Engaging students: Using a truth table

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 Bich Tram Do. Her topic, from Geometry: using a truth table.

Funny video to engage student that a university professor made in class.

Or another clip from the movie “Liar, Liar”

How can you tell if an argument is valid or invalid? In this lesson, we will learn about the truth table and technique to detect the validity of any simple argument.

A2. How could you as a teacher create an activity or project that involves your topic?

I could split students into a group of three students and hand each group 3 bags of different colors cards with printed statements on each one. For example:

Bag 1 has statements such as:

If you are a hound dog, then you howl at the moon.

Bag 2 contains conditions:

You don’t howl at the moon.

Bag 3 has conclusions:

Therefore, you aren’t a hound dog.

In each group, the teacher gives a poster/ construction paper that students must search for the correct responses, match them up, and paste them on the construction paper on the left side. On the right side of the paper, the students are asked to answer the question whether the arguments are valid or not and their reason by making a truth table.

Students will have total of five sets and given about twenty minutes to finish. When the students have all finished, I will ask each group coming up with a new example, state their reasons and present to the class. I might have the students volunteer to be 3 judges and vote for the group with the best example. The activity is fun and helps students to apply what they learned as well as their mastery of the materials.

D1. What interesting things you can say about the people who contributed to the discovery and/or the development of this topic?

According to Shosky (1997), the truth table matrices was claimed to be invented by Bertrand Russell and Ludwig Wittgenstein around 1912. However, there was evidence shown that the logician Charles Peirce (1839-1914) had worked on the truth table logic (1883-84) even before the other two mathematicians worked on the same logic. However, Peirce’s unpublished manuscript did not directly show as a “table”, but the “truth functional analysis”, and was in matrix form. Peirce used abbreviations v (for true) and f (for false) and a special symbol ―< to connect the relationship between statements, say a and b. Later, Russell and Wittgenstein (1912) claimed the first appearance of the truth table device, causing doubts if they worked together or separated and evidences needed to make the claim. In short, the invention of the truth table was credited to Charles Peirce in “The Algebra of Logic” (around 1880) and the “table” form was developed to be clearer and easier for understanding, along with many important contributions of Russell, Wittgenstein based on their knowledge of matrix, number theory, and algebra.

E1. How can technology be used to effectively engage students with this topic?

The truth table topic doesn’t have many engaging activities for students to learn even though it has many applications, especially in digital designing, electrical systems. However, we can include some use of technology so that students who finished group activities early or students who needed more practice can find. This website is an interactive activity for students to do so:

http://webspace.ship.edu/deensley/discretemath/flash/ch1/sec1_3/truthtables/tt_control.html

There are different conditions represented by p, q, and r on the first three columns. The next columns, students are asked to fill out the answer (True or False) to each corresponding condition. When they are done with one column, just click on the statement “I’m done with this column”, and then the students will be directed to another one to try. In addition, they can always click on the pink rectangular box in the bottom to change to a different truth table.

Source:

http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1119&context=russelljournal&sei-redir=1#search=%22truth+tables+history%22

http://www.math.fsu.edu/~wooland/argumentor/TruthTablesandArgs.html

http://arxiv.org/ftp/arxiv/papers/1108/1108.2429.pdf

Engaging students: Radius, diameter, and circumference of circles.

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 Nataly Arias. Her topic, from Geometry: the radius, diameter, and circumference of circles.

D3. How did people’s conception of this topic change over time?

In order to calculate the circumference of a circle we must multiply the diameter by . The diameter of a circle is the length of the line through the center and touching two points on its edge. In simpler terms the diameter is two times the radius. To get the circumference of a circle we have to work with its radius or diameter and . So the more important question is, what is and how does it relate to circles? Pi or π is a mathematical constant which represents the ratio of any circle’s circumference to its diameter in Euclidean geometry. It is the same as the ratio of a circle’s area to the square of its radius. This can be seen as far back as 250 BCE in the times of Archimedes. Archimedes wrote several mathematical works including the measurement of a circle. Measurement of the circle is a fragment of a longer work in which is shown to lie between the limits of $3 \frac{10}{71}$ and $3 \frac{1}{7}$ . His approach to determining consisted on inscribing and circumscribing regular polygons with a large number of sides. His approach was followed by everyone until the development of infinite series expansions in India during the 15^th century and the 17^th century in Europe. The circumference of circles was found in the works of Archimedes and is now reflected in our math textbooks. This topic has been seen for many centuries and is still seen today. It has become an important part of math and has become an important part of the mathematics curriculum in schools.

