# 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 Emma White. Her topic, from Algebra: multiplying binomials. How could you as a teacher create an activity or project that involves your topic? What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Around 600-700 AD, the Hindu mathematicians had taken the Babylonia methods of approaching equations a step further when it came to introducing unknowns, sometimes more than one unknown in a single problem. It wasn’t until the Medieval times did the Islamic mathematicians discuss the variable x and how important it was. This is when the binomials theorems evolved. Furthermore, the Islamic mathematicians were able to use many operations on polynomials and soon binomials, such as multiplication, division, finding roots, and more! One thing I find highly fascinating is the Islamic mathematicians advanced the study of algebra, which “flourished during the golden age”. Evermore so, private collections were found in a lost Islamic library, which was destroyed in the 13th Century. These private collections “altered the course of mathematics.” An example of a concept that was furthered studied was the Fibonacci sequence (which is, in my opinion, one of the most fascinating things in math history and how it relates to the world and finding mathematics around us, but that is for another time…). All I can say is the Babylonians, the Hindu and Islamic mathematicians were a driven and mathematically inclined people and it blows my mind how far these people brought the world of mathematics. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

When it comes to finding ways to use technology for multiplying binomials, I truly believe visuals are essential. I’m a little biased since I was introduced to a way of multiply binomials just last semester in one of my teaching classes and it BLEW MY MIND. I wish I knew how to do this earlier in high school!  Essentially, this online source allows the student to use algebra tiles without having them physically in front of them. Therefore, they can use this source if they have technology capable of doing so (such as a phone, computer, tablet, etc.). This source is visual and easy for students to understand and manipulate. The student starts by placing the corresponding tiles for one binomial across the top like a table (would be 4 x-tiles and 2 1-tiles). Along the left side, the other binomial is represented (long ways/up-and-down). You then multiply corresponding values and where they meet in the open area (example: where an x-tile and another x-tile meet, it would become since x times x is ). Algebra tiles can also be used for upcoming topics the students would learn, such as completing the square. For a student who may have trouble grasping the idea of multiplying binomials and struggling to understand the concept of abstracts, using algebra tiles will hopefully help with the misunderstandings and confusion. All I’m saying is if this concept of online algebra tiles assisted a college student and made the topic MUCH easier to visualize and explain, I’m sure most high school students will find the use of technology in their math class interesting. Who knows, some students may come to love math more because of it!

Reference(s):

“Multiplying Binomials by the FOIL Method” by Professor Dave Explains:

“History of Polynomials”: https://polynomialshistory.weebly.com/history.html

“How modern mathematics emerged from a lost Islamic libray”: https://www.bbc.com/future/article/20201204-lost-islamic-library-maths

Algebra Tiles: https://technology.cpm.org/general/tiles/

# 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 Lydia Rios. Her topic, from Algebra: defining a function of one variable. How does this topic extend what your students should have learned in previous courses?

From Prekindergarten and up, students have been practicing skills that prepared them from the concepts of a function. By counting they knew that they were adding that same number to every other number in the same sequence. By doing 1,2,3,4,5,… counting by ones they realized that every left number was being added by one to get the right number. They were taking the input 2 and doing the operation of addition by 1 to get the output of 3. The same thing was happening for other counting sequences, or even general operation statements such as 1+7=8. They have been building up to the idea of functions without recognizing that they were. You can use this no simple idea that’s been installed in them to understand what functions are. You can build them up from here and then start giving them statements with a missing component so they can find a missing variable. Then finally, building them towards defining a function where you give them similar statements with a missing component so that they can start writing out their own equations. *Don’t forget to introduce input and output and that are function represent the relationship between out input (x) is having this operation done to it to get our output (y).

Mathematics Vertical Alignment, Prekindergarten-Grade 2 (texas.gov)

Introduction to Functions | Boundless Algebra (lumenlearning.com) How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

You could use different appearances in pop culture to get students to understand input and output, such as when you are playing video games you are putting your input on the controller to get the output on the screen. However, this may not have an association with function unless you want to start getting into detail about programming. Therefore, to bring about the topic of functions I would just use a word problem that associates with pop culture. You could also bring the business side of pop culture into the class, such as setting up an equation that shows how the more tickets bought makes and increased revenue for the production of a movie. For example, lets say a ticket cost \$8.50 and the production get’s 40% of the profit. Then you could set up the equal as 0.40(8.5X)=Y with 0.40 representing 40% of the profit that the production team will receive of the \$8.50 tickets. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

