Category: Guest presenter

Engaging students: Finding the domain and range of a function

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 Brittany Tripp. Her topic, from Precalculus: finding the domain and range of a function.

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

One of my favorite games growing up was Memory. For those who haven’t played, the objective of the game is to find matching cards, but the cards are face down so you take turns flipping over two cards and have to remember where the cards are so when you find the match you can flip both of the matching cards. To win the game you have to have the most matches. I think creating an activity like this, that involves finding domain and range, would be a really fun way to get students’ engaged and excited about the topic. You could place the students in pairs or small groups and give each student a worksheet that has a mixture of functions and graphs of functions. Then the cards that are laying face down would contain various different domains and ranges. In order to get a match you have to find the card that has the correct domain and the card that has the correct range for whatever function or graph you are looking at. You could increase the level of difficulty by having functions, graphs, domains, and ranges on both the worksheet and the cards. This would require the students to not only be able to look at a graph of a function or a function and find the domain and range, but also look at a domain and range and be able to identify the function or graph that fits for that domain and range.

These pictures provide an example of something similar that you could do. I would probably adjust this a little bit so that the domain and ranges aren’t always together and provide actual equations of functions that the students’ must work with as well.

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

Finding the domain and range of a function is used and expanded on in a variety of ways after precalculus. For instance, one way the domain and range is used in calculus is when evaluating limits. An example is the limit of x-1 as x goes to 1 is equal to zero, because when looking at the graph when the domain, x, is equal to 1 the range, y, is equal to zero. Finding domain and range is something that is applied to a variety of different type of functions in later courses, like when looking at trigonometric functions and the graphs of trigonometric functions. You look at what happens to the domain of a function when you take the derivative in calculus and later courses. You work with the domain and range of different equations and graphs in Multivariable calculus when you are switching to different types of coordinates such as polar, rectangular, and spherical. There are also multiple different science courses that use this topic in some way, one of those being physics. Physics involves a lot of math topics discussed above.

 

 

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

I found a website called Larson Precalculus that technically is targeted toward specific Precalculus books, but exploring this website a little bit I found that is would be a super beneficial tool to use in a classroom. This website has a variety of different tools and resources that students could use. It has book solutions which if you weren’t actually using that specific textbook could be a really helpful tool for students. This would provide them with problems and solutions that are not exactly the same to what they are doing, but similar enough that they could use them as examples to learn from. This website also includes instructional videos that explain in depth how to tackle different Precalculus topics including finding domain and range. There are interactive exercises which would give the students ample opportunities to practice finding the domain and range of graphs and functions. There are data downloads that give the students to ability to download real data in a spreadsheet that they can use to solve problems. These are only a few of the different resources this website provides to students. There are also chapter projects, pre and post tests, math graphs, and additional lessons. All of these things could be used to engage students and help advance and deepen their understanding of finding domain and range. The only downfall is that it is not a free resource. It is something that would have to be purchased if you chose to use it for your classes.

 

References:

http://esbailey.cuipblogs.net/files/2015/09/Domain-Range-Matching.pdf

http://17calculus.com/precalculus/domain-range/

http://www.larsonprecalculus.com/pcwl3e/

Engaging students: Half-life

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 Brianna Horwedel. Her topic: working with the half-life of a radioactive element.

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

Half-Life of radioactive elements in Pre-calculus is generally used when introducing exponential decay. However, its main application is in the field of Chemistry and Archeology. If students go on to take any type of chemistry, they will definitely learn more about the half-life of radioactive elements and how long it takes to get rid of certain nuclear elements. The half-life of Carbon-14 is especially important in Archeology. Carbon-14 dating is a method used to determine the age of archeological artifacts of a biological origin using the half-life of Carbon-14. This process can date bone, wood, cloth, plant fibers, and more that are up to 50,000 years old. The way it works is as follows: as soon as a living organism dies, it stops taking in new carbon. The ratio of carbon-12 to carbon-14 is the same as every living thing. However, when an organism dies, the carbon-14 starts decaying with its half-life of 5,700 years. The carbon-12 does not decay. When an organism is found, they look at the ratio of carbon-12 to carbon-14 to determine the age based on the half-life of carbon-14.