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?

If I was teaching a middle school class on radius, diameter, and circumference of circles I would incorporate technology in my lesson. I have seen firsthand how effective technology can be when teaching your students. I showed the class a video clip during the lesson. As soon as I pulled down the screen projector they sat up, their eyes lit up and they were excited. This is why I have chosen this video clip from YouTube for this particular topic. I think it’s important to change it up and not always stick to a particular teaching style. Some students learn more visually and watching a video instead of listening to a lecture might be more entertaining for students. I know that teachers can’t rely on only technology to teach their students but using things like YouTube can certainly help and be beneficial. I chose this clip because I liked that it used and went over several examples and related circles to things students see every day like a pizza, tire, and table. I also like that it went over definitions in a clear and easy to understand way for students. It explained what a radius and diameter is and how to find it. This helpful video discusses the calculation of the circumference and its area. It also explains the relationship between and the circumference. This 8 minute clip could be used as part of your explain section of your lesson or could even be used to help students review the topic before a test or quiz.

D4. What are the contributions of various cultures to this topic?

When dealing with the radius, diameter, and the circumference of circles there is no escaping pi. Pi represents the ratio of any circle’s circumference to its diameter and is one of the most important mathematical constants. It’s used in many formulas from mathematics, engineering, and science. In math we use to solve for the circumference of a circle with formula $C=2\pi r$ . Sometime in early history someone discovered the relationship between the size of the circumference and the diameter of all circles was a constant ratio. This was seen and presented in the earliest recorded mathematical documents of Babylon and Egypt over 2000 years ago. At this time they did not use the symbol that we use today it wasn’t till much later. They had established that the ratio was equal to $\frac{C}{D}$ , where C is the circumference and D is the diameter of any given circle. At this stage, the Egyptian and Babylonian mathematicians came up with numerical approximations to $\frac{C}{D}$ which is the number we now call pi. Their methods are still unclear and unknown today. In their time period there was no modern number system. They didn’t even have pencil and paper. It has been predicted that they used a rope and sticks to draw circles in the sand and that they also used the rope to measure how many diameters made up a circumference of a circle.

References

http://www.britannica.com/EBchecked/topic/458986/pi

http://www.britannica.com/EBchecked/topic/32808/Archimedes

http://www.youtube.com/watch?v=Yb1HYyBfLfc

http://www.ms.uky.edu/~lee/ma502/pi/MA502piproject.html

http://www.ams.org/samplings/feature-column/fc-2012-02

The Lighter Side of Teacher Evaluation

More about W. James Popham can be found here: http://www.corwin.com/authors/508599

Trig identity

Engaging students: Finding the volume and surface area of pyramids and cones

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 Laura Lozano. Her topic, from Geometry: finding the volume and surface area of pyramids and cones.

green line

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Now days, pyramids have appeared almost all over pop culture because of the illuminati conspiracy. Famous artist like Katy Perry, Kanye West, Jay-Z, Beyoncé, and many others are believed to be part of this group that practices certain things to retain their wealth. Since it’s a conspiracy, it might not be true. Although that’s another topic, they all use an equilateral triangle and pyramids to represent they are part of the illuminati group. They display it in their music videos and while they are performing at a concert or awards show.

In Katy Perry’s new music video, were she portrays herself as a Egyptian queen, for some weird reason, she has a pyramid made out of what looks like twinkies.

LL1

To make this, the base and height had to be measured to create the surface area of the pyramid.

Also, the picture below is from Kanye West’s concerts. He is at the top of the pyramid.

To make this, they had to consider the size of the stage to fit the pyramid. So the size of the base depended on the size of the stage.

The most famous cone is the ice cream cone. When most people think of cone they initially think ice cream! Ice cream cones are made using the surface area of a cone and taking into consideration the volume of the cone. The bigger the surface area, the bigger the volume, the more ice cream!

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)

Some musical instruments have the form of a cone. For example, the tuba, trumpet, and the French horn all have a cone like shape.

The sound that comes out of the instrument depends on the volume of the cone shaped part as well as the other parts of the instrument. The bigger volume of the cone shaped part is, the deeper the sound, the smaller the volume of the cone shaped part is, the higher pitched it is.

Pyramids can be used in art work. Most of the art work done with pyramids is paintings of the Egyptian Pyramids. But, they can also be used to make sculptures of abstract art. Here is one example of an abstract sculpture made from recycled materials.

If the sculpture is hallow, then to make it you would only need the surface area. If it’s not, then you would also need to calculate the volume to see how much recycled material was used.

