Additional Math Pages & Resources

Monday, January 31, 2011

Where DON'T you use math?

I keep thinking about where we use math, and why learning math is important for grade-school kids, and why we go to all the effort to produce and sell a math curriculum. That's the core of this blog - explaining how math we learn when we are young can be used later, when we mature (get old).

So I thought I'd reflect a conversation with myself about places I might NOT use math:

 1. Body-surfing in the Pacific is a place to forget math, right? Sadly, NO.  As I sit out there in the surf, I often think I'm seeing a new set of waves coming in on 10 second intervals, about 3 feet high. About a third of those are "significant waves" one-third higher than the average; about one tenth of the total are "set waves" five-thirds the size of an average wave; and one out of a thousand waves are twice the average. Math. Click here.

 2. My wife and I are pulling weeds in the back yard. No time for math, right? Alas, NO. We need to know how to check a calendar, we need to know the climatic zone we are in, and we need to know the speed at which weeds grow. I read recently that weed control is dependent on pulling them at the right time, and if you plan on applying chemical weed killers (we don't) the timing depends upon the growth rate of the individual weed species. We have some weeds that are taller than we are.  Click here.

 3. I think I'll do a little maintenance on my car. Grease and grime - but for math, no time. Right? NO. I have to check the tires (air pressure in lbs/sq in or bar), check the oil (one quart low), replace a light bulb (a long catalog search tells me it's an H4 in the USA), etc. etc. Nothing but math, it seems. Click here.

4. Maybe I will just read a book. No math there? Whoops. My mistake. I choose a science fiction book by Jules Verne - From the Earth to the Moon. It's filled with math about planets and speed of light, and tons of gunpowder and diameter of projectiles being shot to the moon ....  Click here.

5. How about a healthy snack? Wrong - there are ingredients on the label, calories of the food, price tags on the package, serving size (to be exceeded), number of items. It's all math. So I grab a tangerine instead. But there's a tiny little label on the back with a code number so the distributor can track it back to the origin. As I peel off the skin I toss the label in the trash. But later I discover there is a website just for fruit label identification and decoding. Ouch. More math. Click here.

6. Looking around my office, I think maybe I will ride my 15-speed bike. Oh no, I made a mistake just in describing it. And I'd have to pump up the tires to 100 psi. At least there's no shift indicator, trip computer or other electronic gadget on my old-school bike. But it is a 57 cm frame with 700 mm wheels and 25 mm tires. And so on. Click here.

I give up. I'm just going home. I'm not going to look at the odometer or speedometer or fuel gauge. I'll avoid math. Although I already know the distance is between 20.6 and 26.0 miles, depending on how I go. And 31-37 minutes. How do I know? Math. Click here.

Friday, January 28, 2011

Math for cooking, Part V

This ends my week to talk about math used in cooking. I'm sure you are tired of the rice and sweet potato recipe by now. So we'll clean up our mess before heading off for the weekend.

After some contemplation, I've come up with these math skills (taught in Excel Math curriculum) that we are likely to need while cooking:
  • putting things in sequential order
  • measuring liquid volume, measuring weight
  • measuring temperature of an item
  • standard and metric units and conversion back and forth
  • conversion of units from volume to weight (1 cup flour by volume = 5 ounces by weight)
  • telling time; number of minutes in an hour; adding minutes
  • calculating elapsed time in minutes (across the 12) on the clock
  • evaluating information to see if it is sufficient; ignoring extraneous information
  • estimating; cost-per-item (unit cost)
  • calculating one-half of an item; fractions
Hopefully you have picked up these skills through school or life experience, so when you go into the kitchen you feel confident in the math part of the cooking process. We'll do what we can to make sure your kids learn math, so they can participate too.
As for actually performing the skills of cooking (cutting, mixing, baking, etc.) you and the kids just need to get in there and give it a try. We don't teach cooking skills in Excel Math.
If you do know how to cook - especially from basic ingredients, please consider sharing your talents with young people. Help them develop a life-long capability that's just as important to their overall health as math.

Thursday, January 27, 2011

Math for Cooking, Part IV

This week's math blog has centered on the math used in recipes. So far the versions I've offered were inadequate. Both the short one (no units) and the symbol version (few words) were unusable, in my opinion. This was not an accident, but to prove a point.

Let's try an enhanced version, including all the math I can think of. In this example I have added conversions to metric units, specified how many people we can feed, how long it will take to prepare, the tools required to do the cooking, an illustration, etc.

Rice and sweet potato, with coconut and cranberries 
  • serves 3-4 adults as a side dish or 2 adults as a main dish
  • active effort 15 minutes; total preparation time 60 minutes 

1 cup (300 g) roasted and skinned (or canned) sweet potato, mashed
1 cup (180 g) Basmati rice
2 cups (500 g) boiling water
1 tbsp (15 g) olive oil or butter
4 oz (100 g) dried sweetened coconut
4 oz (100 g) cranberries (frozen, fresh or dried)
1 tsp (6 g) sea salt
1 tsp (6 g) black pepper

Large (12-inch) skillet with oven-safe lid
Spatula or spoon for stirring rice in skillet
Pot or kettle to boil 2 cups of water
Oven large enough for skillet
Measuring cup and spoons or food scale
Hot pads or towel

