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Without geometry, life is pointless!

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Geometry

Demystifying The Grain Silo Enigma

Farmer John stores grain in a large silo located at the edge of his farm. The cylinder-shaped silo has one flat, rectangular face that rests against the side of his barn. The height of the silo is 30 feet and the face resting against the barn is 10 feet wide. If the barn is 5 feet from the center of the silo, determine the capacity of Farmer John’s silo in cubic feet of grain.

**Here we are presented with a geometric problem that does not immediately showcase its numerous facets. Instead, it demands the creative extension of the given shapes. It also requires the scholar-practitioner to discern how the new geometric relationships and their attending formulas can be used to solve the problem. As such, I intend to approach things in a deductive fashion, albeit with concrete visuals (Flores and Regis, 2003).**

Specifically, in our posited problem, we are asked to consider the following. Farmer John stores grain in a large silo located at the edge of his farm. The cylinder-shaped silo has one flat, rectangular face that rests against the side of his barn. The height of the silo is 30 feet and the face resting against the barn is 10 feet wide. If the barn is 5 feet from the center of the silo, determine the capacity of Farmer John’s silo in cubic feet of grain (Laureate Education, 2007).

This brainteaser would have been less enigmatic if Farmer John had built a right cylinder - shaped silo. Unfortunately for the problem solver, who is charged with deciphering this mystery, our agrarian friend decided to rest one side of the silo against the barn. This resulted in a flat rectangular shaped side where the silo meets the building. This idiosyncratic assembly has affected the base of the cylinder, which, instead of the *de rigueur* circle, has become a segment - challenged three quarter circle. Consequently, these deviations will influence the total volume of the grain silo.

As my initial step in solving this dilemma, I decided to rephrase the problem in my own words. The formula for the volume of a right circular cylinder is area of the base times the height (Van de Walle, 2007). I note that the length of the flat area that rests against the barn is ten feet. A five foot broken line connects the center of the circle (silo) to the midpoint of this ten foot border, dividing it into two, five foot segments. I realized at this stage that, if I am to compute the base of this silo, I will eventually have to calculate the area of an entire circle. I will then have to subtract the area of my supplemental shape to find the area of the base. Accordingly, I commandeered a compass and completed the circle, superimposing it over the barn area. This allowed me a visual representation to support my process. My goal is to create a pair of right isosceles triangles out of the existing shape. This will lay the groundwork for utilizing the Pythagorean Theorem, which will consequently lead me to the hypotenuse, which is also the radius of the circle. Once the length of the radius is discovered, the dimensions of the area of the superimposed circle will quickly follow.

I begin this portion of my quest by labeling one end point of the barn-silo border (A), the midpoint of that border (B), the other end point of the border (C), as well as the center of the circle (E). I will also extend the five foot broken line that connects the center of the circle and the barn. It now reaches the arc line of the laid over circle. I will label this point on the circle (D). Perusing my latest iteration, by connecting points A and C to E, I find that I have created a pair of congruent, isosceles right triangles. I will label these triangles AEB and CEB. Considering the first of these triangles, I notice that the length of the two legs is five feet. Utilizing the Pythagorean Theorem (a squared plus b squared equals c squared), I can now discover the length of the hypotenuse, which also happens to be the radius of the circle. Twenty-five square feet plus twenty-five square feet equals fifty square feet. This also equals the hypotenuse squared. Therefore, the hypotenuse of the triangle, and the radius of the circle, is the square root of fifty, or about 7.07 feet. I can now calculate the area of the superimposed circle by plugging my number into the formula for the area of the circle: pi times the radius squared. This computation, 3.14 times five times the square root of two squared (about 50), results in a circle area of 157 square feet.

