Engaging students: Radius, Diameter, and Circumference of a Circle

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 Michelle Contreras. Her topic, from Geometry: radius, diameter, and circumference of a circle.

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

A way to relate circumference, radius, and diameter to my student’s real life would be by incorporating an end of unit project where they find 5 circular things that are cool to them in their home. I will ask the students to measure those 5 circular objects and find the radius, diameter, and circumference: rounding to the tens place. The students will also be asked to divided the found circumference by the diameter for every object and estimate the number they find in the hundredths place. The students should keep getting the same estimated number and realize how the estimated number for π was discovered. The students are to label the 5 objects with their radius, diameter, circumference, and to present all their finding to the class including the special number they found when dividing the circumference by the diameter. I would also participate with my students in this project by finding 5 objects around my house and present it to the class as well. There is so much the students can gain by this project, not just mathematically. Students will get an opportunity to show their classmates a little bit about themselves as well as gaining confidence in their perception about their knowledge of this topic.

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

Way before William Jones, observations were made regarding the circumference and diameter of a circle. The human race became curious about the circle and made some discoveries; the people saw a relationship between π (pi), the circumference and the radius. The people observed that every time you tried to see how many times the diameter goes into the circumference a similar number was computed. There was talk about the special number being around 22/7 or 355/113 making it seem that the special number was a rational but Jones believed it was an irrational number. Not only Jones but many others before him saw that this special number approached but never quite reached a specific number because it kept going. William Jones introduced the symbol known today for this special number: π in 1706. Though there is a belief by many that Leonhard Euler was the first to introduce and talk about the symbol π, Jones however, published his second book Synopsis Palmanorum Matheseos in 1706 using the symbol π. William Jones was a self-taught mathematician that was born in 1675 that only had a “local charity school education”. Interesting enough Jones was served for the navy before becoming a math teacher. William Jones would charge a fee to those who come to a coffee shop and listen to his lectures in London. Based on the website historytoday.com William Oughtred “used π to represent the circumference of a given circle, so that his π varied according to the circle’s diameter, rather than representing the constant we know today.” The symbol pi is an important irrational number that connects the circumference to the radius and diameter of a circle. There has been many mathematicians who have contributed in some way to this symbol pi regarding circumference.

 

 

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

YouTube is a great tool to use as a classroom teacher, there are many educational videos that can be beneficial to the student’s education. There are videos that include examples and visuals that your students may or may not relate to. Particularly for the topic regarding circumference and its different features radius and diameter there were many intriguing videos that I came across with. There was one video that I felt I would totally use in one of my lesson about circumference. The video is called “Math Antics: Circles, What is Pi?” I would only show about 3 minutes of the video as an engage at the beginning of the lesson, I really liked however how they explained the definition of a circle with visuals. I believe this video will be very beneficial for my students before starting the unit over the circumference, it also does a very good job at capturing the attention of the audience and explaining pi. I have always believed YouTube to be a great tool for educational purposes but there is a website called mathisfun.com which is my go to for a better explanation or summary of certain concepts. This website gives you really good real world examples that anyone can relate to and great ideas for short engaging activities. The definitions are simplified so any middle school student can understand a concept. The website not only have great examples that you can talk about in your classroom as an engage but also have easy to follow explanations. I would definitely use this website when having trouble explaining, in a simpler form, a certain topic to my students.

References:
“William Jones and his Circle: The Man who invented Pi”. July 2009 http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi

“Math Antics: Circles, What is Pi?” https://www.youtube.com/watch?v=cC0fZ_lkFpQ

https://www.mathsisfun.com/geometry/circle.html

 

 

 

 

Engaging students: Using Euler’s theorem for polyhedra

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 Megan Termini. Her topic, from Geometry: using Euler’s theorem for polyhedra.

