Cucumber Day in Calculus, 2014 Edition

FracTad's Fractopia

If I’ve used an activity for three years in a row, I guess it’s a tradition! My calculus students have done this one for several years now, and it’s always been a hit.

To introduce the concept of volumes of rotated solids, I have my students slice cucumbers into disks and measure the volume of each slice. Then they add them up to approximate the total volume. It’s a nice way to help them visualize objects in 3D. You can read about what we do in a couple of earlier posts, here and here.

Here are the basics of the lesson:

1. Review the formula for the volume of a cylinder:

Volume Formula

2. Pair up the students, and distribute the cucumbers.

I used small seedless cucumbers that came six to a pack, and were relatively similar in size:


Have a discussion on whether we can use the cylinder volume formula to…

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It’s Time to Connect With John Wesley


Disconnect-cover InsideOut Music  recently signed John Wesley to its label, and his new album, Disconnect , will be available March 31 in Europe and April 1 in the U.S.  I’m not pulling an April Fools’ joke when I say that it is my favorite album of 2014 so far (despite stiff competition from  the likes of John “KingBathmat” Bassett , Gazpacho , and Transatlantic ).

Who is John Wesley? Hailing from Tampa, Florida, he’s an enormously talented guitarist and vocalist who has toured with Porcupine Tree, Fish, and Steven Wilson. Check out Porcupine Tree’s DVD, Anesthetize, to see how integral he was to their live show. As a matter of fact, after watching that DVD, I wondered why Steven Wilson didn’t go ahead and make Wesley an official member. His guitar playing and vocals added a new and exciting dimension to Wilson’s songs.

Approaching Wesley’s new solo work, I had low…

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Integrity’s Minstrel: John Bassett. Unearth (2014)


Unearth-Album-Cover A review of John Bassett, Unearth ( Stereohead Records ; release date: March 31, 2014).

I’m honestly not sure if my admiration for John Bassett knows many—if any—bounds.

When we first announced progarchy’s birth in the fall of 2012, Kingbathmat’s label reached out to us immediately.  As objective as I’m trained to be in my own actual day-to-day profession (though, I’ve become firmly convinced that so-called objectivity is highly overrated), it’s hard not to be grateful when someone, some band, or some label contacts us.  After all, it’s automatically a profound sign of trust, though always based on a leap of faith.

As reviewers and lovers of music, we’re, of course, not for sale.  Still, we are rather human.  Kindness and relationships make a difference in the ways we perceive artists.  In no genre of music is this more true than in prog, as the audience matters so deeply to…

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The Five Biggest Myths About Secession


With the ongoing tension over Russia’s annexation of Crimea, now would be a good time to talk about the biggest myths people believe about the origins of secessionist movements around the world (even though Crimea is a case of irredentism not secessionism).

  1. Myth: Secession is contagious. Back in the 1990s, journalists worried a lot that the disintegration of the Soviet Union, Czechoslovakia, Yugoslavia, and Ethiopia heralded some broader worldwide trend toward the splintering of the state. Even some scholars indulged talk of our “neomedieval” future of microstates. Now, with secession referendums in Scotland and Catalonia on the docket, a secessionist party gaining support in Quebec, that online referendum in Veneto, and recent events in the post-Soviet space, I’m seeing similar questions about whether this is a new “trend.”

    Fact: Secession happens because of particular circumstances, not contagion. Scholars have looked at the evidence every which way, and in no…

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A Couple of Reading Recommendations

FracTad's Fractopia

I’ve been on Spring Break this week, and we’ve stayed home this year, which has been nice. I’ve had a chance to catch up on some reading, and I really enjoyed two books in particular – one nonfiction, and one fiction. Both will appeal to teachers and lovers of mathematics.

