Engaging students: Negative and zero 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 again comes from my former student Gary Sin. His topic, from Algebra: negative and zero exponents.

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

The idea behind negative and zero exponents is to basically go backwards in our method of obtaining answers to positive exponents. I can create an activity where the students will begin by applying their knowledge on positive exponents represented on a number line and how every exponent increase in 1 multiplies the previous number by the base. I can then ask the students to point out a pattern they notice between the answers as the exponents increase. The students will realize that the answer is always the previous answer multiplied by the base.

Now I will ask the students what will happen if we went backwards down the number line instead. The students will then realize that going backwards meant dividing the next answer by the base. With this realization, I will guide the students all the way back to the first power and ask them what will happen now if we kept dividing by the base. The students will figure out that the zero exponent of a base would be 1. I will continue by asking the students what will happen now if we kept going and dividing by the base. The students will finally realize that negative exponents will meant dividing the answers repeatedly by the base. I will conclude by asking the students to go forward down the number line so that they will conclude that this logical way of thinking works with how exponents work.

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

Exponents are easier ways of representing the multiplication of a base by itself. The students will grasp the concept of exponents once they realize zero and negative exponents are obtained the same way positive ones are obtained, except going backwards.

Therefore, the grasp of exponents is important as they progress towards algebra 1 and 2 where variables are represented with exponents. This is very important as it represents a leap from linear equations to quadratic equations and subsequently cubic equations. Polynomials also greatly utilize exponents and learning how exponents work will allow the students to simplify complicated polynomials by combining like terms. Students learning negative exponents will also allow them to represent polynomials in fraction form which is sometimes easier to manipulate.

The knowledge of exponents is very important once they reach advanced math courses like pre-calculus, calculus and future college math courses. Differentiation and integration both heavily involves exponents.

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

Understanding how negative and zero exponents work depends on basic knowledge of arithmetic and manipulating fractions. Also the students must have prior knowledge on how positive exponents work.

Exponents is the next level after arithmetic. Arithmetic begins with understanding counting, then learning how to add. Multiplication is derived from addition and it is basically the simplification of adding large groups of the same number. We can see that exponents is the next step after multiplication. The simplification of multiplying large groups of the same number.

However, discovering how zero and negative exponents are obtained requires the use of division. Students will apply their knowledge on how to divide and how to represent division as fractions. E.g. 1 divide by 2 can represented as ½.

Of course this requires the basic knowledge on how exponents themselves work and understanding how the exponent depends on the number of times we multiply the base.

Engaging students: Laws of 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 Jesus Alanis. His topic, from Pre-Algebra: the Laws of Exponents (with integer exponents).

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

I would create a project where the students would have to create a “poster”. First, you would give each student a strip that contains one of the laws of the exponent. On the strip, there will be 3 expressions for them to solve that involves one of the laws and have a blank space for the student to create a “rule” for their law. This is where you will let your students find out what law they got. Once they figured out their law they will create a poster that will have the name of the law, the rule of the law (by the rule I mean just using variables, for example, the Product of Powers it would be x^m \cdot x^n = x^{m+n} ), a complete sentence which explains the rule in their own words, and an example of the law which can be one of the expressions from the strip. For the poster, you would want students to use color and decorate the way they want. This will let the student’s inner artist out and creativity shine. You can have your students present their law, or you can have a gallery walk so they can look at all the different laws.

The purpose of the project is for the student to play with the expressions causing them to question which law they received and letting them create a rule that makes them understand how the law works. The sentence on the poster will demonstrate if the student understood the law. This is a project that can be used to let students find out for themselves or this could be a project to help students remember what they learned.

Something extra but you can also make this a relay race by using the strip or the whole paper where the students must at least do one expression from each of the laws of the exponent. In the end, each student in the group has at least done all three laws that were on the page. With the page from TEA, there are only three laws on there, but you could add the rest on there to make the race a little longer.  The goal is to have each student have practice with each law that is on the page, they are in a group so they can help each other and familiarize themselves with the laws and peers.

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

The way students can use the Law of Exponents in the future is that it will help write or type very large numbers towards using fewer numbers. This will not cause the value of the number to change but will be less to write. For example,

2,357,000,000,000 = 2.357 \times 10^{12}.

The law of exponents will also help with loan interest rates that can be used to predict how much you will have to pay in a certain time frame. Exponents are used to determining the pH level of substances, see the growth of bacteria, see the population of a city, and how much has it increased or decreased, and many more.

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How has this topic appeared in pop culture?