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

It is important for us as teachers to create an activity or project that is fun and engaging for the students. One activity that I found using Euler’s Theorem for Polyhedra is a math riddle activity by Albert J. and B. Michael (Reference A). The activity requires students to work in groups and work as a team to create four different types of convex polyhedra using Toobeez. The teacher will first build two Toobeez models, one of a polyhedron and one of a polygon. Then the students will be divided into groups and will build the four different types of convex polyhedra. After building each model, they students will fill in a table for each model the number of faces, vertices and edges. Using previous knowledge, the students will look at what they have recorded in the table and name each figure. Then they will discuss as a group to try to determine the relationship between F, V, and E. Give them some time to all come to a conclusion and then discuss as a whole class their discoveries and compare to Euler’s formula. This is a great way of having the students work together in groups and having them discover Euler’s Formula on their own, instead of just giving it to them.

 

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

Using technology as an activity for your lesson is a good way of keeping your students engaged and wanting to learn about the topic. When learning about Euler’s Theorem for Ployhedra, I found a great website, called Annenberg Learner, for students to use to apply the equation and even find it. After the lesson, this would be a great way for students to apply what they have just learned in the lesson. The activity asks you to choose a 3D shape and it will show you the net of that shape. Then the student will fill in the table on how many faces, vertices and edges that shape has. It will have you try again if you get a number wrong on the table. After you fill in the table, you then will get asked if you see a pattern and any relationship between F, V, and E. Once they see the pattern, they then get to try to fill in the parts of the equation (Reference B). This is a fun and engaging way of students being able to see the table and discover Euler’s Formula on their own or in groups.

 

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

This is a very important topic for students to learn at the beginning. As they continue their mathematics or science education, they will see these 3D figures a lot and will have to find more than just the faces, sides, and vertices. This topic will help students when learning about Platonic Solids. There is a video that I found that would be a great way for them to not only remember Platonic Solids, but also Euler’s Formula! (https://www.youtube.com/watch?v=C36h00d7xGs) It is one of my favorite videos of all time and it will get stuck in your head after listening to it a few times. They also will be using Euler’s Theorem in future geometry classes and even classes in college. They will have to solve for volume and area of these 3D figures and more. So, this topic is the start to what is more to come. This is also an important topic when it comes to jobs like building and constructing buildings and bridges. They would need to know how many faces, vertices and edges the building needs.

References:

A. “Euler’s Formula | Leaning Math Through Fun Activities.” TOOBEEZ Activity Central, 2 June 2014, http://www.toobeez.com/activities/eulers-formula/.
B. “Euler’s Theorem.” Annenberg Learner, http://www.learner.org/interactives/geometry/eulers-theorem/.

Engaging students: Defining the terms collinear and coplanar

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 Kerryana Medlin. Her topic, from Geometry: defining the terms collinear and coplanar.

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As a quick assessment of knowledge and to get the students up and moving, you could set up the class like this (tape on the walls and floor representing lines):

(Bear in mind that you do not have to put crosses up on your walls. The strips of tape can be of any orientation.)

You have different students have their name on a card with a piece of tape on the back so they can stick it to the wall.

We place my name card on a “line” (a piece of tape) on a “plane” (a wall or floor).

We then draw a name from and I ask them to put their name somewhere coplanar/ collinear/ not coplanar/ not collinear to mine. This can be randomly generated by whichever method I choose.

The student then places their name in relation to mine as specified. Then the students confirm the placement with thumbs up or down.

Then we do the same with the next selected student but in relation to the previous student until either all have gone.

You can always minimize the number of students by selecting them randomly and giving them different colored stickers or something similar.   

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How has this topic appeared in pop culture?

In Dungeons and Dragons, there are other planes in which other beasts live. So, demons and the like live on another plane of existence. So, these creatures are not coplanar to humans, elves, dwarves, and such. The following is a chart of the different planes of existence in dungeons and dragons:

This is not the only example of alternate universes and planes, but it is one of the few pop culture references to different planes which is not gimmicky. For example, both The Simpsons and Family Guy have sent their characters to alternate planes of existence.

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How has this topic appeared in high culture?

