Mathematical Journeys: Road Trip Around A Problem

Let’s go on a road trip!

When I teach geometry, especially geometry involving angle measures like this problem, I like to describe the process of solving a problem as taking a little road trip. I describe it this way because this is how I personally feel when solving a problem like this – my eyes rove around the figure from one intersection to the next, and I hop in my little math car and drive along lines and stop at intersections to figure out where I am. Geometry is a very visual discipline, and as a visual learner, I have the most fun when I can trace a physical journey around the problem, solving things as I go. So let’s hop in our math car and chase this problem down!

The first step in any problem like this is to sort out what our givens mean. This usually involves a bit of deciphering the image as well; making sure it unpacks and makes sense in your head before you progress further. You have to be able to read the map if you don’t want to get lost in the middle of Illinois with nothing but cornfields for miles! So, in this figure we’ve got two parallel lines, l and m, that are cut by two parallel transversals, n and o. The figure in the center of the lines is a parallelogram – which may or may not be relevant to the problem, but it’s a good thing to state at the outset. My approach is always to state everything you can deduce from the givens as soon as the givens appear; you can always ignore the information later if it’s not needed.

We’ve also been given the values of two of our angles, angle 1 and angle 13, but they’re written as algebraic expressions rather than straight-up numbers. This is where I always ask the student, “Why would they give us the information like this? Why not just tell us the numbers? What are they expecting us to deduce from this?” If the student can answer those questions, they’re halfway through a proof already. In this case, the angles given are a pair of corresponding angles – the upper-left corner of two of the intersections. What do we know about corresponding angles? They’re congruent, which means their measures are equal. So we should be able to simply set those two expressions equal to each other and solve for x. That’s a good place to start, so let’s hop out of the car and do that.

10x – 7 = 7x + 29
10x = 7x + 36
3x = 36
x = 12

Okay, so x is 12. That’s progress, right?


Well, how does that help us figure out angle 12? Is there some way we should be able to figure out angle 12 just by knowing some of the other angles? Let’s hop back in the car and head over that way to look closer.

Well, looking just at line o, we see another set of corresponding angles – angle 13 should be congruent to angle 9. That’s good, we’re at least at the correct intersection now. Now to get our bearings. Given a single intersection, we basically have two angle measures repeated twice – you can think of them as the pairs of vertical angles, or the pairs of supplementary angles, whichever makes more sense to you. Either way, angle 9 should be congruent to angle 11, and both 9 and 11 should be supplementary to angle 12 and angle 10. That means, whatever value angle 9 is, we can subtract that from 180 to get the value of angle 12.

Okay, sweet. We have a plan. Let’s hop back in the car and go figure out what angle 13 is so we can bring that with us to angle 9.

*road trip montage*

Well, we know what x is now, right? So let’s just plug that back in to our given and see what we get.

X = 12
angle 13 = 7x + 29
7(12) + 29
84 + 29

Okay, so angles 1, 13, and 9 should all be 113. (Incidentally, so should angle 5, but that’s for another problem.) Let’s throw 113 in the trunk and drive back over to angle 12!

*changes the radio station*

Let’s call angle 12 ‘y’ for clarity. Angle 12 is supplementary to angle 9, so the two should add up to 180.

180 = 113 + y
y = 180 – 113
y = 67

So angle 12 should be 67 degrees. Problem solved!

Now, for a bit of extra credit…

These types of problems often involve one image that has several different problems to go along with it. If you had other problems to solve with this same figure, you’d be thrilled to know that you now know the values of each and every angle in the figure. They’re all either 113 or 67, set up in supplementary and/or vertical pairs with the existing ones. Finish your road trip and jot down the values of all the other angles in case you need them later!

Angle 12 is 67, which makes angle 10 also 67 and angle 11 113.
*road trip montage*
Corresponding angles makes angles 5 and 7 113, and therefore angles 6 and 8 are 67.
*pit stop for gas and snacks, road trip continues*
And so on until your road trip is finished and your figure is entirely labeled. That’s the power of parallel lines!

Ellen’s Choice: Applying my rules of Effective Time Management to the SAT, Part 3: Mix and Match!

Ellen’s Rules For Effective Time Management, Part 3

5. Mix up your subjects.

Spending all day working on the same project can lead to feelings of frustration and inadequacy. Mixing up your subjects helps the brain to stay engaged, since it can’t fall into the trance of working on the same thing for hours. If you’re writing a paper and starting to feel annoyed or frustrated with it, take a break and work on your math for a bit. You’ll sit back down to the computer feeling refreshed and relaxed, even if you haven’t stopped for more than fifteen minutes at a time all day.

6. Make the delineations between subjects clear and firm.

When mixing up your subjects, keep them distinct and separate from each other. Take a short break between subjects, or place the rest of your notebooks on the other side of the room so that you’re forced to get up and move around in order to change subjects. Give your brain several minutes to clear and reorganize for the next subject before you dive back in.

The same goes for organization—if you’ve got six subjects all sharing the same notebook, use dividers or get a five-subject spiral notebook to keep them all separate. If you use a three-ring binder, be very picky about where you put your notes. Take a few minutes at the end of each day to sort through your notes and handouts, punch holes in anything that needs it, and arrange them in the correct places in your binder. Keep everything you need for a certain subject in one place. It’ll only take a few minutes per day, but you’ll be rewarded with a much more manageable notebook and far less time spent sifting through loose-leaf paper looking for the prompt for your essay or that one sheet of notes about summations.


These rules apply to the SAT too, though they now apply differently since the SAT’s 2016 Redesign. Before the redesign, the SAT was comprised of nine short sections that alternated subjects between math, verbal, and writing. That format synched up nicely with rule #5, keeping you on your toes by shifting subjects frequently. Not so with the new SAT. Now it more closely resembles the ACT, with one longer section for each subject. So you can’t really mix up your subjects in the same way, rather, you should be practicing staying focused for a solid hour of question-answering on a given subject.

