(created by Ms. Devi, Ms Xuncax, and Mr. Narasimhan)
In this lesson, students will learn mathematics skills to evaluate student loan offers and financially plan their college education.
I. INTRODUCTION: (10 minutes)
Start conversation with students about where they want to go to college. Have them pick a few schools and do research on the tuition and fees for those places. Have them fill out some tables in a worksheet.
II. REVIEW FINANCIAL DOCUMENTS
Give them a typical financial award packet and have them dive into it and underline what they think is important and what they don't understand. Teacher can note where they seem to be most interested. After a few minutes, give them a list of vocabulary words to go find in the documents, and have them read those sections and fill out a vocabulary list.
Examples of vocabulary list: Promissory Note, Grants, Loans, Work-Study, Subsidized v. Unsubsidized, Late Charges,
III. THE MATH OF LOANS
-Using Cumulutative Indebtedness Loan Repayment Schedule
- Teach them about exponential growth using that P (1+r/t)^N
- Set up spreadsheets for them to change different numbers in the repayment plan to see how much more money that ends up costing them.
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IV. DISCUSSION
- What if you don't pay on time?
- Principal and the Interest (exponential growth, optimal payment plans, calculating times of repayment)
- VOlunteer Cancellation Service and Teaching awards
Questions: How much money do you need to repay for this particular Financial Award. Would this financial award work for your different colleges? If not, what other sources could you look at?
Friday, May 1, 2009
Thursday, April 23, 2009
Empowering Women in Math
An interesting article about an educational non-profit in New York working to help women get higher paying jobs and better opportunities through math education:
http://www.nytimes.com/2009/04/19/jobs/19math.html
I actually hate using rote techniques when teaching math, but there is something to be said for being able to execute an arithmetical calculation correctly over and over again.
I thought the part in the article about differences in spatial reasoning were interesting, particularly the part about spatial reasoning being teachable. If you are looking for challenging spatial reasoning questions, a good place to look is Perceptial Ability section of the DAT, or Dental Admissions Test. From speaking to some dentists, being able to look into people's mouths upside down and identify what is going on takes a lot of spatial reasoning, and so they are tested on this in their entrance examinations. I've only found a few links to practice questions like here.
I thought the part in the article about differences in spatial reasoning were interesting, particularly the part about spatial reasoning being teachable. If you are looking for challenging spatial reasoning questions, a good place to look is Perceptial Ability section of the DAT, or Dental Admissions Test. From speaking to some dentists, being able to look into people's mouths upside down and identify what is going on takes a lot of spatial reasoning, and so they are tested on this in their entrance examinations. I've only found a few links to practice questions like here.
Tuesday, April 21, 2009
Monday, April 20, 2009
All the Water In the World
I am trying to get the Google Earth plug-in working on a page somewhere, but for now here is the KMZ file of a demonstration that shows the volume of the water in the world compared to the volume of the Earth in a set of aspiring spheres. There is another demonstration I also found here, that illustrates the amount of CO-2 emissions geographically in a cool spherical bar chart. Reading graphs is an important skill learned in a mathematics education, and it is interesting that new forms of information management tools, like Google Earth, are also enabling new types of graphs and visualizations.
Upon seeing the graphic above, I was first struck by how small the largest sphere of water was to the total size of the Earth. If this relationship is actually to scale, it shows how spreading a substance evenly over the surface of an object that is much larger its size occupies far less volume than the object itself. A question to this end would be how to figure out the average depth of water it would take to cover the Earth if it were evenly spread over its whole surfaceusing the numerical values from the demonstration.
The relationships between volumes of geometric objects can be very useful to intuit when making rather ordinary calculations. It can help you decide which pail you should use to get sand to fill up a sandbox, or the size of a pot to use to make Jambalaya when all the sausage and rice are done cooking. In understanding this relationship, it is helpful to know how the formulas for volumes and areas vary with quantities like length, height and radius.
We discussed how to determine volumes from areas when a shape has a uniform cross section int the post "Rapelling Down Fissures" -- there we learned that for some objects, their volume changed linearly with the height of the solid. So if we were planning on building a new water tower in our village when our population had almost doubled, we could build a water tower with the same circular cross section but just twice as high. If we decided to change the radius of our tank, however, and keep the height the same, would we need a new tank whose radius was twice as big as our previous one? What do you think?
To understand how much our radius must grow to accomodate the new required volume of water, it is helpful to remember that while the volume of our cylindrical water tank will vary linearly with its height, it will vary quadratically with the radius of its circular base. So, increasing the radius by a certain amount would actually lead to a change in volume corresponding to the radius squared, so we wouldn't need to adjust our radius in a one-to-one proportion to our required volume.
Q1: Use the numerical values from the graphic to determine the radii of each of the spheres representing different quantities of water.
