Surface Area and Dimensions of a Tipi

Figure A

 My daughter received a tipi as a gift from her grandfather for her 16th birthday.  This tipi is used for our native american cultural ceremonies.  Time and time again, I observed my relatives set it up for these purposes and I never knew the surface area or the dimensions of this tipi.  The only information I knew was that the diameter is 26 feet and 19 feet in height.  I had no idea how they set it up but I never saw anyone with a calculator or figuring out a math solution for this tipi.

Well... I decided to investigate the set up of this tipi when I learned about dimensions and surface area of a cone.  The surface area of a tipi is shaped like a cone other than the front part that they call ears where smoke escapes from the fire that is built inside.  As you can see in Figure A, I figured about the dimensions by using the Pythagorean theorem.  Then I used the surface area formula to get the results of the surface area.  The end result of what it actually looks like after it is set up (Figure B).

Sources:

Surface Area of a Tipi, Home. Personal photograph by Miranda Tachine-Benally. 2016.

Native American Tipi, Pinon, Arizona.  Personal photograph by Miranda Tachine-Benally. 2016

Tachine-Benally, Miranda.  Personal Measurement Investigation. 08 Oct. 2016. Home. Mesa.

Figure B

Farmer Hank and his Pigs

Farmer Hank

The first day of Math 257 class, our instructor gave us a warm up problem.  I believe the intention behind this was to get our critical thinking going and prepare us for the weeks ahead.  The warm up problem was about a farmer named Hank who had fewer than 100 pigs on his farm.  If he groups the pigs five to a pen, there are always three pigs left over.  If he groups the pigs seven to a pen, there is always one pig left over.  However, if he groups the pigs three to a pen, there are no pigs left over.  What is the greatest number of pigs that Farmer Hank could have on his farm?

I approached this problem by highlighting all the givens first.  Then, I started at the highest number if grouped into five and three pigs were left over which was 98 pigs.  However, 98 pigs didn't equal out with seven and one left over.  So, I moved down to 93, 88, 83 pigs and none of these numbers equal out to seven and one left over.  Once, I reached 78 pigs, it worked out!  The number 78 came out to: 15 groups of 5's with 3 pigs left over, 11 groups of 7's with 1 pig left over, and 26 groups of 3's with no pigs left over.  Therefore, the greatest number of pigs that Farmer Hank could have on his farm that was fewer than 100 pigs was 78 pigs.  

We had a similar problem in my Math 256 class about, A Farmer and her Broken Eggs.

Sources:

http://clipart.coolclips.com/480/v

Tachine-Benally, Miranda. Farmer Hank Warm Up. 03 Oct. 2016. Classroom Worksheet. Mesa Community College. Mesa.   

Cylinder and Cone Experiment

Cylinder and Cone with Rice

Class was fun today! We discussed formulas for a right prism, pyramid, cylinder, cone, and sphere.  The two that I chose are the cylinder and the cone.  The equation for the cylinder is

V

=

π

r

2

h  

and the cone is 

V

=

π

r

2

h

3

The purpose of this activity was to develop the relationship between the volume of a cylinder and its related cone. First, we compared the bases and heights of each one and how they relate to each other which was that they were identical in base and height.  Then, we had to estimate how many of the cones would fill the cylinder up with rice.  My group guessed that two cones of the rice would fill up the cylinder and we were wrong.  The correct answer was that three cones would fill up the cylinder with rice.  After that, we had to write a ratio that compared the volume of the cone to the volume of the cylinder which was 3:1.  The last step was to write a rule or formula to determine the volume of a cone based on the formula for the volume of a cylinder which we wrote as :V=πr2h/3.     

Sources:

Cylinder with cone and rice, Mesa Community College. Personal photograph by Miranda Tachine-Benally. 2016.

Tachine-Benally, Miranda. Chapter 13- Concepts of Measurement. 03 Oct. 2016. Pyramids and Cone Worksheet. Mesa Community College. Mesa.   

