Engaging students: Midpoint 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 again comes from my former student Christine Gines. Her topic, from Geometry: deriving the midpoint formula.

How has this topic appeared in pop culture?

Finding the midpoint between two points is a fairly common situation we find ourselves in daily. Take for example cutting a sandwich into two equal halves. Here you are estimating the midpoint between the ends of the sandwich. Maybe you want the bigger half of the sandwich though. In this case you first find the middle and then move slightly away. Whether we realize it or not, finding midpoints happens all around us and bringing this to students’ attention is crucial for their development of connections.

One way to aid these connections is to demonstrate how midpoints appear in our cultures. In particular, I found a popular music video “Meet Me Half Way” by The Black Eyes Peas. The video/song is about Fergie and Will.I.Am being apart and missing each other. Fergie’s solution is “Can you meet me halfway? Right at the borderline. That’s where I’m gonna wait… for you.” Fergie and Will.I.Am’s beginning locations are the endpoints in this scenario and they will meet at their midpoint. In the video, Fergie has already reached midpoint. Here, her lyrics are “Took my heart to the limit, and this is where I’ll stay. I can’t go any further than this.” This can be interpreted as a unique midpoint. If Fergie goes any further, she will no longer be at the midpoint. Her limit is the one midpoint. At the end of the video, Fergie and Will.I.Am are reunited at their midpoint.

After this connection is made, it could be reinforced by giving students specific coordinates of Fergie and Will.I.Am and asking students to find their midpoint. For example, Fergie and Will.I.Am were shown to be on different planets in the video. So, the teacher could give them the coordinates to Jupiter and the earth. If they succeed with this problem, a follow up could be to find the endpoint when you have Will.I.Am’s endpoint and their midpoint.

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

A common issue students face regarding formulas is memorizing them without fully comprehending the formulas. They say, “give a man a fish and you feed him for a day; teach a man to fish and you feed him for a lifetime.” So, let’s not just give students a formula, but teach them how to derive the formula by letting them explore the concepts for themselves. A good activity to let students do this is as follows:

In this activity students will Investigate finding the midpoint of a line segment and derive the formula for the midpoint of two points on a coordinate plane.

Have students work in groups of 3 or 4. Each group will have a sheet of large graph paper, markers, a ruler, dice and a penny.

Procedures:

Students will find two points by rolling dice and tossing penny (Dice represents number and penny represents positive or negative) and plot them.
They will draw a line to connect these two points.
Next, students can use the ruler to estimate where the midpoint should be.
Have students investigate ways to accurately find the midpoint of the segment and challenge them to find a formula as well.

Students can create several graphs so that they can recognize the patterns. By letting them draw and plot their own graph, students will more readily realize that the midpoint is exactly in between the two x-values and the two y-values. This will then hopefully lead students to recall how to find the average of two numbers, which is essentially what the formula is. It is important that students make this connection to their previous knowledge and to guide students through this exploration, teacher can ask leading questions such as:

What could you use to represent the numbers so you can write a formula?
How did you find that midpoint?
Are you sure that is really the midpoint?
How can you find the number in between two different numbers?

I don’t know about you but I’ve always thought the best educational games are the ones that actually feels like a game and not just something your teacher is making you do. This is exactly how the game “Entrapment” by The Problem Site feels like. Entrapment is actually a puzzle game. The object of the game is to create line segments such that all the given dots are midpoints to these segments.

More specifically, every red dot must be the midpoint of a line segment connecting two gray dots on the playing field. In the image above, the player is one move away from finishing since there remains one red circle which is not a midpoint. This puzzle is not only addicting, but it teaches students to recognize the relationship of x and y (individually) to the midpoint. After completely only a few of these puzzles, this relationship becomes part of your strategy, which in turn pushes students further away from memorization and brings them closer to comprehension. This puzzle brings all these educational benefits, yet it just feels like you’re playing a game!

http://www.theproblemsite.com/games/entrapment.asp#.UxF5ImJdXHQ

Engaging students: Defining the terms acute triangle, right triangle, and obtuse triangle

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 Brittney McCash. Her topic, from Geometry: defining the terms acute triangle, right triangle, and obtuse triangle.

