Additional Math Pages & Resources

Wednesday, March 31, 2010

Math of Nuts and Bolts

You might think I made a mistake in the blog title. I hear you saying "It ought to be the Nuts and Bolts of Math." Not today. As you can see, these are real bolts (no nuts in this photo, though).

NOTE - Bolts may also be called fasteners, screws or other terms.

Bolts are identified in a number of ways. All using NUMBERS. We can describe the:
  • diameter of the bolt (thickness of the shaft)
  • length of the bolt
  • length of the thread portion
  • form or shape of the thread
  • angle of the thread
  • pitch of the thread
  • size and depth of the thread
  • type of head on the bol
  • handedness (rotation to tighten)
  • material
  • grade or strength

Only one bolt in this set is designed to screw into wood. It cuts its own threads as it is turned into a pre-drilled hole. All the other bolts are designed to screw into a matching set of threads.

Two of the bolts have fine threads while the rest have been cut with coarse threads.

Two of these bolts may be called socket cap bolts, or more commonly, Allen bolts. These are used in places where a regular wrench wouldn't easily fit.

Five bolts have a non-threaded portion at the top of the shank of the bolt, just below the head. That leaves space for a washer, or for the bolt to pass through and retain a separate piece of material.

Here are a few more. The color differences reveal the fact that some of them are of different material. One has no head, but has a collar in the middle. One has a conically-tapered head that fits a matching recess. Another has a domed cap and a square shape above the threads. Another has a lock washer, large washer and nut. Finally, one nut with a rounded bottom end is here too!

Here are some of the bolt heads. You can see that different tools must be used to turn them, and also some code markings on them. The markings reveal the strength of the bolts.

You can see that without a math education it would be very hard to distinguish between all these bolts!

They are everywhere - more than 200 billion are used every year in the USA.

In case you are wondering, No, Excel Math is not a hardware store. But our pal Joe accumulates lots of interesting bolts in his business, and he loaned me a few of them for today's blog. Thanks, Joe.

Tuesday, March 30, 2010

Can You Resist Playing Hookey?

Yes, I mean Playing Hookey, not playing hockey.

Hockey is a sport involving pucks and sticks. Hookey comes from a Dutch phrase that means hide and seek or an Olde English term Let's Hook it, meaning to run off and escape, or another phrase by hook or by crook.

Other similar terms are skipping, ditching or cutting. A stricter, more legal term is truancy. Truancy even sounds very serious, whereas playing hookey conveys a sense of youthful whimsy, rather than criminal evasion.
    Playing Hookey implies you are skipping out of school or work to do something else less responsible, like going fishing! Do you know Mark Twain's characters Tom Sawyer and Huckleberry Finn? Or can you recall that other famous character called Huckleberry Hound™? (Click to see Huckleberry deal with truant kids).

    You may be asking, where's the math in this? We find it in the aptly-named Play-Hookey Website. This site's purpose statement says:

    While you're out of the classroom, there's no reason why you can't learn a little something and enjoy yourself while you're doing it.

    This site is packed full of math, computer operation/repair/programming, electronics, physics, chemistry, and optics. The author explains resistors, capacitors, transistors, rectifiers, diodes, inductors, transformers, logic, etc. Don't run! It's not going to hurt you to learn something new!

    Here comes our math. Resistors are simple components that have the ability of resisting the electrical current flowing through a circuit. This property is measured using in units called Ohms.

    Here are some resistors. Because they are tiny, it's tough to print legible resistance values on them. So ages ago they began to be labeled with colored stripes.

    Let's look at the 5-band resistor in the chart below. Using the first three stripes, we create a 3-digit number. In this case it's 237. The third stripe tells us how many zeros to add ( 000 ). That makes it 237,000 ohms.  The fifth stripe tells us how close to its specification this resistor is guaranteed to be. In this case the tolerance is 1%. [You may also find 4 or 6-band resistors.]

    This resistor color code chart was created by Peter Veness at Bournemouth University in Dorset, England.

    While we were trying to play hookey today, we learned about math, color codes and electricity.

    Monday, March 29, 2010

    Analog, or can you lend me a hand?

    The last blog posting subject was the word DIGITAL. Today's is ANALOG (or analogue).

    Analog is derived from the term analogous which means proportionate, or similar - in other words, something that is similar enough to another thing that we can use it to make a comparison.

    Why is this math? We start teaching analog in the First Grade when we describe a clock that has hands to point out the time. Most do not have hands in the sense that this Mickey Mouse watch has hands, but they are similar to arms and hands. We call this an analog clock to distinguish it from a digital clock which has digits to indicate the time (and no hands).

    Here are a few sample analog clocks and watches. We provide an analogous grouping in First Grade.

