Engaging students: Geometric mean

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 Matthew Garza. His topic, from Geometry: the geometric mean.

How has this topic appeared in high culture?
Crockett Johnson was an artist, writer, and mathematician who worked as an art editor for McGraw-Hill in the 1920s. By the 1930s, he was making cartoons; in the 40s he was known for his “Barnaby” comic strips, which appeared in several American Newspapers. He wrote “Harold and the Purple Crayon” in 1955, which may be one his most famous works. In the 1960s he created a series of more than 100 paintings to honor of geometry and geometric mathematicians. Among them was this painting of a construction of the geometric mean of two numbers – line up the lengths and use that as the diameter of a circle, and draw a line from where the two lengths meet up to the circle. If the students know the Pythagorean theorem, they could try to prove that. Crockett Johnson also created a new construction of a regular septagon, using a compass and marked ruler (and trigonometric identities). I found another one of his mathematical paintings on the Smithsonian’s website, of a golden rectangle, and laid it over the geometric mean painting. It seems he included the golden ratio in his work, although I could not find anything verifying this. In general, Crockett Johnson is an interesting person, and that should help engage students.

Wikipedia page: https://en.wikipedia.org/wiki/Crockett_Johnson
Painting at Smithsonian: http://americanhistory.si.edu/collections/search/object/nmah_694664
Another Bio: https://divisbyzero.com/2016/03/23/a-geometry-theorem-looking-for-a-geometric-proof
Regular septagon proof: http://www.jstor.org/stable/3616804?seq=1#page_scan_tab_contents

How have different cultures throughout time used this topic in their society?
Finding much information on the history of the geometric mean is pretty difficult. Pythagoras seems to be generally credited for “discovering” the geometric mean, and the Greek mathematicians are famous for the three means – arithmetic, geometric, and harmonic. Although one not-necessarily credible source explained the word “geometry” comes from words meaning “land measurement.” From this, we can easily consider the task of land management – to find a square plot of land of equal area to a rectangular one, the side length of the square should be the geometric mean of the two sides of the rectangle. For this reason, I believe the geometric mean of at least two numbers must have been used as far back as math has been used for commerce; so pretty close to as far back as math has been used (I wouldn’t be surprised if Egyptians, or even Babylonians, were at least aware of such a relationship, whether or not a constructive proof existed).

http://hsm.stackexchange.com/questions/3057/what-is-the-history-of-the-meanings-behind-the-word-geometric

How has this topic appeared in the news?
Geometric mean is extremely useful for rates and values on varying scales. Rates are used as products – consider something like an investment with a varying return rate for each year. The regular arithmetic mean of the different rates would not give correct results – after one year at rate a, a quantity k becomes ak; after a second year at rate b, the original k is now bak. The yearly average, if taken arithmetically, gives [(a+b)²/4]k after 2 years; if the geometric mean is used, it gives (√ab)²k = abk, so it’s more appropriate. With regard to values on varying scales, it prevents a top-heavy average. Clearly, geometric mean is very useful, which is why finding news will work in a pinch, like if you forgot to plan. Just do a google news search for geometric mean and you find several articles. It’s mostly economic news. The following were not. Alternatively, a search in a scholarly database gives plenty of examples of geometric mean in action, although the technical writing may be difficult for students to get into for an engage.

Geometric mean to measure water quality: http://www.lajollalight.com/sd-beach-water-advisories-20161004-story.html
To measure general wellness of a nation: http://247wallst.com/healthcare-economy/2016/09/22/obesity-violence-helps-push-us-to-no-28-in-global-health-ratings
College sports stats: http://www.usatoday.com/story/sports/ncaaf/2016/10/11/sec-dominates-college-football-computer-composite-rankings/91910190/

Engaging students: Dilations

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 Marissa Arevalo. Her topic, from Geometry: identifying dilations.

How does this topic extend what your students should have learned in previous courses?

As teachers, we want to create connections from prior knowledge to help and assist them create a sort of base or foundation for future courses. Dilations refer to the scaling of shapes that can create similar and/or congruent shapes.