My next step is to determine the area of the sector of the circle, which is the pizza sliced shape AECD. Since the triangles are isosceles, and have common right angles at point B (midpoint of the barn-silo border), their other angles must be valued at 45 degrees. Combining the two 45 degree angles at point E, will yield a resultant value of 90 degrees for the arc (AC). Subsequently, I used the formula for the area of a sector, angle AEC (90 degrees) divided by 360 degrees, times pi times the radius squared (Gustafson and Frisk, 1991). After plugging in the numbers, I calculated the sector’s area to be 39.25 square feet. My overarching goal in this strategy has been to find the area of segment (ABCD) - the curved shape at the top of the circle. Once this area is determined, I can subtract it from the area of the full circle, thereby ascertaining the base of the silo. Thus, my next step is to find the area of the triangle AEC within the sector, and subtract it from the total area of the sector. Using the formula, one-half the base times the height, or five feet times five feet, I arrive at 25 square feet for the area of the triangle. Subtracting this from sector ADCE gives me an area of 14.25 square feet for the segment ADCB. This area represents the entirety of my original add-on. Now all I will need to do is subtract the area of this added-on segment from the area of the superimposed circle, and I will have discovered the area of the base of the silo. Subtracting 14.25 square feet from 157 square feet gives me the base of the silo, or 142.75 square feet. Multiplying this by the height of 30 feet gives me the final volume of the silo, 4283 cubic feet.

Looking back on my work, I find that I was initially stymied by the unfamiliar geometric shape that represented Farmer John’s silo. It was only after deciding to complete the circle that my plan of attack began to take shape. Using an approach that combined concrete drawings, and the extension of existing shapes, with formulaic processes, allowed me to discover new, meaningful relationships among the geometric entities. Furthermore, once I activated their attending formulas in sequential order, the missing pieces fell into place. By implementing this strategy, I was not only able to crunch the numbers, but also deepened my understanding of capacity and volume as it relates to cylinder-type shapes.

In conclusion, we can initiate effective problem solving strategies by first studying the given information, and then rephrasing the wording so that we thoroughly understand what is being asked. In addition, by redrawing, extending, or adding lines and shapes to the original figures, we can create conduits to correlating formulas that will help solve the problem. Since mathematics is, if nothing else, a logical discipline, it therefore makes sense to problem solve in a similar manner.

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References

Flores, A., & Regis, T. P. (2003, March). How many times does a radius square fit into the

circle? *Mathematics Teaching in the Middle School, 8*(7), 363–368.

Gustafson, R., & Frisk, P. (1991). *Elementary geometry*. Hoboken NJ: John Wiley & Sons Inc.

Laureate Education, Inc, (Producer). (2007). [Motion picture]. Baltimore MD.

Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching

developmentally (6th ed.). Upper Saddle River NJ.

Art Deco Steamer Trunk Label

Figure 1

Are four colors always enough to color any map so that no two countries that share a border (in more than single points) have the same color?

It is easy to show that you need *at least* four colors, because Figure 1 shows a map with four countries, each of which is touching the other. But is four sufficient for any map?

Francis Guthrie made this conjecture in 1852, but it remained unproven until 1976, when Wolfgang Haken and Kenneth Appel showed that it was true!

Also, quite interestingly, this proof required the assistance of a computer to check 1,936 different cases that every other case can be reduced to! To date no one knows a quick short proof of this theorem.

**Presentation Suggestions:**

Draw a few pictures to illustrate why the problem is difficult, and why (as some might ask) it is not valid to try to "check by example".

Fractions

** The mastery of fractions is a
fundamental skill associated with mathematics, but its related concepts are not
easily understood by the majority of students. Whether it be ordering,
comparing, recognizing equivalencies, or understanding that a fraction
represents a single number, students rarely have the opportunity to develop
these concepts before they are asked to solve fraction related problems. That is
why an inductive approach, using concrete models, and energized with a
combination of closed and open ended questions, initiated by the instructor, is
so effective. The purveyor of this method, however, must be prudent with both
its pace and scope. The overall unit should consist of a series of activities
which incrementally unveil basic concepts, with each succeeding concept laying
the foundation for mastery of the next. This gives the child invaluable time to
play with concrete manipulatives, reflect upon derived meanings, and extend
their learning with a variety of models, and in an assortment of contexts. In
this manner they will systematically uncover and absorb essential concepts and
knowledge before applying this learning to more recondite pursuits. **