    We often use the word analog to refer to things that are done in the old way rather than the new digital way. For example, a pencil might be analog and a computer keyboard digital. This is a messy way to use the words, but most people would probably understand what we mean. Shall we compare a few items?

    keyboard pencil
    solid state vacuum tubes (valves)
    switch rotary knob
    touch-tone rotary dial
    numerals hands
    digits moving needle
    on/off variable
    CD and DVD vinyl records
    looking through a window screen looking through an open window

    Here's a way to combine analog and digital to display the time:

    Some clever Swedish software guys figured this out. You need an array of 6 externally-controlled analog clocks for each digit. And it takes some fancy thinking to work out the controls!

    How many clocks are needed to show 24 hours, minutes and seconds?
    Do any of those analog clocks need to have 3 hands?

    Friday, March 26, 2010

    Digital Digital Digital (repeat 677,000,000 times)

    Digital is a favorite word nowadays.
    • My Google search turned up 677 million results.
    • My Bing search turned up 424 million results.
    What is digital? What does digital mean? 
    • Does it mean anything with a battery? 
    • Or anything that is programmed
    • Or anything that older people don't understand?
    The English word digital comes from a Latin word digitus meaning digit or finger. As in counting on your fingers rather than drawing with your finger in the sand.  As a part of speech, digital is an adjective. It modifies some other word. Here's one definition:  

    A digital system is a data technology that uses discrete (discontinuous) values whereas non-digital (or analog) systems use a continuous range of values to represent information.

    That doesn't help very much, does it? And since digital has been an English word for 400 years, it's certainly not the only definition. Here are some examples of digital communication that even old folks will understand:
    • smoke signals
    • Morse code - SOS
    • semaphore (optical telegraph) signals
    • Braille
    • ship flags
    • hand stuck out the window to signal a turn, etc.
    Many newer forms of communication are generally described as digital, because
    • they have an electronic, programmatic basis
    • they indicate values using digits (numbers, not fingers!) rather than a needle or graph
    • they can be read or manipulated or stored by a computer rather than the "naked eye"
    Facebook is one of those new digital communication mechanisms, and we at Excel Math have a Facebook account in addition to this blog and our website. Come on over and see us sometime.

    NOTE - When your first name is EXCEL and a large outfit in Redmond, Washington uses EXCEL for one of its favorite products, there are sometimes difficulties! Even if you have been around since the beginning of the digital era ...

    You might think digital means lightweight, portable, inexpensive, etc. For users that's often true, not necessarily for the techies who have to make everything work back in the home office. Here's a nice little digital communication system ... for the captain of a ship who wants to talk to people on shore without everyone else listening in.



    Thursday, March 25, 2010

    How do you rate?

    Rate is a math term that implies one measured quantity is being placed in relationship to or judged by a second measured quantity.

    For example, speed means distance traveled over a period of time. A common unit for rate of speed is mph or kph, which means miles or kilometers (measured by odometer) per hour (measured with a watch).

    Since speed is a very common term, we usually don't bother to say rate of when we speak.

    A "pulse rate" is the number of beats (counted) per minute (measured with a watch). Since measuring our heartbeat is very common, we usually don't bother to say "rate" when we speak, we just say pulse.

    Doctors, nurses and emergency personnel need to measure heartbeat all the time. And they may not have a computer or calculator handy. In the field, they don't even usually wait one entire minute. They count for 15 seconds and multiply their result by 4 to get beats per minute, or count for 10 seconds and multiply by 6.

    Another approach is to count a few beats, then look at a scale to tell what the rate would be per minute.

    Here's a watch that makes it easy for emergency medical personnel. Notice that this watch has a 4-ended second hand. That way you never have to wait more than 15 seconds for a hand to come along! I've never seen another watch like this.

    Instructions are indicated on the dial. When a second hand (in this case, the orange hand) comes by the pointer at the bottom (below 6 o'clock), you start counting respirations (breaths). When you have counted  5 breaths, the orange hand will be pointing at a number on the outer scale. That indicates the respirations per minute. This scale shows a respirations range of 10-60 a minute. Normal for an adult is about 15-20.  In this picture, my respiration rate was about 14.

    You do the same thing to get a pulse, but using the top and right side of the watch. Put your finger on a spot where you can feel a pulse. When a hand passes the top pointer (above 12 o'clock) you start counting. In this case, you count 15 pulsations, then look at the moving hand. If it's where this hand is pointing, it means 70 beats per minute. This scale shows a pulsations range of 50-180 per minute.  Normal for an adult is about 60-100. My pulse rate was about 70.

    As I said earlier, this watch is designed to save the wearer from having to do any math. It's easy enough to count 15-20 respirations in a minute, but much harder to count to 70 or 100 in a minute. So doing a 15 second pulse count and multiplying by 4 is the normal process.