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

I think this topic lends itself nicely to a project. Firstly, I would come up with several half-lives and place them in a bowl. Each student would pick a half-life and have to make up an element. Using poster-board, they would give a brief description of what their element is and then create a graph illustrating their particular half-life. They would then present it to the class explaining how they graphed their line and what equation they used. They could also include a table of input and output values. This would be a great refresher on graphing exponential decays along with allowing a little creativity. I think the students would have a lot of fun with this type of project.

 

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

I found this really great web-site (https://jeopardylabs.com/play/exponential-growth-decay) that has an exponential growth and decay form of Jeopardy. It allows you to pick how many teams there are and then it sets up a Jeopardy board. This would be a really fun way to review at the end of a unit over exponential growth and decay. To make the students more engaged, I would offer extra credit to the team with the highest score at the end. Because it is in a game form, students are more likely to pay attention to this type of review.

 

Engaging students: Solving logarithmic equations

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 Anna Park. Her topic: how to engage Algebra II or Precalculus students when solving logarithmic equations.

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Application:

 

The students will each be given a card with a) a logarithmic equation solution and b) a new logarithmic equation. The student that has a number one on the back of their card will begin the game. The student will stand up and tell the rest of the class what they have for b) the Log equation they have, then the student with the corresponding card will read their solution a) to the first students problem. If that student is correct they will read part b) the new log equation. Then another student that has the logarithmic solution will stand up and say their solution a) and then read their new log equation b). This will continue until the last student stands with their new equation and it loops back to student number one’s solution. This will end the game. This game requires students to solve logarithmic equations and recognize how to rewrite a logarithmic equation. There will be an appropriate amount of time before the game begins so the students can work backwards to find their logarithmic equation that matches their solution.

 

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History:

 

John Napier was the mathematician that introduced logarithms. The way he came up with logarithms is very fascinating, especially how long it took him to develop the logarithm table. He first published his work on logarithms in 1614. He published the findings under “A Description of the Wonderful Table of Logarithms.” He named them logarithms after two Greek words; logos, meaning proportion, and arithmos, meaning number. His discovery was based off of his imagination of two particles traveling along two parallel lines. One line had infinite length and the other had a finite length. He imagined both particles starting at the same horizontal positions with the same velocity. The first line’s velocity was proportional to the distance, which meant that the particle was covering equal distance in equal time. Whereas the second particle’s velocity was proportional with the distance remaining. His findings were that the distance not covered by the second line was the sine and the distance of the first line was the logarithm of the sine. This showed that the sines decreased and the logarithms increased. This also resulted in the sines decreasing in geometric proportion and the logarithms increasing in arithmetic proportion. He made his logarithm tables by taking increments of arc (theta) every minute, listing the sine of each minute by arc, and the corresponding logarithm. Completing his tables, Napier computed roughly ten million entries, and he selected the appropriate values. Napier said that his findings and completing this table took him about 20 years, which means he probably started his work in 1594.

Resource: http://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms

 

 

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Technology:

 

I have found that when it comes to remembering rules, sometime the cheesiest of songs help student’s to remember the rules. It is also a very good engage before the students start with the lesson. The chorus is typically the most important content for the student’s to remember. Here are two videos that would help the student’s to remember how to compute logarithms.

The first video is a song from Youtube set to the song Thriller by Michael Jackson. The song is produced very well and is very engaging throughout the whole song.

The Second video is of a student’s project  on Youtube of how to remember how to compute logarithms to the song Under the sea by the little mermaid. Though the production isn’t as good as the first video, the young girls do a good job at explaining how to solve logarithms.

 

 

Engaging students: Synthetic Division

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 Amber Northcott. Her topic, from Precalculus: synthetic division.

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

 

Synthetic division takes a little to get used to, especially after learning long division with polynomials. One thing is for sure and that is once the students get how to do synthetic division they sometimes prefer it over long division because it is a faster and easier way to divide polynomials. However, the first step is to learn it and there are many different ways to learn it. One way is to create an activity the students can do that will help them learn it.

An activity or project idea is to have the students write their own steps on how to solve synthetic division. Make sure to let the students know that they must put it in their own words. Then put students in groups of three to four and have them share their steps with each other. Let them give each other feedback on their steps and the feedback must be turned in. Once the teacher looks at the feedback, the teacher can give it back to the students and give their feedback to the student as well. Then have the student take the feedback into consideration and change their steps if needed. This activity will allow the student to see how they view synthetic division and what steps they take to solve it. By sharing their steps, they can get an idea of how everyone solves synthetic division and learn from each other.