First, the nonfiction book: Amanda Ripley’s The Smartest Kids in The World and How They Got That Way. I’ve read many books about education, and how American schools are failing our students, but this one is the most eye-opening and refreshing take on that subject I’ve seen. Ms. Ripley shares her data and conclusions through the personal stories of three American high school students who participate in foreign exchange programs. Kim leaves Oklahoma to spend a year in Finland, which has been in the news lately because of its students’ extraordinary performance on the PISA (Program for International Student…

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Math Is So Romantic

FracTad's Fractopia

I’ve been using a new review technique with the students in my Calculus class – speed dating! I wish I could take credit for it, but one of the incredibly creative teachers in the Harpeth Hall math department, Maddie Waud, introduced me to it. The first time I tried it, I was very pleased with how seriously my students took it, and after they finished, they all agreed it was helpful.

To set up a classroom for a speed dating session, divide all the desks into pairs facing each other. I put mine in a large circle.

Speed Dating1

Speed Dating2

At each pair of desks, place a problem for the partners to solve.  There are 18 students in the class, so I wrote up 9 problems. If there an odd number of students, the teacher can fill in and give whoever the solo student is some hints to solve her problem.


Give the students…

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Why So Little Decentralization?


Many scholars (forinstance) have noted a trend around the world of greater decentralization, at least on certain dimensions. Many non-federal, unitary states have tried to devolve some spending and decision-making authority on local or regional governments. Virtually every democratic government nowadays at least feigns some interest in decentralization.

Yet what strikes me is how little decentralization there has been, especially in the developing world. Some developing democracies that are sometimes described (or describe themselves) as “federal” or “semi-federal” include Mexico, India, Indonesia, Brazil, Argentina, Venezuela (before it went authoritarian some time in the 2000s), South Africa, Malaysia, Pakistan, Iraq, Nepal, and Nigeria. Yet none of these countries, other than Mexico, affords its constituent state or regional governments autonomy commensurate with that found in federal and semi-federal “Western liberal democracies” like Spain, Canada, the U.S., Switzerland, Belgium, Germany, Austria, Australia, and Italy. For instance, in Brazil, states do…

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A Sadness of Crowds?

Peter C. Blum

It does not seem as though any influence could induce a man to change his nature into a termite’s. No doubt he will always defend his claim to individual liberty against the will of the group. A good part of the struggles of mankind centre around the single task of finding an expedient accommodation — one, that is, that will bring happiness — between this claim of the individual and the cultural claims of the group; and one of the problems that touches the fate of humanity is whether such an accommodation can be reached by some particular form of civilization or whether this conflict is irreconcilable.

(Sigmund Freud, Civilization and Its Discontents)

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American Founding (H301) Midterm Study Guide

Midterm Study Guide, 2014

American Founding


Two of these will appear as options on the midterm examination this coming Friday.  You will answer (your choice) one of the two offered.  Enjoy studying!

  • Using the Cato Letters assigned and Cato: A Tragedy as the basis of your evidence, explain the ideas of the Commonwealth Men.
  • Define the nature and importance of republican thought and theory for the period leading up to the writing Declaration of Independence.
  • Explain the Protestant nature of the American character leading up to the writing of the Declaration of Independence.
  • Define Common Law rights as well as Natural Rights as understood by Americans in the period leading up to the writing of the Declaration of Independence.

Thomas Jefferson’s Original Declaration of Independence

H301 students, Pauline Maier (RIP) did a stunning job of presenting Jefferson’s original Declaration, complete with Congress’s edits.  I’m having a hard time scanning and posting it.  I’ll have photocopies made for you for Friday.  Thanks for being patient.


Crises Points, 1763-1774

Lexington, Mass.

Lexington, Mass.

“Crises Points, Leading to Revolution, 1763-1774”

Opening to Crisis, 1763-1765

  • Parliament passed the Sugar Act of 1764.  Taxed molasses and sugar at half the rate of the Navigation Acts.  Parliament pleased with itself—after all, they saw it as a tax cut.  But, in reality, that tax had never been enforced!  So, this tax was huge to the settlers.
  • In 1765, Parliament passed the Stamp Act.  Every single legal transaction had to have a stamp on it (meaning it had been taxed).  Pervasive and intensely bureaucratic.
  • In 1765, Parliament also passed the Quartering Act (another tax), forcing average citizens to pay for soldiers who needed housing.  Especially hit New York, the headquarters of British forces in America.

