# Engaging students: Defining a function of one variable

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 Phuong Trinh. Her topic, from Algebra: defining a function of one variable.

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

The understanding of functions is crucial in the study of both math and science. Not only that, some functions, especially function with one variable, are often used by everyone in their daily life.  For example, a person wants to buy some cookies and a cake. The person will need to figure how much it will cost them to buy a cake and however many cookies they want. If the cost of the cake is \$12, and the price for each cookie is \$1.50, the person can set up a function of one variable to find the total cost for any number of cookies, expressed as c. The function can be written as f(c) = 1.50c + 12. With this function, the person can substitute any number of cookies and find out how much they would spend for the cookies and cake. Aside from the situation given by this example, function with one variable can also be used in various different scenarios.

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.)

Function with one variable can be used in many real life situations. Word problems can be derived from every day scenarios that the students can relate to.

Problem 1: John is transferring his homework files into his flash drive. This is the formula for the size of the files on John’s drive S (measured in megabytes) as a function of time t (measured in seconds): S (t) = 3t + 25

How many megabytes are there in the drive after 10 seconds?

This problem allows the students to get familiar with the function notation as well as letting the students work with a different variable other than x.

Problem 2: (Found at https://www.vitutor.com/calculus/functions/linear_problems.html )

“A car rental charge is \$100 per day plus \$0.30 per mile travelled. Determine the equation of the line that represents the daily cost by the number of miles travelled and graph it. If a total of 300 miles was travelled in one day, how much is the rental company going to receive as a payment?”

Besides giving the students practice with finding a solution from a function, this problem let the students practice setting up the equation. This also shows the students’ understanding of the subject.

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

There are multiple resources that can be used to help the students understand what a function is as well as how they should approach a problem with function. One of the resources can be found at coolmath.com. The layout of the website makes it easy to locate the topic of “Functions” under the “Algebra” tab. By comparing a function with a box, Coolmath defines a function in a way that can be easily understood by students, while also showing how a function can be thought of as visually. The site also provides the explanation for function notation with visuals and examples that are easy to understand. On Coolmath, the students will also have the chance to practice with randomly generated questions. They can also check their answers afterward. On other hands, the site also provides definitions and explanations to other ideas such as domain and range, vertical line tests, etc. Overall, coolmath.com is great to learn for students in and out of the classroom, as well as before and after the lesson.

http://www.coolmath.com/algebra/15-functions

References:

“Linear Function Word Problems.” Inicio, www.vitutor.com/calculus/functions/linear_problems.html.

“Welcome to Coolmath.” Cool Math – Free Online Cool Math Lessons, Cool Math Games & Apps, Fun Math Activities, Pre-Algebra, Algebra, Precalculus, www.coolmath.com/algebra/15-functions.

# Engaging students: Finding x- and y-intercepts

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 Lissette Molina. Her topic, from Algebra: finding $x-$ and $y-$intercepts.

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

Using this topic, students can now use word problems that involve two variables in our everyday lives. One problem that many scientists often use is population growth. In population growth, we can usually see a trend of a line and determine the slope. We initially begin with a certain population in a certain year, this is considered the y-intercept, since we start at the initial year that we consider to be at x=0. Using the slope of the line when we are speaking in terms of population decay, we may then set our y=0 to find when a population would be equal to zero. We can also consider other examples such as the depreciation of a car, or when a business’s grows out of debt and begins to profit. Word problems include, but are not limited to, problems that involve a trend and wanting to find where that trend will lead to at a certain point, x, when we are given an initially amount or reverse this operation.

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

This topic crosses multiple courses in mathematics. In general, knowing the x and y-intercepts of equations help students start outlining what the graph of the function might look like. This gives part of the visual representation needed to complete part of the graph. These intercepts usually also give a prediction of what the shape of the graph may look like. A fun assignment would be giving a student two points on the graph and along with the intercepts of that equation that the points belong to. Along with this, these intercepts give us the solutions of the equations. When there are not x or y-intercepts, we would now know that the solutions do not exist or at least are imaginary. Overall, x and y-intercepts help us get a better understanding of what the graphs of almost all equations must look like. This is essentially especially when we are graphing by hand.