**Developing the Concept of Fractions with Concrete
Manipulatives**

**Learning Objectives**

**Students will manipulate pattern blocks to develop the concept of fractions.Students will compare, order, and orally name a series of fractions.Students will discuss and justify the reasons for their answers.Students will use the word***one whole*to identify a single unit.Students will use the words*halves, thirds*and*sixths*to identify fractions.**Students will recognize written representational names of these fractions.**

** My first learning activity
involving fractions utilizes a method that encourages students to take an active
role in the learning process. While using area models, such as pattern blocks,
the students will respond to a series of questions as they solve several
problems designed to introduce the concept of fractions. **

** During this activity, the
students will compare, order and orally name a number of single unit fractions
such as halves, thirds, and, sixths. As they work with the area models, they
will learn a number of meanings associated with fractions. Students will not
only see that fractions are equal parts, but will also understand their relative
size by comparing them to the unit whole. Towards the close of the lesson, after
these concepts are established, the students will be introduced to the
convention of how fractions are represented in written form. To begin, I will
name the hexagon, for instance, to be one whole. I will then ask the children to
manipulate a set of pattern blocks containing trapezoids, rhombuses and
triangles, as they respond to a series of questions: "How many equal parts
(triangles) are needed to make one whole?" (Six). "What is the value of one
equal part (triangle)?" (One-sixth). "What does that mean?" (It takes six of
these equal parts to make one whole). "What would two triangles be called?"
(Two-sixths). How many of these equal parts (rhombuses) are needed to make a
whole?" (Three). "What would two rhombuses be called?" (Two-thirds). "Show me
three equal parts." (Student holds up three rhombuses). "How many equal parts
(trapezoids) are needed to make one whole?" (Two). In this way, the students
quickly learn the concept of relative size, and the spoken terminology of
fractional parts. To reinforce the concept of relative size, as well as formally
introduce the more abstract written names and conventions, I will then ask the
children to place the three shapes in smallest and largest order. As they do
this, I would write the names of the fractions on the board. Next, I will ask,
"Check your blocks and tell me what you notice about the numbers?" (They get
larger). "Look at the board and tell me what you notice about the denominators?"
(They get smaller). As one can see, by utilizing this approach, the teacher is
able to transition from the concrete, to the abstract, in a seamless manner. **

**Correct Shares**

**Learning Objectives**

**Students will identify examples and non examples of designated fractional parts.Students will use conventional terms for fractions (one-fourth, etc.)Students will explain the reasoning behind their answers.Students will connect the concept of fractions and equal shares.****Students will identify and explain the connection between division and fractions.**

** In order to build on concepts
developed in the preceding activity, as well as advance the idea of fractions as
equal shares, I will undertake an activity called Correct Shares. Using an
overhead, I will show the students a number of circles, triangles, and polygons
that are divided into four parts. Some examples will be divided equally, while
the non examples will contain four unequal parts. The children will be asked to
identify those figures that are equally divided. They will also be asked to
explain why the non examples are not showing fourths. Finally, they will, with
guided questioning, explain the nexus between fractions and equal shares. **

**Fractions as Equal Shares**

**Learning Objectives**

**Students will identify, label and understand the concept of one whole, one-half, one-quarter, and one-eighth. Students will compare the relative size of these fractions.Students will explain the connection between division, fractions, and equal shares.****Students will recognize conventions such as written representation of fractions.**

** The activity I have included
below incorporates the ideas presented in the previous activity with a new
concept- enhancing motivation by presented exercises that utilize relevant, real
world materials. This particular lesson continues to develop the idea of
fractions as part of a whole, as well as their relative size. It also connects
fractions with the operation of division into equal shares. Finally, after these
two ideas have been absorbed, it ties into the more abstract concepts, such as
written representational conventions regarding fractions. **