The Scorpion and the Cricket

Figure A

Figure B

 I honestly had a hard time with a problem in class today.  This particular problem was a homework assignment in which we (students) had to figure out the shortest route the scorpion can crawl to get to the cricket.  As you can see in my drawings, I have labeled the scorpion and the cricket.  The scorpion is seated in the center of the right wall, 1 feet from the ceiling.  The cricket is seated in the center of the left wall, 1 feet from the floor.  The dimensions of the rectangular room is labeled as the base being 30 feet (a), the height being 12 feet (b), and the width being 12 feet (Figure A).

In Figure B, I unfolded the rectangle into a net and implemented the Pythagorean theorem.  Then I figured out my new dimensions and labeled them accordingly.  Then, I calculated my figures using the Pythagorean theorem which is a² + b² = c².  As shown in Figure B, the end result was 40 with the conclusion that the shortest route the scorpion can crawl to get to the cricket was 40 feet.

Sources:

Net of a Rectangular Room, Mesa Community College. Personal photograph by Miranda Tachine-Benally. 2016.

Tachine-Benally, Miranda. The Scorpion and the Cricket. 03 Oct. 2016. Classroom Worksheet. Mesa Community College, Mesa.

Surface Area Activity

Trapezoidal Prism

 Activity in my MAT 257 was somewhat interesting today because we got to find surface areas engaged in a hands-on group activity.  My group got the more difficult of five shapes, Trapezoidal Prism.  The other shapes were hexagonal prism, rectangular prism, truncated square pyramid, and a cylinder.  We were instructed to examine the shape, identify the shape, develop a plan for finding the surface area of the shape, perform calculations showing all steps, and label with proper units.  The tools given to us by the instructor was a worksheet, a ruler, a post-it poster, and a marker. 

Our first approach after identifying the shape was to identify how many faces our shape had and label it with an alphabet.  Our shape had six faces so we labeled it A through F.  Then, we had to determine what kind of shapes we would reveal if we could unfold it.  We determined that we would have two trapezoids and four squares.  Next, we had to figure out what formulas would accommodate us in figuring out our surface areas.  We came to the conclusion that our A and B sides, shaped like a trapezoid, would use the trapezoid formula of:  A=1/2h(b1+b2).  Our C, D, E, and F would be calculated using the rectangle formula which is, A=bh (Area = base X height).  As you can see, we had fun with it and we nailed it!! :)


Surface Area of a Trapezoidal Prism

Sources:

Surface Area Gallery, Mesa Community College. Personal photograph by Miranda Tachine-Benally. 2016.

Tachine-Benally, Miranda. Surface Area Activity. 28 Sept. 2016. Mesa Community College, Mesa.



Gumdrop Polyheras

Cube

Square Pyramid

Pentagonal Prism

 Building gumdrop polyhedrons with colored toothpicks and spicy gumdrops was a blast!  This project was part of a unit in our Geometry chapter of our mathematics book.  The goal of this "Centers" activity was to build polyhedrons while paying close attention to the number of faces (the flat surfaces), vertices (corner points), and edges (line segment which joins two vertices).  Furthermore, we had to give an example of, "Where in the World" these polyhedrons exist.  Although, it looks simple, it was actually a brainstorming activity for me as I have not been aware of the many polyhedrons that exist.  The three pictured here are a few of many!

So... for the cube, I indicated there was six faces, eight vertices, and twelve edges.  "Where in the World" can this be found is in our kitchen cupboards with the label "Sugar Cubes."  And for the square pyramid, I indicated there was five faces, five vertices, and eight edges.  It can be found in Giza as an Egyptian Pyramid.  Can you figure out the faces, vertices, and edges for the pentagonal prism?  And, "where in the world" it can be found?  There is a formula called Euler's Theorem that is very helpful.  This formula is V+F-E=2.  Meaning Vertices + Faces - Edges = 2.

Sources:

Gumdrop Polyhedras, Mesa Community College. Personal photograph by Miranda Tachine-Benally. 2016.

Tachine-Benally, Miranda. Resource Sheet 4 a. 7 Sept. 2016. Classroom Worksheet. Mesa Community College, Mesa.