Rule 6 still applies, though, because you will still need the ability to switch subjects quickly. Generally you are given between 2 and 5 minutes’ break in between sections on the SAT, and you’re often not allowed to leave the testing room during that time. So it becomes extremely important to learn to clear your mind quickly and reset for the next subject. Close your eyes, take some deep breaths, whatever you feel you need to do to let go of the previous section and recenter your mind for the next one. If you can, get up and walk around the testing room (or out in the hall if you’re allowed) for a minute or two; shaking the kinks out of your body helps to clear your mind. Try a quick observation exercise – see how much you can see, hear, smell, or feel about your surroundings. Opening up your awareness will help to get you out of the test for a moment. Try not to get too distracted, though – you’ll want to rest your mind while you can and prepare for the next section.

Stay tuned for part 4: Procrastination Sucks!

Writing Rundown: Persistence Pays Off in “Peripheral Presence”

This was a really hard essay to write. Not because I couldn’t figure out what to write about; I knew almost from the moment I read the prompt that I wanted to write about Dracula. On the contrary, it was hard because I had TOO MANY ideas for this essay – I had so many thoughts buzzing excitedly around in my head that my outlines kept coming out really scattered and disorganized. I went through, no joke, at least FOUR different outlines for this essay – and I refused to even start writing a draft until I’d sorted out what precisely was wrong with my outline, scrapped it for the third time, and started over from scratch. I went through several different organizational schemes, starting with one centered around a favorite Hitchcock quote about suspense that was a good idea, but ultimately, had no place in this particular essay. My outline eventually settled on the format that probably should have been obvious from the start – the one that related most closely to the prompt. My outline finally – FINALLY – took on the shape of an essay composed of a paragraph each for the three parts of the prompt (action, theme, and the development of other characters).

Once I had a draft written, it hovered on my computer missing only a conclusion for a long time, because I couldn’t shake the feeling that I was still missing something…some key component of my thesis that I hadn’t addressed thoroughly enough. I had actually surprised myself with a really solid working thesis, even though it was technically two sentences instead of one:

While he rarely physically shares the same space as our heroes, their knowledge of Dracula’s constant scheming and plotting spurs them to act decisively, take risks, and in a variety of ways give their life’s blood in service of his destruction. In a sense, Dracula is the one who turns them into heroes.

Reading through my draft, I realized that other than an initial paragraph about Jonathan Harker scaling the walls of Castle Dracula, I had neglected that second sentence of my thesis – the idea of Dracula being the one who turns them into heroes. THAT was arguably the most important part of my argument – how does his rarely-seen presence become significant? He turns the protagonists into heroes, and brings about his own undoing!

Once I finally figured it out, the essay clicked together pretty quickly. The whole idea of an archetypal hero is a big, complex topic that could probably have been its own series of essays, so I didn’t want to dive too deep into the idea of ‘what makes someone heroic.’ That’s for another essay. So I kept it to a few extra sentences at the end of each paragraph to point out the heroic nature of the actions taken by the protagonists, and the hand Dracula has in prompting all of those actions.

The moral of the story here is that persistence pays off – and that it’s far less intimidating to get the bulk of your brainstorming out in the prewriting phase than it is to rework an essay that’s already drafted. Scrapping an outline – or three or four – is easy. Scrapping three pages of drafted essay can be traumatizing. But sometimes drastic measures are necessary – take a deep breath, clear your mind, and come back to your prompt.

Literature Spotlight: Peripheral Presence

In some works of literature, a character who appears briefly, or does not appear at all, is a significant presence. Choose a novel or play of literary merit and write an essay in which you show how such a character functions in the work. You may wish to discuss how the character affects action, theme, or the development of other characters. Avoid plot summary.
~AP Literature Open Essay Prompt, 1994

Peripheral Presence

Having your name in the title of a book doesn’t mean you get to be in the spotlight. Take the classic 1897 gothic horror novel Dracula, by Bram Stoker. The eponymous vampire appears in person surprisingly little, and only once after his initial conversations with Jonathan Harker. Despite this, he still very much deserves the honor of the novel’s title. His actions set the events of the novel in motion, and the main characters talk of nothing else but him. While not directly seen, his actions leave tangible consequences on Lucy, Mina, and Renfield, and his offstage plotting leaves the heroes struggling to keep up, to determine his motives and his next movements before he can execute on them. While he rarely physically shares the same space as our heroes, their knowledge of Dracula’s constant scheming and plotting spurs them to act decisively, take risks, and in a variety of ways give their life’s blood in service of his destruction. In a sense, Dracula is the one who turns them into heroes.

At the start of the novel, solicitor Jonathan Harker is brought to Dracula’s estate in the Carpathians to help close the sale of some British land to the Count. Harker is welcomed into Dracula’s castle, sight unseen, put up in one of the castle’s gorgeous rooms and treated as an honored guest. At home in his castle in the mountains, Dracula clearly feels comfortable enough in his power to invite a random human into his home. He must not see Harker as much of a threat – and, truth be told, he is correct. Harker is not a threat – at first. Through repeated contact with Dracula and his genteel but immutable scheming, Harker becomes more and more aware that he is being held prisoner in the castle, and more and more determined to escape his gilded cage. Finally, he summons up the courage to scale the outside wall of the castle in an attempt to catch Dracula unawares. “The chances are desperate, but my need is more desperate still. I shall risk it. At the worst it can only be death; and a man’s death is not a calf’s, and the dreaded Hereafter may still be open to me.” (P. 93) Dracula’s presence and true nature have given Harker a new perspective on risky behavior, namely, that death is not the worst thing that could happen to him, and that he would actually rather die doing a courageous thing than be slaughtered to the vampires like livestock.

