# Engaging students: Finding the area of a square or rectangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Christian Oropeza. His topic, from Geometry: finding the area of a square or rectangle.

How can this topic be used in your students’ future courses in mathematics or science?

Finding the area of a square or rectangle comes up again later in Geometry when solving for the surface area of a prism. The reason behind this is because any n-sided prism, where n is the number of sides the base of the prism has, will have n many squares or rectangles. Therefore, in order to calculate the surface area of any prism, the use of finding the area of a square or rectangle is required (Reference 1). Another course that involves this topic is Calculus. When approximating the area under a curve, one strategy is to use left or right endpoint approximation which is just the sum of the areas of the rectangles under or over the curve (Reference 2). This topic is also used in physics when covering measurements. The idea of finding the area of a square or rectangle in the measurements section is to precisely and accurately find the area. How has this topic appeared in the news?

Steiner Ranch is a hair studio that just recently added 1600 square feet, thus bringing their total to 3468 square feet. With the addition of more space the studio now holds: 19 stylist chairs, 8 shampoo bowls, 3 restrooms, and a color mixing room. All in all, this could not have been done without the use of finding the area of a square or rectangle because then the owner, Brian Charles, would not know how much of each studio equipment would be able to fit in a way that was fitting for him (Reference 3). In other news, state deputies of the Legislative Assembly of Rondonia decided to try creating 11 new protected area in the Brazilian Amazon, which amounted to a total of 2,316 square miles. Therefore, the use of the area of a square was used to determine how much area would go to the new protected areas. However, the bacanda ruralista agribusiness lobby opposed this decision and passed a bill that did not allow the process of making the protected areas (Reference 4). How have different cultures throughout time used this topic in their society?

During 570-495 BC, the use of finding the area of a square impacted math in Greek culture. More specifically, a man by the name of Pythagoras created what is known now to be the Pythagorean Theorem. He discovered this theorem by noticing that the area of the square created by the hypotenuse of a right triangle is equal to the sum of the area of the squares created by the other two sides of the same right triangle (Reference 5). Also, there were different cultures who had discovered the same formula as the Pythagorean Theorem, but were not the first to publish their findings. These different cultures include: Mesopotamian, Indian, and Chinese (Reference 6). Finding the area of a square or a rectangle comes up immensely in computing the cost for installation of hardwood floors. The cost is computed by charging the customer for the price of each square foot of wood used and the labor for each square foot of wood installed (Reference 7).

References:

# Engaging students: Finding the area of a square or rectangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Trent Pope. His topic, from Geometry: finding the area of a square or rectangle.

How could you as a teacher create an activity or project that involves your topic?

On this website I saw that the Kelso High School GIC students went out and built a home as a class project. They were able to get a \$13,000 grant from Lowe’s Home Improvement and blueprints from Fleetwood Homes to go out and build a physical house. I like the idea of having students use geometry in a real world application, as a teacher I would bring this idea to paper. Students would design and create blueprints for their dream house using squares and rectangles. I would start by giving them the total area their house will be. For example, I would tell them to make the blueprints for a 400 square foot house. They could have anywhere from 5 rooms to 20 rooms in their house. They will be responsible for showing the measurements for each room. After creating the layout of the house and calculating the areas of each room, students will be given a set amount of money to spend on flooring. They will then calculate the cost to put either carpet, wood, or tile in each room. This is to have students decide if they would have enough money to have a large room and if so what flooring would be best. There are other aspects you can add to this project to make it more personalized, but as teachers we just want to make sure we are having students find the area of square and rectangles.

http://tdn.com/news/local/geometry-in-construction-class-finishes-building-first-home/article_74143492-d7a9-11e2-995a-001a4bcf887a.html How has this topic appeared in the news?

There are many instances of where area made the news. I found multiple websites that talk about how schools are having building projects for geometry and construction classes. These students are building homes from 128 square feet to 400 square feet. Teachers are having students make these homes so that they can see that geometry is in the real world. By having a range of sizes, students have to adjust their calculations. When creating a house or mobile home you need to accommodate for walking space in each room. In order for students to know if there is enough space, they must find the area of each room. Teachers are using this project because blueprints for houses only use squares and rectangles, making it easier for students to practice solving for area of these shapes. This is just the start of teachers making the concept of area more applicable.

http://design.northwestern.edu/projects/profiles/tiny-house.html How have different cultures throughout time used this topic in their society?