    Remember that we started with the term rate. Other common uses of the word are with birth rate, unemployment rate, interest rate, exchange rate, inflation rate, and so on. And of course at this time of the year many of us may be thinking about the tax rate!

    Wednesday, March 24, 2010

    Push Button, Cars Stop, Emergency Vehicles Go

    Yesterday we discussed the button on a pole that pedestrians use at the crosswalk (zebra crossing). It makes the lights change sooner, and stops the cars a bit longer so pedestrians can walk across.

    Today we'll talk about a different button. A button in a bus, fire truck, ambulance or police car. It's called the traffic signal preemption button.

    For a bus, this magic device can HOLD a signal GREEN a bit longer, so buses move more quickly through heavy traffic. The bus system delivers a message to a sensor at the traffic signal. It's like shouting Hold that door! as you run towards an elevator.

    A slightly-different device is in emergency vehicles to speed their response to an accident or fire. If the signals in front of the vehicle are GREEN, the emergency vehicle can go more quickly and safely. This type of device CHANGES a signal from RED to GREEN. 

    Once the traffic signal has changed all other directions to RED, it gives the emergency vehicle a bright white light to show that all is clear. As soon as the traffic signal it loses its traffic preemption notice, the lights return to their normal pattern.

    How does it work?

    There are many systems. One uses bright strobe lights flashing at 10-14 cycles per second. Another depends upon an infra-red signal, like your TV remote. It's directed up and forwards towards the traffic signal. Another system uses a limited-range radio transmitter which isn't as susceptible to fog, rain or other atmospheric conditions that could block the visible or infra-red lights.

    A more complex model has the vehicle communicating with the lights, and the lights warning those along the route that a vehicle is coming. That takes real intelligence! It's covered in patent 7116245, in case you care to read up on it ... here's one of the diagrams.
    Yet another system relies on a GPS location signal emanating from the vehicle. It's covered under patent 7327280. With this approach, a central depot can see the vehicle moving along a screen and change the signals for the emergency vehicle. This system is also at the mercy of signals bouncing around through tall buildings, trees, etc.

    Note - Both these systems were invented by Aaron Batchelder, of CalTech in Pasadena, California.

    Traffic signal preemption is used at railway crossings too, to make sure all vehicles have cleared a set of railroad tracks and that no new vehicles are allowed to approach too closely. If possible, the signals allow nearby non-crossing traffic to keep moving, even "out of cycle" from the normal pattern.

    I know pedestrians are wondering - if you are anxiously pushing a button at a crosswalk, do you have to wait for the emergency vehicle?  YES. It takes priority over your request, as well as priority over the cars.

    You might be the impatient type who always wants a GREEN light, but I would advise you NOT to try to buy or build your own light-changing device. It is illegal for drivers or pedestrians to use them.

    Tuesday, March 23, 2010

    Push Button, Cars Stop, Pedestrians Walk

     It's time for you to confess! When you are walking along and come to an intersection, do you:
    1. run across
    2. wait for drivers to stop for you
    3. press a BUTTON on a pole, then wait for the signal to indicate it's safe
    4. don't bother to go to an intersection but cross wherever you want
    Here are a few of those BUTTONS, in case you aren't sure what I am talking about.

    There's some debate about what the BUTTON does. Some people think the BUTTON does nothing at all but give the pedestrian a good feeling. Others insist the BUTTON makes the signal change sooner, and others are certain it allows more time at that signal for a slow person to get across.

    What about repeated pressing of the BUTTON? Does pounding the BUTTON mercilessly make the signal come faster?

    In our city, pressing the BUTTON often starts lights flashing, buzzing and beeping tones, numbers counting down, etc. It definitely throws the signals OUT of synchronization. And it introduces a challenge to that huge moral point best articulated by Mr Spock, in Star Trek:

    "The Needs of the Many Outweigh the Needs of the Few or the One."

    What do you think? When it comes to pedestrians attempting to use crosswalks or zebra crossing, should the cars rush on, or stop? Should we who are in cars live by the contrary philosophy stated by Captain Kirk?

    The need of the one outweighs the needs of the many?

    I'll leave that up to you to decide for yourself, but sometimes when I am in a mile-long line of cars inching along due to the zealous after-school BUTTON-pressers, I feel less generous than I do at this moment.

    In case you were wondering where is the math?, try to calculate how much longer your favorite signal takes to change with a BUTTON pressed compared to not pressed. Then see if you can determine whether the signals near you were re-timed when daylight savings time changed!

    If you want to study this subject further, you can check out this site: Accessible Pedestrian Signals or you can study this patent for a solar-powered crossing system.

    Monday, March 22, 2010

    Five dollars here, ten dollars there - soon it's real money, Part 2

    In my last post I posed the question of what does it cost to donate to a charity using your mobile phone. Then I didn't give you the solution. I know it wasn't very nice to push the answers into a second post, but I didn't want the first one to be too long. And it gave you a chance to calculate an answer for yourself. But I didn't want to hold out too long, so here's the second post with the answers.

