Student Guest Blog Post: Drawing Mandalas with Compasses and Protractors! Geometry Art!

Here is the first guest post from my student Morgan! Morgan was in my Honors Problem-Solving Seminar class, and her final project was to learn how to draw a mandala using math! Enjoy! For the rest of the posts, visit: sasproblemsolving.blogspot.com

Hi! My name is Morgan, and because of my love for yoga, I decided to learn how to draw a mandala!

Mandalas represent an imaginary place that one's mind travels too when he or she meditates. Each object one observes in that place has significance, embodying an aspect of wisdom or reminding the meditator of a guiding principle. The mandala's purpose is to help transform ordinary minds into enlightened ones and to assist with healing just as yoga has for me with anxiety. The different movements and yoga positions leave me feeling relaxed and allow me to clear my mind. I felt a similar feeling of relaxation when I drew my mandala.

Having never drawn a mandala before, I had predicted it to be a long, complicated process that would be draining. However, I felt calm and stress-free as I allowed my mind to unravel and draw my mandala. At first, it was a little challenging to know what shapes and lines to draw after I had created the skeleton of the mandala, but by the end, I didn't even have to think much. I went with my intuition and let my mind do what it felt in the moment.

Here is a video of my first mandala drawing, showing the whole process and what goes into creating a mandala.

My first mandala completed

A close up shot of my mandala

Materials needed to draw a mandala:

• Compass
• Protractor 
• Ruler 
• Pencil
• Black pen with felt tip (thin sharpies will do)
• Blank notebook or 8.5 by 11 sheet paper

Procedure:

1. Draw “skeleton” in pencil.
• As I showed in my video, use your compass to draw a small circle in the center of the paper then continue outward drawing circles until you reach near the end of the piece of paper. 
TIP: rotate the paper while using the compass as shown in this video. 


• Once your circles are complete, use your ruler to draw straight lines to divide the circle up into several different sections. 

22.5 cm trick shown above
• In addition, although I didn't show it in my video, you can use a protractor and make markings at every 22.5 cm then use a ruler to draw lines so that your mandala is more precise.
• Also, the circles do not need to be exactly the same amount apart from each other. It actually comes out better if you organically draw some circles close together and and some further apart. 
2. Then begin using your black felt tip pen (or Sharpie) to draw the mandala. 
• Some commons designs and shapes include:
• Petals 
• Longer and thinner
• Shorter and rounder
• Triangles
• Various line lengths 
• Circles with dots 
• Squares 
• Symbols such as: 
• Flowers
• Suns
• Moons
• Trees
  
  Basic designs
• Tip: Begin with drawing simpler shapes then use dots and smaller designs to fill in shapes and the areas surrounding them!
3. After you finish drawing and designing with the black felt tip pen, erase the pencil marks underneath and admire your mandala! 
• You can also add color to your mandala to brighten it up if you want!

After learning how to draw a mandala, I shared the process with some students who are part of Saint Andrew's Mu Alpha Theta, which is a math club at my high school. Here below are some photos of them and their mandalas.

Designing the mandala

Beginning the design after creating the skeleton

Working on the first step with compasses

A colorful mandala!

A finished mandala with a unique design: a different design for each half

Another finished mandala

I hope my blog inspires you and teaches you about mandalas! I have linked some mandala designers and tutorials that I found helpful below as resources for you. :) 

Here are some accounts that I found inspiring when creating my mandala!
@courtneybetts was super helpful in getting me started with my mandala and giving me tips. She has a super cool art Instagram story on her page that features beautiful mandalas. I came across @mandalabybhagya 's Instagram page when looking for inspiration for designs for my mandala. She has lots of intricate colorful mandalas, too! 

One of @mandalabybhagya's colorful mandalas

Another one of @mandalabybhagya's beautiful creations

Lastly, here are some tutorials that I recommend to help you on your mandala journey!
https://www.youtube.com/watch?v=OiSzGBguPm0
https://www.youtube.com/watch?v=QcHDIK0E5KY&t=573s

Thanks for reading! :)

My Problem Solving Students Guest Blog: A Unit on Investigating Figurative Numbers

Our math teacher created a new course elective at Saint Andrew’s School called Honors Problem Solving Seminar. The class is interesting and has taught us everything from grit to fun little math tricks. So far we have discussed visual and figurative numbers, which are numbers that can be represented by a regular geometrical arrangement or sequence of evenly spaced points. They are most commonly expressed as regular polygons, for example, triangles, squares, pentagons, hexagons, etc. and also known as polygonal numbers. The class was a nice addition to the schedule providing us with a very relaxed way to enjoy math. This is what we have learned in our first big unit about figurative numbers.