  1. Heat the oven to 275°F or 140°C  [Gas Mark 1 if you are in the UK]
  2. Put the skillet on a large burner. 
  3. Set burner to medium-low heat. Let the pan heat for 2-3 minutes.
  4. Using a pot or tea kettle, start heating 2 cups of water to a boil.
  5. When the skillet is hot, add the olive oil or butter and give it a moment to heat. 
  6. When the oil is hot, stir in the Basmati rice. 
  7. Stir the rice for 5 minutes, or until a few grains begin to turn a toasty brown color. 
  8. Add sweet potato to the pan. 
  9. Stir the mixture to integrate the rice and sweet potatoes.
  10. While continuing to stir, slowly pour the 2 cups of boiling water into the skillet.
  11. Add the coconut, cranberries, salt and pepper.
  12. Turn off the burners for skillet and boiling water.
  13. Put the lid on the skillet. Put the covered skillet in the oven.
  14. Leave in oven for 40 minutes. Do not open lid during this time!
  15. Carefully take hot skillet from the oven using hot pads, and remove lid. Turn off oven.
  16. Fluff and mix the rice with a fork, and allow to cool slightly before serving.
Was that too much? Did I need to tell you to turn off the stove-top burners and the oven? Remind you to turn on the lights in the kitchen if it was dark outside? Specify that you buy ingredients at your local market? Recommend favorite brands? Insist on Ocean Spray™ cranberries? Warn that I meant the actual fruit and not the Irish rock band?
Where do we stop? While developing this blog my sister sent me a recipe for fried mackerel. After the author finished describing the cooking method, he added a postscript,

"I forgot to tell you to open the windows to reduce the fishy smell ... although I suppose you worked that out for yourself."

In any writing endeavor, you have to decide how much to say, and how much to leave the readers to work out on their own.That applies to recipes, and to math problems.

    Wednesday, January 26, 2011

    Math for Cooking, Part III

    This week we've been looking at math for cooking, particularly as it's expressed in recipes. It's almost impossible to tell someone else how to cook a dish, without using some math.

    I gave you a brief recipe for a rice dish, with no units or numbers. Here it is again:

    Boil water. Put olive oil in skillet. When oil is hot, add Basmati rice. Stir until rice turns light brown. Add sweet potato, coconut and cranberries. Add boiling water, cover and put in oven. When rice is done, fluff and serve.

    Most people would agree that is not enough information to successfully prepare the food. Would it be better if I could offer a symbolic, graphical version? About the same number of words, but significantly more data?

    I have created unique symbols for the main elements of the recipe. Here is a key to the symbols. They are based on letters in a font called we are alien!!
    Every recipe in my cookbook would use these symbols, so you could easily learn them, even if they don't resonate with you today.

    Here is the recipe itself:

    What do you think? Do visual representations of a task appeal to you? Does this look math-friendly? Do the symbols detract? (I didn't feel like using tiny measuring cups, etc.)

    We could add more illustrations for the cranberries, coconut and salt and pepper. But they are only the garnish, not the main ingredients - should we label them optional?

    Rather than illustrations, would you prefer more descriptive text and numbers with multiple units of measure? Tomorrow we'll see how verbose a recipe can be, and whether "words, words, words" help a cook more than my drawing.

    Tuesday, January 25, 2011

    Math for Cooking, Part II

    As a math publisher, I consider myself qualified to write this series of blogs on recipes because I have been cooking out of cookbooks for 40 years. And writing highly-technical manuals for most of that time. Technical writing of any kind is a lot like writing recipes, as is math curriculum.

    I have also done some research about recipe-writing this week. I appreciated a chapter in With Bold Knife and Fork by M.F.K Fisher, entitled the Anatomy of a Recipe. Another nice source was a newspaper column in the Financial Times called The Art of Recipe Writing. There were some other articles that didn't help me much, but focused on writing style, choice of food, etc. I did like some material on a website run by the Association of Food Journalists ("a networking system created for journalists who devote most of their working time to planning and writing food copy for news media").

    In general, authors complain about the difficulty of writing a proper cookbook. Not only should the recipes leave no cook's question unanswered, they should also appeal to the diner, provide instruction to the assistants, support the creation of a shopping list, and even contain reference to amusing or amazing meals (or diners) for which the recipe was created. And offer wine suggestions, specify table decorations and special serving dishes or silverware. And so on.

    Fine. But as a math curriculum person, I also want mathematically-correct instructions. What do I mean by this?
    • a cookbook author (or assistants) should have prepared the recipe multiple times, using the chosen instructions and ingredients to confirm that everything works as it should
    • the title should be descriptive
    • specified cooking times should be totaled and provided up front, for the cook's planning and to confirm the food will not be undercooked or burned
    • a recipe should designate the number of people who will be fed
    • the units of measure, heat or time should be familiar to the cook and match the customary usage and appliances in that country [Celsius, Fahrenheit, Gas Mark]
    • the units of measure for the ingredients should be a reasonable, not impossible, degree of precision [i.e. 6 grams, not 6.250 grams]
    • the units for the ingredients should match the way the ingredients are sold in the market (three carrots, one 6 oz. can of tomato paste; a 5-lb. sea bass)
    • the recipe should say when you measure the ingredients - for example 3 ounces of raw shrimp in the shell is different than 3 ounces of shelled, cooked shrimp
    • any unusual ingredients should be highlighted in a list at the beginning, to prevent cooks from getting part-way through and having to give up ( i.e. 3 pinches of saffron; 2 shavings of white truffle, etc. )
    • the size and/or capacity of the cooking equipment (pots, pans, mandolin, blender, oven) should be stated, to ensure they are available
    • the required measuring instruments should be commonly available [cup, spoon, scale]
    • the recipe should be given in the sequential order that a cook would follow; if there are parallel steps (as in my recipe) the author must present them clearly
    So we have confirmed that this is not adequate:

    Boil water. Put olive oil in skillet. When oil is hot, add Basmati rice. Stir until rice turns light brown. Add sweet potato, coconut and cranberries. Add boiling water, cover and put in oven. When rice is done, fluff and serve.