    NOTE: I'm happy to highlight two major contributions that the mobile phone carriers have made regarding Haiti earthquake relief donations. The 90-day holding period is set aside temporarily, so the funds can arrive sooner, and messaging charges are being refunded (or not imposed) on donors. These are excellent developments.

    Here are my results:

    One-Time Fees are $500
    Monthly Fees of $1000 must be paid for at least 12 months, so  = $12,000
    Outgoing promo campaign fees are 3.5¢ per message x 100,000 = $3500
    Transaction Fees on $5 are ( .035 x $5 ) = 17.5¢ + 32¢ = 49.5¢ x 5000 = $2475
    Transaction Fees on $10 are ( .035 x $10 ) = 35¢ + 32¢ = 67¢ x 5000 = $3350
    Message Updates 5000 x 4 messages/month x 6 months = 5000 x 4 x 6 x 3.5¢ = $4200

    Sub-Total $500 + $12,000 + $3500 + $2475 + $3350 + $4200 = $26,025 in fees

    5000 x $5 = $25,000  and  5000 x $10 = $50,000 for a Gross Income of $75,000

    $75,000 - $26,025 = $48,975

    26025 ÷ 75,000 = 0.347 or 35%

    The cost to the donor is $5 or $10 plus at least 5 text messages. Here's how it works: 
    1. Donor texts the donation word to the donation number.
    2. Donor receives a text back asking for confirmation; done by replying with YES.
    3. Donor sends YES. Donors cannot cancel after sending this YES.
    4. Donor receives text that the donation was successful. 
    5. Donor receives text asking for another YES if they want updates from the charity. If this message is ignored, everything stops here.
    6. Donor replies with YES for text updates. 
    7. Donor receives up to 4 messages a month from the charity.
    8. Donor may send a message at any time saying STOP. The charity must respond.
    9. Charity's final message says "We got your instruction to STOP".

    In our calculations, let's assume a user just gets the first 5 messages, and pays $1 for them in total. By this pricing assumption we will under-estimate the costs incurred by the half who want updates (10 or more messages), and over-estimate the cost to those who have a all-inclusive messaging plan. But this is enough detail for us today.

    Now we see that the real cost to our donor is $11. The charity receives the same NET INCOME amount no matter what a donor pays for messaging. We can create a new formula that includes the messaging costs in our gross fees and income. That changes the percentage of the total that goes to fees.
    10,000 x $1 = $10,000

    36025 ÷ 85,000 = 0.424 or 42%

    NOTE: Every calculation of this type is an approximation. The set-up fee is actually $399 but I rounded it to $500. We know that we assumed half the donors want text message updates but later made another assumption that they didn't (in order to accommodate an unknown number who may have free messaging).  Regardless of the amount of fees paid and to whom, this probably represents a large new chunk of income to the charity - some of which is used for its internal operations (
    marketing or legal or accounting costs) and doesn't go directly to the intended relief effort. Could you do better by slipping a $10 bill into the mail? That's another question!

    Five dollars here, ten dollars there - soon it's real money, Part 1

    It's becoming popular to use your mobile phone to send a small sum of money to a charity. It's fast. It's easy. You may have made one of these donations yourself.  And you might wonder, How does it work?

    An established charity registers with a processing service, such as mGive. This service company sets up an account for the charity, with a unique number/keyword combination for their campaign. Notice is broadcast to prospective donors, through advertising or text messages to phones.

    To respond, donors send a text message to a special number. That indicates they want to make a $5 or $10 donation. The phone carrier adds that amount to the donor's phone bill, marked as a donation.

    The mGive Foundation (TMF) collects this money from the carriers. Every 90 days, the foundation wires all money that has accumulated in the account to the charity (meanwhile, donors have paid their phone bills). If the charity disappears, TMF is obliged to give the funds to a similar charity.

    Interesting process, eh? There are safeguards along the way to reduce fraud and risk.

    Using only Excel Math elementary school arithmetic, can we calculate:

    1. How Much Does This Cost?  and  2. Who Is Getting The Money?

    Let's do a case study:

    We decide to set up a charity for astronaut relief in case a meteorite hits the moon or space station. We register with the IRS and over the next 12 months do everything necessary to become a legitimate charity. We set up an medium-level, text-donation account. Over the following 6 months, we promote our cause with 106,000 out-going promotional messages (we get 1000 a month free so we still have to pay for 100,000 messages).

    Our charity receives 10,000 donations. Half of them are for $5 and half for $10. Exactly half of these donors decide they want messages from us on a regular basis telling them how we are spending the money. We spend the next 6 months investing in our charity and telling donors how it's going.