After leaving the Carpathians and traveling to Whitby by sea, Dracula begins turning Lucy. We do not see this happen in person; rather we see our main characters responding to Lucy’s curiously declining health. Over the course of two weeks, all four of the male main characters donate blood to her in a series of transfusions. Each one seems to work until the next morning, when she is found to be even paler and sicklier than before, prompting Quincey to exclaim, “Ten days!…that poor pretty creature that we all love has had put into her veins within that time the blood of four strong men. Man alive, her whole body wouldn’t hold it…What took it out?” (P.232) In this way, Dracula has invoked one of the main themes of the novel: the equating of blood and life. By draining her blood, Dracula is quite literally stealing her life – and by extension, that of the four men who love her. Blood transfusions in this time period were a very risky operation, especially when performed without a proper operating theater. These four strong men who love her are giving their life’s blood to save her – a theme that will continue throughout the novel.

Once poor Lucy has been released from her undeath, the thoughts of the men turn irrevocably to Dracula. Only Harker has seen Dracula in person, and yet all five men feel an intense hatred of the vampire and an intense desire to ensure that what happened to Lucy does not happen to anyone else. Their conversations and constant stewing over Dracula’s whereabouts and motives are a large part of what shapes them into heroes, helped along by their connection with Mina. Harker’s wife is read in to their plans and, though they encourage her to shelter herself from the actual dirty work, she assists them through philosophical conversations about the vampire, his motives, and his nature.

“I know that you must fight – that you must destroy even as you destroyed the false Lucy so that the true Lucy might live hereafter; but it is not a work of hate. That poor soul who has wrought all this misery is the saddest case of all. Just think what will be his joy when he, too, is destroyed in his worser part that his better part may have spiritual immortality. You must be pitiful to him, too, though it may not hold your hands from his destruction.” (P. 420)

Through her sage guidance, the men are able to find a way to pity the creature even as they seek to destroy him – a very heroic attitude to take.

Though their attitude may be heroic, their attempts to act the hero don’t always succeed, thanks to Dracula’s extreme guile and cunning. The men initially urge Mina to stay home while they go about the dangerous tasks involved in the pursuit of Dracula, believing it to be inappropriate work for a woman. This plan backfires when it allows Dracula to successfully put Mina under his thrall. The men catch Dracula in the act – in pretty much the only other physical appearance of Dracula in the novel. After chasing him away and regrouping Mina appears to have survived relatively unscathed, though she is shaken, bitten and bloody. However, when Van Helsing attempts to bless Mina with a piece of sacred wafer it instead burns her flesh, leaving a scar on her forehead. Mina exclaims in the aftermath of this incident, “Unclean! Unclean! Even the Almighty shuns my polluted flesh! I must bear this mark of shame upon my forehead until the Judgement Day.” (P. 406-407) Even though Dracula himself is not physically present – and indeed, is not seen again until the very last pages – Mina’s ‘polluted flesh’ serves as a painful and tangible reminder of the creature’s presence.

Dracula’s presence also has a tangible effect on Dr. Seward’s ‘zoophagous’ patient, Renfield. Dracula’s estate, Carfax, is next door to the asylum, and the reader gradually begins to realize that they can tell whether Dracula is home or away based on Renfield’s behavior. Renfield gives up his zoophagous obsession at one point, and based on his ramblings the reader understands that this is because he believes Dracula is coming to rescue him. Dracula comes home to Carfax, and the reader knows this not because Stoker shows him coming home, but because Renfield escapes the asylum. Seward finds Renfield pressed against the door of Carfax’s chapel, muttering “I am here to do Your bidding, Master. I am Your slave, and You will reward me, for I shall be faithful. I have worshipped You long and afar off.” (P.170) In my annotated edition by editor Leslie Klinger, an annotation on this sequence points out that there has been no communication that we know of between Dracula and Renfield, so “this again seems to be confirmation that Dracula in some way transmits his presence to those sensitive enough to detect it.” Just as Mina is linked to Dracula through his thrall and can be hypnotized into seeing through his eyes, Renfield is linked to Dracula in some supernatural way, and this in turn affects his behavior.

While Dracula may not show his face very often, his presence is keenly felt by all six of Stoker’s central characters. Urged on by Dracula’s seemingly-unstoppable offstage scheming, our protagonists perform (and undergo) emergency surgical procedures, cut the head off of a beloved young lady’s corpse, break into houses, and chase gypsies across foothills – all activities that would be practically unthinkable when we first meet them. From the very first castle wall scaled by a mild-mannered lawyer to the final knife wound suffered by a heroic American cowboy, Dracula’s lurking peripheral presence pushes our protagonists to take ever greater risks, and inadvertently shapes them into the heroes who will ultimately destroy him.

Ellen’s Choice: The Problem with Banning Books

Since Banned Books Week happens in mid-September each year, I’d like to talk today about the problem with banning books. Last year, my Bring Your Own Book club’s topic for September was to read a banned or challenged book. We had a great discussion during our meeting about common threads in all of the books we read, common reasons why books get challenged, and how that relates to the education system in general. One of the things that kept coming up was that often, the reason the book was challenged is the entire point of the book itself – of course it deals with that; that’s the main theme of the book! Whether it’s The Perks of Being a Wallflower and Looking For Alaska depicting kids smoking, drinking, and doing drugs, or it’s The Giver depicting a fundamentally broken society masquerading as a utopia (psst – that’s the definition of the genre – it’s a dystopia!), or even a gorgeous picture book called “And Tango Makes Three” telling a true story about a pair of male penguins who raised an orphaned baby penguin, people who challenge books often seem to miss the whole point of the books they want to ban.

Often when people want to ban books, their reasoning is that the book is ‘unsuitable for the age group,’ which seems to be code for “I don’t trust my kid to understand the nuances of this issue.” Other common reasons to challenge books include quote-unquote “adult themes” (often along the lines of drinking/smoking/drugs, gambling, nudity, sex scenes etc.), and these are even more likely to be the issue if the book centers around teenagers – heaven forbid we depict teenagers smoking outside the school building or drinking underage at a party!