The topic of inputs and outputs can be touched on in reference to theatre. Both in lighting and sound, inputs and outputs are used. Therefore, the concept of this can be taught to the students. For lighting, you can talk about DMX which is what LED lights use so that the technology in the lights can pick up the functions that the computer is telling it to do. You connect the DMX in cord to the DMX in into the lighting board and then the DMX out of the lighting board to the DMX out on the lights. The same idea works with audio. However, the inputs are the microphones and the outputs are the speakers. You would take the microphone aux cord and plug that into the inputs on the Sound Board and then you would take the speaker cord and plug that into the outputs on the Sound Board. Therefore, that particular microphone is connected to that speaker and will only come out of that speaker.

“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: Absolute value

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Ethan Gomez. His topic, from Pre-Algebra: absolute value. How does this topic extend what your students should have learned in previous courses?

This topic extends students understanding of distance relative to positive and negative integers. First, students learn the positive integers — the counting numbers. Then, students are introduced to negative numbers. Visually, positive integers are to the right of zero, and negative integers are to the left of zero; students understand that these numbers exist and where they lie relative to each other. Essentially, students start by having a directional sense of numbers. Also, students also have a good understand of distance. With the concept of absolute value, students are able to associate distance with positive/negative numbers. Negative numbers aren’t just randomly placed but are rather a certain unit away from the number zero. For example, the absolute value of -5 is 5. So, -5 is not just a number that happens to be to the left of zero, but it is also 5 units away from zero. We now have a spatial sense of integers along with the directional intuition, making the numbers feel a bit more tangible and less abstract. How can technology be used to effectively engage students with this topic?

Technology can be used to effectively engage students with the concept of absolute value since it allows students to explore its meaning. Students can discover the connection between distance and integers on their own, which reinforces the meaning-making process that teachers strive to provide students. For example, Gizmos has a wonderful tool that displays integers on a number line. On this gizmo, students are provided a visual that portrays the spatial and directional aspect of integers. This gizmo also makes students take note of the similarities between the absolute value of positive and negative numbers, forcing them to think about why they happen to be the same number sometimes.

https://gizmos.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=210 How could you as a teacher create an activity or project that involves your topic?

I could create a documentation sheet for students to keep track of what time they get home from school. They will keep track of this information for a week. The first time that they record will be the “reference time.” Every day after that, the students will document the time they get home, and how many minutes off it was from the first time, as well as if it was earlier or later than the first time. Having students think about “how many minutes off” they were from the first recorded time get them used to the idea of a magnitude, and how the number they are using tends to always be positive; the only difference is in the description of that number, which can be associated with the positive and negative characteristic of integers.

# Engaging students: Adding and subtracting decimals

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 Sydney Araujo. Her topic, from Pre-Algebra: adding and subtracting decimals. How could you as a teacher create an activity or project that involves your topic?

I have been riding horses since I was 5 years old, when I was around 12 years old I got into the equine sport called barrel racing. The sport is an equine speed event. Essentially horse and rider go through a clover leaf pattern as fast as possible. Placings are separated by 1000ths of a second. At competitions, there are different divisions, typically 4-5. These divisions are separated by half a second. For example, if the winning time of the barrel race was 15.536 seconds, then the winning times of the different divisions would be as follows, 16.036, 16.536, 17.036, and so on by simply adding half a second. It was always interesting to compare times and to see where I could possibly stand in different divisions based on my time and the winning time. I could see myself creating an activity that had my students be given different scenarios like being given a winning time and determining the winning times of the different divisions, determining which division a certain time would be in, how much faster or slower at time needs to be to place, and so on. This was an activity I did regularly at barrel races for myself and other people when watching. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

One of the more popular movies I can think of is the movie Hidden Figures. The movie is about a team of African American women mathematicians who work for NASA to help launch an astronaut into orbit. There are several different scenes in the movie where math problems are being solved and this involves the adding and subtracting of decimals. It shows that doing math by hand and math itself is very important in the real world and has helped us make great discoveries and progress. Another movie where adding and subtracting decimals appeared is in the movie called Gifted, where an uncle of an extremely math gifted child suddenly becomes her guardian. She solves several advanced math problems and proofs throughout the movie. The topic also appears in the classic sci-fi TV show Star Trek. It is constantly brought up throughout the series, typically from the character Spock who will make calculations on the spot. As he is a very smart and logical character, he is often the one who must do the required math in the series. How can this topic be used in your students’ future courses in mathematics or science?

