though, I didn't post it this week, I did check both solutions. None of them are to be rejected. 

◾️ Have you ever solved a logarithmic equation like the one above??

◾️ Comment below with your responses and let me know what you thought about this week's problem.

On to WMP! #11 ▶️

Cheers!

The Younge Lady

Weekly Math Problem! #7

Solving A Quadratic Equation. This week's problem is related to WMP! #6. We're working with another quadratic equation, except this time we don't have to come up with it. So, no word problem here. What does happen sometimes, which can be highly annoying, is being forced to solve a problem using a specific method when you have other options that you'd prefer to use. You can definitely see this in textbook exercises across various types/levels of mathematics. Lots of professors do this. I am guilty as charged, myself. At least when I've done it, it wasn't to torture my students...I promise. 👀 It was really to expose them and push their abilities, so that they wouldn't be caught by surprise. 😁

To solve this week's problem in completion, you need to recall the following math skills:

    ✔️  How to solve rational equations

    ✔️  How to solve quadratic equations

    ✔️  Completing the square method

Ready? If not, here goes WMP! #7 anyway...

Happy solving! 

Check back on Friday, June 19th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

⬇️

⬇️

⬇️

⬇️

⬇️

I don't really have much to say. So...without further ado, here's my solution:

Here is a graph of the quadratic with the solution (roots) identified:

*This graph was generated on Geogebra.org.*

◾️ How do you feel about this method for solving quadratics?? Love it? Hate it?

◾️ What's your go-to method for solving quadratic equations?? 

◾️ Comment below with your responses and let me know what you thought about this week's problem.

Up next...WMP! #8 😏

Cheers!

The Younge Lady