Students may not correlate this conceptual idea to scale factors of function transformations. This skill set is from Algebra I and then extended in Algebra II TEKS, which is taught after Geometry. In Algebra I, students are expected to be able to identify what occurs in a function, (i.e. a quadratic function and such). When given the parent function y=x², if you were to change the size or steepness of the parabola you would either multiply the function by a: y=ax²to create a vertical stretch/compression of the function or multiply by b: y=(bx)² to create a horizontal stretch/compression, which make a and b scale factors. By applying this knowledge, students can hopefully work to identify similar figures and proportions of shapes in relation to their sides/angles.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

In photography, prior to digital photography, we had to have photos developed in a dark room, where the only light source is in a corner of the room given by a light bulb. The darkness allows the processing of light sensitive photo material.

Equipment needed for developing photos:

Enlarger

Chemical bath

Running Water

The photo negatives are taken and enlarged through light onto a print by a specific type of transparent projector as the negatives are see-through, light projects onto the negatives which goes through the negative onto the paper on the base. The paper must eventually be developed in a chemical bath to set and hang to dry. The photos must be enlarged, which is a form of dilation by enlarging the size of the photo onto a new surface with the help of scale factors set by the type of enlarger lens on the enlarger (shown on the right). A similar concept is applied with cinema with the projection of a small film strip through a lens with a light onto a large white screen.

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

A really cool project that I found for a class project is called “Scale Up”. It is meant for the entire class to partake in where the teacher is to pick some picture for the class to scale up in size in pieces. The teacher on the website chose the American Gothic picture and copied it onto an 8.5×11 in. copy paper. She then gave coordinates to each square, so as to easily give each student their own square to make in the picture. Every student was given one or two squares and together they each contributed to the bigger picture and eventually created the entire portrait out of sticky notes by either eyeballing the approximate size of the shapes in their square or by actually scaling the actual size the lines had to be inside of the square they were assigned. This project seems like it would be fun and entertaining for the kids to do together, where they have to in the end talk with one another and discuss what it would take to dilate the photo that there were trying to make.

References: http://fasttimesofamiddleschoolmathteacher.blogspot.com/2014/02/scale-up-picture-class-project.html

https://en.wikipedia.org/wiki/Darkroom

http://ritter.tea.state.tx.us/rules/tac/chapter111/index.html

Engaging students: Finding the volume and surface area of prisms and cylinders

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 Madison duPont. Her topic, from Geometry: finding the volume and surface area of prisms and cylinders.

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

A couple activities from my lesson plan attached below were activities I found to be helpful and interesting for my students when introducing surface area and deriving the formulas themselves from area and circumference formulas they already knew. The first activity I’d like to highlight, though it’s simple, was successful in introducing the concept of surface area to students. In our engage, students were shown pictures of cat posts, cylinders, prism-shaped presents and so on and asked how they could determine the amount of materials needed to cover the surface. They seemed familiar with the concept, but not necessarily the mathematical term or procedure for doing such. After getting their pre-conception-based suggestions and asking them the difference between that and the space the shape takes up (volume), my partner and I were able to see light bulbs go off in their minds and we were able to provide them the answer by introducing the concept of the lesson, surface area. The remaining lesson was an activity where they found the areas of the shapes connected in a cylinder’s net in order to find the total area. After the explore, we had them build the cylinder and then try to determine the area using other formulas. During class discussion, we had students present answers and solidify the reason behind the concept of the formula they found emphasizing the use of circumference being multiplied by length (like length x width of a rectangle but the circumference is the “width”) and that we needed to multiply the area of the circle by two because there were two bases on top and bottom. The student-lead activity of the lesson can be extended to deriving the formula of a surface area of a prism using a prism net, constructing the 3D shape, and then determining the areas of each with different strategies. Once surface area is completed with the two shapes volume exploration could be performed in a similar matter and after all is said and done, the differences between volume and surface area could be compared and contrasted using a chart or Venn-diagram. The activities used and extended from this lesson plan seemed beneficial and better than simply giving the student formulas to memorize and explain because the students physically create the surfaces and see the transition for 2D to 3D and respective use formulas they know to conceptually understand a method of finding the surface area or volume in addition to seeing the formulas. This will help students remember formulas and extend surface area and volume of prisms and cylinders to future topics.