But here’s the thing – literature is at its best when it forces the reader to confront uncomfortable subjects and ponder hard questions. That’s especially true of books for young readers; in my opinion we should be encouraging students to read books that will challenge their conceptions and get them to think hard about their own opinions. The crux of the issue here is that, yes, for the most part we all agree that these underage behaviors are bad – but there’s no denying that they do take place. My argument against banning books is essentially the same argument I have against abstinence-only sex education – if we simply remove all evidence of these activities from existence, we eliminate the chance to have important discussions with students about the ramifications of those activities.

As an example, take Looking For Alaska by John Green. This boarding-school story has been challenged many times, and usually it’s due to a single graphic sex scene in the middle of the novel. John Green has spoken about his rationale for including this scene several times. For just one example, visit his YouTube channel, “Vlogbrothers,” and search for the video titled “I Am Not A Pornographer.” He points out in this video that the language used during the scene is very cold, clinical, unemotional, and the participants barely know each other and their connection doesn’t develop any further in the rest of the story. He contrasts that scene with the scene immediately following it, where there is no physical intimacy yet the emotional bond shared between the characters is powerful and heady. His reason for including these scenes back-to-back is to show that physical contact is not necessarily the end-all be-all of romantic relationships, and that two people can share a closeness purely out of emotional connection even without physical intimacy. In John’s own words, “Looking For Alaska is arguing against vapid physical interactions, not for them.”

If you leave Looking For Alaska in the school library (or, better yet, include it in a curriculum!), you can use it to start a group discussion on these themes and explore them further with the very population you’re concerned about, encouraging them to think critically about the difference between physical and emotional intimacy and make reasoned decisions about how to incorporate both into their own lives. It seems to me that most of the worries people have that lead them to want to ban books could be solved by simply talking to their kids about the realities of the depicted material. Conversation is always better than censorship.

Writing Rundown: Deciphering a Prompt for “The Blanks Left Empty”

In today’s Writing Rundown, I want to leave the brainstorming process for a bit and discuss responding to a prompt. Take a look at the prompt I used for my last Literature Spotlight, “The Blanks Left Empty”:

AP Literature Open-Ended Prompt, 1975, #2:
Unlike the novelist, the writer of a play does not use his own voice and only rarely uses a narrator’s voice to guide the audience’s responses to character and action. Select a play you have read and write an essay in which you explain the techniques the playwright uses to guide his audience’s responses to the central characters and the action. You might consider the effect on the audience of things like setting, the use of comparable and contrasting characters, and the characters’ responses to each other. Support your argument with specific references to the play. Do not give a plot summary.

Whew! That’s a lot of information to sift through. Unfortunately, many high school and college-level writing prompts are as complex, if not more so, than this one. The best thing you can do as a student is to practice parsing long prompts into pieces and figuring out how to attack them. This can and should be part of your pre-writing for any prompt-based essay, so it’s a good skill to have. So let’s go through that block of text again, carefully, and figure out exactly how to tackle this assignment.

The first thing I do when confronted with a prompt this long is to scan through the whole thing and highlight, circle, underline, or otherwise mark any specific instructions the prompt gives. A specific instruction is a word or phrase that indicates a concrete thing they want to see you do in your paper. For this prompt, here’s what I’ve got after that stage:

AP Literature Open-Ended Prompt, 1975, #2:
Unlike the novelist, the writer of a play does not use his own voice and only rarely uses a narrator’s voice to guide the audience’s responses to character and action. Select a play you have read and write an essay in which you explain the techniques the playwright uses to guide his audience’s responses to the central characters and the action. You might consider the effect on the audience of things like setting, the use of comparable and contrasting characters, and the characters’ responses to each other. Support your argument with specific references to the play. Do not give a plot summary.

Okay, let’s go through those points one by one and sort out what they mean for our paper.

Select a play you have read

Fairly straightforward here. If we were given this prompt in the context of an English Lit class, it might specify which play we should be using – usually the one the previous unit centered around. But this is an open-ended prompt, so they’re keeping it more general. The important thing, though it seems obvious, is that you have to have read the play in question. Ideally, you will have read it recently enough to remember it well, and you should have a print copy of the play with you while you write, to cite page numbers for your examples. Don’t make the mistake of trying to write a convincing essay about a play you either haven’t actually read, or haven’t read recently enough to remember well. You’ll only bring yourself an unnecessary headache.

In my case, I immediately went to Assassins as one of my favorite plays, but I kept a few other play ideas in my back pocket just in case this first one didn’t work out. Second-choices for me included Death of a Salesman and Waiting for Godot, but I thought Assassins would be a more fun essay to write, so I started there.

The next highlighted instruction is the big one, the actual ‘prompt’ part of the text:

explain the techniques the playwright uses to guide his audience’s responses to the central characters and the action.

There’s a lot in this sentence, so it bears thinking about a bit more carefully. Let’s break it down even further:

explain the techniques the playwright uses

Okay, so this is going to be an essay about the craft of playwriting – the ways the author creates the story on a structural and artistic level. We’re not so much talking about the themes and symbolism within the story, as often happens with English essays, as we are talking about what stylistic tools the playwright uses to create those symbols and themes. This is going to be a nuts-and-bolts essay, rather than a narrative exploration essay.

to guide his audience’s responses

All art is about creating a response in the audience in some way – whether that’s an emotional response that causes you to cry at the end of The Fault in Our Stars, or the interminable boredom of the first half of The Good Earth, or the tense fear created through the suspense of not knowing Dracula’s specific movements for most of the novel. So here we’re talking about how the playwright can control what responses his audience experiences – how does he get us to feel that profound sadness and grief that makes us cry rather than the fear that would make us scream? So now our technical discussion has an angle – we’ll only be looking at techniques and tools that help to bring out emotional responses. If an example wants a place in our essay, it has to fit into that criteria.

to the central characters and the action.

This little tail might get overlooked if we’re not careful, but it does bear keeping an eye on. Here we’re refining the criteria for our examples – not just a tool that brings out an emotional response, but one that brings out a response relating to the characters or their actions. We’re talking about character development here, rather than plot points that are entirely out of the characters’ hands. That awesome example you have about the thunderstorm that symbolizes the relationships between the main characters? That should only be included in this essay if it actually shapes our perception of those characters moving forward – if it makes us think about those characters in a different way.