In paintings, to paint three dimensional spaces, artists have to consider objects as being in the foreground or background. This is an example of two planes of a painting. As well as orienting objects in such a way that they are “laid against” a plane. Within the composition of the paintings, there are typically diagonal lines which can be used to create the illusion of three dimensions in a two-dimensional medium.

Tintoretto: Last Supper
A wonderful example of this concept is The Last Supper by Tintoretto. The table defines the planes of the room. The left-most edge (blue) being the background and the right-most edge (yellow) the foreground. We see that for the blue plane that the ceiling, the heads of the people, and the edge of the table are all set in this plane, so the background is clearly set.
Because of this, we can also tell that the middle of the painting has a plane defined by the shadow of the table on the floor, the right edge of the table, and the “empty space” above. This illusion of empty space was created because the background plane exists.

Sources
Hadaller, Scott. (2011). Anethemalon Planes of Existence. Inspired Mythos. Retrieved from
http://inspiredmythos.blogspot.com/2011/02/anethemalon-planes-of-existance.html
Tintoretto. (1592-94). The Last Supper. Encyclopedia Brittanica. Retrieved from
https://www.britannica.com/topic/The-Last-Supper-by-Tintoretto

Engaging students: Area of a triangle

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 Kelly Bui. Her topic, from Geometry: finding the area of a triangle.

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

As an activity, possibly the “exploration” part of the lesson, students will be paired in partners and the instructor will provide each pair with a different rectangle or square. The goal is to find the area of half of a rectangle.

The condition they must follow is that they cannot “draw” a straight line across the shape, they must “draw” a straight line starting from a corner. At some point, it should be evident that you can only draw a straight line from a corner to another corner. By drawing a diagonal line across the rectangle, they will now have two triangles (if that isn’t clear to them at this point, let them realize it on their own or go over it as a class at the end of the activity). Using rulers or meter sticks, they will have to discover on their own what the area of half of the rectangle is along with what the formula for that looks like. Most students will probably take the area of the entire rectangle and divide by 2. Once they come up with a formula for the area of half of a rectangle, it should look like A=1/2 bh, tell each student to raise half of the rectangle they cut, and announce: “congratulations, you have found the formula for the area of a triangle.”

 

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

Students begin to see the formula for the area of a triangle in 6th or 7th grade. They know the formula, but often times they don’t understand where it comes from. It can be useful for future homework/test problems that ask for the area of an irregular shape as well as in algebra with unknown lengths. These types of problems require students to think “outside of the box” in order to find the area of an irregular shape. It is not always evident that the irregular shape is simply made out of polygons.

Additionally, this topic will be useful when students are in algebra and they must solve for the area of a polygon that doesn’t have specific dimensions. For example, the trapezoid below has an unknown height as well as an unknown base. It is good for students to know how to apply the area formula of a triangle to solve for the dimensions as well as the area of the entire trapezoid itself. One important thing that should be stressed in the classroom is that formulas are extremely helpful on their own, but they’re even more helpful when they can be applied to different applications.

image1.jpg

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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?

Most students know the formula A=1/2 bh for the area of a triangle, but many students don’t remember the formula used to find the area when all three side lengths are known. Heron’s Formula: √[s(s-a)(s-b)(s-c)] is briefly mentioned in geometry and is often not used in other math courses in high school. Along with his derivation of Heron’s formula, he contributed greatly to ancient society.
Heron of Alexandria was a Greek engineer and mathematician who was known mostly for his work with geometry. He was also a lecturer at the Library/Museum of Alexandria where he would meet with other scholars and discuss work. Additionally, he wrote Metrica, a series of three books which included his work on area and volumes of different types of figures.
It is no secret that Heron had a brilliant mind, and with his engineering and mathematics background, he was actually ahead of the industrial revolution that would take place centuries later. He invented the “Hero Engine, also known as the “aeolipile,” which was powered by steam. Essentially, Heron was the first inventor of the steam engine.
Another one of Heron’s inventions was the “wind wheel,” which is very similar to the modern windmill.