    Tomorrow we will try again.

      Monday, January 24, 2011

      Math for Cooking, Part I

      The intent of this blog is to show how we can use arithmetic, mathematics and geometry learned in elementary school. That's the kind of math we present in our Excel Math curriculum. I'm not making any promises that you will ever use calculus, trig or other advanced math that comes along later in your higher education.

      Last March I devoted a week to food math - or as it turned out - mostly nutritional math. Calories, costs, labels, etc. [Click here to check it out] This week I've decided to tackle cooking math (not eating or nutrition or economic math).

      One definition of cook is preparing food for eating by the application of heat, such as boiling, baking or roasting.

      This definition implies that we may need to know quantities (of the food and/or people to eat it), temperature (how much heat to apply) and time (duration of heating). We could possibly need to know other numbers, but we'll wait and see about that.

      Let's start with a number-free, math-less recipe for a rice dish I prepared last night:

      Rice and sweet potato, with coconut and cranberries

      Boil water. Put olive oil in skillet. When oil is hot, add Basmati rice. Stir until rice turns light brown. Add sweet potato, coconut and cranberries. Add boiling water, cover and put in oven. When rice is done, fluff and serve.

      An experienced cook could easily make this dish using this feeble, math-free recipe. But most people would demand some more information. Here's another version, using numbers everywhere we can:

      Rice and sweet potato, with coconut and cranberries 
      • serves 3-4 as a side dish or 2 as a main dish
      • active effort 15 minutes; total time 60 minutes
      1 cup sweet potato
      1 cup Basmati rice
      2 cups boiling water
      1 tablespoon olive oil or butter
      4 ounces of dried sweetened coconut
      4 ounces of cranberries
      1 teaspoon sea salt
      1 teaspoon black pepper

      1. Heat oven to 275°F
      2. Put a skillet with oven-proof lid onto a burner set on medium-low heat. 
      3. Using a separate pot or tea kettle, start heating 2 cups of water to a boil.
      4. When the skillet is hot, add the olive oil or butter. 
      5. After the oil is hot, add the Basmati rice. 
      6. Heat and stir for 5 minutes, or until some of the rice grains turn a light brown color. 
      7. Add sweet potato to the pan along with coconut, cranberries, salt and pepper. 
      8. Stir the mixture to integrate the rice and sweet potatoes.
      9. While continuing to stir, slowly pour the boiling water into the skillet.
      10. Cover the skillet and place it in the hot oven.
      11. Leave skillet in the oven for 40 minutes without opening the lid.
      12. Remove skillet from the oven and remove lid.
      13. Fluff and mix the rice with a fork, and allow to cool slightly.
      14. Serve.
      Tomorrow we'll talk about recipes and the measurement units and math involved.

      Friday, January 21, 2011

      How do you measure size? Pipe Organs

      I decided to talk about pipe organs today. This is the last of my small series of postings on how to measure and compare size. I don't play any instruments, but I like to listen to (feel) organ music. When sounds are emanating from 32-foot pipes, they shake the building and the body. I guess I have a thing about any kind of loud sound-generating things, like bells and fog horns and sirens!

      Size refers to how big something is - by length, width, height, diameter, perimeter, volume, area, or any number of other criteria.

      When it comes to measuring pipe organs, there are three main factors  that enthusiasts compare - the number of manuals (keyboards), the number of ranks (sets of pipes) and the number of organ pipes themselves.
      There are other factors for comparing, such as the range of pipe size from large to small, the length of the longest pipe, speed or pressure of the wind blowing through the pipes, etc. Then there are the categories of organ, such as Church Organ, Theatre Organ, Fairground Organ, etc.
      I found a page of specifications that show the largest Organs in the world.

      After some research, I created a picture of the largest organ (by number of pipes). Well, it looks like organ pipes but it's really a table showing the number of pipes within each group or voice on the Atlantic City Convention Hall organ. [Click on the picture for a larger version]

      Here's the console from which the Atlantic City organ is played.

      If you would like to hear my favorite organist, Jan Feher, playing the First Presbyterian Church of San Diego's Casavant Frères with 5100 pipes, click here. She's doing a full concert on January 30th if you want to stop by and hear the organ in person. The name of this piece is Basse et Dessus de Trompette by Louis-Nicolas Clerambault.

      This is the Liverpool Episcopal Cathedral organ. Click on the image to see it in a much larger size!

      Thursday, January 20, 2011

      How do you measure size? Large Organisms

      When measuring, we tend to use different forms of measurement for living things than we do for inanimate objects. We discuss this in our Excel Math curriculum as we introduce kids to various units of measure.

      Yesterday we talked about large organizations - using as our basis of comparison the number of people working there. The top 14 organizations I found have an average of a million employees.

      Just as you might choose market capitalization, profit, revenue, units sold, number of outlets, or other measures for comparing companies, you can choose from a variety of measures to determine size of a living organism.

      Today we ask the question, What are the largest living organisms?

      With a tree, you might choose the
      tallest (Hyperion redwood)
      greatest circumference of branches (ficus / banyan trees)
      number of years that it has been living (Methuselah Bristlecone Pine)
      size and number of connected off-shoots (Utah's Pando Grove)
      or volume and mass (General Sherman Redwood).

      With an animal you might decide to compare on the basis of weight (elephant), height (giraffe), length (whale), number of limbs (millipede can have up to 750!), mass (whale record is 190 tons) or total body volume.

      This week I saw a 13,000 lb. male elephant at the San Diego Zoo. His tusks weigh more than 120 pounds each. They had to trim the tusks recently because they were crossing in the front and almost touched the ground. Each trimmed off piece weighed about 25 pounds!