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

Concepts of volume and surface area of cylinders and prisms will be used in several different courses and topics. The first example is in more advanced math topics such as Pre-calculus and Calculus when they are solving word problems such as determining optimized surface areas for companies to use production materials or the volume of water in a cylindrical tank as water is increasing or decreasing within. Another advanced math course that will utilize the concepts of surface area and volume are the higher calculus courses during which you are expected to find volume (integral of 3D figure) and surface area (using double integrals and partial derivatives) of shapes and also when using cylindrical shell, washer, and disk methods to solve integral problems. The formulas for these methods are largely based off of the concept of surface area and volume. In addition to mathematics, surface area will be discussed in sciences in a more conceptual way. In chemistry, surface area is relevant to chemical kinetics as the rate of a reaction is directly related to the surface area of a substance. In other words, as you increase the substance’s surface area, the rate of the reaction is also increased. Additionally, biology uses surface area concepts when considering the size of an organism and how its surface area affects its body temperature or digestion compared to an organism with a different surface area and volume. Lastly, biology relates to these concepts when learning about the surface area to volume ratio of a cell. This ratio bounds the viable size of a cell as the cell’s volume increases faster than the surface area (Surface Area, Wikipedia, 2016). With the knowledge of what is to be built off of these concepts, understanding surface area and volume of 3D shapes such as cylinders and prisms beyond memorized formulas becomes evidently imperative.

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What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

Archimedes, a Greek mathematician, considered his work with cylinders and spheres to be his “most beautiful achievements” as he was able to discover the volume and surface area of these shapes and even wanted his monument to involve a sphere and cylinder (MathPages). He did so by first exploring the area of a circle, which he did by bounding the upper and lower bounds of the circle according to circumference and radius and inscribed/circumscribed n-sided polygons. He then progressed to exploration of the sphere and derived surface area and then the surface area of a cylinder. After, he considered the volume of each shape using what he discovered from surface area with inscribed/circumscribed shapes. According to Mustafa Mawaldi, Archimedes published findings in a book called The Sphere and Cylinder. The more recent history of surface area occurred at the turn of the twentieth century when Henri Lebesgue and Herman Minkowski used the concepts of surface area to develop the geometric measure theory. This theory studies surface area of any dimensions that make up an irregular object (Surface Area, Wikipedia, 2016). Though this is not a comprehensive timeline of the development of surface area and volume, these facts demonstrate that surface area and volume was relevant even in Ancient Greek times and still allows for exploration today, making the topic more relevant and interesting.

https://en.wikipedia.org/wiki/Surface_area

http://www.mathpages.com/home/kmath343/kmath343.htm

http://www.muslimheritage.com/article/volume-sphere-arabic-mathematics-historical-and-analytical-survey

https://en.wikipedia.org/wiki/Surface_area

Engaging students: Area of a triangle

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 Lucy Grimmett. Her topic, from Geometry: finding the area of a triangle.

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

This topic is perfect for creating a mini 5E lesson plan or a discovery activity. Students can easily discover the area of a triangle after they know what the area of a square is. I would give my students a piece of paper (can also use patty paper) that is cut into a square. The students would be asked to write down the area of a square and from there would derive the formula for a triangle by folding the paper into a triangle. They will see that a triangle can be half of a square. The students will be able to test their formula by finding the area of the square and dividing it by 2 and then using the formula they derived. If the two answers match then the student’s formulas should be correct. The teacher would be floating around the room observing, and asking probing questions to lead students down the correct path.