Students will already know that there were many breakthroughs during the industrial revolution, but some of the machines and inventions implemented in the 1800s were actually ideas that were invented centuries before.

Irregular Shape Image: http://www.softschools.com/math/geometry/topics/the_area_of_irregular_figures/

Trapezoid Image: http://www.dummies.com/education/math/geometry/how-to-calculate-the-area-of-a-trapezoid/

Heron of Alexandria: https://www.britannica.com/biography/Heron-of-Alexandria

Hero Engine (aeolipile): http://www.ancient-origins.net/ancient-technology/ancient-invention-steam-engine-hero-alexandria-001467

The Professor vs. the NSA

I didn’t know this interesting bit of internet history:

“It’s July 1977,” Hellman tells the audience. “Whit and I are involved in a major fight with NSA over the data encryption standard.”

American law banned the unlicensed export of weapons. Makes sense: the government doesn’t want civilians wandering into Moscow with a trenchcoat full of fighter jet parts. The question is: Does this law apply to abstract mathematical ideas? By developing new approaches to cryptography, are Hellman, Diffie, and their collaborators de facto arms traffickers? If so, Hellman says, “then by publishing our papers in international journals, we are in some sense exporting plans for implements of war.”

“I think the penalty,” Hellman recalls, “was something like five years in jail.”

Full story: https://mathwithbaddrawings.com/2017/10/11/the-professor-vs-the-nsa/

Engaging students: Vectors in two dimensions

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 Sarah McCall. Her topic, from Precalculus: vectors in two dimensions.

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

For such an applicable topic, I believe that it is beneficial to have students see how this might apply to their lives and to real world problems. I selected the following word problems because they are challenging, but I think it is necessary for students to be a little frustrated initially so that they are able to learn well and remember what they’ve learned.

1. A DC-10 jumbo jet maintains an airspeed of 550 mph in a southwesterly direction. The velocity of the jet stream is a constant 80 mph from the west. Find the actual speed and direction of the aircraft.

2. The pilot of an aircraft wishes to head directly east, but is faced with a wind speed of 40 mph from the northwest. If the pilot maintains an airspeed of 250 mph, what compass heading should be maintained? What is the actual speed of the aircraft?

3. A river has a constant current of 3 kph. At what angle to a boat dock should a motorboat, capable of maintaining a constant speed of 20 kph, be headed in order to reach a point directly opposite the dock? If the river is ½ a kilometer wide, how long will it take to cross?

Because these problems are difficult, students would be instructed to work together to complete them. This would alleviate some frustrations and “stuck” feelings by allowing them to ask for help. Ultimately, talking through what they are doing and successfully completing challenging problems will take students to a deeper level of involvement with their own learning.

 

 

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

I believe vectors are fairly easy to teach because there are so many real life applications of vectors. However, it can be difficult to get students initially engaged. For this activity, I would have students work in groups to complete a project inspired by Khan Academy’s videos on vector word problems. Students would split off into groups and watch each of the three videos on Khan Academy that have to do with applications of vectors in two dimensions. Using these videos as an example, students will be instructed to come up with a short presentation or video that teaches other students about vectors in two dimensions using real world applications and examples.

 

 

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

Immediately when I see vectors, I think of one specific movie quote from my late childhood that I’ll always remember. The villain named Vector from Despicable Me who “commits crimes with both direction AND magnitude” is a fellow math nerd and is therefore one of my favorite Disney villains of all time. So of course, I had to find the clip (linked below) because I think it is absolutely perfect for engaging students in a lesson about vectors as soon as they walk in the door, and it is memorable and educational. I would refer back to this video several times throughout the lesson and in future lessons because it is a catchy way to remember the two components to vectors. This would also be great to kick off a unit on scalars and vectors, because it would get kids laughing and therefore engaged, plus they will always remember the difference between a scalar and a vector (direction AND magnitude!).  

