Engaging students: Powers and exponents

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 Andrew Cory. His topic, from Pre-Algebra: powers and exponents.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

Exponents are just an easier way to multiply the same number by itself numerous times. They extend on the process of multiplication and allow students to solve expressions such as 2*2*2*2 quicker by writing them as 2^4. They are used constantly in future math courses, almost as commonly as addition and multiplication. Exponential functions start becoming more and more common as well. They’re used to calculate things such as compounding interest, or growth and decay. They also become common when finding formulas for sequences and series.
In science courses, exponents are often used for writing very small or very large numbers so that calculations are easier. Large masses such as the mass of the sun are written with scientific notation. This also applies for very small measurements, such as the length of a proton. They are also used in other ways such as bacteria growth or disease spread which apply directly to biology.

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C2. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Any movie or TV show about zombies or disease outbreaks can be referenced when talking about exponents, and exponential growth. The rate at which disease outbreaks spread is exponential, because each person getting infected has a chance to get more people sick and it spreads very quickly. This can be a fun activity to demonstrate with a class to show how quickly something can spread. A teacher can select one student to go tap another student on the shoulder, then that student also gets up and walks around and taps another student. With students getting up and “infecting” others, more and more people stand up with each round, showing how many people can be affected at once when half the class is already up and then the other half gets up in one round.

 

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

Euclid discovered exponents and used them in his geometric equations, he was also the first to use the term power to describe the square of a line. Rene Descartes was the first to use the traditional notation we use for exponents today. His version won out because of conceptual clarity. There isn’t exactly one person credited with creating exponents, it is more of a collaborative thing that got added onto over time. Archimedes discovered and proved the property of powers that states 10^a * 10^b = 10^{a+b}. Robert Recorde, the mathematician who created the equals sign, used some interesting terms to describe higher powers, such as zenzizenzic for the fourth power and zenzizenzizenzic for the eighth power. At a time, some mathematicians, such as Isaac Newton, would only use exponents for powers 3 and greater. Expressing things like polynomials as ax3+bxx+cx+d.

References:

Berlinghoff, W. P., & Gouvêa, F. Q. (2015). Math through the ages: A gentle history for teachers and others.

Wikipedia contributors. (2019, August 28). Exponentiation. In Wikipedia, The Free Encyclopedia. Retrieved 00:24, August 31, 2019, from https://en.wikipedia.org/w/index.php?title=Exponentiation&oldid=912805138

 

Engaging students: Absolute value

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 Haley Higginbotham. Her topic, from Pre-Algebra: absolute value.

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

A fun activity to do would be to have a graph on the floor taped out and divide people into pairs and give them sets of points to graph. Then, they would have to measure how far away they were from origin. I would ask if it mattered that the x and y values were sometimes negative, and why or why not. Hopefully they’ll respond that since they were measuring distance, and distance isn’t negative, then it didn’t matter if the x and y values were negative. And that would lead into the idea that absolute value refers to the distance from origin, and it doesn’t just “make the negative a positive number.” If I were to teach absolute value, I would very much want to emphasize this point because even though it seems like the absolute value just magically gets rid of negative signs, it is important to know what it actually is.

 

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D2. How was this topic adopted by the mathematical community?

Originally, the term absolute value came from Jean-Robert Argand’s term ‘module’ (unit of measure in French). The term wasn’t commonly used in English until about 1857. The standard notation of vertical bars came from Karl Weierstrass in the time intermediate time. Now, the notation of vertical bars is used for different purposes in other areas of mathematics, like determinants and cardinality, which don’t relate to distance. However, the idea of absolute value (or magnitude) extends to the realm of physics, and science in general. Generally, when you want to know how far an object has traveled, but it has returned to its original position, you take the magnitude of the distance. In physics, you often want to find the magnitude of a vector, in order to know the distance. It’s also helpful because you can extend this idea into multiple dimensions, even though the calculations can become longer than just removing the negative sign.

 

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E1. How can technology be used?

GeoGebra’s graphing calculator is fantastic for math in general because it has a wide range of functionality besides just graphing. In terms of absolute value, you can graph the absolute value function easily and it will actually pop up with the vertical bars next to it and not just abs(x) which is good since then student can get more familiar with the notation. GeoGebra allows you to measure distance between points, which is really the important tool in this case. You can easily plot different points and measure the distances to verify more accurately that the distances are the same regardless of sign. GeoGebra is also fairly intuitive to use, which is good if you have students who aren’t very familiar with using technology. Plus, it’s just plain fun to play with and students will love the fact they don’t have to graph a bunch of points and functions by hand.