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

Finding the area of a triangle is important for many different aspects of mathematics and physics. Students will discuss finding the surface area of a figure, finding volume, or learning further about triangles. When discussing surface area and volume students will have to find the area of a base. In many examples a base can be a triangle. For examples, if a figure is a triangular pyramid and students are finding the surface area they will have to find the area of 4 triangles (3 of which will be the same area if the base is equilateral.) The Pythagorean theorem is also a huge aid when finding areas of non-right triangles. Mathematics consistently builds on itself. In physics triangles are very often used to find the magnitude at which force are being applied to an object. They use vectors to show this relationship and then use trigonometry functions (derived from the area) to find the magnitude of the force.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Finding the area of a triangle can be performed using different methods. Heron (or Hero) found Heron’s formula for finding the area of a triangle using its side lengths. Heron was considered the greatest experimenter of antiquity. Heron is known for creating the first vending machine. Not the type of vending machines we have today, but a holy water vending machine. A coin would be dropped into the slot and would dispense a set amount of holy water. The Chinese mathematicians also discovered a formula equivalent to Heron’s. This was independent from his discovery and was published much later. The next mathematician-astronomer who was involved in the area of a triangle was Aryabhata. Aryabhata discovered that the area of a triangle can be expressed at one-half the base times the height. Aryabhata worked on the approximation of pi, it is thought that he may have come to the conclusion that pi is irrational.

Information found from: https://en.wikipedia.org/wiki/Hero_of_Alexandria

https://en.wikipedia.org/wiki/Area#Triangle_area

Engaging students: Defining the words acute, right, and obtuse

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 Lisa Sun. Her topic: how to engage geometry students when defining the words acute, right, and obtuse.

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

I believe a scavenger hunt will be a great activity for the students to help concrete their knowledge of acute, right, and obtuse angles. It will be a take home activity rather than an activity that they’ll complete in school. I’ve created this scavenger hunt to take place outside of the classroom so students will understand that what we learn in math class takes place in our everyday lives outside of the walls of school.

This scavenger hunt activity requires students to observe their surroundings everywhere they go. I want them to find 10 acute angles, 10 right angles, and 10 obtuse angles. Along with that, they must take a picture or sketch accordingly to which angle the image has. (For example, picture/sketch of a corner of book shelf – right angle). To spark some motivation and interest, I will announce to the students that if they are able to find 15 of each angle instead of 10, I will add 2 points to their next exam grade.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Archimedes and Euclid are the mathematicians who have discovered and developed the idea of the types of angles that we have today. As a student, when my teachers related the topic with the brilliant minds who made such discoveries, I felt that the topics that I was learning were more relatable and I had gained a deeper understanding of the topic. I hope to do the same for my students with this topic. Here are the following interesting facts about Archimedes and Euclid to keep the students enlightened for geometry.

Interesting facts about Archimedes:

1 of 3 most influential and important mathematician who ever lived (other two are Isaac Newton and Carl Gauss)
Rumors that he was considered to be of royalty because he was so respected by the King during his time
Invented the odometer

Interesting facts about Euclid:

“Father of Geometry”
His book “Elements” is one of the most powerful works in history of mathematics
His name means “Good Glory” in Greek

How can technology be used to effectively engage students with this topic?

Above is a link that I would present, on replay, as students are walking into my classroom to set the tone of the classroom for the day. Once they are all seated, I will tell them to get out their interactive journal and write at least 5 facts that are new to them as I play the video for them once more. By doing so, we’re keeping the students engaged as they are reinforcing what they just heard in writing. Once students are done with this task, I will select students randomly to state one fact that they had just learned from the video. Guide the students to know and remember the “take home message” which are the following:

Definition of Angle: The amount of turn between two rays that have a common end point, the vertex
Angles are measured in degrees
Angles are seen everywhere
Acute angles: 0 – 89 degrees
Right angles: 90 degrees
Obtuse angles: 91-180

References:

https://www.mathsisfun.com/definitions/angle.html

http://www.yurtopic.com/society/people/archimedes-facts.html

http://www.10-facts-about.com/Euclid/id/382

Engaging students: Deriving the Pythagorean theorem

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 Jillian Greene. Her topic, from Geometry: deriving the Pythagorean theorem.