Adding and subtracting decimals is constantly used in both mathematics courses and science courses throughout high school and eventually college. We see adding and subtracting decimals in some trigonometry concepts when solving for theta and using different trig functions. Students will also see this very often in algebra when dealing with real world situations that forces them to have to use decimals. It appears quite a bit when students approach quadratic equations as once, they learn the quadratic formula to solve quadratic equations that don’t have integers, they will run into many decimals and having to add and subtract. Looking even further into the future of student’s math courses, we often must add and subtract decimals when evaluating different limits and integrals. Adding and subtracting decimals also appears in physics courses. Students will often see many decimals in physics when solving problems using force, density, displacement, and so on. You often see more imperfect numbers and situations in physics as it is more often seen in the real world.

# 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 Ashlyn Farley. Her topic, from Pre-Algebra: powers and exponents.

One class activity that will engage students while reviewing and/or teaching Exponent/Power concepts is “Marshmallow and Toothpicks.” This activity can be used for teaching the basic of exponents, as well as exponent laws. The idea is that the toothpicks are different colors, and the different colors represent different bases, thus the same color means it’s the same base. The marshmallows represent the exponent, i.e. the number of times the student needs to multiply the base. By following a worksheet of questions, the students should be able to solve exponent problems physically, visually, and abstractly. This activity, I believe, is best done with partners or groups so that the students can discuss how they think the exponents/exponent laws work. After the activity, the students are also able to eat their marshmallows, which encourages the students to participate and complete their work. Exponents are used in functions, equations, and expressions throughout math, thus having a deep understanding of exponents and their laws is very important. By fully mastering exponents and exponent laws, the students will be able to more easily grasp more difficult material that uses these concepts. Some specific ideas that use exponents and/or exponent laws in future math courses are: multiplying polynomials, finding the volume and surface area of prisms and cylinders, as well as computing the composition of two functions. Exponents are also used in many other situations than just math, such as in science or even in careers. Some careers that consistently use exponents and/or exponent laws are: Bankers, Computer Programmers, Mechanics, Plumbers, and many more.

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

An easy way to introduce students who have never seen exponents or exponential growth before is to use a graphing calculator. By plugging in an exponential function into the calculator and viewing the graph and zooming out, students can easily see how quickly numbers start to get The website Legends of Learning focuses on creating educational games for students in kindergarten through 9th grade. One game that goes over exponents, as well as the exponent laws, is Expodyssey. This game has the students solve problems to “fix” a spaceship to get back to Earth. The problems are built upon each other, so it starts by having the student answer what an exponent is, then what multiplying two exponents same base is, and keeps building from there. Each concept has multiple problems to be solved before moving on so that the students can show their mastery of the content. I believe that this game also helps improve cognitive skills by having the students do various activities simultaneously, such as calculating, reading, maneuvering elements and/or filling answers as required.

References:
Blog: Number Dyslexia

# Engaging students: Finding prime factorizations

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 Bri Del Pozzo. Her topic, from Pre-Algebra: finding prime factorizations. How could you as a teacher create an activity or project that involves your topic?

An activity that I would create for my students involving Prime Factorization is based on an example that I saw on Pinterest. I would prepare an activity where students would be given a picture of a tree and assigned a two-digit number. I would then have students decorate their tree and at the base of the tree, they would write their assigned number. Then, as the roots expand down, students would be able to write the factors of their number as a factor tree until they are left with only prime factors (based on the image from https://www.hmhco.com/blog/teaching-prime-factorization-of-36). In the example from Pinterest, the teacher focused on finding the greatest common divisors between two numbers and used the factors trees as guidance. For my activity, I would assign some students the same number and emphasize that some numbers (such as 24, 36, 72, etc.) can be factored in multiple ways, so the roots of the trees could look different depending on how the student decides to factor their number. How can this topic be used in your students’ future courses in mathematics or science?

