Engaging students: Parallel and perpendicular lines

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 Sivado. Her topic, from Algebra: parallel and perpendicular lines.

D.1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

I would take my students back to the time of Euclid of Alexandria, around 300 B.C., and his great book The Elements. Little is known about Euclid except the book he left behind which is the foundation of geometry, algebra, and number theory, still to this day. Euclid wrote this book in an axiomatic way, this means that he assumes common notions, definitions, and postulates to be true and then bases all his propositions and axioms on these assumptions. Does this sound like the way that we do mathematics today? To understand how influential and enduring the Elements is I would present this incredible fact; other than the Bible, Euclid’s Elements is the most published, translated, and studied of all books in the world.

Now we would put on our Euclid caps and turn to Proposition 12 and Proposition 31. These propositions tell us how to draw parallel and perpendicular lines based only on the definitions, common notions, and axioms of Euclid. We would do the constructions step by step, straight out of Euclid’s Elements.

http://www.britannica.com/biography/Euclid-Greek-mathematician

http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html

A.2: How could you as a teacher create an activity or project that involves your topic?

To engage the students in the lesson on parallel and perpendicular lines, instead of sitting in class and listing real world examples of parallel and perpendicular lines, I would take the students out of the classroom and take a tour through the school like a bird watching group except our goal is to list all the parallel and perpendicular lines inside and around the school. We could go to the cafeteria, the gym, and walk around the outside of the building. When we got back to class we could create a long list of all the parallel and perpendicular lines that we see to hang on the wall during this unit. After we list the examples, I could ask some thought provoking questions:

“Why are these parallel and perpendicular lines important?”
“How would the world be different without parallel and perpendicular lines?”

E.1: How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

A great activity I found on parallel and perpendicular lines involves using a graphing calculator to discover the similarities in slope between parallel and perpendicular lines. First, you give the students a list of equations to graph on their calculator. Next, you ask them to compare the graphs and identify which lines are parallel and which are perpendicular. Last, you ask them to compare the slopes of the parallel and perpendicular lines. Hopefully, they will discover that parallel lines have the same slope and perpendicular lines have the opposite reciprocal slope. This activity can be done easily because the students should already be familiar with graphing calculators, slope, and y-intercept. The activity would not take much time and can easily be differentiated based on the skill level of the students in your class. You can give some students difficult numbers or more lines to analyze if they finish the initial activity quickly. Also, you could take this one step further and give the students large sheets of graph paper and let them draw the lines and present their findings in front of the class.

Engaging students: Completing the square

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 Diana A’Lyssa Rodriguez. Her topic, from Algebra: completing the square.

A2. How could you as a teacher create an activity or project that involves your topic?

Using Algebra tiles is a great visual way for students to understand completing the square. The students start with the tiles that correspond to the given problem. The unit tiles are then flipped and moved to the other side of the equal sign. The remaining tiles are positioned into a square shape. The corner piece that appears to be missing will be filled unit tiles. What you do to one side, must be done to the other. Therefore the amount of unit tiles added to the square will also be added to the other side of the equation. Find the zero pairs and take them away. Then, find the corresponding tiles that will outline the square, so when multiplied together equals the equation.

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

Around 815 – 850 AD, a mathematician Muhammad ibn Musa al-Khwarizmi was hard at work discovering algebra. He was actually the first person to write a text about algebra. His focus for a lot of the text was the dimensions of a square. At the time it was not called completing the square, but Muhammad was the one who came up with it because it is exactly what he did in order to solve the different equations he had at the time. Very similar to the process described using algebra tiles, Muhammad also saw the equations in terms of actual shapes. One of the original problems he tried to solve was x²+10x=39. He looked at x² as a square with length x and width x. He then created a rectangle with length 10 and width x. The area would equal 10x. To make his theory work he broke up the 10x rectangle into two squares with length x and width 5. Muhammad combined the x² square and the two 5x pieces into an L shape. This partial square must equal some square with the value of 39. So he came to the conclusion that he had to fill in what was left of the L shape to make it a square. But in order to do that he had to add that same value to the other side. In this case he added 25 (which is 5×5). Muhammad’s final answer was (x+5)²=64, x=3. This was his method but at the time he couldn’t prove that it always worked. So if and when students participate in the algebra tiles activity, they are partaking in a small piece of history.