References:

  1. https://www.khanacademy.org/math/precalculus/vectors-precalc/applications-of-vectors/v/vector-component-in-direction
  2. https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwj42PaGqojXAhXKSiYKHTvLD8oQFgguMAE&url=http%3A%2F%2Fwww.jessamine.k12.ky.us%2Fuserfiles%2F1038%2FClasses%2F17195%2FVector%2520Word%2520Problems%2520Practice%2520Worksheet%25202.docx&usg=AOvVaw1IHTinEQtGK4Ww1_JkBhHf
  3. https://www.youtube.com/watch?v=bOIe0DIMbI8

Engaging students: Graphing Sine and Cosine Functions

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 Saundra Francis. Her topic, from Precalculus: graphing sine and cosine functions.

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

To engage students you can have them record a song using the WavePad app. Have students choose a clip of their favorite song to record. Once they record the song they choose, the app will display the sound waves compiled that are used to create the sounds in the song the song. Students will realize that sound waves are in the form of the sine function. This will engage students since you would have related the topic of graphing sine and cosine functions to their favorite song. You could also have students create their own sounds and record them with the app to see the graph associated with the sound they made. Students can look at their sound and other classmates sounds are recognize differences in the waves, you can relate this to the equation f(x)=asin(bx+c)+d. You can them work with students to discover what the constants terms mean in relation to the parent function of sine.

 

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

Sine waves are the basis of sound. Have a piece by Beethoven playing while students are entering the classroom. Tell students that Beethoven was able to create music while he was in the process of becoming deaf. Ask students how they think Beethoven was able to create music in spite of that set back. After you have students share some answers show them the video above which explains how Beethoven’s music (all music) is related to sine waves. The music of his “Moonlight Sonata” is explained using math in the TED-Ed video. While Beethoven did not use this method to create his music he said that he knew what the music looked like. This will show students an example of how sine graphs are used in real life and get them interested in graphing sine and cosine functions.

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Students will be given a TI-Nspire calculator in order to discover how changing the amplitude, period, horizontal shift, or vertical shift changes the equation of sine. Students will start with the graph of f(x)=sinx. They will then manipulate the graph on the calculator to change the function. Have them move the function up and down, right and left, and work with the slope of sinx and the slope of the x. Have students write dawn some of their new functions and sketch the graph. They will then compare how changing the graph effects the equation of f(x)=sinx. Introduce f(x)=asin(bx+c)+d . Give students some time to compare the functions that they created to the formula and describe how each constant changes the graph. Students will hopefully discover how the function f(x)=asin(bx+c)+d relates to amplitude, period, horizontal shift, and vertical shift.

References
1. https://education.ti.com/en/timathnspired/us/detail?id=4E9BA7808CA74F6599BD5EA2037C088A&t=C52AEC55A39243D182772F76318B901C
2. https://www.smore.com/gy9h4-sine-waves-and-music
3. https://www.youtube.com/watch?time_continue=16&v=zAxT0mRGuoY

Engaging students: Dot product

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 Precalculus: computing a dot product.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

This website gives an example of a word problem that students could solve, and it has real-world applications. It is not a complete worksheet for students to work on. The teacher would have to create more word problems incorporating the idea of this website. The example on this web page is that you are a local store owner and are selling beef, chicken, and vegetable pies 3 days a week. The owner has a list of how many pies he sells a day and how much they cost. The cost of beef pies are $3, chicken pies are $4, and vegetable pies are $2. On Monday he sells 13 beef, 8 chicken, and 6 vegetable pies. Tuesday he sells 9 beef, 7 chicken, and 4 vegetable pies. Finally, on Thursday the owner sold 15 beef, 6 chicken, and 3 vegetable pies. Now, let’s think about how we can solve for the total number of sales for Monday. First, we would solve for the sales of the beef pies by multiplying the price of the pie and the number we sold. Then we would do the same for chicken and vegetable pies. After finding the sales of the three pies, we would add up sales to get the total amount for the day. In this case, we would get $83 of sales on Monday. The students would do the same thing for the other days the store is open. This is an example of the dot product of matrices in a word problem.