American Responses, 1765

  • Pamphlet war begins.
  • Protests. mass meetings, parades, bonfires.  Groups began to call themselves the Sons of Liberty.  Would meet under “Liberty Trees” and erect liberty poles, often with effigies hanging from them.  Sons of Liberty led a mob against the Massachusetts home of Lt. Governor Thomas Hutchinson.  Tore the house down and drank his wine.
  • Under the leadership of Patrick Henry, the Virginia House of Burgesses (in minority) passed a series of Resolves against the Stamp Act.  Henry even spoke of regicide, the assassination of King George III.  When told that was treason, he replied “If this be treason, make the most of it.”
  • John Adams is the first American to call for Independence in the Braintree Resolves.
  • Edmund Burke makes inaugural “maiden” speech, January 1766.  Pro-American, throughout the entire conflict.

British response to American Protests

July 1765, Rockingham, pro-American, made Prime Minister.  Rockingham Whigs chose Edmund Burke as their main advisor.

  • Repealed the Stamp Act in 1766 and reduced the Sugar Tax to one penny (less than the cost of a bribe)
  • But to pacify the opposition, Parliament passed the Declaratory Act–stated that Parliament had control of the colonies in all matters.

New Parliament, 1767

In the spring and early summer of 1767, the Parliament passed the Townsend Acts which

  • Suspended the New York Assembly for refusing to supply homes and assistance to British Troops
  • Placed a tax on colonial imports—glass, lead, paints, paper, and tea
  • Set up an office in Boston to monitor smuggling
  • John Dickinson offered first serious response in twelve “Letters of a Pennsyvania Farmer”

Boston Massacre, 1770

On March 5, 1770, a crowd gathered and began throwing snowballs at the soldiers, protesting the standing army as tyranny.  Soldiers opened fire on the crowd.  Americans began calling the army the “corrupt arm of British despotism.”  The soldier were tried, defended by John Adams.  All were acquitted but two who received light punishments.


  • Off of Rhode Island, a British ship looking for smugglers (the Gaspee) ran aground.  Colonists stormed the ship, kicked the sailors off, and burned it.  Parliament established an independent commission to investigate–had power to try out of normal courts.  Colonists viewed this as a violation of rights as it denied a jury of one’s peers.
  • In November 1772, Sam Adams formed the first of the Committees of Correspondence.  They soon developed throughout the colonies—information distribution; relay stations.


In Spring of 1773, the East India Company experienced a real drop in business.  Had 17 million pounds of tea rotting in British warehouses.  Parliament granted the company a monopoly on tea—even allowed to sell it cheaper than the colonists could get it from the Dutch.  Thought the colonists were motivated by material considerations.  All the colonists could see was the monopoly granted to the East Indian Co.

  • Under the leadership of Sam Adams, a group of men dressed as Mohawk Indians boarded a British ship and dumped its tea into the harbor.  Meanwhile, large crowds on the docks cheered.  With the exception of one destroyed padlock, the tea party made sure to violate no property rights beyond the tea.

Coercive Acts, March 1774

  • Closed the Boston Port
  • Set up martial law under the Governor
  • Close the courts in trials of British
  • Soldiers could lodge anywhere—including private homes

Quebec Act, Summer 1774 (considered fifth “Intolerable Act”

  • First Continental Congress called, September 1774
  • Independence from Parliament declared, September 1774

Charles Carroll of Carrollton’s Second CX Letter

American Cicero (ISI Books, 2010).

American Cicero (ISI Books, 2010).

SOURCE: CX [Charles Carroll of Carrollton], Dunlap’s Maryland Gazette; or, the Baltimore General Advertiser (April 2, 1776).

To tamper, therefore, in this affair, or try prospects merely upon the credit of supposed arguments or philosophy can never be the part of a wise magistrate, who will bear a reverence to what carries the marks of age; and though he may attempt some improvement for the public good, yet he will adjust his innovation, as much as possible, to the ancient fabric, and preserve entire the chief pillars and supports of the constitution.