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

As the students are walking into class, I will already have a picture of just a standard (acute) triangle on the board (be it Promethean or white). As class begins, I will pose the question of, “Who can tell me what we are looking at?” Of course, the students will tell me a triangle. I will then proceed to show two more triangles, an obtuse and then a right triangle and ask the same question. The answer will be the same for each. After I show all three, I will put a picture of all three together and ask the students what some of the differences are in each. Once we state the obvious (That there are angles of bigger and smaller sizes in each), I will then post a picture of Euclid. I will ask if anyone knows who this is. More than likely no one will. I will then proceed to tell my students that in 300 BC this man, Euclid, wrote a book called Elements. In this book. We had 4 sub books that consisted of mainly triangles. When telling this fact I will put emphasis on the word “whole” to show how insane that is. By now, the students should be in awe that someone could write so much about triangles. Then I would state that inside this book, Euclid proved that there were 3 different types of triangles. There is obtuse, right, and acute. We could then discuss as a class what we think each triangle presented at the beginning of class is just by sheer guess, and then see if they were right by then going into the actual discussion of the definition of each term. This is a fun and knowledgeable way to bring in some historical background of what they are learning. This shows that it’s just not going to go away, that it has been around for a while, and is still being thoroughly discussed in classrooms, like ours.

A1. What interesting word problems using this topic can your students do now?

An interesting topic that is still around today, is the Bermuda Triangle. After a brief introduction of the definition of an obtuse, right, and acute triangle, I will pose this problem: (There will be a picture of the Bermuda triangle with points labeled, (posted below).)

You are captain of the ship Euclid and are sailing straight for the Bermuda triangle. Hearing of all the bad things that can happen inside the “triangle,” you want to avoid it as best as you can. Luckily for you, you have a super power. You are able to shift one point of triangle wherever you would like. Using your super power and the knowledge of triangles we discussed previously in class, decide which point you would move, and into what triangle so that you can sail past smoothly. You will need to draw your final result with a justification of why you chose that triangle and point.

This question is not only engaging, but it makes them think abstractly. They have to use their knowledge of triangles and produce a result that fits our discussion. Then not only do they have to draw it, they will need to discuss it as well. Talking about why they chose the method they did, helps students retain and process the information better. Take into account, there are multiple ways to answer this question.

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

Triangles are such a widely used topic, that it is almost guaranteed you will see them again. Not only will you use them later on in our course (Geometry), but you will be using them in pre-calculus and so forth. The main triangle you will work with in the future is the right triangle. That is why it is so important for you to learn the difference now between the triangles. Later, you will be discover the different ways you can solve for sides and angles with a right triangle, you will be discovering the different properties that come with each triangle, and how you can draw them using circles. But before you can do any of that, you have to start with the basics, like knowing which triangle is which and their definitions. I would then go in to explain that now only would triangles be used in classes, but in the real-world as well. They are everywhere we look, literally. Every time we look, we are looking at a specific angle. In the video games we play, we are always making decisions based off of the angles we can use, it’s how we build things; it’s everywhere! To have a basic understanding of something so usable in our world, would be essential to success!

Resources:

http://en.wikipedia.org/wiki/Triangle

http://www.livescience.com/23435-bermuda-triangle.html

Engaging students: Dilations

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 Candace Clary. Her topic, from Geometry: identifying dilations.

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

Dilations are types of transformation. One activity that I could create for my students is a matching game. I can create cards with index cards, or sheets of paper that have been cut up, that have pictures on them. Each one will be labeled and the students must classify them as dilations, why they are considered dilations, and how they were dilated. As a follow up to this activity, I could assign a topic to create their own city, or small town. They would be required to draw out their town, as well as model it using common crafts. After they do this, they will need to be able to dilate the buildings, and other such things, to make a life size city. They will not have to make the city with a model, but instead, they will need to make a blue print using their model in mind. On this blue print, they will need to inform me of the size of the dilations.

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

Dilations can be used in many different subjects. Dilations can be used to find sides of a triangle when learning about the triangle congruence theorems. These can be useful in algebra when finding side lengths of figures. This may not happen very often, but it is displayed in algebra. Another place that it will help, although it may not be math, it will help in math classes for architectural students, as well as help people in construction. Many science classes require science projects that work and simulate something real. Dilations can be used when making these projects because you can’t make a real river, but you can structure something that is a smaller figure to the real thing, same thing as a volcano. With architecture, dilations can help with making blue prints and can help in building these blue prints with dilations in mind. With construction, those are blue prints too. I’m not saying in order to build something you must know how to dilate something, but it will help tremendously.

How has this topic appeared in pop culture?