How can technology be used to effectively engage students with this topic?

Geometers Sketchpad is a fantastic resource to be able to more intuitively explore aspects of geometry without the approximation that often comes from using a graphing calculator or a pencil and paper. There is an exploratory activity that can either allow students to discover the Pythagorean Theorem in a different way, or just to reinforce the relationships between the sides. Have students create of a line segment AB with a length of one unit, whatever the measurement might be. Then create a right isosceles triangle using AB as the two equal sides. Now the students will build off of this triangle, making more right triangle (not necessarily isosceles) using the hypotenuse as one of the legs of the next triangle, and the other leg having the same length as AB. Do this 6 times and find the length of final triangle’s hypotenuse. Now explain what the pattern is, and how the relationships work. The final product should look like this:

The final side should be sqrt(7), and the hypotenuses should go sqrt(2), sqrt(3), sqrt(4)…all the way up to x. Hopefully students will be fascinated by the relationship!

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The Pythagorean Theorem was first theorized by Pythagoras, right? Wrong! There’s a very rich history that comes with this theorem that finds a relation in the sides on right triangles. Actually, there were clay tablets indicating an understanding of this theorem found in Babylonian settlements from more than 1000 years before Pythagoras. The Yale tablet, depicted below, has numbers written out in the Babylonian system that give the number “1.414212963” which is very close to √2 = 1.414213562, indicating an understanding of the 1-1-√2 relationship.

Similarly, there are relics from the Chinese and the Egyptian people having either the relationship between the legs figured out, or the existence of 3-4-5 triangles, or a “Pythagorean triple.” The Egyptians made sure their corners on their buildings were 90 degrees by using a rope with 12 evenly spaced notches to make a 3-4-5 triangle. So where does Pythagoras come in? Pythagoras was the first one to formulate a proof in regards to this theorem. So where are his proofs? Well, Pythagoras felt strongly against allowing anyone to record his teachings in any way, so there is no physical proof left behind. However, from what we know about Pythagoras, it is safe to assume that he approached it geometrically.

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

Hello Detective, thank you for coming in to help today. Scar Tellub, 24 year old male, brown hair, green eyes, was found shot early this morning. He was shot for an unknown purpose, but is luckily recovering now. However, we are determined to find this shooter. We know from eye witness testimonies that the gunshot came from overhead, from the top of a nearby building. We know from where the bodies were found, Mr. Tellub was standing perfectly in the center of three buildings, specifically he was 9 feet away from each building. From the entry and exit of the bullet, we can tell the gun was shot from 15 feet away. We have three possible suspects that could be the culprit, but we need your mathematical prowess to help us nail the bad person.

These are the possible shooters:

Madison Bloodi: 19 years old, blonde hair/blue eyes, babysitter. Spotted atop the first building, Trump Tower (20 feet tall), at the time of the shooting.
Hunter Kilt: 34 years old, brown hair/brown eyes, landscaper. Spotted atop the second building, the Eiffel Tower (6 feet tall), at the time of the shooting.
Winston Payne: 26 years old, black hair/green eyes, lawyer. Spotted atop the third building, the Leaning Tower of Pisa (12 feet tall), at the time of the shooting.

Again, thank you for your time, Detective. We know full well that you won’t let us down. Please draw us a photo and show us your work for all three suspects so we can provide them to the judge. Happy mathing!