Q2: What is the volume of saltwater on the Earth? What would be the radius of a sphere used to contain this much water?
Q3: Draw a graph of the volume of each of the quantities portrayed in the aspiring spheres verses the radii obtained in Q2. Does your graph make sense? Why or why not?
Upon seeing the graphic above, I was first struck by how small the largest sphere of water was to the total size of the Earth. If this relationship is actually to scale, it shows how spreading a substance evenly over the surface of an object that is much larger its size occupies far less volume than the object itself. A question to this end would be how to figure out the average depth of water it would take to cover the Earth if it were evenly spread over its whole surfaceusing the numerical values from the demonstration.
The relationships between volumes of geometric objects can be very useful to intuit when making rather ordinary calculations. It can help you decide which pail you should use to get sand to fill up a sandbox, or the size of a pot to use to make Jambalaya when all the sausage and rice are done cooking. In understanding this relationship, it is helpful to know how the formulas for volumes and areas vary with quantities like length, height and radius.
We discussed how to determine volumes from areas when a shape has a uniform cross section int the post "Rapelling Down Fissures" -- there we learned that for some objects, their volume changed linearly with the height of the solid. So if we were planning on building a new water tower in our village when our population had almost doubled, we could build a water tower with the same circular cross section but just twice as high. If we decided to change the radius of our tank, however, and keep the height the same, would we need a new tank whose radius was twice as big as our previous one? What do you think?
To understand how much our radius must grow to accomodate the new required volume of water, it is helpful to remember that while the volume of our cylindrical water tank will vary linearly with its height, it will vary quadratically with the radius of its circular base. So, increasing the radius by a certain amount would actually lead to a change in volume corresponding to the radius squared, so we wouldn't need to adjust our radius in a one-to-one proportion to our required volume.
QUIZ
Q1: Use the numerical values from the graphic to determine the radii of each of the spheres representing different quantities of water.
Q2: What is the volume of saltwater on the Earth? What would be the radius of a sphere used to contain this much water?
Q3: Draw a graph of the volume of each of the quantities portrayed in the aspiring spheres verses the radii obtained in Q2. Does your graph make sense? Why or why not?
Tuesday, April 14, 2009
Dynamic Currency Conversion
There is a very interesting article up on Washington Post that talks about dynamic currency conversion. Basically, this involves an additional transaction when using your credit card overseas to make purchases. It is supposed to be "convenient" for the consumer so that the cost can be seen in the consumer's home currency. But what is the cost of this convenience? That's what the article takes a look at. Here is an embedded currency converter:
Here are some ideas for a lesson plan around this article and currency conversion:
(1) What country would you like to visit most? Estimate the costs of a flight there, food, lodging, transportation, souvenirs, etc. This could all be done in dollars. This could be enhanced by looking up actual rates online, as this will give practice of finding and interpreting deals online for traveling.
(2) Compare the costs for each of the items with how much it will cost with each of the different modes of converting currency. These include, but aren't limited to, using direct cash conversion, using traveler's checks, credit card without DCC, and credit card with DCC. Remember that each of these come with their own conversion commissions. See which one offers the best price overall, and what the price difference is. What is the "cost for convenience?"
(3) Find out what the foreign exchange fees are for yours or their own credit card and compare with the average that's given in the article (~3%). How much money does this amount to?
(4) An investment type of question that requires research: Imagine that you had the equivalent of $10,000USD in a major (or maybe not so major) foreign currency. Based on the current economy and historical fluctuations (click here for viewing historical data), do you think it would be better after 2 years to have exchanged it back to USD (with the exchange fee) or not exchanged it back at all? How does this compare with how 2 years ago would compare to the present?
Here are some ideas for a lesson plan around this article and currency conversion:
(1) What country would you like to visit most? Estimate the costs of a flight there, food, lodging, transportation, souvenirs, etc. This could all be done in dollars. This could be enhanced by looking up actual rates online, as this will give practice of finding and interpreting deals online for traveling.
(2) Compare the costs for each of the items with how much it will cost with each of the different modes of converting currency. These include, but aren't limited to, using direct cash conversion, using traveler's checks, credit card without DCC, and credit card with DCC. Remember that each of these come with their own conversion commissions. See which one offers the best price overall, and what the price difference is. What is the "cost for convenience?"
(3) Find out what the foreign exchange fees are for yours or their own credit card and compare with the average that's given in the article (~3%). How much money does this amount to?
(4) An investment type of question that requires research: Imagine that you had the equivalent of $10,000USD in a major (or maybe not so major) foreign currency. Based on the current economy and historical fluctuations (click here for viewing historical data), do you think it would be better after 2 years to have exchanged it back to USD (with the exchange fee) or not exchanged it back at all? How does this compare with how 2 years ago would compare to the present?
Calculating Carbon Footprints.