If you have an awesome example in mind that doesn’t seem to fit, take some time with it and see if you can make it fit these criteria with a little love. A word cloud can be helpful for this kind of pre-writing – the thunderstorm I mentioned above, from my Wuthering Heights essay, clarified as the central theme of my response only AFTER quite a bit of turning it around as it related to the other points I was making.

So already we’re starting to see the type of essay we’ll be writing – a very technically-focused piece about the playwright’s craft in character development. We’ll be using one play as our lens to focus this broad topic into something workable, and this is where the last two highlighted portions come in:

Support your argument with specific references to the play.
Do not give a plot summary.

These are both good things to keep in mind as we write, as they often get lost in the shuffle.

Specific examples

We’ll need to have some exact quotes and some summarized examples, but we should be able to point to specific aspects of the play to support our arguments. When pre-writing, wether with a word cloud, outline, or some other method, if it doesn’t have at least one example from the play, discard it!

Do not give a plot summary.

This one is important as a way to let your teacher know that you actually read the prompt. It’s very tempting to start off an essay by explaining the plot – you want the reader to come to the work from the same place you do, after all. Right? Well, actually, in a situation like this, it’s better to assume the reader is familiar with your subject material already. If you were in an English Lit class, the teacher would be, and even in an open-ended prompt like this, you very rarely need to know the detailed ins and outs of the plot to understand the writing techniques you’ll be discussing. All a plot summary does is eat up a paragraph or two of your (probably tight) word or page count, making it harder to fit the actual meat of your essay into the rest of your space.

This point was easy for me to hit because Assassins doesn’t really have a plot in the first place – it’s quite abstract as musicals go. All I gave in the way of summary in my essay was a single sentence to explain the theme of the play – “Assassins, which is based on an idea by Charles Gilbert, Jr., is an abstract musical centered around the men and women who, over the years, have killed or tried to kill the President of the United States.” I went on to explain the central theme of the play in a bit more detail, but only because the rest of my essay hinged on articulating how the playwright’s tools ‘guide the audience’s response’ to the central theme as it relates to the characters. Assassins is a very character-driven play in the first place, so that made it a perfect choice for this essay.

And that brings me to another important point – when you’re fortunate enough to be able to choose your own subject material, choose carefully. Pick a play that will support the type of essay you need to write intrinsically, so that you don’t end up fighting with your examples to make them stay on topic. For me, an abstract musical was an obvious choice for this one, since musicals tend to involve more suspension of disbelief for the audience, and the more abstract the staging, the easier it would be to point to the tools used.

So this gives us enough of a sense of what kind of essay we’re writing to begin some pre-writing proper, using any of the techniques we’ve talked about already. For this one, I used some sketchy outlining to get my thoughts in order. And now we need to add one more piece to the puzzle – this one is one that I often add in after I have a basic outline going. You may have noticed that I skipped one sentence of the prompt when I first tackled it above:

You might consider the effect on the audience of things like setting, the use of comparable and contrasting characters, and the characters’ responses to each other.

This type of sentence is VERY important whenever it appears in a prompt. Starting off a sentence with the phrase “You might consider” makes it sound like this is an optional step, something to get you going if you need it, but often this is the teacher’s way of telling you what they’ll be looking for in your paper when grading. It’s always a good idea to try to address as many of these example ideas in your paper as you possibly can – as long as you can find some examples from your source material.

In our case, this means the paper should focus its supporting arguments on three specific playwriting tools: setting, comparing and contrasting characters, and the way the characters respond to each other. When I went to revise my outline, I started with a body paragraph for each of these tools and refined from there. My final paper included a discussion of ‘limbo’ (setting), the use of limbo to show parallel and contrasting aspects of the different characters’ stories without them showing each other (comparable and contrasting characters), and the way the assassins interact with the Balladeer (the way the characters respond to each other). Once these three were addressed, I was free to include any extra examples I wanted – in my case, a paragraph about the importance of direction as a tool and the different audience response caused by making one small directing choice that differed from the script as written.

So there we have it – a pretty decent handle on the type of essay we’re going to write, the angles we’re going to come at it from, and the specific examples we want to include. Finish up your outline, and get writing!

Literature Spotlight: The Blanks Left Empty

AP Literature Open-Ended Prompt, 1975, #2:

Unlike the novelist, the writer of a play does not use his own voice and only rarely uses a narrator’s voice to guide the audience’s responses to character and action. Select a play you have read and write an essay in which you explain the techniques the playwright uses to guide his audience’s responses to the central characters and the action. You might consider the effect on the audience of things like setting, the use of comparable and contrasting characters, and the characters’ responses to each other. Support your argument with specific references to the play. Do not give a plot summary.

The Blanks Left Empty

Narration is often the crux of the novelist’s art. Through skillful use of narration and point of view, a novelist can make his readers acutely aware of not just the events of the novel, but the characters’ opinions of those events. This makes it easy for a skilled novelist to deftly control how his readers respond to the work. The playwright, on the other hand, has no such tool at his disposal. Plays very rarely have a narrator or any sort of explanatory text that is not part of the dialogue or stage direction. How, then, does a playwright guide his audience’s responses to the action and characters? How does he communicate his theme or commentary without the use of narration?

The playwright has numerous other tools to convey his point, all of which stem primarily from the fact that theater as a medium is more open to abstraction. Much of theater, no matter how grand the production, depends on the audience’s imagination to fill in the blanks. A single ornate desk might represent a general’s field office; a simple black ladder might represent a second-floor balcony. The playwright relies on the unspoken agreement that the audience will fill in the rest of the details with their imagination. Even if the general’s office is completely built out in shining gold paint, there is one wall missing, the one through which the audience sees the action. And the building outside the office does not exist. Because of this unspoken agreement, audiences are prepared to see events play out that would not, strictly speaking, be possible, and the suspension of disbelief is even stronger when the play is observed live. Assassins, by Stephen Sondheim and John Weidman, is an example of a play which uses this implied agreement to devastating effect.