References:

http://jwilson.coe.uga.edu/emt668/emat6680.f99/challen/pythagorean/lesson4/lesson4.html

http://www.ualr.edu/lasmoller/pythag.html

(I did a similar activity to the murder one with students before, but I cannot find it online again, so I wrote a new one kind of similar to what I remember)

Engaging students: Introducing proportions

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 Jessica Martinez. Her topic, from Geometry: introducing proportions.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

I would show a slideshow of classical artwork and architect such as the Mona Lisa, the Parthenon and the Pyramids in Egypt and then lead an open discussion of what qualities make these popular icons aesthetically appealing. Then I could lead into what makes a specific shape attractive, such as a rectangle, and have my students draw their ‘best looking’ rectangle, which would then lead into a discussion about the Golden Ratio. The Golden Ratio, also known as the “divine proportion”, appears in thousands of artworks and architecture pieces around the world. I would like to show a piece of artwork or architecture divided into the Golden Ratio rectangles and have my students calculate the Golden Ratio to the nearest hundredth -it’s about 1.618. Then I could go back to my original examples of classical artwork and show them with the Golden Ratio proportions drawn. I could also mention how famous painter Leonardo Da Vinci even illustrated a book about the Golden Ratio called De Divina Proportione, which talks about how mathematical and artistic proportions are used to create artwork and design architectural structures.

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

Continuing on the idea of C2 above, I would like to create a project that involves my students using the Golden ratio and proportions to create their own architecture. I could have them research some more famous examples to get ideas, and then have them design their project blueprints, explaining what measurements, ratios and proportions they used to develop it (I would also include a requirement of using the Golden Ratio somewhere in their design). Then in teams of 2-3 students, I would have them create a small scale model of their design using materials found at home or at a craft store. To make it more authentic to the students and possibly a longer, more intensive project, I could give them a scenario of designing their own dream homes; by adding a budget and some time to research, my students could use proportions to calculate the cost of materials needed to build their home (here I would most likely exclude materials not needed for the actual building; plumping, electric and air conditioning would make it more complicated but will most likely not fit in the time frame of my teaching).

How have different cultures throughout time used this topic in their society?

In American culture (and many other cultures), society considers beauty and attractiveness of high importance; its valued so much in some industries that people will go to the lengths of paying thousands of dollars altering their faces and body shapes to something more ‘aesthetically pleasing’ through plastic surgery. What is interesting to know is that plastic surgeons use proportions in order to create a more attractive look for their clients. Plastic surgeons will photograph a client’s face from the front and side views and divide their face into sections in the picture. Then they will make corrections or marks on the photo (and on the client) of where the client wants surgery using specific proportions that create a look that is more symmetrical and ‘pleasing’ to look at. An example of this is shown below in the proportions of the nose to rest of the human face. (I would probably also remind my students that no one’s face is perfectly proportional and it’s a good thing because otherwise we would all look the same, and that’s boring.)

References

Obara, S. (n.d.). Golden Ratio in Art and Architecture. Retrieved October 13, 2016, from http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Obara/Emat6690/Golden Ratio/golden.html

Zimbler, M. S., & Ham, J. (n.d.). Aesthetic Facial Analysis. In Cummings Otolaryngology. Retrieved October 13, 2016, from http://www.marczimblermd.com/plasticsurgeonnyc/ResearchPublications/CummingsOtolaryngology.pdf

Engaging students: Finding the area of a right triangle

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 Jessica Bonney. Her topic, from Geometry: finding the area of a right triangle.

What interesting word problems using this topic can your students do now?

Since students have learned the area of a rectangle, we can use this previously learned knowledge to help students better understand the area of a right triangle. To start off the class you could say that a farmer needs our help developing his pasture into two hay meadows, one for warm-season grass and the other for cool-season grass. The large, rectangular pasture measures 250 yards wide and 600 yards long. Hancock Seed Company sells bahia grass(warm-season grass) seed for $140 per 50-lb bag per acre and ryegrass (cool-season grass) seed for $25 per 50-lb bag per acre. Have the students initially calculate the area of the pasture, then the area of the area of each triangle. From there the students can calculate how many acres are in each triangular section of pasture to determine how many pounds of seed the farmer will need. This activity allows the students to investigate and see the relationship between the area of a triangle compared to the area of a rectangle in a real world setting.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Khan Academy has a great tool for showing students the area of a right triangle (https://www.khanacademy.org/math/geometry-home/geometry-area-perimeter/geometry-area-triangle/a/area-of-triangle). This tool allows students to see how the area of a triangle correlates to the area of a rectangle. By clicking on the dot and dragging it, the user can see why the formula for the area of a triangle works. Students should have previously learned that the area of a rectangle is the base multiplied by the height (A=bh). This interactive tool shows students that the area of a triangle is one half the area of a rectangle (A= ½ bh). Through further interactions on the website the students then can transform the triangles to rectangles and solve to find the area of the triangle. For further explanation of the formula, Khan Academy has a video demonstrating and proving the area of a triangle using methods from Euclid’s Elements, but in a much simpler form so that students will be able to follow along.