To get the students engaged in the topic, I could bring up the Disney channel movie ‘Honey I Shrunk The Kids.’ This will bring up a discussion with the kids when I ask them what the dad did with his shrink ray. Some ideas that may come up will be that he made them smaller, and then at the end of the movie he made then bigger, back to normal. But in the people were still the same people, they didn’t change, only the size did. At least I hope that is what happens in the discussion. I could then instruct the students into pretending that they had a shrink ray and ‘shrink’ some shapes, as well as other students. This activity, and their answers will be recorded on a chart that they will turn in at the end of class. They, themselves, can decide what size they want to shrink to, but they have to remember to bring the student back to normal at the end of class. I think this activity will be fun for the kids because they will never forget what a dilation is, since they have been ‘dilated’.

Engaging students: Finding the area of a right triangle

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 Andrew Wignall. His topic, from Geometry: finding the area of a right triangle.

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.)

To introduce the topic of the area of a right triangle early in a lesson, we can first examine the area of a rectangle, which students should already know how to do.

Say you have a large rectangular garden, 60 feet wide and 10 feet long. Home Depot sells sod (which is a pre-grown grass on a net that can be spread on the ground) at a rate of $3/square foot. What is the area of the garden, in square feet? How much sod should you order? How much would it cost to cover the entire garden with sod?

Instead of having the entire garden covered with sod, suppose you wanted to cover part of the garden with sod and leave the rest as soil for planting flowers. To make it more visually interesting, you decide to set the sod as a triangle? The sod triangle will have a base of 60 feet and a height of 10 feet. What is the area of this triangle in relation to the area of the entire garden? What is the area of this triangle? How much sod should you order? How much would it cost to cover the triangular area with sod?

Through this activity, we can investigate a relationship between right triangles and rectangles, and also the relation of the area of a triangle compared to the angle of a rectangle.

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

One tool to show the area of a right triangle quickly and easily is the Area Tool on Illuminations (http://illuminations.nctm.org/Activity.aspx?id=3567). With trapezoids, parallelograms, and triangles available, you can click and drag the three vertices of a triangle and instantly see how the area is affected. You can create a quick table and keep a running tally of the base, height, and area, so you can recalculate in front of the class.

Illuminations has a sample lesson plan available online for discovering the area of triangles, and integrates this tool into the plan. If not using this tool as part of a similar plan, we must understand that this tool will not be great for introducing the lesson, as there is no button to lock onto a right triangle. However, there is a button to lock the height, so when you move the vertex opposite the base, you can see how the area does not change, see how the height can be outside the triangle, and extend the formula for the area of a right triangle to the area of any triangle. This tool can then be used in further lessons when discussing the area of parallelograms and trapezoids.

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

Since triangles are one of the most basic shapes, the area of triangles comes up time and time again. Triangles will also be used to find the area of more complex polygons, such as hexagons and irregular polygons, by breaking down complex shapes into simple triangles and quadrelaterals. Trigonometry uses right (and non-right) triangles extensively; in Precalculus, we will revisit the area of triangles, and learn how to find the area of triangles without explicitly being given the base and height.

Outside the classroom, the area of a triangle is used extensively in architecture, as triangles are strong, and triangular trusses and frames are used in many steel structures. As the inside empty area of the triangle increases, then the stress on the triangle increases, and architects must take this into consideration.

Triangles are also used in 3d computer graphics, as the 3d shapes they design actually consist of lots of little triangles, and they have to fit textures of a certain size (say 512 pixels x 512 pixels) onto a few triangles, so it is important that they know how and where for these textures to lie.

References

Math is Fun, “Activity: Garden Area”. http://www.mathsisfun.com/activity/garden-area.html

Illuminations: Resources for Teaching Math, “Discovering the Area Formula for Triangles”. http://illuminations.nctm.org/Lesson.aspx?id=1874

Illuminations: Resources for Teaching Math, “Area Tool”. http://illuminations.nctm.org/Activity.aspx?id=3567

Home Depot, http://www.homedepot.com/p/StarPro-Greens-Centipede-Southwest-Synthetic-Lawn-Grass-Turf-Sold-by-15-ft-W-rolls-x-Your-L-2-97-Sq-Ft-Equivalent-RGB7/202025213

Math is Fun, “Heron’s Formula”. http://www.mathsisfun.com/geometry/herons-formula.html

Maths in the City, “Most stable shape – triangle”. http://www.mathsinthecity.com/sites/most-stable-shape-triangle

Andre LaMothe, “Texture Mapping Mania”. http://archive.gamedev.net/archive/reference/articles/article852.html

Engaging students: Defining the terms perpendicular and parallel

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 Allison Metlzer. Her topic, from Geometry: defining the terms perpendicular and parallel.