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

This video from Khan Academy is a great tool for completing the square. This video explains why we have to take half of the b value and square it (when looking at ax²+bx+c) to obtain the c value. When the students understand why we do something in math, they are more likely to be interested in the topic. The different colors that are used to write out the process allows the students to organize and understand completing the squares better. This particular video is also just long enough to capture the attention of the students but not so long as to lose it. Also, after hearing the same person explain math all the time, students may not understand it as well as they possibly could. So what is said in this video can easily be explained by the teacher but students sometimes need to hear a different voice explain a concept so they can gain a new perspective on the topic.

https://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/ex1-completing-the-square

Resources

Engaging students: Expressing probability as a fraction and as a percentage

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 Zacquiri Rutledge. His topic, from probability: expressing a probability as a fraction and as a percentage.

Probability involves any kind of situation where the outcomes are known, but are not 100% certain. Examples of this could be things as simple as flipping a coin, to trying to predict the next card while playing blackjack. However, for a student who is just beginning to understand probability, I thought a word problem involving the rolling of a die would be just challenging enough. “You and your best friend have been playing Monopoly for hours. After several times around the board, you own a large amount of the properties and your friend is nearly bankrupt. In fact, your friend does not have enough money to survive landing on your Boardwalk property in the corner of the board. In order for your friend to land on this space, he/she would need to roll a 12. First, calculate the odds that one die will roll a 6 and express it both as a fraction and as a percentage. Then, calculate the odds that both dice will roll a 6 and express it both as a fraction and as a percentage.”

After learning the basics of probability as fractions and percentages, students can then begin to learn about how to change them into “odds”, or the probability of a series of actions. Since probability is simply the ratio between the desired number of outcomes and the total number of outcomes, only knowing how to write ratios will not help the student in calculating odds. By changing the probability ratio into a fraction, this will allow the student to easily apply the multiplication principle to a series of actions to find the larger probability ratio. From there the student will be able use previous experience of changing probability into a percentage to state how likely or unlikely a situation is.

Probability is found in many areas of culture and science. However, one of the most widely known forms of probability is in gambling. People all over the world gamble for money, for fun, and sometimes even for sport. A few of the common games people play in casinos are Roulette, Blackjack, and Texas Hold’em. Each one of these games has its own way in which it uses probability to make it more difficult for a player to make money.

To play a game of Roulette, all a player has to do is bet on which number, color, or set of numbers they think might win. On the table are the numbers 00, 0, and 1 through 36. 00 and 0 are both their own color, but 1 through 36 alternate between red and black. After the player bets, the rest of the game is controlled by “The House” or the casino. A ball is placed on a rotating circle that has all of the numbers listed one time on it with a slot in the middle for each one. As the ball rolls, it slows down and drops into one of these holes. This is where probability comes into play. Depending on the player’s bet, they have an x/38 chance of winning. If they select one number, it gives 1/38 or 2.6% chance; for 1-12 its 6/19 or 31.5%. By using a combination of bets, a player can increase their probability of winning by selecting more than one number. Due to Roulettes simplicity, it would make a good beginning topic for a student who is beginning to learn about probability.