References:

Khan Academy: https://www.khanacademy.org/math/geometry-home/geometry-area-perimeter/geometry-area-triangle/a/area-of-triangle

Trigonometry for Physics: http://www.lshsstem.com/uploads/3/9/1/4/39145399/phy_1_trig_for_physics.pdf

Khan Academy: https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles

Radford University: http://www.radford.edu/rumath-smpdc/Units/src/Poles_Sports.pdf

Hancock Seed Company (Bahia Grass Seed): https://hancockseed.com/hancocks-pensacola-bahia-grass-seed-50-lb-bag-4.html

Hancock Seed Company (Ryegrass Seed): https://hancockseed.com/hancocks-ryegrass-seed-50-lb-bag-14.html

Engaging students: Finding the inverse, converse, and contrapositive

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 Heidee Nicoll. Her topic, from Geometry: finding the inverse, converse, and contrapositive.

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

I would start this lesson with if-then statements that were not math related. I would use simple examples such as “If it is raining, then my mother will not let me play outside.” Students will be in groups, and will each group will have a set of cutouts, with each set containing two copies of the word “not”, a card with “if” and a card with “then,” and each “if” statement and each “then” statement on separate cards. They will also have a worksheet that gives them space to write the sentences that we come up with as a class. As the teacher, I will have a set of cutouts that will have either magnets or tape on the back that I will have on the board. I will show them an example, before having them work on their own. I will have the cards, for example “If” “It is raining” “then” “my mother will not let me play outside” on the board. Then I will put a “not” card in front of each statement and ask the students what this statement means. It will say “If” “not” “it is raining” “then” “not” “my mother will not let me play outside,” which translates to “If it is not raining, my mother will let me play outside.” The students will copy the grammatically correct statement onto their worksheet. I will ask them if it is a true statement. Then, I will put the statement back in its original form, and then will switch the “if” and “then” statements, which would result in “If” “my mother will not let me play outside” “then” “it is raining.” The students will copy down this sentence and will discuss whether or not it is true. Lastly, we will do the contrapositive of the statement, and switch the “if” and “then” statements and add the “not” cards. The students will then do several sentences on their own, moving around the cards to form the statements, copying the sentences onto their worksheets, and talking as a group about whether or not the statements are true. This will help the students see the concept behind these different statements before having to learn the names inverse, converse, and contrapositive, and without having to think about them in terms of geometry.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This clip from some Star Trek episode shows an example of times in the English language when it might be hard to decode exactly what someone is saying because of the word “not” or the use of double negatives.

I would show the students the clip and ask them what the man meant by “nobody helps nobody but himself” and if that was a true statement. If they decide that it is not true, then I would ask them what they would change about the sentence to make it true. Although this clip does not explicitly use the ideas of inverse, converse, or contrapositive, it shows the importance of being able to take a somewhat confusing or ambiguous statement and understand it logically. In order for students to understand inverse, converse, and contrapositive, they need to understand that the phrase “this is not an odd number” also means “this is an even number,” or that “this polygon does not have an uneven number of sides” means that “this polygon has an even number of sides.” I would show them examples such as these, and have the students share what they think the statements mean. We would have a class discussion about how language can be confusing at times, and how we need to be able to decode it.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

I would use Kahoot! to create an online quiz. I would have questions such as “which of these statements is logically true?” and “which of these statements is logically false?” Each answer choice would be a short statement, some math related, such as “if a number ends in 2 it is even” and some not related, such as “if the sun is out today, then it is warm outside.” I would also include statements that were the inverse, converse, or contrapositive, such as “if it is not warm outside, then the sun is not out today.” The students would have to read all the answer choices and pick the one that was true or false, depending on what the question was asking. This would get them thinking about whether or not certain statements are true, and would give them practice logically decoding words and phrases. Kahoot! keeps track of the students that answer correctly and quickly and keeps points, so it would be a small competition, which students normally enjoy.