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

The concepts of perpendicular and parallel will be implemented in many of my students’ future mathematics courses not only in high school, but also in college. In algebra, the students are asked to find the slope or the rate of change. In looking at the slope, students are asked to find if it’s parallel or perpendicular to another function’s slope.

In geometry, many shapes have properties that define them as having parallel or perpendicular sides (i.e. squares, rectangles, parallelograms, etc.). Also, in order to decide if triangles are similar, their corresponding sides must be parallel. In order to use the Pythagorean Theorem, the triangle must be right angled or have the two legs perpendicular to one another.

In calculus, students are asked to find orthogonal vectors which are also defined as perpendicular vectors. Also, calculus incorporates concepts from algebra and geometry which in turn, include parallel and perpendicular lines.

Therefore, many, if not all of my students’ future math courses will use the topics parallel and perpendicular. Thus, it would be important for me to teach them the two concepts correctly now so that there wouldn’t be any misconceptions in the future.

C3. How has this topic appeared in the news?

One big thing the news talks about every two years is the Olympics. Using the concept of parallel and perpendicular, the constructions are made for all of the different events. Apparent examples of events incorporating parallel lines are track, speed skating, and swimming. The one I will focus on is swimming, namely because it is a very popular Olympic event and one of my favorites. Pictured below is an Olympic swimming pool of 8 lanes. Do the lanes appear to be parallel? Two things that are parallel are defined as never intersecting while also being continuously equidistant apart. One can clearly see the lanes of the pool never intersect. If they did, then the contestants could interfere with one another. Also, because the Olympics is a fair competition, the lanes are equidistant in order to give each contestant a fair and equal amount of room.

Because the Olympics is a well-known event featured in newspapers, articles, and on TV, the students will be able to understand this real world application of parallel and perpendicular.

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?

Before I would play the video, https://www.youtube.com/watch?v=vnnwfcDcNlY, I would first ask the students to think of as many examples they can of parallel and perpendicular in the real world. After about a couple of minutes, I would tell them to keep those in mind and see if the video included any they didn’t think of. I would play the video from 1:25 to 3:05 which is the portion that displays all of the examples. It has clear pictures of recognizable objects which incorporate parallel or perpendicular lines. Also, the video has labels on the pictures to even more clearly describe where the components of parallel and perpendicular lines are. I believe that the initial brainstorm along with this video would get the students thinking about the importance of parallel and perpendicular lines. Also, I would make the connection that those examples would not be considered parallel or perpendicular unless they met the following definitions. Then I could explicitly define both parallel and perpendicular.

Thinking of real world examples, and seeing pictures of them will help the students understand what parallel and perpendicular lines should look like. After they have this initial understanding, they then could get a better grasp of the definitions. Also, they would recognize the importance of following the definitions to correctly construct objects involving parallel and perpendicular lines.

References:

Detwiler, dir. Intro to Parallel and Perpendicular Line. YouTube, 2010. Web. <https://www.youtube.com/watch?v=vnnwfcDcNlY >.

Engaging students: Inverse Functions

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 Brittney McCash. Her topic, from Algebra II: multiplying binomials.

C3. How has this topic appeared in the news.

For the engagement on this aspect of my topic, I would bring a binomial cube with me. I would pose the question, “What do we do when we multiply two binomials together?” The students, of course would not know the answer. I would then say, “Well let’s what one man did that they even did a news article about him!” This in itself catches the students attention because they are piqued about what exactly I am talking about. I would then pass out a copy of this news article so that the students could read. After popcorn reading out loud, we would discuss the article and about how we could use the binomial cube. I would then take out my cube (If possible, put students in groups and give each group a binomial cube to work with) and ask the students, “How in the world did he use this cube to multiply those binomials (points to equation on board)?” I would give them the hint that they have to add up the sides of the square and solve for the perimeter, and see what they can come up with. This is a great engagement for the kids because not only is it hands on, but the article brings in outside aspects of what they’re learning so that they realize they are not the only ones having to learn the material. It’s also a great way to introduce multiplying binomials because it starts at the beginning of adding variables (which they already know how to do), and it’s a visual representation of concept that is sometimes hard to grasp. It’s also a great way to lead into the FOIL, Box, etc…methods to take it into a deeper explanation. For those that have not heard of the binomial cube, here are some pictures of what the students will be working out.