    How much does all this cost?
    1. It costs around $500 to do paperwork, set up an account, get the keyword MOON, etc.
    2. There's an on-going fee of $1000 per month for a minimum of 12 months
    3. There's a 3¢ or 4¢ fee for each message sent out (over our allowance of 1000 a month)
    4. There's a per-donation charge of 3.5% of the gift 
    5. There's a per-donation fee of 32¢
    6. If we need a new disaster keyword like SPACE STATION, that's another $200 for a year
    7. There may be a charge to the donor for the text messages, of 20¢ or 25¢ each
    8. There may be programming and/or accounting costs for the us (the charity)

    By now, you are probably saying, "Show me the money!" 

    As the title of the blog indicates, when you get a few thousand of this and a few thousand of that, pretty soon it adds up to real money. Go ahead and do your math. I'm working on it from this end, and we can compare notes later today or tomorrow.

    Friday, March 19, 2010

    The CheckAnswer™

    The CheckAnswer™ is a special part of Excel Math curriculum. It is used in throughout Excel Math student lesson sheets for Grades 2-6. This process enables students to check their work, and verify for themselves that they understand the concepts in the day's Guided Practice or Homework.

    Here's how it works:

    All three problems in block A are shown here, with the work and the solutions in red. The results of problems 1, 2 and 3 are added in the bottom right corner of block A, and the sum of those numbers (2377) equals the CheckAnswer shown in the box at the top next to A.

    The first example in block B asks students to select an operation symbol. Since these symbols don't have a numerical value, they cannot be added for the CheckAnswer. To get around this, we show four possible choices and an arbitrary value for each. The + sign is the correct choice, and its value is 25, so 25 is part of the CheckAnswer sum. We use this wherever answers are symbols, true/false, yes/no, etc.

    (In case you are wondering, it is possible for a student to get multiple wrong answers which add up to the correct CheckAnswer. But it's not very likely. And when we insert numerical value choices, as discussed above, we make sure that a wrong choice won't make a correct CheckAnswer.)

    Remainders, fractions, decimals, money and time can be included in a CheckAnswer. When we build the Lesson Sheets we are careful not to co-mingle these with any other type of math in a CheckAnswer! We don't try to resolve remainders, or round seconds and minutes up or down. Notice the examples:

    Here are our original three blocks A, B, C with all the work and the answers provided. This is the view that the teacher would normally see in the Teacher Edition answer key.

    The Associative Principle tells us that it doesn't matter in what order the students do the problems, or in what order they arrange the answers when adding to get the CheckAnswer.  If any answers are wrong, the result is like this—they get the wrong sum and have to go back to recheck their work.

    A few people feel students can exploit the CheckAnswer. They say that students can just solve the two easiest problems, do a subtotal, subtract that subtotal from the CheckAnswer, and use the remainder as the answer for the final problem.

    Yes, perhaps this happens once in awhile. We still ask students to show their work. If a student is stuck, and creative enough to go through all the sub-totaling work, then able to reconstruct a problem solution backwards,  I think that demonstrates ability!

    Learn more about Excel Math and see sample lessons at

    Thursday, March 18, 2010

    Cubist Art Math?

    What are these? 

    They are a series of 11 figures (called nets) that when folded up become a cube. This kind of stuff shows up on math tests, where we ask students to tell us which geometric figure will be revealed if a shape is folded up correctly.

    I found them on a nice German website with a whole page dedicated to cubes. There many views and animations of cubes on this site, and it's worth a close look.

    The title of this blog uses the word CUBIST, which doesn't mean a person who favors cubes. It means a 20th-century school of art pioneered by Pablo Picasso and Georges Braques. Here's a print of a 1917 work by Picasso. My mother gave it to me years ago, and I rarely notice it anymore. It was created a bit after he gave up pure cubist art, but you can get the general idea.

    Picasso sometimes made collages of sketched or cut-out cubes, and assembled them into images. Architects have done the same thing - assembling cubes in various ways to make buildings rather than paintings. Here's a new museum designed by I.M. Pei in the Cubist vein:

    The Geisel Library at UCSD is a prime example of cubes and brutalist architecture - where the concrete structure predominates. It's a group of cube shapes held up in the sky by massive arms.

    I went around our offices today and took some pictures. Our office is neither cubist nor brutalist. But it turns out there are only a few squares or cubes in our building - most of the shapes are rectangles and diagonals. We'll investigate WHY in another blog.

    Above you see the surface of the loading dock; below are the windows in the warehouse.

    The following are our water bottles in crates, waiting for their turn to be used.

    And finally here's a view of the trusses under our warehouse roof.

    If all this talk of cubes leaves you cold, here's a nice image to end with.
    (I'd like to shoot it straight on, but the glass is too reflective.)
    I bought this a decade ago in England and carried it home.
    It's a multi-media collage of architecture prints, ink lines, paint, and wax. 