Other activities or projects also include having the students write down the steps to solving synthetic division. This time though they can use their imagination and get creative. The activity or project can be to make up a poem or acrostic or a story to help them remember how to solve synthetic division. Then have them present their poem or acrostic or story in front of the class, so other students can learn those ideas as well to help them remember how to do synthetic division.

 

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

 

Synthetic division is first seen Algebra II. Students tend to learn it right after learning how to do long division with polynomials. After taking Algebra II students don’t see synthetic division for a while until pre-calculus and calculus. This is because when you hit Pre-Calculus and Calculus you see algebra topics within them a lot more than you would a Geometry and Trigonometry class. This doesn’t mean you can’t see them in Geometry or Trigonometry. This is because like all math subjects and topics they intertwine with each other, so you are bound to see synthetic division in quite a few places in mathematics.

 

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How was this topic adopted by the mathematical community?

 

Synthetic division is also called Ruffini’s Rule, but we don’t see this title very often in textbooks. The reason why it was called Ruffini’s Rule is because of the Italian mathematician Paolo Ruffini, who brought synthetic division to life around 1809. Paolo Ruffini, like all mathematicians, wanted to find a simpler way to do a mathematic topic. This can also be because mathematicians are known to be a bit lazy.

The mathematic topic he wanted to find a simpler way to do was dividing polynomials, so by creating this system we all know as synthetic division he found a cleaner, simpler, and faster way to divide polynomials. Of course, it has certain conditions to follow in order to be able to do synthetic division, but it’s the option is there.

 

 

Resources

 

Click to access 06-05-02-synth-div.pdf

 

 

 

 

Engaging students: Defining intersection

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 Tramashika DeWalt. Her topic, from Geometry: defining intersection.

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

 

I would create a Kahoot to define intersection for my students. I would begin with the basic definition, which is, where lines cross over, meet, or have a common point (Unknown, Math is Fun, 2016). Thereafter, I would display pictures that visually portray intersection and that do not portray intersection. Within the same Kahoot, I would provide the students with the more advanced definition of intersection, intersection sets, “The intersection of two sets A and B is the set of elements common to both A and B” (Unknown, Math is Fun, 2016) according to MathIsFun.com. Like before, I would follow the definition up with pictures for the students to determine if the set intersects or not. After the Kahoot, I would have the students to get into groups of 4, with a large piece of paper, to come up with intersections from their daily life. Finally, the groups would display their findings and we will discuss the results as a class.

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

 

This topic can and will be used in my students’ future math courses. As I mentioned above, the basic definition of intersection will be extended to intersecting sets. In set intersection, the student will have to determine what elements each set has in common (that intersect) in order to determine where the sets intersect. The student will also have to know that the elements that are not common for both sets are not included in the intersection of the two sets. Intersection is used throughout math, so students can encounter it in high school, calculus, functions and modeling, real analysis, abstract algebra, etc. Not only will my students’ encounter intersection in future math courses, but they will also encounter intersections in life. For instance, when they are at a stop light (intersection), at a four-way stop sign (intersection), or even walking around UNT (students’ paths and sidewalks intersect all the time here).

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How can technology 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.

 

As mentioned above, I would create a Kahoot, on kahoot.it, to effectively engage my students with technology to define and solidify the definition of intersection. I would layout my Kahoot by starting with the definition of intersection. Then I would have a variety of picture that would either display a form of intersection, or that would not display a form of intersection. Kahoot is awesome because it allows students to use their cell phone, iPad, or tablet to respond to questions created by the teacher. I feel the Kahoot will be very engaging because it allows the student time to play on their phone (so that the teacher doesn’t have to confiscate them for inappropriate use), listen to cool background music as they solve their problems, and learning about the particular topic at hand, all while having fun.  Now Kahoot even has a podium at the end of the Kahoot that displays the top three point earners.

 

References

 

Kahootit! (n.d.). Retrieved from Kahoot!: create.kahoot.it

https://play.kahoot.it/#/?quizId=8648bc78-08d2-4ea8-9cb8-d23df904ebca

Unknown. (2016). Math is Fun. Retrieved from Math is Fun: http://www.mathsisfun.com/definitions/intersection.html

Unknown. (2016). Math is Fun. Retrieved from Math is Fun: http://www.mathsisfun.com/definitions/intersection-sets-.html

 

Engaging students: Deriving the distance formula

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 Asmar. Her topic, from Algebra II: deriving the distance formula.