(Note: this blog was collaboratively written by the entire class, originally in a Google Doc. All handouts are posted in my blog here.)

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We first started with oblong numbers.  Oblong numbers are the number of dots that can make up a rectangle with its length one more than its width.

To get to an explicit formula that can represent the development of the patterns of the number of dots on each rectangle, we first started by observing the rectangles. The first one consists of two dots; two for its length and one for width. The second has three for length and two for width. The third has four for length and three for width. We found out that as the pattern keeps going on, the new rectangle has one more dot on both its length and width than its previous term. So first, we constructed a recursive formula, using the number of dots on the previous term to define the number of dots in the current term. Since the length and width are both added by one each time, we thought that the current term is the resulting number of rows would be n and the resulting number of columns would be n+1.

Thus, the explicit formula for oblong number is n(n+1)

Then we were given triangular numbers. Triangular numbers are the dots grouped together that make up an equilateral triangle. Initially, we were trying to come up a formula based on the pattern and number of dots in each term: 1, 3, 6, 10... Obviously, the pattern does not appear directly based on the number of dots in each consecutive term. However, with Mrs. Winer’s small hint of drawing a diagonal, we found out that each term of the triangular number is exactly half of the oblong number. So the formula is basically the formula for oblong number divided by 2.

The explicit formula for triangular number is n(n+1)/2

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Now we can use what we have learnt from the oblong and triangular numbers to apply it to finding the sum of n counting numbers. If we pay attention to the triangular number, we will notice that the formation for triangular number is:

So we can find out nth triangular numbers means the sum of the nth counting numbers.

Now, for the sum of nth even numbers, (not counting zero), the first even number, 2 is two times bigger than one, the second even number, 4 is two times bigger than two, and the third 6 is two times bigger than 3.The pattern is the same for their sums. While the sum for all the counting numbers are triangular numbers, by studying the even numbers, we learned, that oblong, the combination of two triangular numbers, is the sum for the even numbers. We also can find it out by investigating the oblong numbers diagram.

After learning about the sum of n even counting numbers, we asked what the sum was for n counting odd numbers. We found out that you could find the sum of the odds just as fast as the sum of the evens. By looking at the diagram below (square numbers), we can conclude that the explicit equation for the sum of odd numbers is  n2 because in the diagram the number of dots in each picture is a perfect square of the nth term.

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To mix it up a bit, our math teacher presented to us the Pizza Problem.

Watch the video below for the Pizza Problem, plus an explanation of the Method of Finite Differences.

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Moving right along, we now faced the challenge of pentagonal numbers, which served to extend and solidify the concept of triangular numbers.

           

In a pentagon diagram, we can divide it into three parts (shown in the picture): red, blue and green. Count the dots in each part, and we can see the red part and the green part are the same; they are both the nth triangular number. In the middle, the number of dots in the blue section is also triangular number, but instead of nth term, it is the (n-2nd) term. Then we add all three parts together, but we count the first dot at the bottom twice, so we subtract 1. Simplify the equation, and we get the pattern for the pentagonal numbers.

RED:

1     1       1

2     3       1+2

3     6       1+2+3               -->    

4     10    1+2+3+4

5     15    1+2+3+4+5

…     …     …

GREEN:

1 1          1

2 3          1+2

3 6          1+2+3              -->

4 10       1+2+3+4

5 15       1+2+3+4+5

… …        …

BLUE:

1 0          0

2 0          0

3 1          1                    -->

4 3          1+2

5 6          1+2+3

…    …         …

Final Simplification

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Seeing that we were able to complete each task given so far, Mrs. Winer then challenged the class with yet another problem that once again would require an understanding of both the oblong and triangular numbers. She asked us,

“ What’s the difference between the sum of the first 2015 even counting numbers and the sum of the first 2015 odd counting numbers.”

At first the class was very confused and did not know how to solve this problem, but after much hard work, our class proudly could say that we discovered not one but TWO! very different methods to solve the problem.