    Le Pavilion II by Kevin Blackham

    Wednesday, March 17, 2010

    Box and String, Part 2

    I'm going to carry on with the theme from yesterday (but the boxes are green for St. Patrick's Day).

    The length of the string is different in each of these three cases. I think we could probably experiment with boxes (or a drawing) and eventually we could prove that the string is the same length in all 3 cases ONLY IF the box is a cube - all three sides are of equal length.

    We would have to prove that:

    (2D) + (2W) = (2H) + (2W) = (2H) + (2D)

    First we would divide everything by 2 to simplify it the formula. A cube's side must then be consistent with this:

    D + W = H + W = H + D

    Let's try it with our existing box. If we have sides of 1, 2 and 3, the formula's results give us

    2 + 3 = 1 + 3 = 1 + 2 

    BUT when we solve it, 5 ≠ 4 ≠ 3  This box with unequal lengths doesn't fit the formula.

    If we use a cube with each side 2 units long, then the results are like this

    2 + 2 = 2 + 2 = 2 + 2 or 4 = 4 = 4 

    That works.

    Here's a flattened cube.  


    And a rotating one.

    Tuesday, March 16, 2010

    Box and String, Part 1

    Math is not just calculations. It is often thinking and THEN calculations. Then some more thinking.

    Here are 3 boxes. The dimensions of the boxes are the same - 1 high, 2 deep and 3 wide. A string or ribbon is drawn around the boxes, but in each case it goes a different way around.

    However, since the string goes around the whole box each time, could the length of the string remain the same? How would you calculate the length of each string?

    How would you calculate the length of each string? Besides measuring it!

    We could try creating a formula or writing equations.

    We have H, D and W dimensions to worry about. In each case the string goes across two different sides, twice.  So there are 4 lengths of strings added together. Or two lengths multiplied by two.

    To make sure we don't forget anything, we'll include the third side in each formula, but multiply its length by zero (the product will equal zero and not affect our answer).

    The string's length on box 1 is the sum of (0 x H) + (2 x D) + (2 x W)

    The string on box 2 is the sum of (2 x H) + (0 x D) + (2 x W)

    The string on box 3 is the sum of (2 x H) + (2 x D) + (0 x W)

    Now solve!

    Click Here For The Answers.

    Why would you care? Well what would happen if you bought tape for a busy company, and one of your packers wrapped the tape around as shown on box 2 (above), and the other wrapped as shown in box 3?

    Even if they did the same number of boxes, they would use vastly different amounts of tape! Think how much sleep would be lost by the Supreme Warehouse Commander in your company! Our Supreme Warehouse Commander is very precise about this sort of thing.

    Just putting the tape gun sideways on the box for the photo might earn me a "What are you doing there?" Could he have been trained as an engineer?

    Monday, March 15, 2010

    Mental Math

    Excel Math creates math curriculum for kids who are about 5-12 years old. We've been in this business for 30 years. One of the original features of our curriculum is called Mental Math. It's a tool for helping kids (or adults) to improve with simple math calculations. Download a copy of it if you want to have a look.

    Mental Math starts out being quite easy and progressively gets harder. Each page has 12 columns of digits, separated by dotted lines. Here's how it works.

    1. Choose a page from the selection of three pages.
    2. Draw a line under one of the digits on the page. They are in groups that total 10, 20 and 30.
    3. The example shows a red line - above it the numbers total 15.
    4. You call out digits above the line, so students can add a running total in their heads.
    5. When you get to the line, ask for the answer.

    Note: If you chose another column, the same horizontal line will give you totals of 13, 15, 17, or 15.

    Here's part of the third page, where the totals go to 100.

    There's nothing magic about mental math. Except the fact that with some practice, we can do these things in our heads instantaneously.

    Last week I went to the store and bought 2 items at $1.99 each. The tax rate is 8.75%.

    I knew the price would be about $4 for the items and about $.15 for the tax. I gave the cashier a $5.00 bill. She handed me back $1.84 cents. I took the change and handed back the dollar. She offered me the dollar again - I refused, saying It's not my dollar. It's yours.

    She looked at the cash register. It said $0.84 on the screen. But now she had a dollar in her hand. She looked at the dollar, at the screen, at me. She counted out $.16 and offered it to me. I refused it.

    She blushed, we all laughed and I walked away. The customer behind was getting irked...

    Will the cashier be insulted if I give her a copy of mental math next time I go to the market?

    Friday, March 12, 2010

    Which is lower and why?

    Savings rates indicate many things. Not just how much money people are putting away. What can we learn about savings with only elementary math to rely upon?

    Saving rate is estimated by subtracting household expenses from disposable income

    Disposable income is income from employment and personal businesses, plus interest, dividends and government benefits -- minus any income taxes and social security withholding.

    Household expenses are cash spent on consumer goods and services and rent (or cost of a home).