References:
en.wikipedia.org/wiki/Absolute_value
geogebra.org/graphing

Engaging students: Arithmetic series

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 Eduardo Torres Manzanarez. His topic, from Precalculus: arithmetic series.

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A1) What interesting (i.e., uncontrived) word problems using this topic can your students do now?

One interesting word problem to ask students to get them thinking about the idea of an arithmetic series, specifically a finite arithmetic series, is to have students come up with the total sum of the first 100 positive integers larger than 0 (i.e., 1 to 100) without actually adding all the integers up. Students will probably not figure out the total sum without adding the integers up one by one but if students are shown these numbers physically as cards labeled then a few might notice that the numbers taken at each end form pairs that add to the same sum. Turns out that the total sum is the number of pairs multiplied by 101. It can be explained to students that the 101 results from taking the first term and the last term (i.e., 1 and 100) and seeing that the sum is 101. This is true when we add 2 and 99, 3 and 98, 4, and 97, and so on. Hence, we will have 50 pairs since we have 100 numbers and so we have 50*101 as our sum. This problem can be extended to the story Gauss and how he apparently solved this problem as a child relatively fast and the teacher pointed out this question to them because he was apparently lazy. Now, this can be extended to adding all the integers from 1 to 200 and so on and having students come up with a general formula. Students can then think about an odd number of integers and see if that formula holds. Lastly, the connection between adding a number of terms with the same difference between each term is defined as an arithmetic series and so all the problems they have been doing are arithmetic problems in disguise.

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B2) How can this topic be used in your students’ future courses in mathematics and science?

This topic is heavily used when discussing convergence in calculus. It provides insight into the validity that every series has a total sum that can be written as a number. Turns out this is true for all series that are finite but when discussing infinite series, it can be true of false that it converges to an actual value. So, students will have to ponder this idea for infinite arithmetic series in the future. Also, arithmetic series can be used to model certain situations in science within biology and physics. Thinking about arithmetic series provides information in tackling other types of series such as geometric in terms of behavior and solution. How does a geometric series behave? Well, each term increases with a common ratio instead of a common addition. Does the finite series converge? Yes, we know that every finite series does and this one basically behaves like the arithmetic in which we can easily find the total sum using a formula. Does the infinite series converge? Well, just like an arithmetic series it depends on the situation and the terms within the problem.

 

 

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

This topic has appeared in a particular movie called “All Quiet on the Western Front” which was released in 1930 and is an adaption of the novel that was published in 1929 by Erich Remarque. Within this movie, there is a scene in which a soldier states the formula for finding the sum of an arithmetic series. The soldier specifically states the formula S = A + N*(L / 2) and this corresponds to arithmetic series in accordance with the area of a rectangle and the area of a triangle. This is in a way a longer version of the short-hand formula we use today. One particular statement made from the soldier is that he mentions how beautiful the formula is. For some students, they can probably relate to the idea that something so complicated as adding 100000 terms that have a constant difference can be found using a short formula. Many problems in mathematics seem complicated at first in accordance with doing “grunt work” but many of them have beautiful solutions to them.

 

The perfect masks for my family

If you’d like these for yourself, I bought these at https://www.teachloveinspire.net/campaigns/-/-/stores/math-teacher/tc-d-28062018-m-math-funny?retailProductCode=7684B076E6A1E6-B5149C444C3B-S3CFM750-TPS1900-BLK&fbclid=IwAR3QMQawI3OJ2T81eXxobbNq61iRjEQ5O-3oOm2b0Eml7mM2alIwlpjgMrE

Garbage math

I really liked xkcd’s take on numerical analysis and error propagation:

Source: https://xkcd.com/2295/

A good mathematical explanation of this comic can be found here: https://www.explainxkcd.com/wiki/index.php/2295:_Garbage_Math

Engaging students: Computing logarithms with base 10

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 Andrew Sansom. His topic, from Precalculus: computing logarithms with base 10.

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D1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The slide rule was originally invented around 1620, shortly after Napier invented the logarithm. In its simplest form, it uses two logarithmic scales that slide past each other, allowing one to multiply and divide numbers easily. If the scales were linear, aligning them would add two numbers together, but the logarithmic scale turns this into a multiplication problem. For example, the below configuration represents the problem: 14 \times 18=252.

Because of log rules, the above problem can be represented as:

\log 14 + \log 18 = \log 252

The C-scale is aligned against the 14 on the D-scale. The reticule is then translated so that it is over the 18 on the C-scale. The sum of the log of these two values is the log of their product.