https://www.mathsisfun.com/algebra/matrix-multiplying.html

 

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

An idea I was able to see in an actual classroom during observation this week was the use of Fantasy Football in matrices. A teacher at Lake Dallas High School has her classes in a Football Fantasy League competing against each other. The way they started this activity is that the students have to keep up with the points that their teams are earning. They are doing this by the information the teacher gives them about how to score their players. Each class chooses one quarterback, running back, wide receiver, kicker, and defense to represent their team. The point system is the same as in the online fantasy. For instance, Aaron Rodgers, quarterback for the Green Bay Packers, throws for 300 yards, two touchdowns, and one interception. The points Rodgers earns you for the week comes from taking the several yards and multiplying by the points earned for each yard. Then, do the same for touchdowns and interceptions. After computing this, you will then add the numbers up to get the total points you receive from Aaron for the week. This is using dot product because we have two matrices, which are the stats that the player receives in the game, and the points you get for those same stats. By doing this activity, the students would be working on this aspect of pre-calculus for the entire football season.

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Graphing calculators would be a great way to use technology to teach this topic. When computing the dot product of two matrices, there are two ways to do it. One is by hand and the other is a calculator. As the teacher, it would be more efficient for you to see how students are learning the material by having them compute it by hand, but no student wants to do that with every problem. A way the teacher could incorporate solving for the dot product using a calculator in an engaging way would be to have students complete a scavenger hunt. In the scavenger hunt, students will have to solve problems of the dot product to get the next clue and move on to the next. The idea of this would be for the students to show that they can work the calculator and actually get answers. You could have anywhere from five to ten questions for them to solve and decoy answers throughout the room with little mishaps. This would get the students up and moving for this activity

 

Decimal Approximations of Logarithms (Part 5)

While some common (i.e., base-10) logarithms work out evenly, like \log_{10} 10,000, most do not. Here is the typical output when a scientific calculator computes a logarithm:

In today’s post, I’ll summarize the past few posts to describe how talented Algebra II students, who have just been introduced to logarithms, can develop proficiency with the Laws of Logarithms while also understanding that the above answer is not just a meaningless jumble of digits. The only tools students will need are

To estimate \log_{10} 5.1264, Algebra II students can try to find a power of 5.1264 that is close to a power of 10. In principle, this can be done by just multiplying by 5.1264 until an answer decently close to 5.1264 arises. For the teacher who’s guiding students through this exploration, it might be helpful to know the answer ahead of time.

One way to do this is to use Wolfram Alpha to find the convergents of \log_{10} 5.1264. If you click this link, you’ll see that I entered

Convergents[Log[10,5.1264],15]

A little explanation is in order:

  • Convergents, predictably, is the Mathematica command for finding the convergents of a given number.
  • Log[10,5.1264] is the base-10 logarithm of 5.1264. By contrast, Log[5.1264] is the natural logarithm of 5.1264. Mathematica employs the convention of that \log should be used for natural logarithms instead of \ln, as base-10 logarithms are next to useless for mathematical researchers. That said, I freely concede that this convention is confusing to students who grew up thinking that \log should be used for base-10 logarithms and \ln for natural logarithms. (See also my standard joke about using natural logarithms.) Naturally, the 5.1264 can be changed for other logarithms.
  • The 15 means that I want Wolfram Alpha to give me the first 15 convergents of \log_{10} 5.1264. In most cases, that’s enough terms to provide a convergent whose denominator is at least six digits long. In the rare instance when this doesn’t happen, a number larger than 15 can be entered.