The topic of finding area of squares and rectangles is used throughout many cultures. In the Native American culture along with todays, we see it in growing crops. A farmer must know how big their crop is so they can figure out how much food they will have at harvest. An instance used by many cultures is creating monuments in the shape of square pyramids. In order to build it the right way, you must know the area of the bottom base to build on top of that. A final use of it in our culture is in construction, when we decide how we want to build a building. The concept of area is something that many cultures use today because of how easy it is to calculate. This creates a great way for cultures that are less educated to become familiar with the same concepts as other cultures.

References

Geometry in Construction class finishes building first home. n.d. <http://tdn.com/news/local/geometry-in-construction-class-finishes-building-first-home/article_74143492-d7a9-11e2-995a-001a4bcf887a.html&gt;.
Tiny House Project. n.d. 6 10 2017. <http://design.northwestern.edu/projects/profiles/tiny-house.html&gt;.

# Geometry and Halloween Costumes

From a friend’s Facebook post (shared with her permission):

For every time a geometry student asks, “When am I ever going to use this in real life?” Well, if your child ever asks you to make her a Harley Quinn costume, and there is no pattern, so you have to draft your own, you will need to find the sides of a square using the measurement of the diagonal…

[I]f you need to have a square patchwork of different colored fabrics which line up on diagonal points for a specific measurement so that you have four colored diagonal squares from the shoulder to just below the waist, you would need to find the measurement of the four equal sides of each square. Then you would add seam allowances so you could cut the squares out of the different colored fabrics and sew them together in exact lines to line up just right so you could make a top that looks like the top the character wears. And since this character is only a cartoon character who has been made into a little doll, not many people out there in the world have yet attempted an actual costume to be worn by a real live girl. Of course, a person could just take a pencil and a ruler and draw squares, but without using math, that person could not put together a patchwork of colored fabric squares with this result. The finished product: # Engaging students: Finding the area of a square or rectangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Juan Guerra. His topic, from Geometry: finding the area of a square or rectangle. E1.       How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The website below contains an activity that relates both perimeter and area. In particular, the activity stimulates the student’s mind by making them think of a way to get the amount of fencing that they would need in order to build the stable for animals. After the character in the game learns about perimeter, he is made to think about the area that would be created from the stable. Then the activity mentions the different possibilities of getting the same perimeter but at the same time, the area of each different possibility is also analyzed. The activity makes students realize that even though all stables have the same perimeter, the area was different most of the time. The activity also has the students practice taking measurements and finding the perimeter and area of rectangles. This activity targets multiple objectives and skills because students learn about perimeter, area, and go over measuring the sides with a virtual ruler. This website contains more interactive games that target multiple skills, which will be helpful to the teachers when planning a lesson. Aside from having interactive games, the website also contains videos on tutorials for some basic computations or definitions of terms in math.

http://www.mathplayground.com/area_perimeter.html F3.   How did people’s conception of this topic change over time?

Ancient civilizations have known how to compute the area of basic figures including the square and the rectangle. These civilizations include the Egyptians, Babylonians, and Hindus. The Babylonians actually had a different formula for the area of a square or rectangle. The formula we know today is a*b, where a and b are the lengths of the figure. The Babylonian formula for multiplying two numbers, which was essentially the same as finding the area was [(a + b)2 – (ab)2]/4. Looking at the formula, it is clear that they had a different perception of what it was to find the product of two numbers and also the area of a square or rectangle. It turns out that the Babylonians were the only ones who used a different formula for the area of a rectangle or square, which means that they saw area differently than the other two civilizations. Another person that represented area was Euclid. In his book named Euclid’s Elements, he showed how multiplying two numbers would look geometrically, which was by taking a segment with length a and another segment with length b and putting them together so that they form a right angle at the ends and completing the rectangle by adding the other two missing sides. This method was used for visualizing the multiplication of numbers but it was also the representation of what area looked like geometrically although Euclid did not mention in his book that this was called area.

http://en.wikipedia.org/wiki/Egyptian_geometry

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_mathematics.html B1.   How can this topic be used in your students’ future courses in mathematics or science?