Most modern students have never seen a slide rule before, and those that have heard of one probably know little about it other than the cliché “we put men on the moon using slide rules!” Consequently, there these are quite novel for students. A particularly fun, engaging activity to demonstrate to students the power of logarithms would be to challenge volunteers to a race. The student must multiply two three-digit numbers on the board, while the teacher uses a slide rule to do the same computation. Doubtless, a proficient slide rule user will win every time. This activity can be done briefly but will energize the students and show them that there may be something more to this “whole logarithm idea” instead of some abstract thing they’ll never see again.

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

Computing logarithms with base 10, especially with using logarithm properties, easily leads to learning to compute logarithms in other bases. This generalizes further to logarithmic functions, which are one of the concepts from precalculus most useful in calculus. Integrals with rational functions usually become problems involving logarithms and log properties. Without mastery of the aforementioned rudimentary skills, the student is quickly doomed to be unable to handle those problems. Many limits, including the limit definition of e, Euler’s number, cannot be evaluated without logarithms.

Outside of pure math classes, the decibel is a common unit of measurement in quantities that logarithmic scales with base 10. It is particularly relevant in acoustics and circuit analysis, both topics in physics classes. In chemistry, the pH of a solution is defined as the negative base-ten logarithm of the concentration of hydrogen ions in that solution. Acidity is a crucially important topic in high school chemistry.

 

 

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A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Many word problems could be easily constructed involving computations of logarithms of base 10. Below is a problem involving earthquakes and the Richter scale. It would not be difficult to make similar problems involving the volume of sounds, the signal to noise ratio of signals in circuits, or the acidity of a solution.

The Richter Scale is used to measure the strength of earthquakes. It is defined as

M = \log(I/S)

where M is the magnitude, I is the intensity of the quake, and S is the intensity of a “standard quake”. In 1965, an earthquake with magnitude 8.7 was recorded on the Rat Islands in Alaska. If another earthquake was recorded in Asia that was half as intense as the Rat Islands Quake, what would its magnitude be?

Solution:
First, substitute our known quantity into the equation.

8.7=\log I_{rat}/S

Next, solve for the intensity of the Rat Island quake.

S \times 10^{8.7} = I_{rat}

Now, substitute the intensity of the new quake into the original equation.

M_{new}=\log (I_{new}/S)

=\log(0.5I_{rat}/S)

=\log (0.5S \cdot 10^{8.7}/S)

= \log (0.5 \cdot 10^{8.7})

= \log 0.5+ \log 10^{8.7}

=\log 0.5+8.7

=-0.303+8.7

=8.398

Thus, the new quake has magnitude 8.393 on the Richter scale.

References:
Earthquake data from Wikipedia’s List of Earthquakes (https://en.wikipedia.org/wiki/Lists_of_earthquakes#Largest_earthquakes_by_magnitude)

Slide rule picture is a screenshot of Derek Ross’s Virtual Slide Rule (http://www.antiquark.com/sliderule/sim/n909es/virtual-n909-es.html)

 

 

 

Engaging students: Using right-triangle trigonometry

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 Cody Luttrell. His topic, from Precalculus: using right-triangle trigonometry.

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A.1 Now that students are able to use right triangle trigonometry, there is many things that they can do. For example, they know how to take the height of buildings if needed. If they are standing 45 feet away from a building and they have to look up approximately 60 degrees to see the top of the building, they can approximate the height of the building by using what they know about right triangle trigonometry. Ideally, they would say that the tan(60 degree)= (Height of building)/(distance from building = 45). They can now solve for the height of the building. The students could also use right triangle trigonometry to solve for the elevation it takes to look at the top of a building if they know the distance they are from the building and the height of the building. It would be set up as the previous example, but the students would be using inverse cosine to solve for the elevation.

 

 

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A.2 An engaging activity and/or project I could do would be to find the height of a pump launch rocket. Let’s say I can find a rocket that states that it can travel up to 50 feet into the air. I could pose this problem to my students and ask how we can test to see if that is true. Some students may guess and say by using a measuring tape, ladder, etc. to measure the height of the rocket. I would then introduce right triangle trigonometry to the students. After a couple of days of practice, we can come back to the question of the height of the rocket. I could ask how the students could find the height of the rocket by using what we have just learned. Ideally, I would want to here that we can use tangent to find the height of the rocket. By using altimeters, I would then have the students stand at different distances from the rocket and measure the altitude. They would then compute the height of the rocket.

 

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D.1 In the late 6th century BC, the Greek mathematician Pythagoras gave us the Pythagorean Theorem. This states that in a right triangle, the distance of the two legs of a right triangle squared added together is equal to the distance of the hypotenuse squared (a^2+b^2=c^2). This actually was a special case for the law of cosines (c^2=a^2+b^2-2ab\cos(\theta)). By also just knowing 2 side lengths of a right triangle, one may use the Pythagorean Theorem to solve for the third side which will then in return be able to give you the six trigonometric values for a right triangle. The Pythagorean Theorem also contributes to one of the most know trigonometric identities, \sin^2 x+\cos^2 x=1. This can be seen in the unit circle where the legs of the right triangle are \sin x and \cos x and the hypotenuse is 1 unit long. Because Pythagoras gave us the Pythagorean Theorem, we were then able to solve more complex problems by using right triangle trigonometry.