** I start this exercise on
fractions by passing out a candy bar (made of construction paper) to each child.
Each candy bar is divided into eight segments. As we discuss the activity, I
model each step in the procedure. We next examine the candy bar together. As a
group we reach the consensus that it represents one whole. We also note that it
is divided into eight equal parts. We refer to these equal parts as fractions of
the whole, or just fractions. Next, we cut the bar in half and discover that
there are four pieces in each half. Therefore, four pieces equal one-half. I
draw a picture of the fraction on the board (using a fraction strip measurement
model). Then I label the fraction using traditional representation. As we
progress through the lesson I continue to think out loud, identifying each
fraction, and reminding myself how many equal parts of the whole it represents.
I next call on a certain student to come up with his candy bar and scissors. We
talk about him/her splitting the whole with a friend so that each would have
equal shares (four, or one-half). I call on individual students to explain what
equal share might mean (each child gets the same amount). After the partition,
these two call two other friends to come up. They have to split each half again
into equal parts (two each or one-fourth). As the lesson progresses, I continue
to elicit evidence of understanding, or confusion from the class. I respond to
these "battlefield assessments" with appropriate feedback. As we create each new
equal share, I continue to draw and label a fraction strip, and verbally
reiterate the term "equal share." Finally, four more friends join the group,
looking for equal shares. So each child now has a single piece, or one-eighth.
Once we have divided the candy bar into eight segments I use the drawings on the
board to demonstrate how we went from one-half to one-fourth to one-eighth. I
ask the children to reorder them from smallest to largest and use the
accompanying drawings to explain the relative size of each fraction in real
terms. At the conclusion of the lesson, each child gets a real candy bar to
enjoy.**

**Fractional Parts Counting**

**Learning Objectives**

**Students will count equal shares of manipulatives in order to understand the numerator, or top number, as representing the number of things counted.Students will recognize the denominator, or bottom number, as representing what is being counted (fourths, sixths, etc.). Students will recognize fractions that are less than one whole.****Students will recognize fractions that are more than one whole, both as fractions and mixed numbers. (One or more wholes plus a part of the fraction).****Students will understand representational conventions of fractions and mixed numbers.**

** I have already presented the
students with a foundation for understanding the concept of fractions, as well
as the concept of fractions as division of the whole into equal shares. It is
now time to expand upon this knowledge with activities that focus on both parts
of the fraction as less than, and greater than, one whole. In order for students
to comprehend this concept, they will not only call upon their understanding of
unit fractions, developed in the preceding activities, but will also be asked to
count a number of equal parts of a whole, and explain how their resulting
fraction compares with that whole. For instance, I would designate each area
model piece as being sixths. I would present them with seven of these pieces and
ask them to count all (one sixth, two sixths…seven-sixths). I would then ask,
"If we have seven sixths, is that more or less than one whole?" "Why?" I would
then give them fourteen-sixths and ask them to count. I would ask, "Is
fourteen-sixths more than two wholes?" "Why?" I can also use this activity to
establish the groundwork for mixed numbers. "What is another way to say
fourteen-sixths?" (Two and two-sixths).**

**Comparing Fractions**

**Learning Objectives**

**Students will use length models and paper and pencil activities to estimate and compare the relative size of like fractions.Students will name and order these fractions.Students will discover patterns associated with their results.****Students will discuss, analyze, and synthesize these results into a general rule.**