ARTICLE: News Article about Binomial Cube

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

A great way to start off with this engagement would be to take the students back to sixth grade. Start off with asking students, “Who remembers when we had to learn how to add and subtract fractions?” Most, if not all, of the students should raise their hands. You can then ask, “Okay, good. So does anyone remember what the next step was after we learned how to add and subtract fractions? What did we learn how to do next?” The answer I am looking for here is multiplying and dividing. After that is established, you can lead in with, “Okay, so who can tell me what the next step would be with what we have previously been learning (adding and subtracting binomials)?” The answer is multiplication and division. Make sure to let them know that you will only be focusing on the multiplication aspect for now. Then you can pose some questions like, “What does multiplying binomials look like? How do we do it? Is there more than one way?” You can then go into a deeper exploration of multiplying binomials and the different ways you can do so. This is a good way to introduce multiplying binomials because not only did I bring in one concept students were already familiar with, I brought in two. I utilized something they already knew (even if subconsciously) back in middle school, and applied that same order to something more complex. It showed them that there was a purpose for learning what they did, and why there is a reason we go in the order that we do. Then you have the aspect of taking something they had been previously working on this semester and extending it further. This helps the students connect with what they are learning and realizing there is a purpose. Because multiplication is repeated addition, we are taking something they have previously learned, and extending it further. Another reason this is a good plan is because you start off with such a basic question, that every student knows the answer. This allows for immediate attention because all the students know what you are talking about, the more they understand, the more likely they are to participate in classroom discussion.

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

In multiplying binomials, technology is a wonderful thing. It can allow students the opportunity to learn in new and interesting ways. When thinking of an engage for this topic, I thought of the 9^th grade Algebra 1 class I am currently teaching. High School students are sometimes the hardest to keep entertained, and I think I found the perfect video to help keep there attention. This video is a group of students who did a rap about the FOIL method. What better way to relate to students then students themselves! I would start class off by telling the class, “Today we are going to start of by watching a fun video over something we will be learning today.” Proceed to play the video, and observe how every student is watching. The video is fun while also informing. It describes the method, though not thoroughly, but it gives the students an idea of what will be coming. This video helps show that other students all over the state/world are learning the same thing, and are bringing a fun new aspect to the learning of the material. After the video is played, you might ask the class to try and guess at what exactly you will be covering today. It’s always good to see their minds work and try to figure it out. This question also allows them to connect the video back to the classroom environment and settle down. You can then begin your lesson on multiplying binomials. At the end of the lesson, I would bring up the video again, and ask the class if they can recall what FOIL stands for and to give me an example. I would probably make this their exit ticket for the day and have them write it down on a piece of paper. (This video runs a little long, and I would recommend editing some parts out for time sake. )

Resources:

http://www.youtube.com/watch?v=MG-c7NWFS8U

http://www.noozhawk.com/article/santa_barbara_montessori_school_open_house_binomial_cube_20140118

http://montessorimuddle.org/2012/02/02/using-the-binomial-cube-in-algebra/

Engaging students: Inverse Functions

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 Allison Myers. Her topic, from Algebra II: inverse functions.

CURRICULUM

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

Functions are a composition of one or more actions that maps one object onto another (each input maps to one output). Inverse functions are a composition of reverse actions that “undo” the actions of the original function.

Inverse functions have real-world applications, but also students will use this concept in future math classes such as Pre-Calculus, where students will find inverse trigonometric functions. Inverse trigonometric functions have a whole new set of real-world applications, such as finding the angle of elevation of the sun, or anything which models harmonic motion.

Students will also see this concept again in Calculus, where they will differentiate inverse trigonometric functions to solve real-world applications involving rate of angular rotation or the rate of change of angular size.

TECHNOLOGY

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

In the past, I taught a lesson where the Explore portion of the lesson utilized dry erase markers and transparency sheets to allow students to discover what happens graphically when computing an inverse (trigonometric) function. My goal was for my students to understand why we compute inverses the way we do. To my horror, my theoretical 15-minute, super insightful Explore became messy, full of problems, and confusing to my students.