In the future, students will need to know the concept of area in general in order to solve other types of problems in courses like calculus. To illustrate a better example, suppose you have the equation $y = x$. What if you wanted a student to find the area of the triangle formed on the interval from 0 to 5? It would seem obvious that when the student graphs it and creates the triangle from that interval, he or she would use the formula for the area of a triangle once they are able to find the base and the height of the triangle. Another example where they would have to find the area of a rectangle would be when they have an equation like $y = 5$. Let’s say that you wanted to find the area of the rectangle formed from 0 to 4. The student would naturally use the formula that has been known to them for a long time and plug in the numbers. So what if we asked them to find the area of the function $y = x^2$ from 0 to 10? Would the student be able to use the formulas for area that he or she knows? This is where the concept of integration can be introduced to the student. The student might develop the curiosity of wanting to find out how it would be possible to find the area under a curve since the formulas for area that he or she has known all along do not apply. This is only one example where area can be seen in future courses but it seems like an activity like this would naturally lead into integration in a calculus class.

# Engaging students: Finding the area of a square or rectangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Kayla (Koenig) Lambert. Her topic, from Geometry: finding the area of a square or rectangle.

B) Curriculum: How can this topic be used in student’s future courses in math or science?

Finding the area of a square or rectangle can be applied in many other subjects throughout a student’s school career. This topic is learned around 4th or 5th grade, and around this time students will just be using the formulas to find the areas. In middle school, they might be finding the areas by way of more difficult problems, like word problems. The real fun for this subject, in my opinion, doesn’t start until high school. In high school you can use the area of squares and rectangles to find the solutions to many problems. In high school geometry, the Pythagorean Theorem is taught. The area of squares is related to this depending on how the teacher presents this to the student. The Pythagorean Theorem states that “in any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares whose sides are the two legs of the right triangle” (Square-geometry).

In college, possibly high school calculus, students will learn to approximate the total area under a curve (or integral) using the Riemann Sum. To approximate the integral, you find the area of each rectangle, and all of the rectangles areas added together give you the approximated integral. The area of rectangles is also used in Statistics. When creating a histogram, you multiply the height (density) and width of the bars (rectangles).  Then adding the areas (relative frequencies) of all of the bars should be equal to one. Students will also need to use the area of squares and rectangles on college placement exams and standardized testing. C) Culture: How has this topic appeared in high culture (art, classical music, theatre, etc.)?

In my opinion, anything and everything is a form of art, so the area of squares and rectangles can appear in an infinite amount of high culture. M.C. Escher has used squares and rectangles to create tessellations and “portrayed mathematical relationships among shapes, figures and space” (MC Escher). The area of a rectangle was used to Polykeitos the Elder who was a Greek sculptor. He used the area of a rectangle to create the perfect ratio for the human body. Painters also needed to figure out how to depict 3D scenes onto 2D canvas during the Renaissance (Mathematics and Art).

However, one of the more well-known applications of mathematics in art is the Golden Rectangle, which just so happens to involve the area of squares and rectangles. The Golden Rectangle is the area of the original rectangle to the area of the square, which is also the Golden Ratio. In other words, the Golden Rectangle is a rectangle wherein the ratio of its length to its width is the Golden Ratio (Golden Rectangle). Many ancient art and architecture have incorporated the Golden Rectangle into designs. The Golden Rectangle was used in the floor plans and design of the exterior of The Parthenon, which was a Greek temple dedicated to goddess Athena in 5th century BC (Mathematics and Art). Leonardo DaVinci also used the Golden Rectangle in his work. When painting the Mona Lisa, he used this to “draw attention to the face of the woman in the portrait” (Mathematics and Art). DaVinci also used the Golden Rectangle in the Last Supper using it to create a “perfect harmonic balance between placement of characters in the background” and also used it to arrange the characters around the table (Mathematics and Art). D) History: Who were some of the people who contributed to the development of this topic?