D5. How have different cultures throughout time used this topic in their society?

In ancient history, the Egyptians used to build pyramids to build a tomb for pharaohs and their queens to protect their bodies after their death. The pyramids were built to last forever. No one knows exactly how they built the pyramids but people have had theorys on how they were built.

The most famous pyramids are the Pyramids of Giza. The pyramids are Pyramid Khafre, Pyramid Menkaure, and Pyramid Khufu. It is the biggest and greatest pyramid of Egypt. This pyramid used to measure about 481 feet in height and the base length is about 756 feet long. However, because the pyramid is very very old, erosion causes changes in the measurements of the pyramid. When scientiest and archeologist had to find the differrent measurements they most likely used the formula to find the volume and surface area of the pyramid. However, back then, the formula was probably not discovered yet.

An example for cones is the conical hat. Used by most the Asian culture, conical hats, also know as rice hats, or farmers hat, were worn by farmers, and they are still somewhat used today. There are many types of conical hats that can be made today. Some are widder than others, and some are taller than others. To make the hats, the maker of the hat has to consider the surface area of the hat to make the hat properly.

Resources:

http://www.history.com/topics/ancient-history/the-egyptian-pyramids

http://www.thelineofbestfit.com/news/latest-news/kanye-wests-yeezus-stage-show-includes-mountains-pyramids-and-jesus-impersonator-139788

http://www.youtube.com/watch?v=0KSOMA3QBU0

http://earthmatrix.com/great/pyramid.htm

Averages and percentages

Source: http://www.xkcd.com/937/

Engaging students: Finding the volume and surface area of spheres

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 Allison Myers. Her topic, from Geometry: finding the volume and surface area of spheres..

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

Show students pictures of the Personal Satellite Assistant (PSA). Tell students they are going to investigate how the surface area and volume of a sphere change as its radius changes.

Explain that they will also determine how big the PSA is in real life.

Remind students that NASA engineers have created a 30.5-centimeter

(12-inch) diameter model of the PSA, but they want to shrink it to 20 centimeters (8 inches) in diameter.

Use a 30.5-centimeter (12-inch) diameter globe and let students know the globe is roughly the size of the current PSA model.

Ask students how the PSA might look different if its surface area were reduced by half.

Ask how the function of the PSA might be different if its volume were reduced by half.

Ask students what information they need to calculate its surface area and volume.

If they appear confused, draw three circles of different sizes and ask students how to calculate the area of each of the circles.

The only information they need is the radius of the sphere. Review the properties of a sphere.

Ask students what formulas are necessary to calculate the surface area and volume of the sphere. Write these formulas on the board:

Surface Area = 4 x πx radius x radius

Volume = 4/3 x πx radius x radius x radius

Show students a baseball, softball, volleyball, and basketball. Ask them if they think the surface area and volume of a sphere change at equal rates as the spheres increase from the size of a baseball to the size of a basketball.

Ask students how they will verify their hypotheses.

Curriculum

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

In calculus students will learn that you can revolve a curve about the x or y-axis to generate a solid. For example, a semicircle [f(x) = √(r²-x²)] can be revolved about the x-axis to obtain a sphere with radius r. From this, the different formulas for calculating the volume of a sphere can be derived.

In calculus, students will also learn how to find the surface area of a sphere by integrating about either the x or y axis.

Resource: http://www.math.hmc.edu/calculus/tutorials/volume/

At some point, students may also extend their knowledge of spheres into higher dimensions (hyperspheres), where they will learn how volume changes according the dimensions they are working in.

Resource: http://spacemath.gsfc.nasa.gov/weekly/6Page89.pdf

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

For Volume of a Sphere:

Recent Hubble Space Telescope studies of Pluto have confirmed that its atmosphere is undergoing considerable change, despite its frigid temperatures. The images, created at the very limits of Hubble’s resolving power, show enigmatic light and dark regions that are probably organic compounds (dark areas) and methane or water-ice deposits (light areas). Since these photos are all that we are likely to get until NASA’s New Horizons spacecraft arrives in 2015, let’s see what we can learn from the image!