While reflecting after the lesson, I began to consider how using technology would have better served my students (in their understanding) and myself (in my goals for the lesson). I found Glencoe’s directions for using the TI-Nspire to compute inverse functions (see image below). Using the TI-Nspire, I would start the lesson with a real-world example and data and have my students complete Step 1. Next, I would explain our need to “undo/reverse” the data, and allow the students to come up with different ways to do so. After that, I would ask the students to make conjectures about possible formulas. Using the TI-Nspire would be less messy and time-consuming (as compared to my experience with markers and transparencies), and would also allow the teacher to be within the context of a real-world problem. I believe if we used this (or similar) technology, combined with the constructivist-style teaching, students would come away with not only a better understanding for computing inverse functions but also their real-world applications.

Source: http://glencoe.com/sites/common_assets/mathematics/alg2_2010/other_cal_keystrokes/TI-Nspire/Nspire_423_424_C07L2B_888482.pdf

CULTURE

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

Inverse functions are used every day in real life. For example, when a computer reads a number you type in, it converts the number to binary for internal storage, then it prints the number out again onto the screen that you see – it’s utilizing an inverse function. A basic example involves converting temperature from Fahrenheit to Celsius.

Another example, if one considers music notes on paper to be a function of the sound produced, then the software Sibelius can be considered the inverse function, as it takes a musician’s music and converts it back to music notes.

Engaging students: Introducing variables and expressions

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 Christine Gines. Her topic, from Pre-Algebra: introducing variables and expressions.

APPLICATIONS

As we all know, introducing variables in a Mathematics class often intimidates students. As teachers, we can minimize this by creating activities where students are eased into the new topic in a fun and educational environment. This can be achieved through the following activity that introduces variables:

In this activity, students discover “the value of words.” On notebook paper, have students write the letters of the alphabet in order down the left side of the paper. Down the right side of the notebook paper, have them write the numbers from 0 to 25. The letters should corresponding to the numbers. The numbers are the values to each letter, or variable.

To begin, you could have your students find the value of their own name and last name.

Ex. Chris –> C=2, H=7, R=17, I=8, S=18

= 2+7+17+8+18 = 52

You could ask the following questions:

Which has a higher value – first or last name?
What is the difference in the values of your first and last names?
Find words whose values are equal to 25, 36, or 100.
What is the three-letter word with the greatest value?
Are the greatest values always associated with words that contain the most letters?

You could also pair your students and have them write codes to each other. Furthermore, challenge them to write their code with value restrictions and allowing them to *,/,+,-.

This activity develops algebraic thinking in a concrete manner students can understand without presenting them with an overwhelming amount of new information. It is a very flexible activity in which you could make it your own and get the kids excited about it. For example, the activity could even be competitive by challenging students to write an expression for CAT where the value would equal 2 (C+A*T = 2+0*10). This is definitely something I would use to introduce variables.

More about this game can be found here: http://illuminations.nctm.org/Lesson.aspx?id=1156

CURRICULUM

A variable expression is a combination of variables, numbers and operations. The only new information being presented is the unknown represented as variables and how to solve for that variable. Students don’t know this, but it’s quite similar to what they have been doing in school for years. Take 2x=4 for example. We know x=2 because 4/2=2. This expression is equivalent to just writing 4/2=_, which is a simple division problem that students have seen time and time again.

Variable expression are not always given, though. Students will learn how to construct them by analyzing word problems for key clues. This is where the vocabulary students have been working with comes into play. Common words that they will see are sum, difference, quotient, product, etc.

A key rule to +/- fractions is “Whatever you do to the top, you have to do to the bottom.” This theme directly correlates with solving expression with respect to the left and right side of the equal sign. Therefore, we can conclude that variable expressions are a combination of skills that students have learned previously with the exception of written variables.

TECHNOLOGY

With the fast growth of technology, more and more useful sources are becoming readily available to us and it’s important to take advantage of this. Math Play is a website that provides a variety of interactive online games organized by content and all grade levels.

One game in particular, Algebraic Expressions Millionaire Game, serves perfectly as an introduction to constructing variable expressions. The game has the theme of “Who wants to be a millionaire?” and challenges students to chose an equivalent representation of an expression written in words. The problems increase in difficulty as you progress, using clues such as less than, difference, sum, product, quotient, etc. This Algebraic Expressions Millionaire Game can be played online alone or in two teams. The link to this game can be found below:

http://www.math-play.com/Algebraic-Expressions-Millionaire/algebraic-expressions-millionaire.html

This game is a great way for students to develop a conceptual idea of what variable expressions represent. It also builds a foundation for solving and constructing word problems. Try pairing students to compete against each other to add motivation. You could even hold a tournament!