    Because saving is a remainder calculated by subtracting expenses from income (both of which are estimates) savings estimates are relatively inaccurate.

    Household saving rates vary between countries - partly due to differences in how health care, unemployment and retirement are handled in each country. The older the population, the less money is saved. Retired people are not growing their savings, but depleting them.

    Saving rates have been stable or rising in Austria, France, Italy, Norway and Portugal. They have been falling in most other countries, such as Australia, Canada, Japan, the United Kingdom and the United States (until very recently). 

    Some countries have Negative Saving Rates – which means current household expenses exceed their income. This group includes countries like Australia, Denmark, Greece and New Zealand.

    Who cares about this? The Organisation for Economic Co-operation and Development cares.  This group compares statistics across the most developed countries (the G7). In a fascinating research paper released this week, they found savings rates change due to these factors:
    • Interest rates go up – savings in most countries increase (except in the UK and US!)
    • Inflation increases – savings go up 1.4-1.8 times as much as the reported inflation rate change
    • Government debt increases – savings go up slightly in the US and France (fear for the future?)
    • Unemployment increases – savings change inconsistently; withdrawals by the unemployed are sometimes but not always offset by increased savings of those who fear losing their jobs
    • Housing prices increase – the savings rate may decrease as house prices get higher (US, France, Italy, UK) but in most other countries this does not happen
    • Stock ownership increases – as families buy more stocks they save less and they risk losing those investments (in 2000, households in OECD countries had 17% of a year's income in stock markets)
    I think that's enough of that. The real question is who saves the least, and why? Americans save the least of all the big countries.

    We have saved the least for many years. Was it our rising home prices? Our investments in the stock markets? Now that both sectors have crashed, what happened?

    We are saving more for a rainy day. Are we convinced the rainy day is here? If I knew that, I probably wouldn't be writing math textbooks for a living ...

    Thursday, March 11, 2010

    Which is higher and why?

    Most people are very interested in taxation. Or perhaps I should say we are interested in reducing or avoiding taxes when they are applied to us!

    Many countries use a taxation structure called VAT (value-added tax). VAT is paid by companies who create products, all the way up through the production chain. It is INCLUDED in the sales price of most products, by law.

    VAT was first adopted in France in 1954. VAT was created to help avoid a taxation problem: when obvious taxes exceed 10% of the cost of an item, citizens increasingly seek creative ways to AVOID, REDUCE or EVADE taxes -- by smuggling, lying, "forgetting" and so on.

    NOTE: Tax evasion is illegal, while other activities related to avoiding and reducing taxes are usually legal.

    In the USA we have favored local SALES TAXES. These taxes on goods and services are levied by local governments to pay for local needs. They are added ON TOP OF the listed sales price, so they are easily seen (and resented). Those taxes are approaching 10% in many places.

    Do you want to AVOID or REDUCE your sales taxes? Join the crowd. There are thousands of books, websites and advisors promising to help you save sales taxes, income taxes, capital gains taxes, corporate taxes, etc.
    1. Go to Delaware, Montana, New Hampshire, Oregon or (most of) Alaska - they don't charge sales tax. But your own state might want to charge you if they know what you are buying is going to be brought back home. This loophole-plugging scheme is called USE TAX.
    2. Go to Hawaii. They don't charge sales tax either. Instead they charge EXCISE TAX on almost everything. But a tax is a tax, and you won't save anything there because the excise taxes are high.
    3. Buy on the Internet. Most Internet retailers do NOT collect sales tax if the buyer is outside the state where the Internet retailer is located. If they do have a facility in your state, then they should charge you your local sales tax. Look carefully before you buy. Shipping may be more than the cost of your local tax, and your state may charge you USE TAX if they catch you.
    4. Buy from a retailer who says "We'll pay the sales tax!" This is usually a furniture store or some other business that discounts the price of their products to offset the tax.
    5. Buy overseas. You can reclaim the VAT that you pay in-country, if you can prove that you will be using the item back in your homeland. Remember you may have to declare the purchase to US Customs, may have to pay DUTY on that item, and/or may have to pay USE TAX to your home state.
    6. Buy things that aren't taxed. In many places you can buy food, clothing and books with no or reduced sales taxes.
    7. Don't buy anything.
    Here's a map that shows average state sales taxes in the United States. There are approximately 7000 different sales tax rate-setting districts in the USA! A few database companies collect and re-distribute these figures to accounting systems and cash registers.

    Canada uses a complex system of taxation depending on the province. Taxes include GST (national VAT), HST (harmonized system) and/or PST (Provincial tax). I don't want to try to explain it today.

    Here's a chart that shows average VAT taxes around the world. It appears that Iceland has the highest VAT in the world, at 25.5%. But several other countries are right behind, with VAT at 25%.