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

Graphing calculators is one fun essential way of finding intercepts as well as learning functions on a calculator. When a student graphs a function on a graphing calculator, for example, the sine function, we can ask the student where they believe the graph would intercept with the x-axis. We would then ask them to find the intercepts using the calculator by pressing [2nd][trace][4] function and proceed to find the approximated x-intercepts. The student would then find that the intercepts occur at every npi/2. Essentially, using this function is an interesting way of estimating the intercepts along the graph in an interactive way. Other online graphing calculators may do this as well and give students a better understanding of where the intercepts occur.

# Engaging students: Probability and odds

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 Victor Acevedo. His topic, from Pre-Algebra: probability and odds.

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

There is an online interactive game in which students practice their knowledge on probability. The game is called “Beat the Odds” and it is on PBS’s learning media website. There are two game modes: training and competition. In training mode, students must answer questions about finding the probability of various events. (rolling a die, picking from a deck of cards, etc.) For each correct answer, students earn digital money and the questions scale in difficulty. After the students feel that they have earned enough money, they can switch over to competition mode. Competition mode allows students to bet money against other bot players to see who can answer questions the most accurately. Students are asked various questions and whoever is the closest to the correct answer wins the money in the “pot.”  Students can keep playing either until they lose all their money or until they decide to get out while they are ahead.

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

Probability is an integral part to sports analysis. In baseball, batting averages are used to determine a player’s batting ability by dividing the number of successful hits by the number of at bats. This statistic can be used to determine the probability that a player may hit a ball during their next at bat. For example, a player that has a .400 would have roughly a 40% chance of hitting the ball during their next at bat. By using a player’s batting average and other stats, teams can decide how to set up their line up for going up to bat. Typically, the players with the highest batting averages take up the first 5 spots in the lineup. The first three players need to be able to make it on to a base, while the fourth player needs to be a heavy hitter than can possibly have everyone score runs. Coaches consider every players’ batting averages, as well as other stats, to help them determine their best lineup and chances of winning.

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

Quantum theory is a branch of physics that focuses on studying the different properties of atoms and particles. The most famous application of probability in quantum theory is the concept of the wave-particle duality of light. A thought experiment with Schrodinger’s cat helps to illustrate this idea in terms that most can comprehend. A cat is trapped in a box with a poison gas that is randomly released. As an observer, you cannot tell whether that is dead or alive unless you open the box. Schrodinger theorized that until the box is open, the cat is neither dead nor alive but rather in between. The concept of wave-particle duality states that light and other quantum sized particles can behave as either waves or particles depending on the observer. Theoretical physicists have concluded that this idea of fluctuating realities is an underlying truth of all probabilities. Because of this, physicists believe that either we must accept this as truth and hold true the possibility of multiple universes, or that there may be something wrong with the theory as it currently stands.

References

Fell, A. (2013, February 5). Does probability come from quantum physics? Retrieved from https://www.ucdavis.edu/news/does-probability-come-quantum-physics/

Freudenrich, C., Ph.D. (2000, July 10). How Light Works. Retrieved from https://science.howstuffworks.com/light6.htm

# Engaging students: Line Graphs

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 Tinashe Meki.  His topic, from Pre-Algebra: line graphs.

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

An engaging activity to introduce line graphs is to compare the height of boys v. girls in the classroom. I would pick 6 girls and 6 boys from the class and line each group up separately from shortest to tallest in front of the same board. Then, mark their heights by placing a point above their heads. Connect the points of the height of the boys in one color and the height of the girls in another. After the activity, I would ask students probing question to analyze and compare the data in the graph: Which group had the shortest height? Which group had the tallest height? Which group’s height increased the most? and Which two points has the greatest increase in height?  Then, create a x and y axis to provoke discussion on the naming of the axes.