Solution 1

First line up the first 5 even and first 5 odd numbers

2 + 4 + 6 + 8 + 10

1 + 3 + 5 + 7 + 9

Now you can see that the even numbers are always one greater than the odds. Since there are 2015 numbers, and each time the evens are one number more than the odds, the total difference will be the 2015.

Solution 2

This method uses direct substitution to find the first n counting even and odd numbers.

“N^2+N,” is the equation for sum of the even numbers and “N^2” for the sum of the odd numbers. The difference between the first n even and n odd counting numbers is the difference between the nth oblong number N^2+N and N^2:

N^2+N - N^2

The class then used direct substitution to solve the problem

(2015^2 + 2015) - (2015^2)= 2015

Although the two methods are very different, they both work in finding the answer. The class was split over which one they thought was easier.

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According to Pythagoras,“everything is related to mathematics. Numbers are the ultimate reality, and through mathematics, everything can be predicted and measured in rhythmic patterns or cycles.” So far, Pythagoras has been right about everything we have learned in this class and it is cool to see how everything comes together. Although there may be different ways to solve a problem, when it comes to math there is no ambiguity and everything eventually will fit together to form one solution.

Teaching a Problem Solving Class for the First Time

Math 

School's out for summer, and I couldn't sleep last night. Call me crazy. I was up tossing and turning all night...thinking about what you ask?? About the new Honors Problem Solving Seminar elective that I will be teaching next year. 

My brain was churning throughout the night about what I should and should not do or add. We are going to a very interesting block schedule next year, and I have to consider how I want to teach the course based on this. The course is a semester course, and students can sign up for both semesters...this means I can't repeat what I did first semester...so it really is a full course I have to plan for. What makes this difficult is that there will be students enrolled second semester that did not take it first semester...so I have a LOT of planning to do...perhaps this is why I kept dreaming/thinking about it all night!

Here are some things I'd like to incorporate into the course. The model we will have for our modified block schedule next year is: 50 minute class Monday, and two 90 minute blocks during the week. 

• Hand out a Problem Set a week before actually going over them. This will consist of about 8 questions a week similar to this one that I used when I taught a problem solving class in Hong Kong one summer. This will be gone over the following week (see below) Exeter Style. 
  

• Monday (50 minutes) will be competitions day. I will give students 30 minute competitions like Florida Math League competitions, Hustles, speed math, mental math, and Relays, and then will take the remaining 20 minutes to have students go to the board to explain anything that some students did not understand. 

• Block 1: Students will go over the first problem set. At the beginning of class, students will go to the board (I have whiteboard walls) and will put up solutions to their favorite problem. They will then each explain the problems to the rest of the class and others can add the way that they did the problem. When finished, the next problem set will be handed out.

• Block 2: Part 1: Show a video...this could be a Ted Talk, like Angela Duckworth's Ted Talk on Grit or maybe one of these on the Monty Hall Problem. Yes this is what was going through my brain at 3am last night! 

• Block 2: Part 2: Teach a lesson that is not part of any curriculum that we teach. I am thinking of breaking this up by quarters. I may poll the kids to help with quarters 3 and 4.
• Quarter 1: Sequences--I will likely use problems from this book Finite Sequences and have kids develop formulas ANY WAY THEY SEE FIT for figurate numbers or Tower of Hanoi sequences and will teach the method of finite differences and possibly Proof by induction.
• Quarter 2: Probability--this is the one thing we do not have enough time for in our curriculum and I know we can spend a ton of time on this. 

• Have students blog once per week. There will not be tests or quizzes in this class, so this will be a major part of their grade...I will take the best one and post it each week as a guest blog in my blog...this came to me in my sleep...I can't wait! 

I think the best part of it is that I either have taught all of the students in the course or know them from Mu Alpha Theta, my math club. I am really looking forward to this class!

Follow me on Pinterest to see some of the pins for High School math lessons that I will try to incorporate into this class.

Play/Eat

I am going to try to cook more this summer. Here is something I had when we went to a tapas restaurant in Missouri last week...Baked Goat Cheese in tomato sauce...I found this recipe and recreated it. To make it even easier, I would just use your favorite tomato sauce rather than making your own. It was definitely well liked in the family!

Any ideas for a problem solving class? Please post below :)

~Lisa