Assassins, which is based on an idea by Charles Gilbert, Jr., is an abstract musical centered around the men and women who, over the years, have killed or tried to kill the President of the United States. The central theme of the play is a dark reflection of the ‘American dream’; the idea that if any kid can grow up to be the President, then it follows that any kid can grow up to kill the President. The play explores potential motives for the various assassins and seeks to humanize them, to make them more than just “footnotes in a history book” (to use Sarah Jane Moore’s words from the final scene). The primary way Assassins achieves this is through its use of setting.

Assassins makes heavy use of the theater concept of ‘limbo,’ a neutral stage space with little or no sets and props, that places the characters in a realm devoid of context. With a few exceptions, the people depicted in Assassins lived in completely different times and locations from one another, but the play places them all in a shared space – limbo – that can become whatever is necessary to communicate the theme at hand. The bulk of the play is presented in vignettes, isolated sequences that center around each of the assassins. Their story is portrayed in one or two songs, loosely threaded together by the occasional ‘limbo’ scene with more than one assassin conversing, or the pseudo-narration of the audience surrogate character, the Balladeer. We’ll get to him in a moment.

This extensive use of limbo has a few distinct effects on the audience’s response. First, the abstract nature of the setting makes it possible to portray multiple sides of an event simultaneously in the same space, such as when Guiseppe Zangara’s story depicts his execution over top of a chorus of excited witnesses gushing about the minor fame they received from being nearby. The following dialogue, taken from the final chorus of the song “How I Saved Roosevelt,” happens with the witnesses clustered around a microphone located immediately behind Zangara’s electric chair, so that they all share the same spotlight center stage.

And why there no photographers? For Zangara no photographers! Only capitalists get photographers! No right!
Lucky I was there!

No fair, nowhere – so what?
I’m on the front page, is that bizarre?!
No sorry!
And all of those pictures, like a star!
And soon, no Zangara!
Just lucky I was there!
We might have been left

Who care?
Bereft of F!
Pull switch!
No luck, no more, no…!

Zangara never gets to finish that sentence – the chair is activated, and the witnesses hit their final high note over his convulsing form. This use of juxtaposition in limbo space helps to drive home the way that we tend to downplay the plight of the assassin. We get wrapped up in our own little victories and forget that for someone to be driven to kill, they must have stakes high enough to kill for.

Limbo is also used to compare and contrast the various assassins, showing how their stories mirror and support each other. Portions of different but parallel stories occur in the same space and time, allowing characters to display complimentary facets of their personalities to the audience without showing them to each other. This is true of the song “Unworthy Of Your Love,” wherein John Hinckley and Lynette ‘Squeaky’ Fromme sing a love duet, not to each other, but to their respective infatuations with others who barely notice them. The theme of “find[ing] a way to earn your love” drives both of them to the same desperate act despite their different circumstances.

Of course, limbo is a magical place, and sometimes these characters do interact. Whenever they do it gives the play a sort of otherworldly feeling, a surreal tone that somehow allows John Wilkes Booth and Sam Byck to exist in the same time and place. Two full-company scenes of this type roughly bookend the musical, one at the beginning in a nebulous barroom, and another near the end when the assassins reunite in the Texas Book Depository to persuade Lee Harvey Oswald to join their ranks. During these large group scenes, the assassins bounce ideas and themes off of each other, which allows the audience to see the playwrights’ ideas more directly and begin to form a response. At the start of the play, these ideas are directed at the general unfairness of the world at large, and the themes that emerge are those of desperate attempts to fix it:

“ZANGARA: What should I do?
BOOTH: Have you thought about killing President Roosevelt?
ZANGARA: Do you think it will help?
BOOTH: It couldn’t hurt.”

By the time Oswald is revealed, though, the audience has already watched these men and women go through their own stories, with varying degrees of success and fulfillment. Their ideas and themes thus become driven by the need to convince this meek, frustrated, suicidal man to perform one powerful act that will “revive and give meaning” to all of their stories.

“MOORE: You think you can’t connect? Connect to us!
CZOLGOSZ: You think you’re powerless? Empower us!
BOOTH: It’s in your grasp, Lee. All you have to do is move your little finger. You can close the New York Stock Exchange.”

These types of conversations help the playwrights get across a few potential reasons why these men and women would have done something this extreme, both from the characters’ perspectives and the unsaid ones brought about by their interactions.

Limbo is a powerful enough tool on its own, but in Assassins it is joined by another, equally powerful one: the audience surrogate character. An audience surrogate is a character who presents the perspective that is most likely shared by the audience, who can ask the burning questions the audience might have and force the characters to display their themes more obviously. In Assassins, this character is the Balladeer, a ‘normal’ American who tells the stories of the Assassins through bits of sung narration. Besides introducing the stories, though, the Balladeer has a definite opinion about the assassins; they’re poor, misguided nobodies who have been stepped on and largely forgotten by the country, subjects suitable for pity and derision. He says what most of us would have probably thought about the assassins – if we ever thought about them at all. He dismisses their plights as selfish or small, not worthy of the extreme reactions they had to them:

“BALLADEER: They say your ship was sinking, John,
You’d started missing cues.
They say it wasn’t Lincoln, John,
You’d merely had
A slew of bad

Of course, this being limbo, the assassins won’t let him get away with that kind of abuse without a fight. Even the above stanza, sung as John Wilkes Booth attempts to dictate his indictment of Lincoln, is punctuated by Booth telling him repeatedly to “Shut up!”. Over the course of the play, the audience grows to empathize with the assassins, and starts to hate the Balladeer’s antagonism and dismissiveness of them. By the time the song “Another National Anthem” comes around near the end of the play, the audience is downright relieved to see the assassins finally turn on the Balladeer, overpowering his voice with their combined chorus and driving him off the stage so that they can finish their last chorus uninterrupted. The Balladeer is the embodiment of our tendency to ignore the darker parts of history, the people who ended up on the so-called ‘losing’ side.