Engaging students: Finding the area of a circle

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 Daniel Herfeldt. His topic, from Geometry: finding the area of a circle.

Learning area of a circle as well as the circumference in geometry is beneficial for anyone who is looking to pursue any career involving math. This includes anything from a math teacher to an architect. This helps with most future courses in mathematics. With this said, it will be very beneficial when going into pre-calculus. This is because in pre-calculus you will deal a lot with trigonometry. This includes such things as the unit circle, which is a great deal to pre-calculus itself. Being familiar with the equations of a circle helps to understand why things work in a unit circle. It can help with simple things such as why x²+y²=r². Knowing the area of a circle will make the class easier to understand in all. This topic is also very important to future architects. The reason for this is because if an architect doesn’t know the area of a circle or any other shape, it would be very difficult to construct a building. If one cannot figure the dimensions of a pillar to help support the ceiling of a building, the building will have a possibility to collapse. This causes the structure of a building to rely highly on the dimensions, area, and volume of all shapes including the circle. This proves that the importance of the area of a circle to be very high. Most students will not know that everywhere they go, circles are needed. Informing them about these small details could have the students more eager to learn. Giving them great real world examples might also help the students understand and grasp the knowledge that you are trying to teach them because it relates to them.

Circles will be anywhere you go. They are in your everyday TV show, video games, and movies. Although at first glance you might not actually see them, they really are there. When creating a character for an animated movie or a popular video game, artist first start to draw with simple circles and lines. They need to figure out a certain area of the character’s face to be able to fit the facial features. For example, they need to be able to fit eyes, a mouth, a nose and a few more features. From this, they will go on to the animation of the character. This also includes circles because in an animation, when you are wanting to move one object, you have to move it all. The same process applies when working on the landscape properties. It will mostly start with simple lines, circles, and boxes. From there, it will progress into more advance steps, putting more and more detail into it. When moving a character, it is also necessary to move the landscape and surroundings as well. This would be great to tell a class because students will be able to relate to the subject. Most kids in the high school level will play video games. Whether the game is on their phone or a gaming console, they still require the beginning steps. If the student doesn’t play video games, they can relate to it due to watching an animated movie. This will be a great way to engage the class in the first few minutes of class. Below is a picture of the progression of drawing a Pokémon.

Many ancient civilizations have been fascinated with circles. Circles can be seen in many ancient structures and buildings from the Roman Coliseum to Stonehenge.

One ancient civilization fascinated with circles were the Greeks. The believed the circle to be a sacred divine shape mostly based on its multiple points of symmetry. The Greeks also invented a puzzle called squaring the circle, in which the person had to construct a square with the exact area of a circle a compass and a straight edge. This puzzle has been proved mathematically impossible.

Other instances in which circles played an important role in history were the circles that appeared in the crops in different areas of the world. These crop circles have been argued to be a hoax while others indicate it is not possible for the crop circles to be the work of humans. Regardless, of their origin, these crop circles continue to fascinate us.

Circles continue to have significance today. They are used in logos and other things usually to signify unity and harmony. Even the Olympic symbol is made up of five interlocking colorful rings. The circle is still found today enclosing the all seeing eye over the pyramid in the dollar bill on the US currency.

The significance of presenting this information to students, especially high school students, is to give them background information. I think many high schoolers will be interested to learn how circles have been significant in other cultures throughout history. Students can be given a short introduction in the subject and asked to look for more instances in which circles played a role in an ancient civilization and then bring that trajectory to modern times.

Website: https://nrich.maths.org/2561