There are a few ways that Prime Factorization can be used in my students’ future math courses. Prime Factorization is incredibly useful when learning how to simplify fractions. By practicing Prime Factorization, students become more familiar with the factors of large numbers, which becomes helpful when simplifying fractions. In the instance that a fraction is not in its simplest form, students will have an easier time recognizing such and will feel more confident in simplifying the fraction. Additionally, Prime Factorization prepares students for finding Greatest Common Divisors. Knowing how to find Greatest Common Divisors can be useful when solving real-world problems as well as in simplifying fractions. At a higher level of math, Prime Factorization allows students to practice the skills needed to prepare themselves for factoring things more complicated than numbers. For example, the idea of factoring can be applied to factoring a common factor out of an expression, factoring quadratic equations, and factoring polynomials with complex numbers. 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.

Khanacademy.org would be a fantastic website to engage students in this topic because of the inclusion of multiple representations. This website allows students to work through multiple practice problems where they can find the Prime Factorization of a number. When the student gets the question correct, they can move on to the next question, or they have the option to view a brief explanation on how to arrive at the correct answer. If students get a problem incorrect, they can retry the problem or get help on the question. The “get help” feature also provides students with a brief explanation, with options in video form and picture/written form, of how to solve the problem. Another important feature of this website is the ability for students to write out their thoughts as they work through the problem. Khan Academy allows students the option to use an online “whiteboard” feature that appears directly below the problem. This “whiteboard” feature allows students to write out their work and also offers a walkthrough of how to draw a factor tree.

# Engaging students: Solving two-step algebra problems

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 Chi Lin. Her topic, from Pre-Algebra: solving two-step algebra problems. How could you as a teacher create an activity or project that involves your topic?

There is an interesting activity that I found online. It is called mini task cards. However, I want to rename this activity as “Find your partners” as an engage activity in this topic. I am going to create some two-step equations on the cards and give those cards randomly to the students at the beginning of the class. Each student has one mini card. The students will have 5 minutes to solve the equations and they will find the partners who have the same answers as them (there is 2-3 person in each group). The person who has the same answer with them will be the partner that they are working together with in the class. I will set up the answer as their group name (for example, if the answer is 1, then it means the group name is “Group One”). Here is an example that how the card will look like.

12 Activities that Make Practicing Two-Step Equations Pop How can this topic be used in your students’ future courses in mathematics or science?

Solving two-step equations is the foundation of solving multi-step equations. Solving two-step equations looks easy but it can become very hard. This topic can be applied in lots of areas such as high-level math classes, computer science, chemistry, physics, engineer, and so on. Most definitely, the students will see lots of problems about solving multi-step equations in different high-level mathematics courses in college, such as pre-calculus, calculus 1-3, differential equations, and so on. Also, the students will use the knowledge when they write the code in computer science class. For example, when they write down the code of two-step or multi-step algebra problems, they need to know which step goes first. If they do the step wrong, then the computer program will compute the wrong result. Moreover, the students will use solving two-step equations in chemistry class. For example, the students will apply this knowledge, when they write down the chemical equations and try to balance the equations. How does this topic extend what your students should have learned in previous courses?
First, students should know what linear equations are and how to write down the linear equations. Second, students should know how to solve one-step algebra problems, such as $x+8=16$ or $x/8=16$. Students should have learned that when they solve for the one-step equations (addition and subtract), whatever they do to one side of the equation, they need to make sure they add the same thing to the other side. For example, when they solve the equation $x+8=16$, they can subtract 8 for both sides, which is $x+8-8=16-8$. Therefore, x=8. Also, student should know that when they solve for the one-step equations (multiplication and division), they need to multiply both side by the reciprocal of the coefficient of the variable. For example, when they solve the equation $x/8=16$, they need to multiply the reciprocal of $1/8$ for both sides, which is $x/8*8=16*8$. Therefore, $x=128$. Thus, when they learn to solve two-step equations, they need to combine these rules.

Solving Two-Step Equations

# Engaging students: Solving one-step algebra problems

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 Emma White. Her topic, from Algebra: solving one-step algebra problems. How can this topic be used in your students’ future courses in mathematics or science?