      A fungus (neither plant or animal) may be a colony composed of millions of individual mushrooms (flowering parts) which are what we see above ground. In Oregon there's a fungus that may stretch for 2200 acres. A similar description of "colony" could apply to the Great Barrier (coral) Reef near Australia, and Neptune grass in the Mediterranean.

      Go here if you want to read more on large organisms.

      Go here if you want to watch the San Diego Zoo elephant camera.

      Go here if you want to climb up a tree that's 379 feet tall.

      Wednesday, January 19, 2011

      How do you measure size? Large Organizations

      What makes someone or something BIG or small?

      When we look at organizations, is it the totality of the stock value (market capitalization)? Is it revenue per year? Is it profit? Is it the number of branches or outlets? Is it the number employed? The number of customers? The size of their warehouses or truck fleets?

      When you discuss institutions and organizations, you need to decide what factors are important in your comparison. If your group has lots of employees but makes no money, is that a problem? It might be a non-profit, serving its customers in many ways without having to generate returns for its owners. It could be the biggest, best, non-profit-making group in the world.

      Today we'll look at a group of the world's BIGGEST organizations, measured by the number of employees. All the figures come directly from the company websites or annual reports, checked yesterday. [Click the chart to see a larger version]

      Walmart Worldwide = 2,050,000
      National Health Service UK = 1,700,000
      McDonalds Corp. and Franchisees Worldwide = 1,600,000
      Indian Railways India only = 1,540,000
      State Electric Grid of China = 1,500,000
      People’s Liberation Army China = 1,250,000
      China National Petroleum China = 1,005,000
      Foxconn Technology Group China = 1,000,000
      US Postal Service USA = 584,000
      Carrefour Retail Stores Worldwide = 475,000
      Agricultural Bank of China = 441,144
      Deutsche Post Germany = 420,000
      Siemens AG Germany = 405,000
      United Parcel Service UPS Worldwide = 400,600

      These organizations employ 14,370,744 divided by 14 = an average of 1,026,482 people each!

      You can see that this is a mixed bag, including retailing, state-run organizations, and manufacturing; military, railroads, utility companies, banks, post offices. So it's not appropriate to compare them by revenue or products delivered, or retail outlets!

      Of course there are other ways to measure BIGNESS. Here is a condensed version Wikipedia's article on Market Capitalization:

      Market capitalization is a measurement of size of a corporation equal to share price times the number of outstanding shares purchased by investors of a publicly-traded company. Since owning a share is ownership of a fraction of the company, capitalization represents the public's opinion of a company's value.

      As of today, January 18, 2010, the three largest companies in the world by this measure ( in US dollars ) are
      • $396.9 billion Exxon Mobil
      • $312.5 billion Apple Computer
      • $254.7 billion PetroChina
      Totalling almost a trillion dollars of market capitalization, they are BIG! companies.

      Tuesday, January 18, 2011

      How do you quantify competence?

      How can you measure the competence of a country compared to those around it?

      In the USA, we pride ourselves on being the standard that others look up to - in areas like democracy, human rights, entrepreneurship, etc. We know how to make a buck!

      France sees itself as the center of fashion, culture, etc. The British contributed jury trials, financial expertise, etc. The Swiss do chocolate, watches and trains that run on time. Each country has its own contributions to the global human endeavor.

      Of course, not everyone agrees with us, or the French, British or Swiss. They may see their own country (or culture) as more valuable, more important, more idealistic.

      The problem is that these national and cultural contributions can't be easily measured. They might be visible (more or less) but they are not quantifiable. How do you count influence?

      Since this blog deals with understanding the world using only elementary level math, I wanted to deal with this political-social-economic question in a straightforward way. Today I read some material in the Financial Times newspaper that has helped me a great deal.

      Here's a chart I constructed from the data they presented on China's economic impact on the world. We can see, by the numbers, what influence Chinese trade has. [Click the chart for a larger version]

      This chart displays China's percentage of foreign trade with each country. It includes both imports and exports for the 12 months from August 2009 - August 2010 (and 1992). The chart shows the top 28 countries.

      The USA currently has a bit over 14% of its trade with China. The highest is South Korea at 22.8%, and the lowest on this chart is Belgium at 2.9%.  The average country now has 10% of its trade with China.

      When the same data is analyzed for 18 years ago, the most interdependent country was Japan at 5%.  The average country then had 1.6% of its trade with China.

      This chart is an analysis is of current and past trading patterns. Looking ahead, China Development Bank, one of their top investing institutions, has made loans totalling $65 billion in the past two years, through offices in 150 countries. If you include all the other Chinese banks' activity around the world, China's total investment in developing countries may be $110 billion in two years. This investment will eventually pay off in the future.

      These numbers suggest we are not the only country that knows how to make money!

      Friday, January 14, 2011

      How Fat Am I? Part V

      This is the fifth blog on finding out how much body fat we have. The past 4 days we've looked at the easy methods of estimating body fat. Easy also means relatively inaccurate from a math perspective.

      The definitive approach to determining body composition is called an autopsy. A person dies, someone cuts up the body, and weighs all the parts. Not a method anyone would choose! Although autopsies are often done to determine the cause of death, they are done very rarely for body composition research - perhaps only about 50 times in the past 50 years.

      The best common approach to body composition measurement is called hydrostatic testing. After you breathe out, emptying your lungs of as much air as possible, operators in a fitness or research clinic will lower you into some water and measure the amount of water that is displaced. That will tell the testers precisely what your body volume is, and allow them to calculate how much of you is water.