Finding ways to teach mathematics through the context of environmental awareness and climate change is an interesting and important challenge. Mathematics can be helpful to understand things on scales much larger than many people are used to, particularly geological scales of size and time.
Quantifying climate change is an interesting mathematical challenge, since weather and climate are such complicated systems. There are a number of interesting quantities we can consider, however, to assign quantitative measure to detrimental changes in the Earth's ecosystem. The rates of species extinction and deforestation, as well as the amount of water and food that will continue to be available are interesting metrics to help us quantify the problems we are facing.
Perhaps the most talked about number regarding climate change is a person or organization's Carbon Footprint, or the amount of CO-2 they emit based on their lifestyle. That is what we will consider using the Berkeley Institute of the Environment's Carbon Calculator.
The video below is (I believe) part of the 25 million dollar Virgin Earth challenge, which calls for teams to suggest strategies to specifically remove CO-2 from the atmosphere.
The Berkeley Institute of the Environment has developed a very interesting tool called the CoolClimate Carbon Footprint Calculator which enables a user to enter information related to their lifestyle, and determine how much carbon dioxide they are emitting compared to the national and international averages. There are a number of interesting ways to teach high school level mathematical concepts through this interactive environment.
Some ideas:
(1) Have students use the Carbon Calculator to enter values related to their household. This will likely require them to talk to their parents about their household's monthly utility bills, miles driven and the square footage of their house. This seems like a good way to help promote awareness of issues like home and auto ownership for the students, and facilitate discussion between them and their parents.
(2) Once a student uses the Calculator to determine their household's CO2 emissions, this number will appear in the upper right corner of the carbon calculator. With this number, ask them to compute the size of a building it would take to store all of the carbon they will emit this year. This is an opportunity to discuss the notion of density and volume, how to determine the weight of a molecule (i.e. CO2) using its chemical formula, and converting from units of linear measure (inches, meters, etc.) to units of cubic measure (cubic inches, cubic meters, etc.). To give a standard of measure, you might ask a student to measure the volume of their bedroom, the volume of their school and the volume of a sports stadium given its dimensions. Once they do this, they can compare this to the volume of CO2 they emit.
(3) The calculator also gives bar graphs to indicate how much CO2 is emitted by an individual compared to national averages and international averages. These quantities can be used to discuss percentages and percent increases. A question might be as simple as having them calculate what percentage of these other averages is their emission footprint? What is the percent increase of their footprint compared to the other averages??
(4) Given the average carbon footprint of a US household and that of an international household, ask a student to calculate how much total carbon is emitted by humans in the world, and what percentage of that is emitted by the US?? This would involve them doing some independent research on population sizes, and to realize that they need to know those values to get to the answer.
(5) The interactivity of the Carbon Calculator enables us to explore the underlying functions being used by the BIE to generate these estimates. The equations are given in a document here, for the purposes of helping students check their answer. What I might ask them is to calculate the functions for metric tons of C02 vs. amount of money spent on furniture, food, square footage of their house, etc. This would require them to change those values and see what the resultant change is by the calculator. Since exact values aren't given, and no decimal significant digits are kept for the tons of CO2 emitted, this would require them to make big changes in a value (say 100$ on furniture, 1000$ on furniture, $5000 on furniture) and use these big jumps to estimate a slope of character to the graph they obtain.
(6) An interesting feature in calculating carbon footprint based on your utility supplier (at least for California) could lead to an interesting research project by students to understand what kinds of energy sources are used for the power grid, which utility companies are more progressive in pursuing "green" and non-polluting alternatives, and how where you live and the choices made by your public utilities and local government affect your carbon footprint regardless of your personal efforts of conservation.
(7) Ask students to use the calculator to determine what lifestyle changes they and their family would have to undertake to get to the international average. Are these changes possible? What are some alternative means to reduce our individual impact or community's impact? How many trees would they need to plant to offset their carbon footprint? Do the indirect costs of such efforts override the apparent benefits?
Quantifying climate change is an interesting mathematical challenge, since weather and climate are such complicated systems. There are a number of interesting quantities we can consider, however, to assign quantitative measure to detrimental changes in the Earth's ecosystem. The rates of species extinction and deforestation, as well as the amount of water and food that will continue to be available are interesting metrics to help us quantify the problems we are facing.
Perhaps the most talked about number regarding climate change is a person or organization's Carbon Footprint, or the amount of CO-2 they emit based on their lifestyle. That is what we will consider using the Berkeley Institute of the Environment's Carbon Calculator.
The video below is (I believe) part of the 25 million dollar Virgin Earth challenge, which calls for teams to suggest strategies to specifically remove CO-2 from the atmosphere.
The Berkeley Institute of the Environment has developed a very interesting tool called the CoolClimate Carbon Footprint Calculator which enables a user to enter information related to their lifestyle, and determine how much carbon dioxide they are emitting compared to the national and international averages. There are a number of interesting ways to teach high school level mathematical concepts through this interactive environment.
