The mathematics-based process used by Inmarsat and the UK’s Air Accidents Investigation Branch (AAIB) to reveal the definitive path was described by McLaughlin as “groundbreaking.”
“We’ve done something new,” he said.
Here’s how the process works in a nutshell: Inmarsat officials and engineers were able to determine whether the plane was flying away or toward the satellite’s location by expansion or compression of the satellite’s signal.
What does expansion or compression mean? You may have heard about something called the Doppler effect.
“If you sit at a train station and you listen to the train whistle — the pitch of the whistle changes as it moves past. That’s exactly what we have,” explained CNN Meteorologist Chad Myers,who has studied Doppler technology. “It’s the Doppler effect that they’re using on this ping or handshake back from the airplane. They know by nanoseconds whether that signal was compressed a little — or expanded — by whether the plane was moving closer or away from 64.5 degrees — which is the latitude of the orbiting satellite.”
Each ping was analyzed for its direction of travel, Myers said. The new calculations, McLaughlin said, underwent a peer review process with space agency experts and contributions by Boeing.
It’s possible to use this analysis to determine more specifically the area where the plane went down, Myers said. “Using trigonometry, engineers are capable of finding angles of flight.”
My understanding is that even though the pings from a satellite to the plane and back were occurring, even though the plane’s location was not being transmitted. From the Doppler shift of those pings, the plane’s trajectory could be reconstructed.
Someday, for teaching purposes, I hope that a formal write-up of this procedure is published. The details will probably be over the heads of most students, but this is a eye-catching, though indescribably tragic, example of how mathematics can be creatively used to solve a mystery.
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 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.
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:
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!
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 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.
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.
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 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.
If you have questions or comments about this film please contact @statsdancer #dancingstatistics or dancingstatistics@gmail.com
The BPS runs an annual public engagement grant scheme. Through these grants we aim to help members promote the relevance of evidence-based psychology to wider audiences either through direct work or by organising interesting and relevant communications activities. For press inquiries please contact the BPS Press Office. http://www.bps.org.uk/what-we-do/awar…
IdeasTap is an arts charity for young, creative people at the start of their careers.
Visit their website for more information http://www.ideastap.com.
If you have questions or comments about this film please contact @statsdancer #dancingstatistics or dancingstatistics@gmail.com
The BPS runs an annual public engagement grant scheme. Through these grants we aim to help members promote the relevance of evidence-based psychology to wider audiences either through direct work or by organising interesting and relevant communications activities. For press inquiries please contact the BPS Press Office.
IdeasTap is an arts charity for young, creative people at the start of their careers.
Visit their website for more information http://www.ideastap.com.
Mean Green Math 