From Wolfram Alpha, I see that \displaystyle \frac{22}{31} is the last convergent with a numerator less than 100. For the purposes of this exploration, I interpret these fractions as follows:

  • The best suitable power of 5.1264 for an easy approximation on a scientific calculation will be (5.1264)^{31}. In this context, “best” means something that’s close to a power of 10 but less than 10^{100}. Students entering (5.1264)^{31} into a calculator will find

(5.1264)^{31} \approx 1.009687994 \times 10^{22}

(5.1264)^{31} \approx 10^{22}

In other words, the denominator of the convergent \displaystyle \frac{22}{31} gives the exponent for 5.1264, while the numerator gives the exponent for the approximated power of 10. Continuing with the Laws of Logarithms,

\log_{10} (5.1264)^{31} \approx \log_{10} 10^{22}

31 \log_{10} 5.1264 \approx 22

\log_{10} 5.1264 \approx \displaystyle \frac{22}{31}

\log_{10} 5.1264 \approx 0.709677\dots

A quick check with a calculator shows that this approximation is accurate to three decimal places. This alone should convince many students that the above apparently random jumble of digits is not so random after all.

While the above discussion should be enough for many students, some students may want to know how to find the rest of the decimal places with this technique. To answer this question, we again turn to the convergents of \log_{10} 5.1264 from Wolfram Alpha. From this list, we see that \displaystyle \frac{89,337}{125,860} is the first convergent with a denominator at least six digits long. The student therefore has two options:

Option #1. Ask the student to use Wolfram Alpha to raise 5.1264 to the denominator of this convergent. Surprisingly to the student, but not surprisingly to the teacher who knows about this convergent, the answer is very close to a power of 10: 10^{89,337}. The student can then use the Laws of Logarithms as before:

\log_{10} (5.1264)^{125,860} \approx \log_{10} 10^{89,337}

125,860 \log_{10} 5.1264 \approx 89,337

\log_{10} 5.1264 \approx \displaystyle \frac{89,337}{125,860}

\log_{10} 5.1264 \approx 0.70981249006\dots,

which matches the output of the calculator.

Option #2. Ask the student to “trick” a hand-held calculator into finding (5.1264)^{125,860}. This option requires the use of the convergent with the largest numerator less than 100, which was \displaystyle \frac{22}{31}.

  • Option #2A: Use the Microsoft Excel spreadsheet that I’ve written to perform the calculations that follow.
  • Option #2B: The student divides the smaller denominators into the larger denominator and finds the quotient and remainder. It turns out that 125,860 = 31 \times 4060 + 0. (This is a rare case where there happens to be no remainder.) Next, the student uses a hand-held calculate to compute

\displaystyle \left( \frac{(5.1264)^{31}}{10^{22}} \right)^{4060} \times (5.1264)^0

In this example, the \times (5.1264)^0 is of course superfluous, but I include it here to show where the remainder should be placed. Entering this in a calculator yields a result that is close to 10^{17}. (The teacher should be aware that some of the last few digits may differ from the more precise result given by Wolfram Alpha due to round-off error, but this discrepancy won’t matter for the purposes of the student’s explorations.) In other words,

\displaystyle \left( \frac{(5.1264)^{31}}{10^{22}} \right)^{4060} \times (5.1264)^0 \approx 10^{17},

which may be rearranged as

(5.1264)^{125,860} \approx 10^{89,337}

after using the Laws of Exponents. From this point, the derivation follows the steps in Option #1.

Decimal Approximations of Logarithms (Part 4)

While some common (i.e., base-10) logarithms work out evenly, like \log_{10} 10,000, most do not. Here is the typical output when a scientific calculator computes a logarithm:

To a student first learning logarithms, the answer is just an apparently random jumble of digits; indeed, it can proven that the answer is irrational. With a little prompting, a teacher can get his/her students wondering about how people 50 years ago could have figured this out without a calculator. This leads to a natural pedagogical question:

Can good Algebra II students, using only the tools at their disposal, understand how decimal expansions of base-10 logarithms could have been found before computers were invented?