–Hume’s Essays, Idea of a Perf. Commonwealth

Our present government seems to be approaching fast to its dissolution; necessity during the war will introduce material changes; INDEPENDENCE, the consequence of victory, will perpetuate them.  As innovations then must be made, let them be adjusted according to the advice of Mr. HUME, ‘as much as possible, to the ancient fabric.’  Let the spirit of our constitution be preserved; nay, improved by correcting the errors of our old system, and strengthening its soundest and best supports.  I shall briefly mention, without the least design of censuring past transactions, or calling blame on any man, what appear to me defects in our present government; and shall attempt with great diffidence to point out the proper remedies.—The following are some of its defects:

The principal offices are too lucrative, and the persons enjoying them are members of the Upper House of Assembly.  It is unnecessary, and be thought inordious [sic], to dwell on the mischiefs, which the Public has experienced in consequence of the misunderstanding between the two branches of the legislature, commonly occasioned by a difference of opinion respecting the fees of those offices.  Much time has certainly been consumed in debates, and conferences, and messages on that subject, which could have been usefully employed in other matters.  Is it proper that the same person should be both Governor, and Chancellor?  The Judges of our Provincial Court hold their commissions during pleasure; the duties of their station are most important and fatiguing; the highest truth is reposed in them, yet how inadequate the recompence!  The lessening of lucrative offices, and proportion the reward to the service in all; the exclusion of placemen from both Houses of Assembly; the separating the chancellorship from the chief magistracy; and the granting commissions to the judges of the Provincial Court, QUAM DIU SE BENE GESERRIT, with salaries annexed equal to the importance and fatigue of their functions; it is humbly conceived, would be alterations for the better in every instance.

The constitution of the Upper House seems defective also in this particular, That the members are removeable [sic] at the pleasure of the Lord Proprietary.  To give to that branch of the legislature more weight, it should be composed of gentlemen of the first fortune and abilities in the Province; and they should hold their seats for life.  An Upper House thus constituted would form some counterpoise to the democratical part of the legislature. Although it is confessed, that even then the democracy would be the preponderating weight in the scales of this government.  The Lower House wants a more equal representation, to make it as perfect as it should be.  At present, one third of the electors sends more delegates to the Assembly than the remaining two thirds. It will not be contended, I presume, that what has always been deemed a capital fault in the English Constitution is not one in ours, formed upon that model?  Every writer in speaking of the defects of the former, reckons the unequal representation of the commonality among the principal.  The Boroughs are proverbially stiled the rotten part of the constitution, on account of their venality, proceeding from the inconsiderable number of electors in most of them.  If the people of this Province are not fairly and equally represented; no doubt, in the new modeling of our constitution, great care will be taken to make the representation as equal as possible.  Before I proceed to point out a method for facilitating this desirable reform; I shall state some objections which have been used against the measure, and endeavour to give a satisfactory answer to each of them—

A new representation will offend those counties which now send to the Assembly a greater number of Delegates, than their just proportion; hence divisions will ensue, at all times to be dreaded, but most in the present.  Is the proposed alteration just or not!  If just, then they only should be esteemed sowers of division, who oppose it.  But should such a representation take place, more Delegates would be chosen in some counties, then in others, and should the votes be collected individually on all questions as heretofore practiced, they might frequently be carried against the smaller counties.  What is that but saying that the majority would determine every question, the only mode of decision that can with any propriety be adopted?  The Colonies vote by Colonies in Congress, therefore counties ought to vote by counties in Convention; and therefore it is useless to alter the representation, or to have more delegates from one country than another.  The force of this reasoning, if there be any in it, I could never comprehend.  The colonies are separately independent, and to preserve this independency, it may perhaps, be necessary to vote by colonies in Congress, but surely it will not be said that the counties of Maryland are thus independent, or that there are as many little independent principalities in the Province as there are counties?  All our counties are subject to the same legislature, which within the territory owning its jurisdiction, is supreme.   The inconveniences that may arise from the colonies voting by colonies in Congress must be imputed to the jealousy of Independence; or to use a softer term, to the necessity of securing to each colony its own peculiar government.  To reconcile the strength, safety, and welfare of all the United Colonies, with the entire privileges, and full independency of each, requires a much great share of political knowledge, than I am made of.  But if this imperfection in the general constitution or confederation of the colonies, can not be remedied, does it follow, that the same imperfection should run through the particular constitutions of every colony, or be retained in that of any one?  Different colonies may possibly, in process of time have different interests; and to secure to each Colony its peculiar and local interests; it may be thought most prudent to establish independence of each on a permanent basis.  Have the two shores of Maryland, or can they have[,] a difference of interests?  Should they be really so distinct, as to warrant the suspicion that the lesser part would be sacrificed to the greater; then are they unfit to be connected under one government; a separation ought to take place, to terminate the competition and rivalship, which this diversity of views and interests would produce; and all the consequent evils of two powerful and opposite factions in the same state.  The interests of the two shores are the same; in the imagination of some men a difference may indeed exist, but not in reality.  Is the welfare then of the whole Province to be sacrificed to the whims, the caprice, the humours [sic], and the groundless jealousy of individuals, not constituting a twentieth part of the people?  Let those, who would oppose an equal representation on a supposed contrariety the IPSE DIXIT of no man, or set of men ought to prevail against a reform of the representative, so consentaneous to the spirit of our constitution.