I did not really find where it appeared in pop culture, but I did find a connection of how you can use the clip of SpongeBob to the Law of Exponents. The way you can connect them is that SpongeBob says all the specific rules to blow a bubble. This is to engage students and make sure to activate their prior knowledge that goes with the rules like the way we do with the area of a rectangle we first have to find the length of the sides and then place them in the formula to be multiplied. The small clip is a demonstration that with the Law of Exponents we must “obey” the math operations so that our results are as perfect as the duck bubbles. Also, we must make the connection between rules and laws which are very similar.

References

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

 

How to picture an exponent

While I’m easily amused by math humor, I rarely actually laugh out loud after reading a comic strip. That said, I laughed heartily after reading this one.

Source: https://xkcd.com/2283/

Engaging students: Laws of 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 Lyndi Mays. Her topic, from Pre-Algebra: the Laws of Exponents (with integer exponents) green lineWhile thinking about different activities that I could do with Laws of Exponents I decided to try making a bingo card. I like this idea because it’s a way for students practice on different problems while playing a game. The way I have it set up to use in a classroom, I have questions that I would ask. One example is . I would put this up on the board and the student has to solve it and see if they have the answer on their card. I would tell the students what the answers were until after we were done with the activity so that they’re not just waiting to hear the answer instead of doing the work. If a student got a “bingo” then I would check their answers and if they got them all right then I would have an incentive like 5 extra points on a homework assignment of their choice or something along those lines.

So, if I wrote on the board the equations x^4(x), x^0 y^5, (2x^2-3y^5)^0, and x^5 y^{-2} . If a student received this card, then on these questions they would get a “bingo” on the descending diagonal from left to right. You’ll also notice that I included some wrong answers in a few of the spots. Hopefully the students would notice they were not all the way simplified and would know they couldn’t use those.

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Students can use Laws of Exponents to help them understand Laws of Logarithms. They will use the Laws of Exponents throughout Calculus courses when taking the derivatives or integrals of different problems. It’s important for students to understand these laws so that they can simplify problems and use them to their advantage. One example is when the student is asked to solve \int x^{-4} \, dx. If the student has a good understanding of the Laws of Exponents, then their first reaction will be to change it to \int dx/x^4 = -1/3 x^3 + C. Having this understanding is necessary for this problem and helps when students already know the Laws of Exponents so that they’re not having to learn extra material basically.

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Archimedes is the one that discovered the Laws of Exponents. He did this by breaking everything down as much as possible. To show an example,

3^4 \times 3^2 = (3×3×3×3) (3×3)  We can do this just by know the definition of exponents

= 3×3×3×3×3×3     Once we remove the parentheses we see we’re just multiplying 3 together 6 times.

= 3^6                         This is just the definition of exponents again

Teaching the students the Laws of Exponents this way can show them how a mathematician discovers all these rules that we follow and gives them a better understanding of the laws. Opening up this interest might help the students become more interested in math. Another example that I would show students would be y^5/y^3. From here I would show the students that we could break it down to (y \times y \times y \times y \times y)/(y \times y \times y). Hopefully, then the students would see that you could divide and get rid of the denominator, y×y=y^2, and this is why it is ok to subtract when a term with an exponent is being divided by something with the same base. This is also a really good way to show students why they can NOT use these laws when they’re working with terms with different bases.

References:

Exponentiation. (2017, September 1). In Wikipedia, The Free Encyclopedia. Retrieved

23:05, September 1, 2017, from https://en.wikipedia.org/w/index.php?title=Exponentiation&oldid=798388543

Engaging students: Negative and zero 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 again comes from my former student Austin DeLoach. His topic, from Algebra: negative and zero exponents.

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

The topic of negative and zero exponents is very important when or if the students get to calculus. Although that will be several years down the line, having a solid fundamental grasp on the idea of negative and zero exponents will help them understand derivatives a lot better. Because derivatives of “simple” functions just multiply the coefficient by the exponent and then subtract one from the exponent, it is important for the students to have a good understanding of what negative and zero exponents are. If they do not understand already, they will be confused about why, for example, the derivative of 3x is just 3. It also greatly simplifies derivatives of things like 4/x2, as the students will simply be able to recognize that that is the same thing as 4x-2 and follow standard rules instead of needing to think about the quotient rule and waste time with that. It will also help them in the more near future when they work with simplifying expressions with the exponents written in different terms (i.e. with a positive exponent or with a negative exponent in the denominator), as it will help them recognize what simplifications mean the same thing. Explaining that understanding negative exponents will thoroughly help them in the future may be enough for some students to want to solidify their grasp on the topic.