Engaging students: Fractions, percents, and decimals

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 Billy Harrington. His topic, from Pre-Algebra: fractions, percents, and decimals.

Application:

1) Problems that arise with integrating fractions, percents, and decimals include instances such as shopping during a sale at a certain store or shop. The type of shop does not matter whether it is a flea market, or a high-end clothing store. A sale affects all types of stores in the same way. When an item is (1/3) of its original price, people must convert this into a fraction and then convert to a decimal to find out the whole dollar value which will most likely involve decimals as well as the fractions/percentages indicating the amount of money off the original price.

I used this website as an example of problems:

http://www.bbc.co.uk/skillswise/worksheet/ma18comp-l1-w-problem-solving-with-fractions-decimals-and-pct

Another really good exercise in percentages, fractions, and decimals is budgeting a certain income over a year. Students should calculate the percent of their budget that they spend on a home, food, necessities, and their leisure activities. Some students can be told to start budgeting using fractions, while another group of students is told to budget based on percentages. When the class is done, students can come together for a class discussion, and share the benefits, and obstacles of budgeting using the method they performed.

2) For a full activity, each student will get one full sheet of printer paper, and a pair of scissors (or be split into small groups of 2 to 4 four people in each group to save paper). Each student/group will start by acknowledging that their full page represents 1 part of 1 whole and represent this as a fraction and a decimal. Students will then continue by cutting their paper in half and notice that there are now two pieces in front of them. They will continue to cut their paper in half another five to six times and then represent each stage by a fraction.

Stage 1	1 part of 1	Represented (1/1)
Stage 2	1 part of 2	Represented (1/2)
Stage 3	1 part of 4	Represented (1/4)
Stage 4	1 part of 8	Represented (1/8)
Stage 5	1 part of 16	Represented (1/16)

Curriculum

1) When students get to their upper level math classes or even when they get to college, they must calculate their own grade/GPA. Not all classes or grades are going to be graded equally and on the same scale. Some classes are graded on a 1000 point scale where as some classes are weighted on a 75 point scale. To convert their weighted total number of points to calculate their letter grade, students must either set their percentage total in a proportion and weigh out the actual score on a 100 point scale to calculate their grade based on the letter grade scale. A student may say, “I have a 130 in this class, this must be an A!” This may be great, or it could be terrible depending on the grading scale, that’s why students must weigh it against the total point value, then convert it to a percent to find out their true letter grade and see in fact if their 130 is truly a good grade worthy of passing.

2) Students will always need basic math in their lives, even throughout adulthood. Percentages, fractions and decimals should be part of that foundation of mathematics that they know. A big part of this topic that students should learn is budgeting, even if it is a small allowance they receive on a weekly, or bi-weekly basis. If they’re given $20 every other week, how are they going to spend or save that money over the 2-week period they have? Students could spend it all, save it all, or spend some, and save some. Students could calculate the percent of money they did spend if they decided to spend money and see what fraction, percent or decimal value best represents what money they spent, and/or saved.

Culture

1 & 2) Percentages, fractions, and decimals is actually really important in the media world such as music and film industries. Take ITunes for example as the sole business that sells music, and also a different assortment of films. The consumers are drastically affected by other media sources, such as a television, or even a newspaper. If a “huge hit” is coming from this new movie coming out next Friday, chances are that a huge percentage of people are going to partake in the new film and go watch it at the local theater. If the movie is a success, then chances are that the movie will reach the top of the box office. The box office is determined by profits over a short amount of time when a movie/film is released into theaters. Movies such as Harry Potter and the Hunger Games were big sell-outs in the box office because there was such a huge profit made off of the films. Profits based on ticket sales are depicted by a percentage of average sales, which means the higher the percent of people that went and watched the new movie, means that the profits are going to be higher. Based on these statistics, movies are then ranked in the box office to see which movie was the most successful at the end of the year.

Rank 1 in Box Office for 2013 –

Hunger Games Catching Fire at over $420 million dollars

This concept applies in Theater as well such as Broadway plays they make huge profits on ticket sales

3) A huge way fractions, percents, and decimals has influenced the world and our culture is by our economy and our market system. Our current economic system is currently in shambles and is desperately trying to fix itself through many irregular and unorthodox ways that sometimes turn out for the worse. The economy is not easy to understand and explaining how the market works to an average citizen probably will not go well, so the market and its different branches are represented in simple, yet intricate graphs, percentages, and decimals to represent how the current day has progressed. There are some days where the DOWJONES may be below 13% where as some days the NASDAQ may be up 10%. Different branches of the economy are each shown in simple percentages, if people don’t understand the values of percents, fractions, and decimals; there is almost no hope for that person to understand the current economic situation.