    Who's taxed the least? That's very hard to say. It depends on what you eat, what you drink, how much you drive, and where you go, whether you are a tourist or a local, etc.

    Who's taxed the most? That's also hard to say. It depends on what you buy, how much you make, whether your income is earned from a job, derived from savings, or investments, an inheritance, etc.

    Have you ever wanted to INCREASE your taxes? I can't help you. There are no books on that ...

    In Excel Math we teach kids to calculate sales tax. We don't discuss reducing, avoiding and evading.

    Wednesday, March 10, 2010

    Which is easier and why?

    Here's a problem adapted from one at the end of fifth grade Excel Math curriculum.

    It's not too difficult, but the fact we have to show more than one solution process illustrates a troublesome math situation.

    Are there multiple ways to correctly solve a problem?

    The answer is almost always Yes.

    We may be taught to solve for answers in one or two different ways. Sometimes kids may discover additional methods on their own, or in a team where they learn techniques (or just get answers) from their peers. Once in a while a student will turn in a paper, with no evidence of how the answers were obtained. We hear teachers say,

    I can't give credit for this; it's not right unless you show the work!

    Then some students reply,

    I don't know how. I just know this is the answer. I can see it.

    They can see it?

    The other day when I was researching the blog on Hundred, I read about the Abipones,  a tribe of native people in South American 200 years ago:

    The long train of mounted women was surrounded in front, in the rear and on both sides by countless numbers of dogs. From their saddles the Indians would look around and inspect them. If so much as a single dog was missing from the huge pack, they would keep calling until all were collected together again. I have often wondered how they, without knowing how to count, could tell at once, despite the huge throng, that one dog was missing.

    We normally use the term Number Sense to describe knowledge about numbers and counting. But in this paragraph, Karl Menninger describes a scene where native peoples clearly had a highly-developed sense of quantity without using numbers (as we understand numbers).

    Difficult-to-explain math abilities may be seen in autistic people, and in math whiz-kids. They solve problems so easily that they don't know how to explain it. My wife often sees that at her school.

    I can do some problems this way, but because of my job I've learned how to explain the process(es).
    I'm not in a classroom, so personally, I don't quarrel about the method or showing the work. I'm happy to see a (correct) answer.

    Here are 6 different ways I created to solve the problem:

    Yes, it's useful to know HOW to solve a problem. OK, it's nice to SHARE the PROCESS with others. DOCUMENTATION is important. Written PROOF is desirable.

    But sadly, we have introduced an uncomfortable philosophical question:

    Is it the journey (process) or destination (answer) that matters most in math?

    Tuesday, March 9, 2010

    Which is harder and why?

    Which is harder for you to solve - the first, or the second?  Again, the first or the second?

    Which is harder for you to solve - the first, or the second?  Again, the first or the second?

    What am I asking you? And why? I'm asking about your perception of the difficulty of two math processes that are taught to elementary school kids. Subtraction in the first case, division in the second.

    What makes a problem difficult? We judge them in several ways:
    • My examples (subtraction, division) are what you might call negative operations, in that the result (remainder or quotient) is smaller than a number you started with.
    • A positive operation (addition and multiplication) gives you a result (sum, product) that is larger than a number you started with.
    The negative nature of subtraction or division does not in itself make a problem difficult, even if it might seem psychologically less pleasant than adding. Notice:   

    I have ten dollars. I am given ten more dollars. I now have twenty dollars. Yay :-) 

    seems a bit nicer than

    You have forty dollars. You have to pay ten in taxes. Now you have thirty dollars. Boo :-(

    It's not this negative nature, but the deficit spending component of some problems that makes them harder. You might have to borrow (or carry, or regroup). 

    Some problems require no borrowing. Let's look at $6.77 - 2.67

    a. 7 minus 7 is 0, 7 minus 6 is 1, 6 minus 2 is 4 = $4.10

    Some problems require a lot of borrowing. Look at $93.61 - 41.95

    a. 1 minus 5 is [Hold it! I need 10 cents. Hey 6 can I borrow from you?] now 11 minus 5 is 6.  
    b. Now 6 minus 9 [Wait, the 6 is a 5; I need a  dollar. Hey 3 can I borrow from you?] now 15 tens minus 9 tens is 6 tens. 
    c. The 3 dollars is 2 dollars [due to our previous borrowing], minus 1 dollar leaves 1 dollar. 
    d. Finally, 90 dollars minus 40 dollars is 50 dollars = $51.66

    Some are low levels of multiplication (6 into 7 goes 1 time with 1 left over) and others a higher level of difficulty (6 into 42 goes 7 times with 0 left over).

    These are still easy. Let's go up a notch. More remainders, even in the answer.  More decimal places in the dividend. Decimals in the divisor....

    Do you wonder where financiers and politicians learned about borrowing? We taught them in school.