Mother of all Teachable Moments, Part III

Right now (November 10 2008) the canvass process is going on; this means that the outputs of the mark-sense machines are being added up. The actual ballots are not looked at yet. the mark-sense machines are not networked (probably a good thing) so the results have to be read off of counters and typed into a data base. Also, some absentee ballots are being entered but others are not because of disputes.

This is a canvass, not a recount. The canvass will surely show that a recount must be done.

The difference between Coleman and Franken is now about 200 votes out of 3 million, with Coleman ahead.

The disputes will be resolved later. Also, later, ballots will be examined to see if the intent of the voter is clear from the marked ballot even if rejected by a machine. Later, I will send you examples of mis-marked ballots from the Star Tribune.

Here is about 50 minutes audio of Mark Ritchie, The Secretary of State of Minnesota from Friday November 7, 2008.

http://minnesota.publicradio.org/www_publicradio/tools/media_player/popup.php?name=minnesota/news/programs/2008/11/07/midday/midday_hour_1_20081107_64

Minnesota Senate Election

Answers below:

The 2008 Senatorial Election in Minnesota was predicted to be a "statistical dead heat" and has turned out to be nearly a tie.

The results, according to the Minnesota Secretary of State, http://www.sos.state.mn.us , are:

Coleman, Republican: 1 211 538
Franken, DFL : 1 211 196
Barkley, Independence: 437 376
Minor Parties: 25 164

The eligible voters are: 3 741 514 (not all eligible voters are registered). Minnesota State Law mandates a recount if the difference in the two leading candidates is less than 0.5% of all votes cast. The losing candidate may waive this if he or she does so in writing. I don't think Franken is likeley to do so (although Coleman has suggested it).

Questions: is an automatic recount mandated in this case?

What percentage of the total votes cast did each of Coleman, Franken and Barkley get? Express as a percentage, with three decimal places, for example, 43.xxx%

Compare to pre-election polls, which gave Franken 43%, Coleman 36% and Barkley 16% with a sampling error of plus or minus 5%.

If you go to the Secretary of State website and look at the Recount Manual, you will see (page 8) many examples of mis-marked ballots. If the intention of the voter is clear, such ballots, which optical scan machines will not count, can be counted by hand. In my experience, about 1% of student multiple choice exams are defective. If that is true in this case, how many mismarked ballots might there be?

What percentage of eligible voters actually voted?

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The total number of voters was: 2 885 274
The difference between Coleman and Franken was 342 votes, this is o.012 percent, less than the 0.500 percent required for a mandatory recount.
The percentages were Coleman 41.991%
Franken 41.978%
Barkley 15.158%

Polls predicted Franken would get 38 to 48%, Coleman 31 to 41%, and Barkley 11 to 21%. The sampling error margins are approximate, Coleman was at the high end of the predicted range, the other two candidates within the predicted range.

There might be about about 29 000 mis-marked ballots.

77% of the eligible voters actually voted. The percentage of registered voters is higher, since not all eligible voters are registered.

Travel with $4 per gallon gas

A true story, August 2008.

A family drove from the Twin Cities to Yellowstone Park and return, possibly with some side trips. The family spent $475.90 on gasoline. If they paid $4 per gallon, how many gallons did they consume?

If their car traveled 22.5 miles per gallon, how far did they travel?

The round trip distance from the Twin Cities to Yellowstone Park and return is about 2040 miles. Compare your result to this distance.

Answers: CXIX gallons (rounded to the nearest gallon).
MMDCLXXVII miles (rounded to the nearest mile, and greater than 2040 miles, telling us they made 600 or 700 miles of side trips).

Speeds in meters per second

A typical walking speed is two meters per second. Probably the fastest speed for running (by people) is about ten meters per second -- 100 meters in 10 seconds.

An American Football field is 91.44 meters long, from goal line to goal line. At two meters per second, you can walk that far in 45 or 46 seconds.

Timing yourself with a watch with a second hand, how long does it take you to walk 91.44 meters?
(Mathematical experiment.)

The distance between bases on a baseball field is 27.432 meters. You can time a batter from home plate to first base to estimate the batter's speed. Somebody stealing second base might not start from first base, but if a player "tags up" you can time between other bases.

Walking in kilometers

Today, we walked 4 kilometers (4000 meters) in 33 minutes and 27 seconds. What was our speed in meters per second? Hint: convert minutes to seconds, add 27 to get seconds. Divide to get meters per second.

Answer: very close to 2 meters per second.

Is this reasonable: make chalk marks, 2 meters apart. Start walking at one-Mississippi, hit the first chalk mark at two-Mississippi, can you hit the second chalk mark at three-Mississippi?

Walking in kilometers

Yesterday, it was raining. Umbrellas slowed our walk. It took us 36 minutes to walk 4 kilometers, but these numbers should make computation easy. What was our speed in kilometers per hour?