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

Line graphs are the foundation for many other subjects within mathematics. For example, graphing linear equations in Algebra builds concepts of connecting two or more separate points to form a line. The ability to visualize relationship between points further enhances students’ understanding of linear equations. Understanding how to interpret a line graph based on data prepares students to be able to interpret linear equations. Topics such as slope would be easier to introduce to students who understand the concept of the “change in values”. Students would be able differentiate between increasing or decreasing slope. Although a line graph’s main purpose is to compare data, subtle lessons help students understand algebraic equations also.  Students could apply this line graph to slope by plotting different points on a coordinate plane. The students can randomly connect two points and compare the relationship of the lines they have created. They could differentiate how different lines are increasing and decreasing based on their direction. They could also compare the different rates of change between the lines.

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.

I open the weather app on my phone every morning after getting ready for class. Prior to this assignment, I never noticed how the predicted weather was displayed on my phone. The app uses a line graph to show the different temperature levels during the day and week. Weather apps and websites show students how line graphs can be used for scientific purposes. An engaging activity could be to observe how line graphs are utilized to predict change in different parts of the United States. To make things more interesting, students can be assigned different cities in the U.S to search on the weather website. Once the students have analyzed the graph, they can take turns sharing interesting trends about their cities temperature line graph.

# Engaging students: Circle Graphs

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 Phuong Trinh.  Her topic, from Pre-Algebra: circle graphs.

How has this topic appeared in pop culture?

Circle graphs, or pie charts, are regularly used to visualize data and information. As technology advances, pie charts do not appear only in statistic or scientific documents anymore. They have started to show up more regularly on social media as a mean for the younger generation to express themselves. One can easily type “funny circle graphs” into Google and get back plenty of results on various.

While the students might not be familiar with the formal documents, they can easily put themselves into the situation described in Figure 1. The students can discuss what the colors from the picture represent, as well as the meanings of their proportion. From there, the students can make connection to the data and information from more formal subjects such as statistic or science.  On other hands, showing them a funny example not only will get a chuckle out of them, it can also pique their interest in the topic.

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

Circle graphs can be used in many projects and activity.  An activity that can get the students to engage in the topic is having the students create circle graphs about themselves, more specifically, how they spent their time on an average day. The students will be given a circle graph that is divided into 24 equal sectors. Each sector represents one hour. The students will use different colors to record their activities for one day (24 hours), and provide a key to show which activity is presented by each color. The proportions of each activity will be different, depends on how much time they spent for each activity. Once the graphs are completed, the students will share and explain their circle graphs with their shoulder partner. With this activity, the students will learn how to create and interpret a circle graph while sharing who they are.

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

In this day and age, most students are familiar with technology. It is a great way to engage the students into the lesson.  The National Center for Education Statistics (NCES) provides a good website for getting students to understand the relationship between data and circle graphs (Reference A). The layout of the website is fairly simple and easy to understand with 2 tabs on the left side and 5 on the right. The left tabs include “Help” tab, which provides explanation for each element that appear on the right side of the page, and “Example” tab, which provides examples of how different types of graphs look like. The tabs on the right include “Design”, “Data”, “Labels”, “Preview”, and “Print/Save”. With the pie chart design, the site allows us to adjust the data amount, or “slices”, as well as input data as needed. On other notes, under the “Labels” tab, we can choose the type of value that will be shown (For example, value or % of total). As they explore the site, the students can compare their data with the graphs in order to make connection to how the arc length of each slice is proportional to the data it represents.

References:

# Engaging students: Multiplying fractions

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 Mario Acosto. His topic, from Pre-Algebra: multiplying fractions.

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.)

Word problems can be a good way to make your students start to think about topics. I feel like giving students worksheets isn’t a good way for them to learn new material because it’s just boring and makes the students not be excited about the topic. Some word problems can be very interactive such as the example that I have right here. (https://www.pbslearningmedia.org/resource/mket-math-ee-vidgoliathbeetle/beetle/#.W4nobvZFxu0).

This video shows a great way for students to first visualize on how to multiply fractions and it also gives the students something new to know about beetles. I really love this video because of how detail the images are and gives the students an example problem at the end of the video. For the example that is given at the end of the video, I will make the students pair up and let them try to solve the example for at least 10 mins and then go over it together as a class. If the student has a hard time seeing fractions then this is a good way to see them.