Solving one-step algebra problems strings into many future scenarios the student may (and will probably) encounter. One-step algebra problems infer that there must be two-step algebra problems and three-step algebra problems and so forth. As mathematicians, we know this to be true. While mathematics in my focus of study, I want to show the importance of learning this concept as it will aid in other classes. Stoichiometry is a concept taught in chemistry that has to do with the “relationship between reactants and products in a reaction” (Washington University in St. Louis, 2005). Chemical reactions require a balance. Essentially, once-step algebra expressions require just the same where both sides of the equations must be equal for the expression to be true. An example of a stoichiometry equation one may see in chemistry would be:

_KMnO ${}_4$ + _HCl → _MnCl ${}_2$ + _KCl + _Cl ${}_2$ + _H ${}_2$O

In the blanks, a variable can be placed, such that:

aKMnO ${}_4$ + bHCl → cMnCl ${}_2$ + dKCl + eCl ${}_2$ + fH ${}_2$O

Next, we would apply the Conservation of Mass. This concept deals with the number of atoms that must be on each side for the equation to be balanced. Writing the elements and their balanced equations with the variables, it follows:

K: a = d
Mn: a = c
O: 4a = f
H: b = 2f
Cl: b = 2c + d + 2e

As we can see, there is going to be more expressions and substitutions that must take place. That is something you can solve on your own if you wish. Overall, we see the importance of learning one-step algebra problems because this will be the foundation for solving more complex questions, even more so outside of the math classroom. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Theatre is more than the actors on the stage. While the performance and show are the part most people acknowledge and enjoy, the technical part behind the performance is what allows the show to happen. Algebraic problems are often used in technical theatre, especially when it comes to building a set. A prime example is building a single foundation (usually used in One Act plays where the whole play takes place in one scene). Focusing on a rectangular foundation, if we know the amount of space the actors, set, and featuring décor need, we can use this in an algebraic expression. Furthermore, if we also know dimensions of one of the sides (length or width), a variable can be used for the unknown side (since the area of a rectangle is length times the width). If we want to take this a step further, multiple one-step algebraic expressions can be used when making the foundation. If we know the length and width of the foundation and the length and width of the sheet floorboards to be used, we can write various expressions to determine how many sheet floorboards need to be used lengthwise and widthwise (example shown below). How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The use of technology is on the rise and the involvement of newer generations is greatly rising as well. Because of this, utilizing online resources is an effective way to capture the attention of the students and make math more engaging. Using algebra tiles is a perfect way to resemble this topic, even more so when it can be done online. Therefore, the teacher does not need to buy any materials and the students (especially high schoolers) don’t have to carry paper resources around or even home where, we all know, they will end up in the trash. Online algebra tiles provide a way to visually see the one-step algebra problem and work accordingly. Even so, these tiles can be an introduction and foundation on what is to come (these tiles are also a great source for solving two-step equations, distribution, polynomials, the perfect square, and so forth). Another insight for using online algebra tiles is in some schools where technology such as tablets/computers are provided, the students can share their screens to a projector (or whatever resources the classroom may have) and describe their thinking process to the class. This builds on the idea of students learning, processing, and being able to teach their peers what they learned as well.

References

# Engaging students: Finding points on the coordinate plane

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 Morgan Mayfield. His topic, from Pre-Algebra: finding points on the coordinate plane. C2: How has this topic appeared in high culture (art, classical music, theatre, etc.)?

One popular art/sport high school students may take part in is marching band. I did four years of marching band in high school and I loved it. One has to wonder: “how does each performer know where they should be?” I’ve included a link from bandtek.com that describes the coordinate system marching bands use. It isn’t quite the same as the coordinate plane in a math class. When starting marching band, you learn how to take appropriately sized “8 to 5” steps, which simply means 8 equally spaced steps for every 5 yards on a football field. Each member will receive little cards that have “sets” on them. A set is a specific point on the field where the performer must be at a specific time of the show. Usually, performers will take straight paths from set to set in a specific amount of 8-5 steps. Looking at a bird eye’s view of the football field, one can see a rough coordinate plane. Like a coordinate plane has 4 quadrants, a football field has a rough 4 quadrant system where a performer is assigned to stand a specified amount of 8-5 steps from a specified yard line either on side 1 or 2 for their horizontal position and a specified amount of 8-5 steps from the front/back hash for vertical position facing the home sideline. Side 1 refers to the left side of the field from the home side perspective, Side 2 refers to the right side of the field from the home side perspective, and the front/back hash refers to the line of dashes that cut through the middle of the field horizontally from the home side perspective.