Hint: convert 36 minutes to fraction of an hour, then divide 4 kilometers by this fraction to get kilometers per hour. Dividing by a fraction divides the mathematicians from the non-mathematicians, so if you have any trouble, let me know.

Answer: Between VI and VII kilometers per hour.

In new Minnesota Science standards, all computations are in meters, kilograms and seconds. What was our speed in meters per second? Hint: convert 4 kilometers to 4000 meters and 36 minutes to seconds, then divide. You will get an answer that is roughly 2 meters per second.

This is advanced, but the exact answer is 50/27 meters per second. 50/27 has a repeating decimal expansion, 1.851 851 851 you can write as 1.851 with a bar over the "851"

You could lay out 2 meters on the ground and see if you can walk that far in a second. Elementary-school students may need to jog. Start on two-Mississippi and see if you finish by three-Mississippi.

Wind speeds in Canada are given in meters per second and in the United States, in knots (nautical miles per hour). It is handy that the conversion from one to the other is close to "two;" double meters per second to get knots (approximately).

Beginning Distance, Speed, Time problems

This morning, I walked 2.4 miles in 34.5 minutes. What was my walking speed in miles per hour?
Round your answer to nearest tenth.

Hint: Convert 34.5 minutes to fractional hour by dividing by 60 (60 minutes per hour). For example, 30 minutes is 0.5 hour.

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Answer: faster than IV MPH, not as fast as V MPH.

The $600 Economic Incentive Payment

The Federal Government is sending many taxpayers a $600 check to boost the economy (May, 2008).

One writer, who drives a 1984 Volvo station wagon that gets 18 miles to the gallon, thought about spending the money for gasoline. He lives near San Francisco, CA, where gasoline is very expensive. He pays $4.19 for one gallon. All taxes are included in that $4.19.

To the nearest gallon, how many gallons could he buy?
To the nearest mile, how many miles could he drive with that gasoline?
Just for fun, can you find someplace he can drive to, and return, from San Francisco with that much gasoline?

Answer here:

Word Problems: Hands-on Equations

At Seward Elementary, a public elementary school in Minneapolis MN, a substantial number of 3rd-grade and 4th-grade students are doing linear algebra using the Hands-On Equations method of Henry Borenson, Ed.D.

Lately, we have been doing word problems. Here is a link to Dr. Borenson's site with 11 verbal problems; after you get there, click on Verbal Problems at the top of the page: click here:

In accordance with our theme of money problems, I repeat #11 here:

Curly, Larry and Moe each have a collection of silver dollars. Curly has 5 more coins than Moe, and Larry has double the number of coins that Curly has. The sum of the number of silver dollars in the entire collection is 35. How many coins did each person have? Extra credit: who were Curly, Larry and Moe?

In Borenson's pictorial notation:

Moe M
Curly M + 5
Larry M + 5 M + 5
35

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Solution M = V; Curly = X; Larry = XX

Unit Price: AA Alkaline Cells

The unit price is the price for one item. You find it by division and, perhaps, rounding.

Seward uses quantities of AA "batteries" to power graphing calculators. Strictly speaking, a single AA "battery" is a cell, 4 of them together form a battery. Today, at Home Depot, Energizer brand AA cells are for sale at the following prices:

36 cells for $12.89
20 cells for $10.47
10 cells for $6.87
6 cells for $4.77

Assuming you can use all the cells you buy, what is the price per cell, the unit price, for each package? Round results to the nearest cent.

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A note on standard tests. Standard tests often present a set of prices like the cell prices, and ask: "What is the lowest price per unit?" Often, but not always, the largest quantity has the lowest unit price. You should calculate all the unit prices, then pick the lowest one. Guessing is not a good idea. It isn't obvious, but the typical unit price problem requires four divisions, not much work if you have a calculator.

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Answers: 36 pack, XXXVI cents per cell, 20 pack, LII cents per cell, 10 pack, LXIX cents per cell, 6 pack, LXXX cents per cell.

L= 50, X = 10, V = 5.

Deep Discounts in a "Dollar Store"

In the United States, there are "dollar stores." Everything in these stores has a price of one dollar.

Teachers often go to these stores to buy extra school supplies for their students.

For example, a store might sell rulers in a package of 3 for one dollar.

You should beware of really deep discounts, but I saw a sign in a dollar store that was closing: "Everything must go, 75% off!"

So, I bought 4 packages of rulers, indeed the checkout clerk applied the 75% discount to each package, multiplied the discounted price by 4, and added 6.5% sales tax. The rulers were perfectly usable in school: they have centimeters and millimeters on one edge and inches on the other.

To the nearest cent, what was the price of one ruler?


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Answer: IX cents each. Alas, went I returned to the store to buy more, it was closed forever.