Here’s a trial-and-error technique — an exploration activity — that is within the grasp of Algebra II students. It’s simple to understand; it’s just a lot of work. The only tools that are needed are

  • The Laws of Logarithms
  • A hand-held scientific calculator
  • Access to the Wolfram Alpha website (optional)
  • A lot of patience multiplying x by itself repeatedly in a quest to find integer powers of x that are close to powers of 10.

In the previous post in this series, we found that

3^{153} \approx 10^{73}

and

3^{323,641} \approx 10^{154,416}.

Using the Laws of Logarithms on the latter provides an approximation of \log_{10} 3 that is accurate to an astounding ten decimal places:

\log_{10} 3^{323,641} \approx \log_{10} 10^{154,416}

323,641 \log_{10} 3 \approx 154,416

\log_{10} 3 \approx \displaystyle \frac{154,416}{323,641} \approx 0.477121254723598\dots.

Compare with:

\log_{10} 3 \approx 0.47712125471966\dots

Since hand-held calculators will generate identical outputs for these two expressions (up to the display capabilities of the calculator), this may lead to the misconception that the irrational number \log_{10} 3 is actually equal to the rational number \displaystyle \frac{154,416}{323,641}, so I’ll emphasize again that these two numbers are not equal but are instead really, really close to each other.

We now turn to a question that was deferred in the previous post.

Student: How did you know to raise 3 to the 323,641st power?

Teacher: I just multiplied 3 by itself a few hundred thousand times.

Student: C’mon, really. How did you know?

While I don’t doubt that some of our ancestors used this technique to find logarithms — at least before the discovery of calculus — today’s students are not going to be that patient. Instead, to find suitable powers quickly, we will use ideas from the mathematical theory of continued fractions: see Wikipedia, Mathworld, or this excellent self-contained book for more details.

To approximate \log_{10} x, the technique outlined in this series suggests finding integers m and n so that

x^n \approx 10^m,

or, equivalently,

\log_{10} x^n \approx \log_{10} 10^m

n \log_{10} x \approx m

\log_{10} x \approx \displaystyle \frac{m}{n}.

In other words, we’re looking for rational numbers that are reasonable close to \log_{10} x. Terrific candidates for such rational numbers are the convergents to the continued fraction expansion of \log_{10} x. I’ll defer to the references above for how these convergents can be computed, so let me cut to the chase. One way these can be quickly obtained is the free website Wolfram Alpha. For example, the first few convergents of \log_{10} 3 are

\displaystyle \frac{1}{2}, \frac{10}{21}, \frac{21}{44}, \frac{52}{109}, and \frac{73}{153}.

A larger convergent is \frac{154,416}{323,641}, our familiar friend from the previous post in this series.

As more terms are taken, these convergents get closer and closer to \log_{10} 3. In fact:

  • Each convergent is the best possible rational approximation to \log_{10} 3 using a denominator that’s less than the denominator of the next convergent. For example, the second convergent \displaystyle \frac{10}{21} is the closest rational number to \log_{10} 3 that has a denominator less than 44, the denominator of the third convergent.
  • The convergents alternate between slightly greater than \log_{10} 3 and slightly less than \log_{10} 3.
  • Each convergent \displaystyle \frac{m}{n} is guaranteed to be within \displaystyle \frac{1}{n^2} of \log_{10} 3. (In fact, if \displaystyle \frac{m}{n} and \displaystyle \frac{p}{q} are consecutive convergents, then \displaystyle \frac{m}{n} is guaranteed to be within \displaystyle \frac{1}{nq} of \log_{10} 3.)
  • As a practical upshot of the previous point: if the denominator of the convergent \displaystyle \frac{m}{n} is at least six digits long (that is, greater than 10^5), then \displaystyle \frac{m}{n} must be within \displaystyle \frac{1}{(10^5)^2} = 10^{-10} of \log_{10} 3… and it’ll probably be significantly closer than that.

So convergents provide a way for teachers to maintain the illusion that they found a power like 3^{323,641} by laborious calculation, when in fact they were quickly found through modern computing.