Blackjack is game that uses cards to determine who wins or loses, instead of a ball and a wheel. The object of this game is to get as close to 21 as possible without going over, as well as attempting to beat the hand “The House” is holding. While there are a lot of calculations that must go in when calculating probability in a game of blackjack, it is possible to do it on a smaller scale. To do so, a player would have to look at what cards had come up in the past and then look to see what card it is that they need. Since there are only 4 of each card in the deck, assuming the player nor “The House” is holding the card he/she needs, the probability would be 4/(52 – y), where y is the number of cards that have already been shown and are not the card the player needs. Texas Hold’em uses this same kind of idea, but instead is used when playing against other people rather than against “The House”. This version of poker has become so well known, it is featured on an ESPN sports channel, where people play in a live tournament and compete for millions of dollars. What is significant about this channel is that they show what cards each of the players are holding as well as what cards are on the table. Then, once every player’s cards are seen, the channel shows on one side of the screen a player’s percent chance of winning. This percent is calculated by an analysis of what cards are in the player’s hand, what cards are in everyone else’s hands, and what cards are on the board. After analyzing the cards, it is then calculated what the probability is that the best possible cards the player needs are going to come up. Even though only basic probability is being used here, this is still on a much higher difficulty due to the amount of numbers that must be processed. However, given its complexity and how the probability can change by the turn of a card can make both Blackjack and Texas Hold’em an interesting topic for a student of probability.

Engaging students: Square roots

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 Tiffany Jones. Her topic, from Algebra: square roots.

B.1 How can this topic be used in your students’ future courses in mathematics or science?

One area of mathematics I wish I had more practice with in grade school is numerical reasoning. I feel that, as a student, I was allowed to use my calculator too much and am struggling to remove my calculator crutch. I hope to encourage my students to sharpen their numerical reasoning skills and to not rely on their calculator. Does this number make sense? Is it too high, too low? Is a negative result valid given the scenario of the problem? The following video introduces a method to estimate the square root of non-perfect squares to the nearest tenth by hand:

“Estimating Square Roots To the Nearest Tenth by Hand” by Fort Bend Tutoring

It gives the students another tool for their toolbox of numerical reasoning, practice using formulas, reviews long division by hand, and strongly encourages students to remember the perfect squares.

I think that introducing this idea as an engage could intrigue student to wonder why the formula works and to wonder what else they are able to do quickly by hand.

Fort Bend Tutoring’s YouTube channel offers videos on a wide verity of high school mathematics topics and courses. The videos cover several examples. They are engaging, not dry and there is also a “theme song” to the videos. I feel that these videos can sever as a great addition to lessons as extra help to the students.

D.1 What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The following story was first told to me in a calculus one course. While the telling of the story was to serve as amusement and did not directly relate to the topic of the day, it stuck with me. It comes to mind frequently when working with the Pythagorean theorem and with irrational squares. And when given this assignment, I saw square roots as an option, this story again came to mind. I think having an interesting story cross my mind makes a problem overall more fun. I would want to give that to my students. The article “The Dangerous Ratio” by Brain Clegg does a wonderful job of telling the story, its implications, and gives a mock dialogue so reads can work through the logic. At the end of the article, there is a link to an activity about the proof that the square root of 2 is irrational.

E.1

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

I really like the idea of a flipped classroom and hope to be able to practice it in my classroom. While a completely flipped classroom will take some time to implement, videos such as Math Antics’ “Exponents & Square Roots” will be a great place to start.

This particular video address a previously learned topic, namely exponents and relates it to the new topic. It provides definitions and visuals to remember how the terms relate to each other and how to read the symbols. It goes through several examples of varying level and shows the viewer how to use technology such as calculators to solve hard problems. In addition, the video addresses some common misconceptions such as mistaking the root sign and the division sign. Moreover, it ties everything together with a quick review at the end.

One of my favorite aspects so of flipped classrooms, is that the student can review the video over and over. Math Antics does an excellent job of talking the math out to the viewer. The animations are amusing yet helpful. While a lot of information is covered, the video is not dry.