A Cab ride in Chicago with a Friend

Cab fares in Chicago work like this: when the ride starts, the driver drops the flag on the meter. An immediate charge of $2.25 appears. This charge covers the first 1/9 of a mile. For each additional 1/9 of a mile, the meter adds 20 cents. Hence, for miles after the first, the charge is $1.80 per mile. (What is posted in the taxi is: "$2.25 flag drop, $1.80 each additional mile.")

There is a flat (doesn't depend on distance) "surcharge" for each additional passenger.

The distance from your hotel to The Art Institute of Chicago is 1.38 miles. You could walk that far in about 20 minutes, but it is raining and you decide to take a cab. You take your friend along (one additional passenge). What is the meter charge when you arrive?

Hints: for the first mile, you are charged $2.25 plus 20 cents for each of the remaining eight ninths of a mile. You can then convert 1/9, 2/9 etc. into decimals and find how many ninths you have been charged for when you reach 1.38 miles.

You must add the passenger surcharge to this.

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Answer: V dollars and XLV cents.

Roman Numerals: V=5, L=50, X=10. XL is the same as X + X + X + X, or L minus X.

A Cab ride in Chicago

Cab fares in Chicago work like this: when the ride starts, the driver drops the flag on the meter. An immediate charge of $2.25 appears. This charge covers the first 1/9 of a mile. For each additional 1/9 of a mile, the meter adds 20 cents. Hence, for miles after the first, the charge is $1.80 per mile. (What is posted in the taxi is: "$2.25 flag drop, $1.80 each additional mile.")

The distance from your hotel to The Art Institute of Chicago is 1.38 miles. You could walk that far in about 20 minutes, but it is raining and you decide to take a cab. What is the meter charge when you arrive?

Hints: for the first mile, you are charged $2.25 plus 20 cents for each of the remaining eight ninths of a mile. You can then convert 1/9, 2/9 etc. into decimals and find how many ninths you have been charged for when you reach 1.38 miles.

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Answer: IV dollars and XLV cents.

How Many Miles Per Year Do I Drive?

I first drove my car on August 30, 2006. I suspect that I drive less than 10 000 miles per year. I notified my insurance company of this and they reduced the price of my car insurance.

The insurance company advised me to check how many miles I drove from time to time.

On March 8, 2008 the car told me that I had driven 15 000 miles and I needed an oil change. About how many miles do I drive per year?

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Hints: this is an estimation problem. You could, for example, assume that a year has 12 months, each of 30 days, and even assume that the the 15 000 mile notice appeared on April 30, 2008. Then get the miles per month and multiply by 12 to get miles per year.

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One approximate answer: DCCL miles per month times XII.

A Tax Question

The school district I live in just had a vote to increase taxes to pay for better schools.

Before I get to that, consider a simple example. Suppose 20 people vote yes and 30 people vote no.

What were the respective percentages? You need to get the total first! In total, 20 plus 30, or 50 people, voted. The percentages were 40% "yes" and 60% "no."

In real life, 4702 people voted "yes" to increase taxes and 3554 people voted "no" to the tax increase. What were the percentages of yes and no votes?

There was a second question on the ballot, to increase taxes even more to cover improvements in education and to cover inflation. To this, 4092 voted "yes" and 4150 voted "no." What were the percentages of yes and no votes?

Was the vote on the second question close? Did the same total number of people vote on both questions?

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Note: V=5, X=10, and L=50. In Roman numerals, if a single smaller unit precedes a larger unit, you subtract. So XL=40.

Vote totals: First vote, 8 thousand + CCLVI; second vote 8 thousand + CCXLII
Percentages: rounded to nearest whole percent, Yes LVII per centum, No XLIII per centum.
Second vote: Yes L per centum, No L per centum, but No won by LVIII votes out of about 8 thousand cast; the Romans didn't use decimals but the vote lost by about VII tenths of one percent.



Hint: this is a favorite question on standardized tests. The test doesn't tell you to, but you must add the votes to get the total first (!) then calculate the percentages.

posted by Stephen Margolis @ 6:15 AM

Another Taxi Story

In the Twin Cities, taxi fare works like this: as soon as the passenger gets in, the driver drops the flag and $2.50 appears on the meter. Every mile, including the first mile, costs an additional $1.90. Fractions of a mile are 38 cents for every 1/5 of a mile.

My car wouldn't start and I needed to take an urgent trip. Using Mapquest, I found the trip was 15.6 miles long. What was the likely fare, including a 20% tip? Round the fare plus tip to the nearest dime (10 cents).

I found out in advance that the taxi company accepts credit cards. But, if not, could I have paid with $40 in cash?

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Answer: XXXVIII dollars and LX cents. I could have paid with two twenty-dollar bills.

Note: V=5, X=10, and L=50.