An example bandtek.com uses is, “4 outside the side 1 45, 3 in front of the front hash” which would mean the following position: D1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

René Descartes was a 17th century (1600’s) French mathematician and philosopher. Many people study his work in modern day math and philosophy classes. Some may know him as the man who wrote “cogito, ergo sum” or “I think, therefore I am”. Well, there is a legend about his discovery of the Coordinate Plane. Descartes was often sick as a kid, way before modern medication and technology. He would often have to stay in bed at his boarding school until noon because of his illnesses. This gave him quite a bit of downtime to be observant of his environment. Laying on his bed, he could see a fly crawl around on his ceiling. He thought of ways to describe the location of the fly as it scuttled about the ceiling. Imagine telling a friend where the location of the fly was, “A little to the left of the right wall and a little down from the top wall”. This just isn’t precise enough, nor an easy way to communicate information. However, Descartes realized he could quantify the precise location of the fly from using the distance from a pair of perpendicular walls. Descartes then translated this idea onto a graph where the perpendicular “walls” continued infinitely in both directions and became “axes”. “Flies” then became “points” or “coordinate pairs”. Thus, the coordinate plane was born, and so was a way to describe points in space. Just a little bit of imagination, self-questioning, and observation lead to a fundamental change in Mathematics, a way to tie Algebra and Geometry together. 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? 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 believe that https://www.chess.com/vision could be an effective website to engage students on finding points on the coordinate plane in a class that is being introduced to the idea for the first time. Many students won’t know how a chessboard is setup or even know how to play chess. The cool things are that they don’t need to know the fundamentals of chess and that the chessboard is essentially Quadrant I of a coordinate plane (where a1 is in the bottom left corner). The above website tests the player to locate as many squares (points) on a chessboard (coordinate plane) as they can in 30 seconds, given random chess coordinates. There is a way to toggle settings to also test yourself on moves and squares. In a classroom, I would only toggle the setting to list random “black and white squares” where the board is set with a1 at the bottom left corner. Students could start the day with this website as a precursor to formalizing the idea of finding points on a coordinate plane. This website is engaging (with an exclamation point)! The game can be made into a fun little competition amongst students. The time limit and game-y feeling to it encourages active participation. The game takes minimal explanation from the teacher for students to get the hang of it (no chess skills required). The fact that chessboards have one axis in letters and the other axis in numbers aids students in reading the coordinate plane x-axis first, then y-axis like the chess coordinates. I would only have the students run the game for a few rounds, making the activity in total 7 minutes or less.

References:

https://www.chess.com/vision

https://wild.maths.org/ren%C3%A9-descartes-and-fly-ceiling

https://maths2art.com.sg/2018/01/16/have-you-ever-followed-a-fly

# 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 Austin Stone. His topic, from Pre-Algebra: powers and exponents.

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

“The number of people who are infected with COVID-19 can double each day. If it does double every day, and one person was infected on day 0, how many people would be infected after 20 days?” This problem can be a current real-life word problem that all students can relate to given the times we are in. This problem would be a good introductory for students to see how quickly numbers can get when using exponents. This would be an engaging introductory to exponents and will get the students interested because they can easily see that this can be used in current problems facing the world. This problem could also work later in Algebra if you ask how many days it would take to infect “blank” amount of people. This makes the question more of a challenge because they would have to solve for “x” (days) which is the exponent. How has this topic appeared in the news?

This topic has been the news so far in 2020 if we are being honest. COVID-19 is a virus that has an exponential infection rate, just like any virus. When talking about COVID-19, news reporters and doctors usually use graphs to depict the infection rate. These graphs start off small but then grow exponentially until it slows down due to either people being more aware of their hygiene habits and/or the human immune system getting more familiar with the virus. Knowing how exponents work helps people better understand the seriousness of viruses such as COVID-19 and the everlasting impact it can have on the world. Doctors study what are the best ways to slow down the exponential growth so that a limited number of people contract and potentially die from the virus. To do this, they predict the exponential growth keeping in mind the regulations that may be enforced. Whatever regulation(s) slow down the virus the most are the ones that they try to enforce.

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

An easy way to introduce students who have never seen exponents or exponential growth before is to use a graphing calculator. By plugging in an exponential function into the calculator and viewing the graph and zooming out, students can easily see how quickly numbers start to get massively large. A teacher can set this up by giving the students a problem to think about such as, “how many people would be infected with the virus after “blank” amount of day?” Students then could guess what they believe it would be. After revealing the graph and the actual number, students will probably be surprised at how big the number is in just a short amount of time. After that, the teacher could show a video on YouTube about exponential growth and/or infection rates of viruses and how quickly a small virus can turn into a pandemic. This also has very current real-world applications.
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