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

Knowing that different cultures used multiplying fractions in their own way is so satisfying because each one is so different to where it makes you think that a new math concept of multiplying fractions could come up within the next century. This article gives amazing examples of how each culture used multiplying fractions in their own way but end up having the same mindset.    http://www.math.wichita.edu/history/topics/num-sys.html

To engage my students into learning about this I would make a chart of the different cultures there are in the above link and make the students choose one culture as a group and let them learn about it. After they have learned more about the culture then I will tell them to come up with a way to teach what they have learned to the class. This should take about half of the class time but not all. It’s a good way for students to learn something new while still being engaged.

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.

Technology is the best way to teach students in this generation because many of the students are high tech with their devices. So, having a website that teaches your students about multiplying fractions but in an engaging way. (  http://math.rice.edu/~lanius/fractions/frac5.html )

# Engaging students: Square roots

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 Julie Thompson. Her topic, from Algebra: square roots.

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

When I think of square roots my mind immediately takes me to the very popular movie ‘The Wizard of Oz’. In a scene near the end of the movie, the scarecrow incorrectly states the Pythagorean Theorem. He states it so fast that some people may not have time to process what he is saying is incorrect. The theorem he states is as follows: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” There are a couple things wrong with this statement. First of all, the Pythagorean Theorem is based on right triangles, not isosceles. Secondly, we take the square of two specific sides and set it equal to the square of the third side, not the square root of ‘any two sides’ equal to the square root of the remaining side.

As an engage, I think it would be very interesting to first show the clip of the movie to capture my students’ attention, and then have a discussion about why the theorem is wrong and what the correct theorem actually is!

Also, I found an awesome worksheet from Mathbits that is all about this scene from the movie and goes through a couple examples that shows why his theorem can’t work, and also allows students to prove why it is false!!

https://mathbits.com/MathBits/MathMovies/OzMath.pdf

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

A very engaging website that was actually introduced to me in college: KAHOOT! I first played Kahoot in my TNTX 1200 class here at UNT. It was very exciting and fun for me, as a college student, to play, so I know middle and high school students will love it as well. Kahoot is an online quiz game where students use their own technology to join in to the game with a game pin provided by the teacher. Students get to give themselves a game nickname which makes it fun to be able to see their name pop up on the scoreboard. Then a variety of questions on the topic are asked, one at a time, with a time limit for the students to answer in (usually about 20-30 seconds). This is a quick game that can be used as an engage at the beginning of class to get students thinking and excited about the topic for the day. In this case…square roots! I found a great Kahoot created by ‘remangum’ that focuses on finding square roots of numbers (it throws in a couple cube roots). Once you get passed about 7 questions, they throw some variables into the mix. One of the question asks to find d:

Sqrt (d*d)=9, where * is multiplication. In this case, d=9 because sqrt(81)=9. I like this because it allows the students to think a little harder and problem solve.

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

Many students who enter middle school/ early high school wonder why they have to learn all these pointless concepts such as square roots and the order of operations. They might even think to themselves, “When will I ever need to know this in the future when I have a job?” According to homeschoolmath.net, “The answer is that you need algebra in any occupation that requires higher education, such as computer science, electronics, engineering, medicine (doctors), trade, commerce analysts, ALL scientists, etc. In short, if someone is even considering higher education, they should study algebra. You also need algebra to take your SAT test or GED.” This is very important to let students know, but they may not believe you or care. For instance, they may say that’s true for math and science professions, but they are planning to major in something totally different and they won’t need math. Math actually can be useful in other fields, but for the sake of this question, I will stick to math and science.

In their future classes, such as Algebra II, they will be using things such as the quadratic formula. This will involve plugging in and simplifying things under a radical, as well as dealing with square roots in whole equations rather than just on their own. Also, understanding the nature of square roots will help them in future courses such as PreCalculus when they must study all the characteristics of the square root function. As an engaging aspect to all of this, I may mention that, “Studying algebra also has a benefit of developing logical thinking and problem solving skills. Algebra can increase your intelligence! (Actually, studying any math topic — even elementary math — can do that, if it is presented and taught in such a manner as to develop a person’s thinking.)”