      (If this sounds familiar, it is the method I used to measure a copper ingot last summer. And it's how Archimedes discovered how to tell if a crown was made of silver or gold).

      After you come up out of the water, the hydrostatic testers get to do some math. Lots and lots of math. How much air was still in your lungs, how much bone you have, etc.

      Body Density = dry weight / [((dry weight - wet weight) / water density)- Residual Volume in Lungs - 0.1] 

      I thought it would be simple, but after looking at a bunch of calculations, I decided I'm only a math publisher, not an exercise physiologist. I'm not going to put all the math in here. But here is an illustration and link to an extremely keen interactive way to learn about hydrostatic testing, from the Univ of Vermont.

      While hunting around, I did find a business opportunity to operate a Mobile Hydrostatic Body Fat Testing Lab. It's essentially a big truck, like the one we use to deliver Excel Math. It contains a stainless tank, some scales, a heater, changing rooms, etc.

      There is one more very accurate method - DEXA, or Dual Energy X-ray Absorbtiometry.

      A subject reclines on a body scanner. X-rays come from below and pass upwards through the body to a detector overhead. Photons from two different kind of x-rays are measured. The ratios of each that reach the detector are compared to predict total body fat, fat-free mass, and total body bone mineral. This takes only 10 - 20 minutes and is useful for most people except those too large for the machine.
      The main issue with DEXA testing is that the machines are large, they emit X-rays, and they take a whole room. And they cost a lot of money, say $50,000-150,000+.  Perhaps fine for labs or hospitals, but not for the home.

      I suppose if you got the tank truck franchise you could put a DEXA machine in your truck too, and offer customers a choice of testing methods ... math would be useful to count all the money!

      Thursday, January 13, 2011

      How Fat Am I? Part IV

      This week we've been talking about the math involved in measuring (or estimating) body fat.

      Today we will consider a Bioelectrical Impedance Analysis scale. One simply stands on this scale in bare feet and the display shows weight and estimated body fat. You might ask, How does that work?
      The body conducts electricity. The typical  $50-250 BIA home scale applies a small current through the foot pads which travels up one leg and down the other, back to the scale. The various components of your body (fluid, fat, muscle, bone, etc.) all have differing resistance to electrical current. Circuitry in the scale measures the signal it receives (as modified by your body) and estimates your body fat. This particular unit has handles. If you grip with both hands it can also estimate fat in your upper body.

      NOTE - all of these scales do the math for you. No arithmetic or math skills are required. Sorry.

      Here's a professional model that costs $1500. It has clever features such as subtracting the weight of your clothes, shoes, iPhone, etc. so you don't have to take the measurements in your birthday suit.
      Lab quality testing equipment tends to have electrodes that connect to both arms and legs. It is capable of much more precise results. Here's a professional body composition measuring tool that costs $4000.

      All of these scales come from a company called Tanita

      NOTE: I have no connection with Tanita. I work for Excel Math and no, I haven't thought of a way to buy one of these from my expense account.

      Getting accurate and repeatable readings with any of these neat devices requires good electrical contact at your skin, a consistent level of fluid in your body, constant temperature and time of day, and assumes you are not touching anything metal (or operating a short-wave radio, or flying Ben Franklin's kite, or ...). These scales should not be used if you have a pacemaker or other electronic implant.

      BIA measurements don't usually tell you where the body fat is located, and are not accurate if you have extreme body fat levels (less than 10% or higher than 75%).

      You can get lots of data from a scale like this. Here's a sample that could be saved in a spreadsheet:
      Standard, Male, 30 years old, Height 67.0 in {0, 16, ~0, 1, ~1, 1, ~2, 1, MO, “SC-240”, SN, “00000003”, Bt, 0, GE, 1, AG, 30, Hi, 67.0, Pt, 0.0, Wp, 182.2, FW, 18.7, fW, 34.0, MW, 148.2, mW, 140.8, sW, 19, bW, 7.4, wW, 105.6, ww, 58.0, MI, 28.5, Sw, 140.4, OV, 29.8, IF, 6, rb, 8247, rB, 1971 ,rJ, 12, rA, 29, ZF, 372.8, CS, 2A 
      (You'll have to read the manual to find out what all this means!)
      The best use of a home-quality BIA scale is to compare your own readings from day to day (when you are on a diet, or workout regime). Home scales do not give you accurate figures for comparing yourself to others. Only the professional-level devices are used in clinics that test large populations.

      If you want to do some math today, ask yourself how many times you will measure your body composition, and calculate the cost-per-measure figure for one of these three machines. For example, if I had made a New Year's resolution to track my body fat, I would expect my resolve to last about a month. I'd measure myself about 5-10 times. A $100 scale would cost me $10-20 per test. (But being a techie, I would rather have the $1500 professional model!)

      If you are better at New Year's Resolution-keeping than I am, you might want to look here and get the full kit - body analyzer, food scale, jump rope, etc.

      Wednesday, January 12, 2011

      How Fat Am I? Part III

      Welcome to the Excel Math blog where I talk about things grown-ups can do with math they learned in elementary school - the kind of math our curriculum helps present to K-6 kids.

      My wife is a PE teacher. For 35 years I have watched her measure people and calculate their fitness and fatness. Why does she do this?

      We want to know how much of the volume of our body is made up of fat, and how much is solid material (bone), muscle tissue, water and air. With this information, we have a good idea of how healthy, strong, long-lived, etc. we are likely to be. Fitness professionals can then provide you with suggestions on how to improve your health.

      My wife knows how to use a skin-fold caliper and can poke and pinch you until the truth emerges about your sub-cutaneous (under the skin) body fat. For many years, this was the most cost-effective and accurate way to determine body composition. The problem is, it's hard for an amateur to do accurately, and almost impossible to measure yourself. And the results vary for children, adults, and various ethnic groups.