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

Many high school students complain about why they have to take a math class or that math is not fun. Deriving and even learning the distance formula is not interesting for very many students. One way that I would engage my students would be to take the entire class outside to teach this lesson. We will walk down to the football and I will have a three students go to one corner of the football field while the rest of the class stands at the opposite corner diagonally. I will then hand a stopwatch to three other students. Each of them will have one stopwatch. The three students on the opposite corner will be running to the corner where the rest of the class is standing. The students holding a stopwatch, will each be timing one of the students running. I will ask one student to run horizontally and then vertically on the outrebounds of the football field, one student will run vertically and then horizontally, and the last student will run diagonally through the football field. Once all three students have made it to the corner where the rest of the class is, I will then ask everyone “Who do you think made it to the class the fastest?” I will allow them to say what they think and why, and then I will ask the students with the stopwatches to share the times of each of the students that ran. At the end, this will get the students to conclude that the student that ran diagonally got to the entire class the fastest. This is a short activity, but it changes the atmosphere for the students by taking class outside for a little, and it is fun.

 

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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?

There were three main mathematicians/philosophers that contributed to the discovery of this topic. Pythagoras, Euclid and Descartes all played a roll in deriving the distance formula. Pythagoras is a very famous mathematician. At first, he saw geometry as a bunch of rules that were derived by empirical measurements, but later he came up with a way to connect geometric elements with numbers. Pythagoras is known for one of the most famous theorems in the mathematical world, the Pythagorean Theorem. The theorem touches on texts from Babylon, Egypt, and China, but Pythagoras was the one who gave it its form. The distance formula comes from the Pythagorean Theorem. Euclid is known as “The Father of Geometry.” He has five general axioms and five geometrical postulates. However, in his third postulate, he states that you can create a circle with any given distance and radius. This is represented by the formula x2+y2=r2. The distance formula comes from this equation as well. Last but not least, Descartes was the one who created the coordinate system. When finding the distance between two points on a coordinate plane, we would need to use the distance formula. All three of these men helped form the distance formula.

 

 

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

Students find everything more interesting when they are able to use technology to learn. There is a website that allows students to explore math topics using what is called a Gizmo. A Gizmo can be used to solve for the distance between two points. The students are allowed to pick what their two points are and then use the distance formula to find the distance between the points they chose. When students have control over something, they tend to do what they are supposed to do without any complaints. The Gizmo allows students to explore on their own without the teacher having to tell them what to do step by step. I can even ask the students to plot three points that form a right triangle and have them find the distance of the points that form the hypotenuse. This can allow the students to make the connection between the distance formula and Pythagorean Theorem. There are many applications out there, but I remember using Gizmos when I was in high school and I loved it. It is a great tool to explore a mathematical topic.

 

 

 

 

References:

 

http://www.storyofmathematics.com/greek_pythagoras.html

http://www.storyofmathematics.com/hellenistic_euclid.html

http://www.storyofmathematics.com/17th_descartes.html

 

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

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

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

As a teacher I think that a fun activity that is not too difficult but will need the students to be up and around the room is kind of like a mix and match game. I will give a bunch a students, a multiple of three, different angles. And then I will give the rest of the students cards with acute, obtuse, and right triangle listed on them. The students with the angles will then have to get in groups of three to form one of the three triangles. Once the students are in groups of three, they will then find another student with the type of triangle and pair with them. They will then present and explain to rest of the class why they paired up the way that they did. I think that it would be a good way for the students to be up and around and decide for themselves what angles for what triangles and then to show their knowledge by explaining it to the class.

 

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

The topic of defining acute, right, and obtuse triangles extend what my students should already know about the different types, acute, right, and obtuse, angles. The students should already know the different types of angles and their properties. We can use their previous knowledge to build towards defining the different types of triangles. I will explain to the students that defining the triangles is like defining the angles. If they can tell me what angles are in the triangle and then tell me the properties of the triangles then they can reason with it and discover which triangle it is by looking at the angles.

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

I found an article that I like that was written about a soccer club, FC Harlem. FC Harlem was getting a new soccer field as part of an initiative known as Operation Community Cup, which revitalizes soccer fields in Columbus and Los Angeles. This particular field, when it was opened, had different triangles and angles spray painted on the field in order to show the kids how soccer players use them in games. Time Warner Cable was the big corporation in on this project.

 

References:

http://www.twcableuntangled.com/2010/10/great-day-for-soccer-in-harlem/