Finding the area of squares and rectangles didn’t just come out of the blue; we can thank geometry and ancient mathematics for the development of this topic. One person in particular who contributed to the development of this topic was Euclid, or Euclid of Alexandria, who was a Greek mathematician and known as the “Father of Geometry” (Euclid). He was said to revolutionize geometry and his book The Elements is considered the most influential textbook of all time (History of Mathematics). The collection of his books, all thirteen of them, contain all traditional school geometry (Solomon).

However, Euler wasn’t the only one to contribute to this topic. Pythagoras and his students discovered most of what high school students learn in geometry today (History of Mathematics). In the classical period, Aryabhata wrote a treatise including the computation of areas. From the kingdom of Cao Wei, Liu Hui edited and commented on The Nine Chapters of Mathematics Art in 179 AD (History of Mathematics). There are so many people who contributed to this topic, and people are still contributing and developing to the area of squares and rectangles today!

Works Cited

“Euclid – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia.  20 Feb. 2012. http://en.wikipedia.org/wiki/Euclid.

“Golden Rectangle.” Logicville : Puzzles and Brainteasers.  20 Feb. 2012. http://www.logicville.com/sel26.htm.

“M. C. Escher – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/M._C._Escher.

“Mathematics and art – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia.  20 Feb. 2012. http://www.en.wikipedia.org/wiki/Mathematics_and_art.

Solomon, Robert. The Little Book Of Mathematical Principals, Theories and Things. New York: Metro Books, 2008.

“Square (geometry) – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/Square_(geometry).

“History of mathematics – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 20 Feb. 2012. http://en.wikipedia.org/wiki/History_of_mathematics.

# Engaging students: Finding the area of a square or rectangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Alyssa Dalling. Her topic, from Geometry: finding the area of a square or rectangle.

D. How have different cultures throughout time used this topic in their society? • For three thousand years, the Great Pyramid of Giza was the world’s tallest man-made structure. It is also the oldest structure of the Seven Wonders of the Ancient World. It was built by cutting huge stones into rectangles then placing each stone into place to create the base. It is believed by many that the pharaoh Khufu had his vizier Hemon create the design for the great Pyramids. What is amazing about the design of the Pyramid of Giza is that each of the four sides of the base has an average error of only 58 millimeters in length. Meaning the base is almost a perfect square!
• It would be fun to start the engage with introducing the Pyramid of Giza and explaining the facts above. Then students would be given the dimensions of other pyramids where they would have to find the area of the base to see whether they created a square or rectangular pyramid. This would get them excited about this topic because students would be exploring math that has actually been used in real life. • The Mesoamericans also built pyramids with square and rectangular bases. The picture above is in a city known as Chechen Itza which is located in the Mexican state of Yucatan. It is called El Castillo, and also known as the Temple of Kukulkan. Unlike the Egyptian pyramids though, the Mayan pyramids were usually meant as steps to get to a temple on top. The pyramids consisted of several square bases stacked onto each other with steps up each side. El Castillo consists of nine square terraces each about 8.4 feet tall. The main base of the pyramid is approximately 55.3 meters (181 feet).
• What would be fun to do is have students find the area of each level and compare it to all the levels on the pyramid. I feel students would have fun seeing just how big this type of structure is and understanding the planning it took to create the different levels in this pyramid. B. How can this topic be used in your students’ future courses in mathematics or science?

• Finding the area of squares and rectangles will be used a lot in Algebra and Algebra II. One example in Algebra is when students start solving for unknown variables. A student would be asked to find the area of a square when they have two unknown sides.
• The following is an example engage problem students would use the finding the area of a square or rectangle to solve.

Principal Smith has decided the school needs a new practice basketball court. The current practice court is a square with an area of 144 square feet. She wants the new court to be a rectangle twice as long as it is wide. Find the length of all the sides of both the old court and the new court and find the area of the new court. $x^2 = 144$

So $x = 12$

Then $x(2x) = 2x^2 = 2(12)^2 = 288$

The square court has sides of 12.

The rectangular court has sides of 12×24 and an area of 288 square feet.