      This morning I had my wife take some measurements on me using her simplest skin-fold caliper. Here she is pinching and measuring my waist while I take the photos.

      Notice the caliper has two arrows - you are supposed to press with your thumb until they line up - this ensures you close the caliper adequately. A plastic arm deflects when the jaws are tight enough on your fat. The settings seem a bit tight to me - ouch!

      In the second picture she is measuring at the back of my upper arm. The scale indicates mm of body fat. These are the only two places we measured, but there are many more that can be checked. Go here to see some examples.

      If this was a real test, she would have made multiple measurements around my body. Then using software or a calculator, she would add all the measurements (in mm) and calculate as shown below. This is called the Jackson & Pollock equation:

      MALE Body Density = 1.0990750 - 0.0008209 (X2) + 0.0000026 (X2)2 - 0.0002017 (X3) - 0.005675 (X4) + 0.018586 (X5)
      X2 = sum of the chest, abdomen and thigh Skinfold in mm 

      X3 = age in years 
      X4 = waist circumference in cm 
      X5 = forearm circumference in cm

      FEMALE BD = 1. 1 470292 - 0.0009376 (X3) + 0.0000030 (X3)2 - 0.000 1 1 56 (X4) - 0.0005839 (X5)
      Where X3    = sum of triceps, thigh and suprailiac Skinfold, in mm 

      X4 = age in years
      X5 = gluteal circumference, in cm

      Then using the Siri equation, she would convert body density to % of body fat, like this

      FAT% = ((4.95 ÷ Body Density) - 4.5) x 100

      Does that seem like a lot of work? Yes, indeed it does. But it's elementary math - nothing that the average person couldn't handle with a little care and attention to the decimal places.

      You might wonder how these obscure formulas were developed. People did thousands of these skin fold tests trying to find a correlation to the ultimate body composition method - hydrostatic weighing.

      Before we talk about hydrostatic weighing on Friday, we'll look at another common method known as bioelectrical impedance analysis. Sounds impressive. It's a special scale that you step on, and it gives your weight and an estimate of your body fat.

      Tuesday, January 11, 2011

      How Fat Am I? Part II

      Yesterday I gave you a brief introduction to the Body Mass Index. A magic, mathematical number. Or is it? Even a person who only uses elementary school math can see its shortcomings.

      From a math point of view, the formula (weight over height squared) does not accurately reflect how weight changes with height.  For an extended discussion of this, go here. It also does not reflect the makeup of athletes, because increasing your muscle mass causes you to appear even more overweight (muscle weighs more than fat).

      There are lots of other issues with BMI, so concerned scientists, doctors, etc. have come up with lots of other ways to answer the question How Fat Am I? which is where we started yesterday (the question behind many New Year's Resolutions).

      The US Navy, who measures and employs lots of fit young men and women, has came up with an enhanced version of the BMI test. This was published about 25 years ago, from research work done right here in San Diego. It adds several measurements of circumference (the distance around the perimeter of an object).

      You should aim for accuracy of 1/4" while using a non-stretching, flexible measuring tape around the body. Note that if you cannot easily (or honestly) measure yourself this accurately, you might want a helper.

      Follow this process:
      1. Measure weight in the morning after going to the toilet but before dressing or eating
      2. Measure height with shoes off
      3. Measure circumference of the neck below the Adam's Apple
      4. Men measure waist at the belly button
      5. Women measure waist at point of smallest circumference
      6. Women measure hips at point of largest circumference
      Then you want to find a website that calculates the body mass for you, OR a scientific calculator (because this is a bit above the elementary-level math taught in Excel Math).

      men: body fat % = 86.010 x log10(abdomen - neck) - 70.041 x log10(height) + 36.76
      women: body fat % = 163.205 x log10(waist + hip - neck) - 97.684 x log10(height) - 78.387
      Using this process, I get BMI of 26.9 kg/m2 with a Waist-to-Height ratio of .51
      The web calculator I used suggest that I have 16.9% body fat, and lean body mass of 151 lbs.
      That's a bit better than yesterday's BMI of 27.3 (only because it makes me feel less fat).

      By the way, this procedure is based on testing lots of young active people from European countries, not older, sedentary people (like me), or people with different ethnic backgrounds such as Pacific Islanders, Asians, blacks, etc. But it's more reflective of reality than the simple BMI that does not measure girth.

      There are many other, more elaborate ways to measure, calculate, estimate and understand your body composition and the fat you carry around. We'll look at a few more tomorrow.

      Monday, January 10, 2011

      New Year Questions: How Fat Am I? Part I

      Do you know how your body fat percentage compares to that of your peers?

      The term "Body Mass Index" describes a simple method of representing the amount of body fat an individual carries. It was originally developed by a Belgian researcher, Adolphe Quetelet around 1840. It was meant to compare individuals in large populations. It was not intended to be applied in a "guilt-inducing" way to label someone as overweight.

      Here's a table so you can look up your BMI for yourself. Click on the chart to see it in a larger size, then click the back arrow to return to this page:

      Is the BMI calculation rocket science? No, it's not rocket science, you can estimate your body mass index (as shown on the charts) with elementary arithmetic as taught in Excel Math.

      First, measure your weight and height, then proceed to the calculations.