** This exercise builds of the
concept of relative size of fractions discussed in the prior activity. It also
requires that the students utilize a set of length models (Cuisenaire rods), as
well as graph paper, pencils, and markers, and asks them, at the end of the
activity, to synthesize the knowledge they accrue to formulate a general rule.
The goal of this lesson is to give the students practice comparing fractions
with like denominators and same size wholes. The exercise begins with the
children making a stick comprised of six similarly colored Cuisenaire cubes.
Next, they will outline this horizontal row on graph paper. I will ask them use
one colored marker to shade in the six outlined cubes. After they have done
this, I will ask the students to make a stick of five cubes, all containing the
original color, and then add a sixth cube of another color to it. Ask the class,
"What fractional part of the whole is the different colored cube?" (0ne-sixth).
Have them make another outline and use two markers to color the appropriate
cubes. (Five cubes will contain the original color and the sixth will be shaded
with a second color).**

** Next, ask the class to make
additional sticks of Cuisenaire cubes, with both colors, showing all possible
fractional parts of the whole. Again, have them outline their results on graph
paper. Ask, "What do you notice about the pattern of these fractions?" (Each
block contains one less cube of the original color). "What does this mean?" (The
size of the fraction is getting smaller). Have the students write the fractions
in the form of 1/6, 2/6, etc. Ask "What fraction contains more of the bar,
one-sixth or four sixths?" (Four-sixths). Then note, "We can see that
four-sixths is bigger than one-sixth because it is more of our whole." Next, ask
them, "Can you find another pair of fractions in which one fraction is larger
than the other?" After all the possibilities have been discussed, have the
children, using representational conventions; order the fractions from smallest
to largest. Let them compare these numbers with the outlined blocks of
fractions. Allow them time to discuss and reflect upon their results. Then ask
them to devise a general rule about the data they have accumulated. (Fractions
with the same denominators can be ordered by comparing their numerators). **

**Using Benchmarks of Zero, One-Half, and One**

**Learning Objectives**

**Students will use three benchmarks (zero, one-half, and one) on a number line to estimate the approximate and relative size of the fractions.****Students will demonstrate their number sense by explaining their answers.**

** Understanding why a fraction is
close to the benchmark numbers of zero, one-half or one is a propitious way to
begin developing fractional number sense, as well as sharpen skills involving
correct estimation. This subsequent activity expands, as well as further
refines, the comparison concepts developed in Activity Five. It utilizes the
benchmark numbers of zero, one-half, and one on a number line to help children
visualize the approximate and relative value of their fraction. This offers
students the opportunity to develop a sense of fractional numbers, as they sort
a number of fractions into three separate groups. Each group will be comprised
of fractions sharing close proximity to the same benchmark number. As they
arrange these fractions, the students can use a variety of area, length or set
models to help convey the concept of approximate size. The difficulty of these tasks
depends on the fractions involved. Denominators can range in value from two to
one hundred, and the numerators’ values can, at times, exceed one whole. As a
result of this regimen, the concept of fractions is extended both in range and
scope. Another fruitful result of this activity is that, once mastered, the use
of benchmarks will greatly facilitate the gauging of a fraction’s value, as well
as helping students decide which of two given fractions is larger. To sum up, fractions are often a
difficult concept for young learners to grasp. Starting with concrete models is
a vital first step. Once they have mastered the idea that fractions are simply a
part of the whole, they can then move on to more abstract concepts involving
conventions such as written representation. Introducing the concept of fractions
as division, fractions as greater than one whole, and mixed numbers, are part of
the continuum. These are most propitiously introduced using a variety of area,
length and set models, followed by conventional representation. This step by
step development creates a solid foundation that will make more advanced
concepts easier to assimilate. By using a sampling of area, and
length models, I have provided the students with an array of individual options
to discern what fractions are, how they are related to a unit whole, how to
compare fractional parts to the whole, as well as to each other. Following
mastery of this basic understanding, I have introduced important content
specific vocabulary, as well as numeral representations. I have presented real
world connections to enhance motivation. Additionally, I have laid the
groundwork for comprehending the nexus between fractions and division. Finally,
through the medium of benchmarks, I have striven to inculcate, within the
students, an intuitive sense of the related numbers. **

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