Problem 1

– Using a millimeter ruler, what is the scale of the Hubble image in kilometers/millimeter?
Problem 2

– What is the largest feature you can see on any of the three images, in kilometers, and how large is this compared to a familiar earth feature or landmark such as a state in the United States?
Problem 3

– The satellite of Pluto, called Charon, has been used to determine the total mass of Pluto. The mass determined was about 1.3 x 10²² kilograms. From clues in the image, calculate the volume of Pluto and determine the average density of Pluto. How does it compare to solid-rock (3000 kg/m³), water-ice (917 kg/m³)?
Inquiry:

Can you create a model of Pluto that matches its average density and predicts what percentage of rock and ice may be present?
Resource: http://spacemath.gsfc.nasa.gov/weekly/6Page143.pdf

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 Michelle Nguyen. Her topic, from Geometry: introducing proportions.

A2. How could you as a teacher create an activity or project that involves your topic?

Using the video presented in E1, I would create a project that consists of the students making a poster of their own body with the proportion that they found within their body parts. For example, they would use the measurement of their foot and try to find out the amount of feet needed would create their height. Once they figure out all the proportion in their body, they would make a poster representing their finding. Throughout the project, the students will be able to write the proportion that compared the ratio of their feet to other part of their body. The outcome would similar to the pictures in the video that is shown in the engage. By doing this, the students can refer back to the engage to help them finish their project or use the engage to give them an example of what the project should look like. After the project, the students should be able to understand that proportion is the comparison of two ratios.

B2. How does this topic extend what your students should have learned in previous courses?

In previous courses, students should have covered ratios. Since proportion deals with fractions and ratios, students should be able to learn that proportion is the comparison of two ratios. This topic also extends the idea of comparing two different items to each others. With the ideas of ratios, the students should understand that units are important because they cannot compare two different ratios that are not related to each other. During algebra 1 the students should learn how to solve equations and when dealing with proportions the students may be required to solve for the missing variable in a proportion. With the knowledge of solving equations, the students will be able to cross multiply and solve for the missing variable. In conclusion, ratios, comparison of items, and solving equations should be learned before this topic is introduced. Proportion is the extended idea of ratio comparison.

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?

http://pbskids.org/cyberchase/videos/ecohaven-cse-ep-301

By showing this video in beginning of class, students are able to understand the basic meaning of proportion. This is a good video to engage students because the students are able to test out the real life situation. For example, in this video, the kids found out that the length of their foot is the same as the length of their face. Students can see that there is a proportional relationship with their own body part. With this whole episode of Cyberchase, students are able to see the different proportionality that is present with their own body. As the episode continues, the kids continue to measure different body parts to see how many foot spans would construct another body part. With the use of one type of measurement, the students will see the different proportionality that exists in the human body. During this episode, the kids measure that seven foot span is equal to the arm length and then they also discovered that the height is the same length as the arm length. Students will be able to make their own connection to proportion after seeing all the measurements mentioned in the episode.

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 again comes from my former student Michelle McKay. Her topic, from Geometry: deriving the Pythagorean Theorem.

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

Below I have attached an activity that I like to call “Being Pythagoras for a Day”. To summarize the activity, students are given instructions (with a few guiding images) that leads them to physically manipulate various shapes that demonstrate the relationship between the sides of a right triangle. By the instructions, students will derive the Pythagorean Theorem on their own and come to understand why each side in the equation is squared. Let it be noted that the title of this activity is not just a gimmick. The proof the students will work on in this activity is the same as the one Pythagoras was given credit for using.

Michelle_McKay_BeingPythagorasForADay_A

How has this topic appeared in the news?

Not even a year ago to this day, Coach Jason Garrett of the Dallas Cowboys made a splash in the world of sports and math with his unusual demands of his players: they needed to have a sound understanding of Geometry, including the Pythagorean Theorem. Garrett fully believes that players must understand the Pythagorean Theorem to make better decisions out on the field. The following quote was taken from an interview where Garrett discusses why he feels being familiar with the Pythagorean Theorem can prevent a poor decision:

“If you’re running straight from the line of scrimmage, six yards deep, that’s a certain depth, right? It takes you a certain amount of time. But if you’re doing it from 10 yards inside and running to that same six yards, that’s the hypotenuse of that right triangle. It’s longer, right? So they have to understand that, that it takes longer to do that. That’s an important thing. Quarterbacks need to understand that, too. If you’re running a route from here to get to that spot, it’s going to be a little longer, you might need to be a little fuller in your drop.”

Let this be a wakeup call for everyone who wants to become a professional football player and never thought they would have to use the Pythagorean Theorem outside of high school!

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
People can easily recognize the Egyptian pyramids as one of the wonders of the world. What is not often discussed is how the engineers and architects of the day used the Pythagorean Theorem to lay the pyramids’ foundations correctly. Those primarily responsible for the pyramids’ construction were called “rope-stretchers”. This name came from the inventive method of tying thirteen, evenly spaced knots into a rope. When the rope was pegged to the ground, a 3-4-5 triangle was produced. This allowed them to accurately and consistently map out the bases of the pyramids.