      We'll do this in both standard and metric units. First the standard US units:
      Weight   185 pounds
      Find your height in inches
      Height   69 inches
      Now multiply weight by 703 x 185 = 130,055
      Now multiply height by itself  (height squared) 69 x 69 = 4761
      Divide weight by the height squared   130,055 ÷ 4761 = BMI 27.3

      To do the same calculations in metric units, start with kilograms (or divide pounds by 2.2):
      Weight 84.1 kg
      Then find your height in meters (or multiply inches x .0254)
      Height 1.753 m
      Now multiply height by itself (height squared) 1.75 x 1.75 = 3.073
      Now divide weight by the height squared     84.1 ÷ 3.1 = BMI 27.1 

      Now do we know our true body fat percentage? NO, we have a guesstimate. Finding real body fat IS rocket science.

      Body mass composition is more accurately calculated using hydrostatic density testing - in a laboratory with measurements taken with accuracy of ±.05 cm; using with skin-fold calipers and pinching skin in various specified places, calculation of total body volume by immersing you in a water tank; and tossing bones from a newt into smoldering lotus leaves (optional).

      We'll talk about this more tomorrow.

      Friday, January 7, 2011

      Math Reasoning, Part III: Bugatti Veyron

      Yesterday I posed a question - how much air does a conventional gasoline-engined vehicle consume at 200 mph? If you didn't read yesterday's blog, go back now and check it out so you can follow the rest of this solution.

      I did some research, assembled some figures, and now (using only elementary school math) we have to make progress towards our solution by determining:

      How much fuel will we burn in a minute, or more precisely, how many liters of fuel do we consume at 200 mph?

      That's not so easy to learn! But fortunately when a Bugatti Veyron set the current record for fastest street car (257 mph), its fuel consumption was measured.

      The Veyron achieves about 2.3 miles per gallon at 240 mph. If you divide 240 by 60 you find you are traveling 4 miles in a minute.

      We only care about how much we burn in a minute, so:

      4 miles ÷ 2.3 miles/gallon = 1.7 gallons

      Thus in the Veyron we are burning roughly 1.7 gallons per minute, or (1.7 x 3.78) 6.4 liters.

      Of course this is at 240-250 mph. Let's assume the fuel consumption is a bit less at 200 mph. Care to guess with me? I'm going to say 1.3 gallons per minute, or 5 liters per minute.

      5 x 8500 = 42,500 liters of air per minute. That's more air than the Jaguar's gas turbines!

      The breathe at the rate my math suggests, the Veyron must have large air intakes so it can gulp all that air. Yes, see? They are in the rear of the car over the occupants' heads.

      NOTE: I just read an article quoting TOP GEAR presenter Jeremy Clarkson as saying at top speed the Veyron consumes 45,000 liters of air per minute. Confirmation our math is right!

      Even elementary math can be useful. We might also predict that the Jaguar gets better fuel economy than the Bugatti at the same speed. It's lighter, smaller and more aerodynamic -  and because it consumes less air it must therefore be burning less fuel. Proving this prediction is another day's problem.

      Let's go one step further today. Rather than being the fastest car in the world, the 2000-2006 Honda Insight was the most fuel-efficient gasoline vehicle.


      At 60 mpg and 60 mph, it consumes 1/60 of a gallon per minute, or .063 liters a minute.
      That means .063 x 8500 = only 141 liters of air per minute. 

      Thursday, January 6, 2011

      Math Reasoning, Part II: Jaguar C-X75

      Recently I saw and fell in love with the Jaguar C-X75 concept car. It's gorgeous. Take a look:

      This high-performance hybrid vehicle uses one electric motor per wheel for propulsion. It has a pair of tiny gas turbines that turn generators to recharge the batteries or power the motors directly. The C-X75 accelerates from 0-62 mph in 3.4 seconds and can reach a top speed of 205 mph, says Jaguar.

      What caught my eye was a number in the brochure: 35,000 liters per minute. That's right, the twin turbines consume 35,000 liters of air per minute at top output. Hence it has huge air intakes on its lower flanks, just in front of the rear wheels.

      I wonder, Is that more or less air than a conventional high-performance gasoline-engined car consumes?

      Let's see if elementary math can help us solve this (quite complicated) question. This is a multiple-step, higher-order, critical-thinking exercise and there are many ways to find a solution. Here's the process that makes sense to me.

      I know a gasoline engine burns fuel most efficiently at a ratio of 14.7 parts of air to 1 part of fuel, by mass (weight). This is called the stoichiometric ratio. Look it up if you want to know more. I knew this from my previous career in automotive diagnostics.

      There are the gas turbines!

      Now we can start assembling our data.

      FUEL weighs about 750 grams/liter  and  AIR weighs 1.3 grams/liter. These are standard numbers.

      750 ÷ 1.3 = 576  which means FUEL is 576 times heavier than the same volume of AIR or to put it another way, we will need a LOT MORE AIR by volume than we do fuel (to have the same mass of both of them). Another way to say it is Air is less dense than gasoline which we all know. Now for some more math.

      576 x 14.7 stoichiometric air/fuel ratio = 8467.2 or to simplify things, let's round this to 8500. A conventional car will burn 8500 liters of air per liter of fuel. Any car, at any speed.

      So how much fuel will we burn in a minute, or to be more precise, how many liters of fuel do we consume at 200 mph?

      Stay tuned; we'll finish this tomorrow. In the meantime, would you care to take the wheel?

      Wednesday, January 5, 2011

      Math and Reasoning

      Math reasoning means thinking about a question and using a process to solve it. Presumably there are numbers involved somewhere in there too. The term might suggest a specifically mathematical process of  reasoning, deduction, and inference. Alternatively, it can imply a mix of quantitative analysis and subjective intuition. Or an intentional problem-solving process called trial-and-error.

      Or maybe do we just skip the reasoning part and move directly to guessing?

      Today I am going to list some questions about mathematical reasoning.