Resources:

“Estimating Square Roots To the Nearest Tenth by Hand” by Fort Bend Tutoring – https://www.youtube.com/watch?v=bUh7Hj-3dkw

“The Dangerous Ratio” by Brian Clegg – http://nrich.maths.org/2671

“Exponents & Square Roots” by Math Antics –https://www.youtube.com/watch?v=C_iKTTI1E34

Engaging students: Combinations

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 Taylor Vaughn. Her topic, from probability: combinations.

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

One interesting way that I thought about introducing combinations is bringing in combinations that students do often, but do not really think about. When it is known that the family is going on vacation, as a girl, the first thought is “what am I going to wear?” Being a girl, I was always told that I cant pack as much as I wanted to because I also wanted to bring extra clothes just in case I didn’t want to wear what I had planned for that day. One activity I thought bout is actually bring in a suitcase to class with clothes and try and plan a 3 day vacation and figure out how we, as a class, was going to pack this suitcase. I could include different scenarios such as, if the hotel has a laundry room, and how would being able to wash clothes and put them back in the suitcase change how we pack. Also, what happens if we add shoes and socks? How would this change affect the number of combinations we can have? I think it would be really cool for students to touch and play and bring in ideas that they don’t necessarily think has anything to do with math.

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

Everyone listens to music, but there are so many different types of genres, artists, and songs. Have you ever thought, “Will we ever run out of new music?” Well someone by the name of Michael has. He has done the research of what others say about the math of the order of the notes and how many combinations of these notes can we get that will create a new song

One activity that could be done after the video is given 8 notes, how many different measures could students in the class come up with. Then the whole class could see how many people got the same measure or did everyone get something completely different. Then you could also ask “Did we cover all the possibilities? How do we know? How can we show this mathematically?” Lastly, if there are so many possibilities, why are there so many songs with the melodies? There is a video that has one melody and sings a lot of songs to that one melody. (PG-13 Warning: gratuitous cursing near the end of the video.)

The one thing I didn’t like about the first video is the length and he makes connections about songs that are really outdated. SO this video has songs that will relate closer to this generation of students.

How can this topic be used in your students’ future courses in mathematics or science? In school, students that didn’t like math the way I did always asked, “Well when will we ever use this again?” Well even though we use combinations more than we think, it can also be used in later math classes. Ever thought that combinations had anything to do with Pascal’s triangle? What is Pascal’s triangle, you may ask? Well according to Math Is Fun, it is a pattern of numbers where the starting number is 1 and “each number is the two numbers above it added together (except for the edges, which are all “1”).” Well what if you are asked to find entry 20 of the triangle. The one thing I would do would keep the pattern of the triangle and write out all the entries until I got to that number, but using combinations you can get any entry you would like without writing all the entries out. The formula is where n is the row and k is where it is in the nth row.

$\displaystyle {n \choose k} = \frac{n!}{k! (n-k)!}$

While teaching this, I would definitely talk about factorial and how it relates to the lesson.
References

Stevens, Michael. “Will We Ever Run Out of New Music?” YouTube. YouTube, 20 Nov. 2012. Web. 02 Sept. 2015.

The Axis of Awesome. “4 Chords.” YouTube. YouTube, 20 July 2011. Web. 02 Sept. 2015.

Pierce, Rod. “Pascal’s Triangle” Math Is Fun. Ed. Rod Pierce. 30 Mar 2014. 3 Sep 2015 http://www.mathsisfun.com/pascals-triangle.html

Engaging students: Fractions 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 Samantha Offutt. Her topic, from Algebra: fractions and decimals.

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

Students will use/convert fractions and decimals in a number of ways in future courses in mathematics and science. The best example is percentages. In a probability/statistics class, percentages are a major component for answering many types of problems. For example, in the college course Math 3680, percentages are used frequently. So in a large set of data, one is asked to record the frequency of a certain number, take the frequency and divide it by the total number of entries, and one is almost always ask for them to be written as decimals to the 4^th number. After determining the relative frequency, you can tell what proportions of the data are between certain stipulations. For example, if there were 50 numbers that are between 1 and 20, one can be asked, “What proportion of the numbers are between 7 and 13.” So even to this day in college, students still use this pre-algebra topic.