To come at a fair and equal representation of the whole people, the following method is proposed.

Let the Province be divided into districts, merely for the purpose of elections, containing each one thousand voters, or nearly that number; let every district elect [_____] representatives: suppose, for instance, the whole number of electors should amount to 40,000, then if every 1000 should elect two Representatives, there would be 80 Representatives returned to Assembly.  For facilitating the divisions of the counties into districts, and for the ease and convenience of the inhabitants in other respects such counties as are too large, may be divided into one or more, according to their respective extent.  As our people will in all likelihood greatly increase in the course of fifty or sixty years it may be necessary in order to correct the inequalities which that lapse of time will probably occasion in the representatives to new-model the elections, preserving the same number of Delegates to Assembly, it may be ordained that 1500 return two only.  If this precaution be not taken, the representative may in time become too numerous and unweildy [sic]; a medium should be preserved between a too small and a too large Senate; the former is more liable to the influence of a separate interest from that of their constituents and combination against that interest, the latter unfit for deliberation and mature counsels.  ‘Every numerous assembly (Cardinal de Retz observes) is [a] mob, and swayed in their debates by the least motives; consequently every thing there depends upon instantaneous turns.”  When the Province shall contain as many inhabitants as it will be capable of supporting in its most improved state of cultivation, it will not even then be advisable to suffer the representative to exceed one hundred for the reasons assigned.  If the Delegates should ever amount to that number, to preserve a due proportion between the two houses, the members of the Upper House ought to be encreased to eighteen or nineteen.  It would add to their importance, and give them more weight in the government, if on all vacancies they were to chuse [sic] their own members.  The above alterations of our constitution are no ways inconsistent with a dependence on the crown of Great Britain, and therefore, not justly liable to the censure and opposition of those who wish the colonies to continue dependent.

During the civil war the appointment of a Governor and Privy Council, not to exceed five, must be left to the two Houses of Assembly; and if we should separate from Great Britain, the appointment must remain with them.  The Governor and Council are to be entrusted with the whole executive department of government, and accountable for their misconduct to the two houses.  A continuance of power in the same hands is dangerous to liberty; let a rotation therefore be settled to obviate that danger yet not so quickly made, as to prove detrimental to the State by frequently throwing men of the greatest abilities out of public employments.  Where would be the inconvenience if the Governor and Privy Council were annually elected by the two Houses of Assembly, and a power given by the constitution to those Houses of continuing them in office from year to year, provided that their continuance should never extended beyond the term of three successive years?

A variety of other arrangements scarcely less important, must follow the proposed changes; only the outline of the constitution is drawn, the more intricate parts, “their nice conections [sic], just dependencies” remain to be adjusted by abler heads.  A long dissertation was not intended, a minute detail would be tiresome; and perhaps the author may be justly accused of having already trespassed on the patience of the Public.

–CX [Charles Carroll of Carrollton]


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