Engaging students: Rational and Irrational Numbers

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 Allison Metzler. Her topic, from Pre-Algebra: rational and irrational numbers.

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

The video below is a scene from Star Trek. While most students will not have seen this version of Star Trek or perhaps any version at all, most are familiar with the franchise. Because the students will recognize the popular TV show, this video will immediately grab their attention and keep it for the whole video. The video clearly displays how it’s impossible for the computer to compute pi because it is a “transcendental” number. Thus, since pi is irrational, the computer will never be able to find the last digit of pi, causing it to focus on this insolvable problem forever. This video would provide the students with not only entertainment, but also a way to easily remember what an irrational number is. I would also point out that if Spock would have told the computer to compute a rational number such as any fraction or whole integer, it would have taken a matter of seconds.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Rational and irrational numbers can be found in music theory which is incorporated in classical music. First, Pythagoras was credited for discovering that “consonant sounds arise from string lengths related by simple ratios: -Octave 1:2 –Fifth 2:3 –Fourth 3:4” which are all rational numbers. Rational numbers are also found in the sound frequency and the diatonic scale. In order to get an equal tempered scale, we must get from the note C to the note C’ in twelve equal multiplicative steps we must find x such that x¹²=2. This causes us to take the twelfth root of 2 which produces an irrational number. The benefit of tuning a piano to tempered scale is that (1) “Sharps and flats can be combined into a single note” and (2) “Performers can play equally well in any key.” Rational and irrational numbers can also be found in other areas of music as evidenced below.

”At least one composition, Conlon Nancarrow’s Studies for Player Piano, uses a time signature that is irrational in the mathematical sense. The piece contains a canon with a part augmented in the ratio square root of 42:1.”

Also, when you play a fretted instrument (i.e. guitar, banjo, balalaika, bandurria, etc.), you are playing irrational numbers. According to http://www.woodpecker.com/writing/essays/math+music.html, the reason guitars are so hard to tune is that “our ears don’t like the irrational numbers”. However, they are needed to make “complex chordal music.”

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

The video below displays who discovered irrational numbers while also getting into why the square root of 2 is irrational. I would play the video until the 4 minute mark so that I can keep the attention of the students. I would then go further into who contributed to the discovery of irrational numbers.

The Pythagoreans were set on the idea that all numbers could be expressed as ratios of integers. However, Hippasus of Metapontum, a philosopher at the Pythagorean school of thought, discovered otherwise. He supposedly used the Pythagorean Theorem (a²+ b²= c²) on an isosceles right triangle where the congruent sides were each 1 unit. Using the theorem, he found that the hypotenuse was the square root of 2 which proved to be incommensurable. The other Pythagoreans were so horrified with this discovery, that it’s said they had Hippasus drowned. They wanted to punish him while also keeping irrational numbers a secret. However, it’s hard to prove that this information is true because of the vague accounts of who discovered irrational numbers. Therefore, I would inform my students of this interesting story, but also tell them about the uncertainty of what actually happened.

References:

Discovery of Irrational Numbers (n.d.). In Brilliant. Retrieved February 7, 2014, from https://brilliant.org/assessment/techniques-trainer/discovery-of-irrational-numbers

Hippasus (2014, January 14). In Wikipedia. Retrieved February 7, 2014, from http://en.wikipedia.org/wiki/Hippasus

Pre-Algebra 32-Irrational Numbers. YouTube, 2012. Web. 7 Feb. 2014. <https://www.youtube.com/watch?v=q_wstDWjnKQ>.

Reid, H. (n.d.). On Mathematics and Music. In Woodpecker. Retrieved February 7, 2014, from http://www.woodpecker.com/writing/essays/math+music.html

Shatner, William, and Leonard Nimoy, perf. Star Trek. YouTube, 2009. Web. 7 Feb. 2014. <https://www.youtube.com/watch?v=H20cKjz-bjw>.

Time Signature (2014, February 6). In Wikipedia. Retrieved February 7, 2014, from http://en.wikipedia.org/wiki/Time_signature

Wassell, S. R. (2012, March 29). Rational and Irrational Numbers in Music Theory. In docstoc. Retrieved from http://www.docstoc.com/docs/117428973/Rational-and-Irrational-Numbers-in-Music-Theory