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

Although this is not about the early adoption of negative and zero exponents in the mathematical community, Geoffrey D. Dietz points out more recent bias for or against the use of negative exponents in textbooks in his Journal of Humanistic Mathematics (linked at the bottom of this answer). Dietz brings up the idea of what is considered “simplified” when it comes to negative exponents vs exponents in denominators. He rated over 20 mathematics textbooks from 1825 to 2012 from “very tolerant” of negative denominators in simplified answers to “very intolerant”. Interestingly, his first encounter with an “intolerant” textbook was not until the 20th century, and textbooks began getting more polarized as very tolerant or very intolerant closer to the end of the 20th century and getting closer to today. This is interesting when it comes to adoption by the mathematical community, as there is a significant inconsistency, even today, about whether negative exponents can be considered “simplified” or not. It will be important to point this out to your students so they can be prepared for their future teachers who may have different preferences on simplification from you, as that will help them understand the polarity in the mathematical community on this topic, as well as hopefully make them want to understand what negative exponents really mean. Dietz recommends giving your students practice with not only converting negative exponents to positive exponents, but also from positive to negative, in order to make sure they are prepared for whatever preferences come up as well as solidifying their understanding of what negative exponents mean.

http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1110&context=jhm

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E1. How can technology be used to effectively engage students with this topic?
This video from Khan Academy does a good job at explaining why negative and zero exponents are what they are. Although Khan Academy videos will likely not be the most engaging for all students, this video is short enough to maintain the attention of the class, and it the logic in it is helpful for the students who don’t understand how the definition of negative and zero exponents was decided on. The presenter does well explaining the idea of “going backwards” and dividing by the number when you decrease the exponent. It’s a good way to explain the “why” for students who ask about it, and it also is a good way to change up the pace for students, as playing videos during class could prevent it from becoming stale for the students, keeping them engaged for longer.

https://www.khanacademy.org/math/pre-algebra/pre-algebra-exponents-radicals/pre-algebra-negative-exponents/v/negative-exponent-intuition

 

 

My Favorite One-Liners: Part 42

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

The function f(x) = a^x typically exhibits exponential growth (if a > 1) or exponential decay (if a < 1). The one exception is if a = 1, when the function is merely a constant. Which often leads to my favorite blooper from Star Trek. The crew is trying to find a stowaway, and they get the bright idea of turning off all the sound on the ship and then turning up the sound so that the stowaway’s heartbeat can be heard. After all, Captain Kirk boasts, the Enterprise has the ability to amplify sound by 1 to the fourth power.

My Mathematical Magic Show: Part 9

This mathematical trick was not part of my Pi Day magic show but probably should have been. I first read about this trick in one of Martin Gardner‘s books when I was a teenager, and it’s amazing how impressive this appears when performed. I particularly enjoy stumping my students with this trick, inviting them to figure out how on earth I pull it off.

Here’s a video of the trick, courtesy of Numberphile:

Summarizing, there’s a way of quickly determining x given the value of x^5 if x is a positive integer less than 100:

  • The ones digit of x will be the ones digit of x^5.
  • The tens digit of x can be obtained by listening to how big x^5 is. This requires a bit of memorization (and I agree with the above video that the hardest ones to quickly determine in a magic show are the ones less than 40^5 and the ones that are slightly larger than a billion):
    • 10: At least 10,000.
    • 20: At least 3 million.
    • 30: At least 24 million.
    • 40: At least 100 million.
    • 50: At least 300 million.
    • 60: At least 750 million.
    • 70: At least 1.6 billion.
    • 80: At least 3.2 billion.
    • 90: At least 5.9 billion.

 

 

 

 

 

 

 

Exponents and the decathlon

During the Olympics, I stumbled across an application of exponents that I had not known before: scoring points in the decathlon or the heptathlon. From FiveThirtyEight.com:

Decathlon, which at the Olympics is a men’s event, is composed of 10 events: the 100 meters, long jump, shot put, high jump, 400 meters, 110-meter hurdles, discus throw, pole vault, javelin throw and 1,500 meters. Heptathlon, a women’s event at the Olympics, has seven events: the 100-meter hurdles, high jump, shot put, 200 meters, long jump, javelin throw and 800 meters…

As it stands, each event’s equation has three unique constants — $latex A$, $latex B$ and $latex C$— to go along with individual performance, $latex P$. For running events, in which competitors are aiming for lower times, this equation is: $latex A(BP)^C$, where $latex P$ is measured in seconds…

B is effectively a baseline threshold at which an athlete begins scoring positive points. For performances worse than that threshold, an athlete receives zero points.

Specifically from the official rules and regulations (see pages 24 and 25), for the decathlon (where P is measured in seconds):

  • 100-meter run: 25.4347(18-P)^{1.81}.
  • 400-meter run: 1.53775(82-P)^{1.81}.
  • 1,500-meter run: 0.03768(480-P)^{1.85}.
  • 110-meter hurdles: 5.74352(28.5-P)^{1.92}.