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

I found this awesome project on a teachers blog: http://teachinginroom6.blogspot.com/2012/02/math-social-studies-awesomeness.html, This certain teacher did a social studies spin on colonial era quilts. I think it was awesome. “I then had the students create a 20 cm x 20 cm square (we have cm graph paper available at school). Choosing either 2, 4, or 5 colors, the students created a square that had at least one triangle, quadrilateral, was bright, and symmetrical (Stephanie).” Then the students created fractions by counting how many squares, of the 400 squares, took up each color. Later found the decimal of those fractions, and finally determined the percentage each color owned on their square. The teacher took each square and made a quilt. I’m in love with this project and I think it’d look fantastic in the classroom. Students get to practice multiple skills and are given the opportunity to have their work displayed in the classroom.

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

Technology is a very useful tool for students. Instead of a teacher lecturing, they can find videos of all sorts on the Internet. If a teacher simply wanted to let Khan Academy engage students and do examples for the kids in a short 8 minutes, then they could use this very helpful video (that the students can reference later at home if they have any questions):

Students get to dive right into the topic and see how it is done, but later when they are at home and have forgotten some things, they have access to the exact video. Technology is very useful both in the classroom and at home. Also this video shows more than just one, simple example. I think it’s great the video shows problems of different levels of difficulty.

References

Khan Academy. “Converting Fractions to Decimals | Decimals | Pre-Algebra | Khan Academy.” YouTube. YouTube, 8 Apr. 2007. Web. 04 Sept. 2015.

Stephanie. “Teaching in Room 6: Math + Social Studies = Awesomeness.”Teaching in Room 6: Math + Social Studies = Awesomeness. 3AM Teacher, 5 Feb. 2012. Web. 05 Sept. 2015.

Engaging students: Fractions, decimals, and percents

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 Perla Perez. Her topic, from Algebra: fractions, decimals, and percents.

A2. How could you as a teacher create an activity or project that involves your topic?

This past summer when I worked as a program assistant for TexPrep, we had the opportunity to have a pizza party. How fun! Well it took longer than we thought to pick out a place and figure out how much we all had to pay. I got to thinking about how this could be a great engaging activity for students to get excited about decimals, fractions, and percents.

The activity will go as follows:

Students are split up into groups of four with each group given a pizza place. Every person has one of the following roles: the researcher, the recorder, the calculator, and the presenter (to compare with other groups). Their goal is to find the pizza place that is the cheapest, gives the most pizza, and figure out how much each individual would have to pay. By comparing each other’s work during presentations, students get to compare, contrast, and see the different methods used to solve the problems. This also gives the teacher an opportunity to understand their comprehension level of the subject and see if converting a percentage is difficult for them or not. When all the groups are finished gathering their information they will present. Afterwards (if allowed), we will reward ourselves with eating pizza! Through this activity students will have to come up their own way to solve these problems. It leads them to work with: Decimals, since they must include every penny (including tax); Fractions, when it comes to figuring out how much each individual owes; and Precents, when asked to compare prices between pizza places.

C3. How has this topic appeared in the news?

Decimals, fractions, and percent are used in media to represent a variety of concepts from the percent of the candidate poll elections to percent chance of rain. Now some of these topics might not sound interesting to most students, but current events such as the movement to raise minimum wage to $15.00 can grab their attention. Students can then be given questions such as: How does that affect the regular worker financially? Are employees working the same hours? Do employees get fewer hours and more pay, or do they keep their regular hours? In the Time article “Here’s Every City in America Getting a $15 Minimum Wage”, it mentions how some restaurants are increasing their prices from 4% to 21% which begs to question, is everything in the market going to increase as well? All the answers to these questions can be found in the news and prompt their interest in actually doing the math to find out the answers. The news also gives them the real world application student’s consistently are trying to find. Engaging students about the news and simply prompting them before the lesson allows students to continue thinking about it as they go forth in the lesson.