Some ideas:
(1) Have students use the Carbon Calculator to enter values related to their household. This will likely require them to talk to their parents about their household's monthly utility bills, miles driven and the square footage of their house. This seems like a good way to help promote awareness of issues like home and auto ownership for the students, and facilitate discussion between them and their parents.
(2) Once a student uses the Calculator to determine their household's CO2 emissions, this number will appear in the upper right corner of the carbon calculator. With this number, ask them to compute the size of a building it would take to store all of the carbon they will emit this year. This is an opportunity to discuss the notion of density and volume, how to determine the weight of a molecule (i.e. CO2) using its chemical formula, and converting from units of linear measure (inches, meters, etc.) to units of cubic measure (cubic inches, cubic meters, etc.). To give a standard of measure, you might ask a student to measure the volume of their bedroom, the volume of their school and the volume of a sports stadium given its dimensions. Once they do this, they can compare this to the volume of CO2 they emit.
(3) The calculator also gives bar graphs to indicate how much CO2 is emitted by an individual compared to national averages and international averages. These quantities can be used to discuss percentages and percent increases. A question might be as simple as having them calculate what percentage of these other averages is their emission footprint? What is the percent increase of their footprint compared to the other averages??
(4) Given the average carbon footprint of a US household and that of an international household, ask a student to calculate how much total carbon is emitted by humans in the world, and what percentage of that is emitted by the US?? This would involve them doing some independent research on population sizes, and to realize that they need to know those values to get to the answer.
(5) The interactivity of the Carbon Calculator enables us to explore the underlying functions being used by the BIE to generate these estimates. The equations are given in a document here, for the purposes of helping students check their answer. What I might ask them is to calculate the functions for metric tons of C02 vs. amount of money spent on furniture, food, square footage of their house, etc. This would require them to change those values and see what the resultant change is by the calculator. Since exact values aren't given, and no decimal significant digits are kept for the tons of CO2 emitted, this would require them to make big changes in a value (say 100$ on furniture, 1000$ on furniture, $5000 on furniture) and use these big jumps to estimate a slope of character to the graph they obtain.
(6) An interesting feature in calculating carbon footprint based on your utility supplier (at least for California) could lead to an interesting research project by students to understand what kinds of energy sources are used for the power grid, which utility companies are more progressive in pursuing "green" and non-polluting alternatives, and how where you live and the choices made by your public utilities and local government affect your carbon footprint regardless of your personal efforts of conservation.
(7) Ask students to use the calculator to determine what lifestyle changes they and their family would have to undertake to get to the international average. Are these changes possible? What are some alternative means to reduce our individual impact or community's impact? How many trees would they need to plant to offset their carbon footprint? Do the indirect costs of such efforts override the apparent benefits?
Monday, April 13, 2009
Getting a Piece of the Pie
Providing students with the tools to interpret quantitative data and information they encounter on the Internet or through news sources is an important component to promoting mathematical literacy. I have recently been teaching my students how to read the information found in Google Finance while implementing important concepts in fractions and ratios, as well as scientific notation.
A typical lesson might look like this:
(1) Ask the students about companies they know about, especially related to things they are interested in. Some of my students have picked sporting goods companies (Nike, Reebok, etc.), entertainment companies like Disney, or clothing companies like the Gap. Also ask them about what companies they think might be doing well or not so well in the current financial crisis, and why. Use this information to bring up a few Google Finance summaries of some of these companies.
(2) Given their imaginary bankroll of $10,000, I ask students to decide how much they want to invest in their first company. Using that amount, students can use the share price to determine how many shares they can buy. I then ask them to determine what percentage of the company they now own, given how many shares they have by directing them to the Outstanding Shares entry of the Finance summary. Because that number is very large, I have them compute this percentage without a calculator using scientific notation.
(3) Many students don't realize that the price of a share isn't that important, since different companies are broken up into a different number of shares. This point is well expressed by using Google's stock price vs. a company like Microsoft. Google's price is currently in the high 300's, while Microsoft is just below 20, making it seem like they can buy MORE of Microsoft with the same amount of money. By having them compute the percentage of the company they own for the same amount of money, this fallacy can be debunked, and the idea of market capitalization (i.e. the total value of a company) can be introduced.
(4) In continuing to analyze a particular stock, the idea of positive and negative slope can be introduced in the context of linear regression -- i.e. approximating a trend in a stock price using a line. The slope of that line corresponds to the gain or loss of value of the stock, and can thus help them to understand why we spend so much time discussing the slope. If a stock changes directions many times, this is also an opportunity to introduce how when using polynomials to approximate graphs, the degree of the polynomial determines how many slope changes our approximation can accomodate.
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