References

A Visual Approach to Simplifying Radicals (A Get Out of Jail Free Card). (2012, January 15). Retrieved September 09, 2016, from https://reflectionsinthewhy.wordpress.com/2012/01/15/a-visual-approach-to-simplifying-radicals-a-get-out-of-jail-free-card/

Babylon and the Square Root of 2. (2016). Retrieved September 09, 2016, from https://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/

Buncombe, A. (16, April 4). Square Root Day: There are only nine days this century like this. Retrieved September 09, 2016, from http://www.independent.co.uk/news/world/americas/square-root-day-there-are-only-nine-days-this-century-like-this-a6967991.html

Fowler, D., & Robson, E. (n.d.). Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context. Historical Mathematica, 366-378. Retrieved September 9, 2016, from https://math.berkeley.edu/~lpachter/128a/Babylonian_sqrt2.pdf.

Mark, J. J. (2011, April 28). Babylon. Retrieved September 09, 2016, from http://www.ancient.eu/babylon/

# Engaging students: Solving word problems of the form “a is p% of b”

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 Christian Oropeza. His topic, from Algebra: solving word problems of the form “a is p% of b.”

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.)

Students would be able to answer word problems that involve real world applications. For example, a student could be asked: “Sam went to Academy to buy clothes, sports equipment, and fishing gear. At the register the total of Sam’s transaction before tax is \$141.32. Given that the sales tax is 8.25%, what would Sam’s total be after tax?” These type of word problems would be relatable to students, which would show them the importance of this topic in life. Students always ask the question, “how is this used in everyday life?”, and with these type of word problems students may be able to generalize the concept more easily. When students cannot relate to a topic in math they become easily discouraged, give up, and stop paying attention in class, but with problems like these the students would be able to incorporate the topic into their own lives. Some other problems that students could be asked could involve any type of scenario where there is a percentage to be found between two numbers (Reference 1 & 4).

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

This topic can be used in different scenarios for math and science, but Chemistry is an excellent example. In chemistry, there is a topic that covers calculating percent composition. The basic idea of this topic is to calculate the percentage of each element’s mass in regard to a molecule’s total molecular mass. An example would be, “Calculate the mass percent composition of each element in a potassium ferricyanide, K3Fe(CN)6 molecule.” (Reference 2). These types of problems would help students understand how much a certain element or compound is in a particular molecule. Another example of how this topic can be used, is in math when a student has to convert between fractions, decimals, and percentages in a word problem. An example could be, “Mia has a basket full of fruit. In this basket she has 1/5 apples, 2/3 oranges, and 2/15 bananas. What percent of each fruit does she have in relation to the basket?” Students would be able to work on their converting skills to enhance their understanding of multiple representations of the same number (Reference 3).

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.

Technology is always a great way to engage students especially with the newer generation of students where technology is part of their everyday life. The website mathisfun.com (Reference 4) is an excellent piece of technology to introduce or review this topic to the students because the website goes through visual representations of how a percentage of a whole looks like. Also, the website has a section where a student can input a number and a slider that allows the student to move it around to see what number would represent a certain percentage of the number inputted. Another example of effective technology is the website Khan Academy (Reference 1) because it has real world problems that are relatable. The website also gives hints and step-by-step solutions for each question in case a student is stuck and does not know what to do next. The use of multiple websites is good for students to have a variety to choose from in case one is easier to understand than another.

References:

# Engaging students: Multiplying binomials

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 Sarah McCall. Her topic, from Algebra: multiplying binomials.

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

My hope is that this topic may be easier to understand if student’s can first recall an easier concept that they have already mastered, and then build upon that foundation to learn new skills. For example, at this point students should have already learned the distributive property. To introduce this new concept, I would begin by writing 4(x-5)=4 on the board and asking students what the very first step would be to solve for x. They should know to start by distributing the four to both x and -5, to get 4x-20=4. After completing a few similar examples as a class and/or in groups, then the idea of multiplying binomials would be introduced. This way, students are less intimidated when presented with new material, and they will have a good understanding of how to distribute to each term.