Helpful links:

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?

As we continue to advance in technology, we begin to see how there are many ways a student can learn. The internet is full of different educational games, activities, calculators, and above all videos that are useful to educators. There are videos basically for everything. So what better way to engage students than with a video that knows exactly how they feel like in this one: https://www.youtube.com/watch?v=cGqQOQavbls. The video is a great representation of how a unique activity such as magic can be used to stimulate students in understanding the idea of how fractions, decimals, and percentages relate to one another. Aside from funny videos students also like to interact in games like: http://www.math-play.com/Fractions-Decimals-Percents-Jeopardy/fractions-decimals-percents-jeopardy.html and http://www.topmarks.co.uk/maths-games/7-11-years/fractions-and-decimals. The first game allows students to practice converting fractions, decimals, and fractions from one to another and shows them how they are related. The last website gives teachers a variety of tools to choose from, all of which can help a lot in the classroom.

References:

http://www.topmarks.co.uk/maths-games/7-11-years/fractions-and-decimals.

My Mathematical Magic Show: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show the mathematical magic show that I’ll perform from time to time.

Part 1: Introduction.

Part 2a, Part 2b, and Part 2c: The 1089 trick.

Part 3a, Part 3b, and Part 3c: A geometric magic trick.

Part 4a: Part 4b, Part 4c, and Part 4d: A trick using binary numbers.

Part 5a, Part 5b, Part 5c, and Part 5d: A trick using the rule for checking if a number is a multiple of 9.

Part 7: The Fitch-Cheney card trick, which is perhaps the slickest mathematical card trick ever devised.

Part 8a, Part 8b, and Part 8c: A trick using Pascal’s triangle.

Part 6: The Grand Finale.

And, for the sake of completeness, here’s a recent picture of me just before I performed an abbreviated version of this show for UNT’s Preview Day for high school students thinking about enrolling at my university.

Engaging students: Solving one-step algebra problems

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 Jason Trejo. His topic, from Algebra: solving one-step algebra problems.

A2) How could you as a teacher create an activity or project that involves this topic?

How can I engage my students with solving for a variable? Off the top of my head, I came up with 3 tried and true surefire ways that would not only further my students understanding but also be a ton of fun for them: Algeblocks with accompanying interactive whiteboard, using a balance and counters, and possibly using snacks (e.g. cookies, chips, candies, etc.)

First things first, the Algeblocks:

Essentially, Algeblocks are made of a variety of cubes and rectangles that represent ones, tens hundreds, thousands, and even the variables x and x². Although obscured in the picture, the Algeblocks mat in the back represents a balance where the fulcrum is “=” and each end of the balance represent both sides of the equation. There is even a place that represents negative numbers! Using the problem “x+4=8”, students would have 8 green blocks to the left of the fulcrum and 4 green blocks with an x block. Students would then add or take away tiles to solve the equation. As for problems such as “4x=16”, the students would display the problem using the blocks and then group the green blocks with the x’s to find there answer. Now that I think of it, I would essentially do the same thing but use either a real balance with any type of manipulative.

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

Being able to solve single step algebraic problems is a foundation to algebra in general, correct? This means that this will continue to pop up regardless of what math class (and even science classes like chemistry). There will always be problems given to students where they will need to solve for a variable and the final step of even the most excruciatingly, horrific looking algebra problems is usually adding, subtracting, multiplying, dividing, etc. to get the “x” all alone. In reality, solving an initial value problem (like I currently do in my Differential Equations class) boils down to one step algebraic solutions.

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?