Some argue that the rope-stretchers fully understood the Pythagorean Theorem and used that knowledge to manipulate the ropes, while others argue that they were intuitively using the properties of a right triangle. Due to this area of ambiguity, it is unclear whether Pythagoras was taught the theorem by the Egyptians first, or if, through watching the process, he was able to discover the relationship of a right triangle’s sides on his own.

Interestingly enough, there exist various pieces of artwork depicting Egyptians holding ropes and using them for measurement. Just by looking at the images, it is not clear if the ropes are being used for the construction of the pyramids or for dividing land (another event where the knotted ropes were used to fairly distribute plots of land).

Sources:

Engaging students: Distinguishing between axioms, postulates, theorems, and corollaries

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 Michael Dixon. His topic, from Geometry: distinguishing between axioms, postulates, theorems, and corollaries.

A2. How can you create a project for your students?

A project that I would have my students do to show that they know what the differences between these four logical terms are to ask them to write a story to model each one. There are several subtleties between these terms that require defining. Axioms and postulates are very similar, both are terms to describe something that is held to be true, and neither require any proof. The general idea is that these are supposed to be “obvious”statement that require no argument. Theorems are ideas that are heavily proven to be true, following the axiomatic method. Corollaries, however, generally follow directly as a result of a theorem, usually requiring only very short proofs.

As an example of what the students could come up with, they could write about two different doctors, who happen to be brothers. The first is a successful general physician in a remote village. He studied for many years to become the man in his village that takes care of all the illness and injuries that the villagers suffer from time to time. He is able to take care of almost anything that requires medicine or general care. But occasionally, the physician decides that a villager needs extra care or surgery that he cannot provide, so he sends them to his brother. His brother is just as successful a doctor, but instead of studying general medicine, this brother focused only on learning how to perform any kind of surgery. When the physician sends a villager to the surgeon, the surgeon figures out what needs to be done and then operates on the villager. Between the two of them, the village hasn’t suffered a death due to sickness or injury in several years.

In this example, the physician would model an axiom, and the surgeon would represent a postulate. Both of them are known by everyone to be excellent in their functions, modeling that they are known to be true. But axioms are held to be true in general, across many categories and sciences. A postulate, however, is known to be true, but is specific to one particular field.

C3. How has this appeared in the news?

If I ask you, “who is the most famous mathematician?”what would you say? Its probably not a question that can safely be answered without causing an argument among mathematicians. But to the layman, the best answer would most likely be Albert Einstein. He is famously known for his General Theory of Relativity. After publishing this work in 1905, Einstein steadily rose to fame, for this work and later for his work on the Manhattan Project and his work in quantum mechanics. And even still today, Einstein’s work still influences the scientific community. Recently it has been reported on PBS that a previously unknown theory that Einstein was working on has surfaced that leads to the idea that he might have supported the idea of a steady-state universe. Pioneered by Fred Hoyle, steady-state theory states that the universe is constantly expanding, but not becoming less dense, hence it remains steady throughout time. Einstein even used equations from general relativity to support his theorem. The article states that Hoyle did not know of Einstein’s support, and though Hoyle’s theorem was mathematically sound, it did not become universally accepted. With Einstein’s support, that result could have turned out differently.

D2. How was this adopted by the mathematical community?

When speaking of the axiomatic method and the history of proofs of this nature, naturally the conversation takes a turn towards the ancient Greeks. Most famously, Euclid developed his geometry using postulates, axioms, theorems, and corollaries. No history would be complete without mentioning these facts. In fact, it was Euclid’s Elements and the parallel postulate that led to a focusing on deductive reasoning and a general application of the axiomatic method in the early 19th century, after the discovery of non-Euclidean geometry. When it is assumed that the negation of parallel postulate is true, an entirely different geometry than we are used to comes into being. Logically it can be reasoned and soundly proven using exactly the same method of logic as Euclidean geometry. This led to a mathematical revolution of sorts, where mathematicians began trying to formalize axiomatically all of mathematics into a system. This led to all kinds of interesting paradoxes, including the incompleteness theorem, among others.

http://www.differencebetween.com/difference-between-axioms-and-vs-postulates/

http://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/

http://www.pbs.org/wgbh/nova/next/physics/einsteins-lost-theorem-revealed/

http://www.encyclopediaofmath.org/index.php/Axiomatic_method