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

Teaching students some of the history behind what they are learning can be a great engaging tool. In this case it is helpful to know where the foil method first originated. I would incorporate this by discussing how it first was used in 1929; in William Bentz’ Algebra for Today. In Algebra for Today, Bentz was the first person to mention the “first terms, outer terms, inner terms, last terms” rule. Students should be knowledgeable about the history behind the math they are using, so that they realize the importance of this method. I also believe that it will be cool for students to see how a method developed is still relevant 88 years later. This technique was created in order to provide a memory aid, or “mnemonic device” to help students learn how to multiply binomials. The fact that it is still being used even today proves what an influential concept it was at its time, and throughout the years.

E1. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

I am a huge fan of incorporating technology in the classroom, and YouTube is especially great because most students already use YouTube outside of school. The following clip (stopped at 1:48) provides a clear, concise explanation and demonstration of the FOIL method for multiplying binomials. It explains how factoring and foiling are related, and shows students which order to distribute in (first, outer, inner, last). The acronym FOIL is easy for students to remember, and gives them something that they can write down each time they complete a problem to help them distribute properly. Additionally, the clip is just under two minutes, which is the perfect time to ensure that students don’t zone out or lose interest before the end of the video. I would choose to follow up this video by completing a few examples as a class, emphasizing the four steps of foiling as mentioned in the video and how to use them.

# Engaging students: Adding and subtracting polynomials

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 Algebra: multiplying polynomials.

A2. How could you as a teacher create an activity or project that involves your topic?
The main idea of the activity will be finding an expression to represent the area of the border given the dimensions of the outer rectangle and the inner rectangle. The students will need to know how to multiply binomials and add or subtract polynomials. Therefore, this activity would be towards the end of the unit. Students will be asked to roam around the classroom or the hallway in search of items that already have dimensions labeled. For example, in the hallway there may be a bulletin board with the dimensions (2x² – 7) for the length and (3x – 4) for the width. Inside of the bulletin board, there will be the dimensions of a smaller rectangle. The question will be asked: What expression will represent the area I want to cover if I want to cover the only the border with paper?
Students may work in partners or groups to put minds together to solve this problem. Every object labeled with dimensions will be in the shape of a rectangle and the math involved will require students to multiply binomials and subtract polynomials.

B2. How does this topic extend what your students should have learned in previous courses?
Before students learn to add and subtract polynomials, they learn how to combine like terms such as 3x and 5x. When we add and subtract polynomials, it is very similar to combining like terms in algebraic expressions. Students will need be familiar with the concept of combing like terms before they add or subtract polynomials. To introduce the topic of combining polynomials, it can be set up horizontally.

Such as: (3x² – 5x + 6) – (6x² – 4x + 9)

By setting it up this way, students can determine which terms can be combined and which terms need to be left alone. Additionally, students will build on the concept of combining like terms as it applies to this process as well. Setting it up horizontally will also increase the chance of preventing the mistake of forgetting to distribute the negative sign throughout the second polynomial. Once students are comfortable doing it this way, the addition and subtraction can be set up as a vertical problem where students must now take the step to align the like terms together in order to add or subtract. By taking the step to set up the polynomials horizontally before vertically, it will give the students a deeper understanding of what concept is actually behind adding and subtracting polynomials.

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

The website http://www.quia.com provides a multitude of activities relating to different subjects. The game I chose to correlate adding and subtracting polynomials is identical to the actual game Battleship. The game can be played by anyone with access to the internet and Adobe. This game is interactive because you won’t have to perform math on every single shot fired at the enemy. If the student does hit one of the vessels, in order to actually “hit” the enemy’s ship, the student must successfully add or subtract two polynomials. If a student hits a vessel but is unable to solve the polynomial correctly, the game will highlight the hit area so that the student can try again. This game can either engage the students to see who can sink all of the enemy’s ships first, or it can be assigned as a homework assignment that requires showing work and screenshotting the end result of the game. Lastly, you can choose the level of difficulty of the game. For example, on the hard level, you must determine the missing addend or minuend to the expression, or add or subtract polynomials of different degrees.
The website also offers an option to create your own activities, so if Battleship isn’t panning out as desired, it is possible to create your own game for your students.
Game: https://www.quia.com/ba/28820.html

References:

Battleship: https://www.quia.com/ba/28820.html