Interestingly enough, I have the perfect example that ties both Khan Academy and the “use of a balance” activity I mentioned earlier. A quick Google search for “one-step equations” gives a link to Khan Academy that allows for a digital balance and you are to solve the equation given with the balance. This would be an amazing tool for teachers to use when they don’t have actual balances for their class or even have their students create a profile on Khan Academy and use it to be able to track extra problems the students can do. Besides Khan Academy, there are even some cheesy yet fun games (like “Equations Pong” off the XP Math website) that would give the students more practice with these equations while feeling like a reward since they are playing a game. Plus, students can go head-to-head in “Equations Pong” and a vast majority of students like to best their friends in anything and everything.

References:

Information on Algeblocks: http://www.hand2mind.com/brands/algeblocks

Image of Algeblock Mats: https://cdn.hand2mind.com/productimages/76986_Algeblocks_Mats_BQS-web.jpg

Khan Academy use for subject: https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/why-of-algebra/e/one_step_equation_intuition

Equations Pong Game: http://www.xpmath.com/forums/arcade.php?do=play&gameid=105

Engaging students: Factoring polynomials

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 Irene Ogeto. Her topic, from probability: Venn diagrams.

A2. How could you as a teacher create an activity or project that involves your topic?

As a warm up activity to a lesson on Venn diagrams, I could set up a model Venn diagram made out of tape on the classroom floor or in the hallway outside of the class. The topic for the activity would be comparing the number of students who prefer to play indoor sports versus the number of students who prefer to play outdoor sports. I would ask the students who prefer to play outdoor sports such as soccer, baseball, football or field hockey to stand in the circle that represents outdoor sports. Then I would ask the students who prefer to play indoor sports such as bowling or table tennis to stand in the other circle. Next, I would ask the students who prefer to play both indoor and outdoor sports such as basketball, volleyball or badminton to stand where the circles intersect. Lastly, I would ask the students who don’t prefer to play any sports to stand outside the two circles.

With this activity we can explore these questions:

How many students prefer to play indoor sports?
What is the percentage of students in our class prefer to play indoor sports?
How many students prefer to play both indoor and outdoor sports?
What percentage of students in our class prefer play both indoor and outdoor sports?
What percentage of the students in our class prefer to play sports?

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

Venn diagrams have appeared in children’s TV shows such as Cyberspace. In this episode of Cyberspace which is was aired on PBS in Season 1, the Cyberspace squad uses a Venn diagram to rescue the Lucky Charms. The squad uses the terms “or” and “and” with respect to sets to find the Lucky Charms. Motherboard tells them that the Lucky Charms is both blue and tall. One circle represents the blue bunnies and the other circle represents the bunnies of another color. The area where the two circles intersect represents the area where the tall and blue bunnies are. The squad works together to find the Lucky Charms using applications of Venn diagrams. Venn diagrams can be used to explore possibilities and combinations of things. This video can serve as an introduction to a lesson on Venn diagrams. It enables students to see how math is part of culture, as it is found in television shows.

Episode 112: “Of All the Luck” http://www.pbs.org/parents/cyberchase/episodes/season-1/

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

John Venn (1834-1923) the famous mathematician, devised a way to picture sets by creating what is now known as Venn diagrams in 1881. John Venn was born in Hull, New England, United Kingdom. He was a lecturer, president of a college, and a priest for some of the years in his life. Venn wanted to show how different groups of things could be represented visually. John Venn called Venn diagrams Eulerian circles because they were similar to the Euler circles created by Leonhard Euler. While they share similarities, Euler circles and Venn diagrams are different. Venn diagrams are more sophisticated and are used to represent all possible combinations of classes. Euler circles differ in the sense that the circles do not always have to intersect and do not always represent all possible combinations. Some people still refer to Venn diagrams as Eulerian circles to this day and often some people use the two terms interchangeably. Despite the differences, both diagrams are used in math every day.

References:

http://www.venndiagram.net/the-history-behind-the-venn-diagram.html

http://www.mathresources.com/products/mathresource/maa/venn_diagram.html

http://www.pbs.org/parents/cyberchase/episodes/season-1/