On a related note, with this moment we can see as well the importance of the impact of direction on a play. Plays are meant to be staged and performed, and the production team can make choices that can enhance, detract, or add additional commentary layers onto an already complex production. In the 2004 Broadway revival of Assassins, director Joe Mantello chose to have actor Neil Patrick Harris play not just the Balladeer, but Lee Harvey Oswald as well. During “Another National Anthem,” where the script says the assassins force the Balladeer offstage, this production has them instead surround him, forcing him to the ground and concealing him from the audience as they sing the final refrain, then taking his flannel shirt and leather jacket to reveal the white T-shirt and blue jeans of Lee Harvey Oswald as the song concludes. This directing choice throws a dart into the idea that with enough antagonism, anyone can be turned into an assassin. Even the one person who has been spouting the traditional ‘American dream’ narrative the whole time, pitying the assassins, can be transformed into one by an uncaring world full of hate and vitriol. (Incidentally, Joe Mantello won the Tony Award for Best Direction of a Musical for his work on that production.)

Just because a play does not have a traditional narrator doesn’t mean the playwright has to suffer the uncertainty of whether or not his work will have the desired effect on the audience. He has a plethora of tools at his disposal and can take the full potential of the medium into account when creating a work. I chose to focus primarily on the idea of ‘limbo’ as a setting for this essay, but there are countless other ways to drive home a point with finesse and subtlety through the medium of theater. When you take into account directing, production and acting choices, the playwright never has to fear that his audience won’t get the point. They’ll come into the theater expecting to fill in the blanks, and all he has to do is leave those blanks in the appropriate places.

Ellen’s Choice: Applying my Rules for Effective Time Management to the SAT, part 2: Break Time!

Ellen’s Rules for Effective Time Management, Part 2

3. Know when it’s time to take breaks.
Spending a good chunk of time on one subject is good; it helps you settle into a rhythm and lets your brain get into the correct frame of reference for the subject. But there exists a horizon beyond which no progress can or will be made. It’s the point at which your brain has become over-saturated with the current material, and if you continue on you’ll just end up working yourself into circles of frustration. In paper writing, it’s the point at which anything you wrote would make sense to you regardless because you’ve been reading the same few paragraphs to yourself for hours. In math, it’s the point at which you will just end up confusing yourself more and more as you try desperately to work it out. When that moment arrives, you know it’s time to take your break.

I don’t care how much work you have, there’s always enough time for a fifteen-minute break. The trick is making sure that that fifteen minutes doesn’t turn into two hours. Stick to your schedule, take your breaks when you need them (and take one every couple of hours even if you don’t think you need one), and you’ll stay refreshed and energized much longer.
And on those days when you’re fortunate enough to have plenty of time to finish something, take advantage of the time to take more breaks. Feeling confused about that one paragraph in your paper you can’t seem to get right? Stop writing and go take a walk. Let your mind forget completely about your paper for an hour or so. Then come back and you’ll have a much better idea of how to progress. Sometimes, especially with paper-writing, you just have to give it time to process.

Once again, these rules apply to studying for standardized tests just as much as any other subject, but with a slight twist. My rule of thumb for studying for a standardized test is to simulate test conditions as much as possible as often as possible during the studying process. For a standardized test, this often means timed drills. Before the big SAT redesign of this past year, the SAT was comprised of nine sections averaging 30 minutes each, which meant practicing staying focused for 30 minutes and then quickly switching subjects. After the redesign, though, it works much more like the ACT, with fewer sections that are much longer, keeping each subject confined to one section. Now, the strategy is to practice maintaining concentration for an hour at a time. Take the first of the two points above. That horizon beyond which no progress can be made? We want to train that horizon to happen after the section is over, not in the middle of it. So practicing with a timer, noticing when your personal point of no return is, and adding just a few problems each successive time can help to extend your concentration past the end of the section, when it’s okay to stop thinking about it.

The second of those two points is important as well, though once again, on test day you don’t get to choose when you take breaks. You’ll be given a short break, possibly two, between sections at a predetermined time in the test. Training yourself to shrug off the previous section and start the new one with fresh eyes is important, because strictly speaking you’d want to take more breaks than the test allows you. Learn to give yourself a tiny break, even if it’s just five seconds with your eyes closed, to help re-center in between sections on the test. Even within a given section – if you’re one of the lucky ones who doesn’t struggle with finishing in time, that gives you more time to take mini-breaks. Take a moment between passes through the section or after a particularly daunting problem to shake it from your mind. No use dwelling on question 13 when there’s 45 more of them and you won’t get any new information from mulling it over and over.

Stay tuned for part 3: Mix and Match!

Mathematical Journeys: A Tale of Two Contexts

Sometimes the same procedure shows up in two different contexts. This is especially common in the fields of math and science, as science employs in real-world application many of the techniques we learn in their abstract form in math class. For some reason, the principle as shown in a high-school science class is often much harder for students to understand than it was in the math class. (My personal theory is that science teachers are applying the concept in a way that changes how they explain how it works, and they probably have not collaborated with the student’s math teacher to ensure they’re reinforcing the same terminology.) Last week one of my students ran into this phenomenon in her own work; a concept from last year’s math class showed up in her physics class. To help her understand it, we went back to the original math concept and talked about proportions.

The science homework she was struggling with was the old chestnut about unit conversions; rows and rows of fractions set up in perplexing fashion, with equals signs between each one and confusing scribbles across and between them. When taught in physics, the idea of unit conversions frequently becomes clouded and nonsensical – it becomes a feat of rote memorization to attempt to figure out what the correct procedure is for each specific case. In truth, unit conversions are a simple case of mathematical proportions, and the easiest way for my student to understand it was to connect it to the original context in which she learned about proportions last year – similar triangles.