 

My Favorite One-Liners: Part 120

I used these shirts as props when teaching Precalculus this week, and they worked like a charm.

After deriving the three Pythagorean identities from trigonometry, I told my class that I got these hand-made his-and-hers T-shirts for my wife’s birthday a couple of years ago. If you can’t see from the picture, one says \sin^2 \theta and the other \cos^2 \theta.

After holding up the shirts, I then asked the class what mathematical message was being communicated.

After a few seconds, someone ventured a guess: “We add up to 1?”

I answered, “That’s right. Together, we’re one.”

Whereupon the class spontaneously reacted with a loud “Awwwwwwwwww.”

I was exceedingly happy.

Engaging students: Graphing a hyperbola

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 Biviana Esparza. Her topic, from Precalculus: graphing a hyperbola.

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

Prior to learning about conics and hyperbolas in precalculus, students should be able to identify different shapes and figures and learn to identify cross sections of prisms, pyramids, cylinders, cones, and spheres, from geometry class. In algebra 2, students learn to write quadratic equations and learn vocabulary such as vertex, foci, directrix, axis of symmetry, and direction of opening, all which are used when dealing with hyperbolas as well.

 

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How has this topic appeared in pop culture?

The sport of baseball originates back before the Civil War and has come to be known as America’s pastime. On average, 110 balls are used in a major league baseball game, because the balls are usually tossed out if they’ve touched the dirt. Baseballs have a rubber or cork center, wrapped in yearn, and covered with leather sown together tightly by 108 stitches of red string. The stitches are in a hyperbola shape if looked at from a certain angle and depending on how the pitcher has held the stitches, different pitches are thrown.

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

Desmos is a great, interactive website that has many activities that can be used in the classroom. One of the activities it has is called Polygraph: Conics. The Desmos activity is similar to the board game Guess Who? in which students are in pairs and will ask yes or no questions to guess the graph of a hyperbola or ellipse of their choosing. This activity encourages students to make good questions and use precise vocabulary and academic language when describing conics, specifically over ellipses and hyperbolas, so that they can win the game.
https://teacher.desmos.com/polygraph/custom/560ad28c9e65da5615091ec7

References:
https://teacher.desmos.com/polygraph/custom/560ad28c9e65da5615091ec7
https://en.wikipedia.org/wiki/Baseball_(ball)

https://www.teksresourcesystem.net/module/content/search/item/681138/viewdetail.ashx

 

Engaging students: Law of Sines

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 Tiger Hersh. His topic, from Precalculus: the Law of Sines.

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

This topic can be extended to geometry where students must be able to use trigonometric identities (1) to identify the degree or length in order to use the Law of Sines. The issue about trigonometric identities is that you can only use them on right triangles (2). However, with the Law of Sines, students are able to use the trigonometric identities they have learned in Geometry and are able to draw a perpendicular line across a non-right triangle (3) and then apply the Law of Sines to solve either the height of the triangle, the length of the side of the triangle, or the degree of an angle of the triangle. So, the Law of Sines use the idea of trigonometric identities from Geometry in order to be applicable.

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How can this topic be used in your students’ future courses in mathematics or science? Unit circle calculus / solving for height of triangles

Students are able to the Law of Sines in order to find the height or degree of a triangle on the unit circle in precalculus or to calculator vector quantities in physics. The Law of Sines is prominent in the unit circle which is noticeable in the linked website which will provide students a connection from the Law of Sines to the unit circle. The Law of Sines also connects to physics where vectors used to show motion and direction in two dimensional space. The Law of sines may also be applied in physics where in (2); The vectors form a non-right triangle. The vectors ‘length’ can be determined by identifying the magnitude of each vector and then using the method as described before to use the Law of Sines in-order to find vector r.

https://image.businessinsider.com/51910f38ecad040506000002?width=800&format=jpeg&auto=webp

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

The law of sines has appeared in almost every 3 dimensional video games known to exist that has characters that are rendered with polygons. To note: it’s not just any polygon that can be used to create the characters you see in video games but specifically, they usually use triangles to render the characters. Even some movies that use animation software use these triangular polygons to render the figures in the movie; like for example Woody from the movie Toy Story (as seen below with polygons). We can use the Law of Sines in order to find the length or degree of each triangle on the figure if we were willing so.