For the heptathlon:

 

  • 200-meter run: 4.99087(42.5-P)^{1.81}.
  • 400-meter run: 1.53775(82-P)^{1.88}.
  • 1,500-meter run: 0.03768(480-P)^{1.835}.

Continuing from FiveThirtyEight:

 

For field events, in which competitors are aiming for greater distances or heights, the formula is flipped in the middle: $latex A(PB)^C$, where $latex P$ is measured in meters for throwing events and centimeters for jumping and pole vault.

Specifically, for the decathlon jumping events (P is measured in centimeters):

  • High jump: 0.8465(P-75)^{1.42}
  • Pole vault: 0.2797(P-100)^{1.35}
  • Long jump: 0.14354(P-220)^{1.4}

For the decathlon throwing events (P is measured in meters):

  • Shot put: 51.39(P-1.5)^{1.05}.
  • Discus: 12.91(P-4)^{1.1}.
  • Javelin: 10.14(P-7)^{1.08}.

Specifically, for the heptathlon jumping events (P is measured in centimeters):

  • High jump: 1.84523(P-75)^{1.348}
  • Long jump: 0.188807(P-210)^{1.41}

For the heptathlon throwing events (P is measured in meters):

  • Shot put: 56.0211(P-1.5)^{1.05}.
  • Javelin: 15.9803(P-3.8)^{1.04}.

I’m sure there are good historical reasons for why these particular constants were chosen, but suffice it to say that there’s nothing “magical” about any of these numbers except for human convention. From FiveThirtyEight:

The [decathlon/heptathlon] tables [devised in 1984] used the principle that the world record performances of each event at the time should have roughly equal scores but haven’t been updated since. Because world records for different events progress at different rates, today these targets for WR performances significantly differ between events. For example, Jürgen Schult’s 1986 discus throw of 74.08 meters would today score the most decathlon points, at 1,384, while Usain Bolt’s 100-meter world record of 9.58 seconds would notch “just” 1,203 points. For women, Natalya Lisovskaya’s 22.63 shot put world record in 1987 would tally the most heptathlon points, at 1,379, while Jarmila Kratochvílová’s 1983 WR in the 800 meters still anchors the lowest WR points, at 1,224.

FiveThirtyEight concludes that, since the exponents in the running events are higher than those for the throwing/jumping events, it behooves the elite decathlete/heptathlete to focus more on the running events because the rewards for exceeding the baseline are greater in these events.

Finally, since all of the exponents are not integers, a negative base (when the athlete’s performance wasn’t very good) would actually yield a complex-valued number with a nontrivial imaginary component. Sadly, the rules of track and field don’t permit an athlete’s score to be a non-real number. However, if they did, scores might look something like this…

 

 

Lessons from teaching gifted elementary students (Part 6b)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprising depth of mathematical knowledge.

Here’s a question I once received:

255/256 to what power is equal to 1/2? And please don’t use a calculator.

Here’s how I answered this question without using a calculator… in fact, I answered it without writing anything down at all. I thought of the question as

\displaystyle \left( 1 - \epsilon \right)^x = \displaystyle \frac{1}{2}.

\displaystyle x \ln (1 - \epsilon) = \ln \displaystyle \frac{1}{2}

\displaystyle x \ln (1 - \epsilon) = -\ln 2

I was fortunate that my class chose 1/2, as I had memorized (from reading and re-reading Surely You’re Joking, Mr. Feynman! when I was young) that \ln 2 \approx 0.693. Therefore, we have

x \ln (1 - \epsilon) \approx -0.693.

Next, I used the Taylor series expansion

\ln(1+t) = t - \displaystyle \frac{t^2}{2} + \frac{t^3}{3} \dots

to reduce this to

-x \epsilon \approx -0.693,

or

x \approx \displaystyle \frac{0.693}{\epsilon}.

For my students’ problem, I had \epsilon = \frac{1}{256}, and so

x \approx 256(0.693).

So all I had left was the small matter of multiplying these two numbers. I thought of this as

x \approx 256(0.7 - 0.007).

Multiplying 256 and 7 in my head took a minute or two:

256 \times 7 = 250 \times 7 + 6 \times 7

= 250 \times (8-1) + 42

= 250 \times 8 - 250 + 42

= 2000 - 250 + 42

= 1750 + 42

= 1792.

Therefore, 256 \times 0.7 = 179.2 and 256 \times 0.007 = 1.792 \approx 1.8. Therefore, I had the answer of

x \approx 179.2 - 1.8 = 177.4 \approx 177.

So, after a couple minutes’ thought, I gave the answer of 177. I knew this would be close, but I had no idea it would be so close to the right answer, as

x = \displaystyle \frac{\displaystyle \ln \frac{1}{2} }{\displaystyle \ln \frac{255}{256}} \approx 177.0988786\dots