A proportion is a pair of fractions that are set equal to each other. Remember – one of the definitions of a fraction is a ratio (comparison of two quantities), so setting two fractions equal to each other signifies that these two ratios are equivalent. We use this to understand similar triangles in early algebra, working through the logical reasoning process to grasp the concept that increasing all parts of the figure by the same amount will maintain the relationship of parts to each other. Unit conversions are simply another application of this same idea.

Let’s say your physics problem involves the length of a bridge. The problem states that the bridge is 3200 feet long, and wants to know how many meters long that is. We’re not doing anything to change the bridge itself, we’re simply changing how we measure it. That can be analogous to finding an equivalent fraction – we’re not changing the value of the fraction, just how it’s written. No matter whether we measure the bridge in feet, meters, inches, or miles, the relationship of the various parts of the bridge to each other (the proportions of the bridge) will stay the same. So this is a proportion problem.

For a proportion problem to work, we need two fractions. We only have one number in the problem, so where are the rest of the puzzle pieces? They’re implied, in the form of something called a conversion ratio. The conversion ratio is the fraction which indicates how the two units relate to each other – in our case, how many feet are in one meter. If we know that, it’s a simple similar-triangles-type problem to figure out our conversion.

So what’s the conversion ratio for feet into meters? Well, usually in a problem like this, the conversion ratios are either easy to find out for yourself, or they’re provided for you. A quick internet search reveals that one meter is equal to 3.28084 feet. So, without doing any math just yet, common sense tells us that converting the 3200 feet of the bridge into meters should give us a smaller number than we started with – there are multiple feet in one meter, so we’ll be dividing the number of feet up into one-meter chunks. (I find this kind of common sense flagging is helpful for these kinds of problems.)

So let’s figure out our proportion!

1 meter/3.28084 feet = x meters/3200 feet

I find it helpful to read this sentence to yourself as an analogy: “One meter is to 3.28084 feet as x meters are to 3200 feet.”

We know we have the proportion set up correctly because the matching units are on corresponding sides of each fraction. Just as a similar triangle problem needs the corresponding sides to match up with either the top or bottom of each fraction, unit conversions need to match up the units the same way.

You know what to do from here, right? Cross-multiply!

3.28084x = 3200(1)

x = 3200/3.28084

Hold up there for just a second. Remember a bit further up when I flagged for us that we’d be “ dividing the number of feet up into one-meter chunks”? Check it out – that’s exactly what we’re doing! Taking the total number of feet and dividing it by the number of feet per meter. That’s another way it’s sometimes described in science classes:

total feet [divided by] feet/meter

and when you divide by a fraction, you multiply by the reciprocal, so:

total feet * meter/feet

Sometimes science teachers will make a big show out of placing the fractions like this, so that they can physically cancel out the feet and all they have left is meters. I’ve found this to be helpful in calculations that require multiple conversions (inches to kilometers, for example), since it makes sure you keep the units straight, but it’s just as easy to do the proportion problems one at a time, and these days you can usually find the conversion ratio to go straight from one to the other anyway. A quick internet search tells me the conversion ratio for the example I just gave is 39370.1 inches per kilometer. Boom. Done. Write your proportion and go to town.

Anyway, heading back to our original problem, we see that:

x = 3200 feet/3.28084 feet per meter

x = 975.36 meters

Which does satisfy our initial common sense check, as it’s considerably smaller than the starting value.

In these examples, I find it much more useful to think of them as proportion problems in the vein of the similar triangle problems we practice so much in algebra. The algorithm is the same, it’s just the context that differs, and if that’s the case, why not think of the problem using whatever context makes the most sense to you?

To learn more about similar triangles, check out my blog post about that time my TV broke.

Ellen’s Choice: Applying my Rules for Time Management to the SAT

Way back in 2010, one of my first blog post series on WyzAnt took the form of a five-part series on rules for effective time management.  For the next few Ellen’s Choices, I’ve decided to go back through these rules and apply them to the world of preparing for the SAT (or any standardized test).

So let’s begin with Part 1: All-Nighters Are Evil

Ellen’s Rules for Effective Time Management

1. Never pull an all-nighter.
2. NEVER pull an all-nighter!

Seriously! I mean it. All-nighters are downright useless. Besides the fact that this concept breaks almost all of my other rules for effective time management in one go, all-nighters cause fatigue, stress you out, and just end up producing sub-par work. You can’t write well when you’re tired, and staying up all night studying just means you’ll be yawning all the way through the test the next day. If you haven’t learned the information on the test by the night before, you’re not going to learn it in one fatigue-inducing night of sleep-deprived cramming. If your paper isn’t written by the night before, then it’s not going to be a good paper, regardless of how late you stay up to finish it. Moreover, even if you do manage to retain something, depriving your brain of the rest and reorganization that sleep provides means it will be that much harder to recall it during the test when you need to. Bottom line is, your brain works best when it’s well-rested, so don’t deprive it of the one thing it needs to help you succeed.

This rule works well for prepping for any standardized test. Many times I’ve had a student come to me saying, “I’m taking the test next weekend and I just realized I need help!” Unfortunately, standardized tests don’t function like regular school tests, and just knowing the material covered won’t necessarily get you the score you’re looking for. To really do well on a test like the SAT, you need to know how to take the test itself – strategies and tricks for pacing yourself and figuring out the desired answer even when you don’t know the material in question. And to get used to those strategies and tricks, you need to practice. Prepping for the SAT involves leaving yourself enough time to practice taking the test repeatedly, so that you can analyze your performance and figure out where to focus your efforts to increase your score. If you haven’t gotten to your target score by the night before your test date, cramming won’t change that. In fact, most SAT prep books mention this specifically, recommending that you take the night before the test to rest and refresh yourself with something you enjoy – go see a movie with friends, have a nice dinner out, read a good book. Just don’t stay up too late; you need to get a good night’s sleep before your early test date the next morning!

Stay tuned for Part 2: Take Breaks!