      1. If mathematical reasoning does not involve numbers is it still considered "math"?

      2. Is mathematical reasoning useful to anyone being compelled to learn it?

      3. Is mathematical reasoning an measurable goal in school math classes?

      4. Are classroom teachers capable of teaching mathematical reasoning?

      5. Can mathematical reasoning be taught [or learned] by anyone?

      6. Do calculation skills lead to understanding and reasoning?

       7. Can practicing math help develop mathematical reasoning?

      8. Does "math anxiety" prevent or derail mathematical reasoning?

      9. Do cooperative/group activities enhance or dilute individual understanding and reasoning?

      10. Can calculators and computers increase or decrease mathematical reasoning?

      11. Why do students feel that mathematics is a totally foreign experience for them?

      12. Is knowledge of the context of a problem essential for mathematical reasoning?

      13. Must students construct their own "math knowledge" in order to truly "get it"?

      14. Is capacity for mathematics innate (you either have it or you don't)?

      15. Is elementary school too early or too late to teach mathematics to children?

      Whew! By just listing these questions it is clear to me that math is not only a science. Math and the way we pursue the process of math education rapidly becomes a worldview, a mindset, a philosophical position, a religion.

      A critical question comes to my mind as I am contemplating mathematical thinking and assembling this blog:

      Why do we always show a person holding their head or chin or chewing a pencil when we want to imply thinking? Is there a link between posture, gesture and rational thought?

       As in the The Thinker sculpture by Rodin

      Some day I may try to answer a few of these questions in light of our experience creating and selling elementary math curriculum. But not today.

      Tuesday, January 4, 2011

      How I personally use the math we teach

      I want to list some of the things in my life TODAY that are requiring me to use the skills we present to  elementary math classes with our Excel Math curriculum. And since we are talking about numbers, I'll make a numbered list instead of bullets.

      At this point I should remind you that in addition to numbers (pun) we also cover time, dates, calendars, reasoning and logic, problem-solving, geometry, etc. etc.

      Here we go:

      1. I am working on some use tax issues. I need to add the price of items I've purchased, get a sub-total, calculate 8.75% of the total, calculate 10% of the total, add those two percentages to the sub-total, calculate another 10% of THAT number, and so on.
      2. I'm executor for an estate. I'm required to keep track of all my time spent sorting out its affairs. Once I've done all the work I can multiply the hour total by an hourly rate and present my bill to the lawyers.
      3. Keeping track of my hours could be done easily with a software program. I checked the Apple App store to find one for my iPhone but it turns out there are hundreds available and most score 2 or less out of 5 on the rating system. I want a program that's rated better than 40% and I don't have time or energy to test them myself. 
      4. The assets of the estate include some gold and silver coins and jewelry. I have to get all these things appraised and then decide how to sell them to get the best return.
      5. In addition to the jewelry there are other assets in this estate. I did an inventory. I made a spreadsheet to track the items and get a total value that can be divided among the four heirs.
      6. We went on vacation over the holiday and I could add up all the receipts and see what we spent (or I might not).
      7. Because we went to 4 different countries, I have come home with dollars, pounds and euros. I'd like to know the total value of the cash I have left, so I'll need to count the money (and figure out which coins are which), look up the exchange rates for each currency, then do the conversions and add to get the total.
      8. The maintenance light on my car came on last night. I need to look at the mileage and check the owner's manual to see what tasks must be performed (and whether the work is scheduled to be done at a certain mileage or an interval of days).
      9. It's time to pay some bills, which requires logging onto my bank website, entering lots of numbers and letters, then selecting the bills, transferring the money, and making sure the funds get there on time (and there's cash available to pay them).
      10. I have an appointment later this afternoon that is 20 miles away. I need to calculate an estimate of the time it will take to get there, so I will know when to leave here and head out for the appointment.
      To be honest, I use math all the time, whether I am working on the math curriculum or not. None of this stuff is rocket science, but it uses all the tricks I've learned in math classes as a kid.

      And now it's time for me to go to that appointment!

      Monday, January 3, 2011

      Rock 'n Roll is here to stay; old guys and gals won't go away

      Welcome to 2011. A couple days ago I was filling in a form, and I got to enter 1/1/11. Never again in my lifetime!

      As the new year begins I was reading about 2010 concert tours. Apparently some of the musicians and entertainers made lots of money, while others lost out due to the economic distress in the US and European markets. Using only elementary math skills as taught in our Excel Math curriculum, here's a summary chart I constructed from the data:

      This table shows gross ticket sales in US$; blue for US sales and green for international sales.

      The industry seems to track sales trends by analyzing the top 50 tours.
      • 2010 ticket sales of $1.7 billion were down 15% from 2009's $2.0 billion
      • 2010 tickets sold total of 26.2 million was down 12% from 2009's 30 million
      • 2010 per-ticket prices averaged $65 were down 2% versus 2009's average of $67
      Not all numbers are going down. Age of performers is up and average age of audiences is up.

      I'm sure for some people these are distressing numbers. I don't go to concerts very often - perhaps a couple per decade. But these factoids impress me:
      • the average top touring star is 46 years old (average age of a top recording star is 29)
      • there are more touring performers in their 60's than in their 20's
      • only 2 top performers began touring in the last 20 years (Lady Gaga & Michael Bublé)
      • Leonard Cohen, Tina Turner & Cliff Richard are in their 70's; Chuck Berry is 84
      Finally, here's a pie chart to show you the distribution of ages of the top touring performers. Notice how many grey-hairs are at the top of the chart!

      I wonder what the trends are for readers of this blog? By the way, we now have had readers from 150 countries! Thanks for stopping by and visiting us at Excel Math.