Guessing Techniques

Intelligent guessing is key on the SAT/ACT. Making sensible guesses isn’t always easy, however. Fortunately, a few simple strategies make it easier for students to improve the odds of picking the right answers to questions that have them stumped. 

Joe Average

Meet Joe Average, the typical high school math student. On hard end-of-section questions, Joe always picks an answer he can understand, one that’s easy to get. Since questions near the end are always hard, any answer Joe would pick for these questions will be wrong. Since Joe always falls for trap answers, those too simple to be correct for questions coming near the end of the test, you should eliminate “Joe Average answers” to any question near the end (i.e. in the last third) of the math section.

Hard Questions, Hard Answers

In general, “Hard questions have hard answers.” When guessing on a problem near the end of the math section, after elimination, avoid easy-looking answers (simple numerals or expressions) and pick the hardest-looking answer choice (one involving square roots, parentheses, fractions, etc.).

Imposters

To fool students into picking wrong answers to hard questions, correct answers are often hidden among similar-looking answer choices. When guessing, you should favor “imposter” answers, those trying to impersonate the others (i.e. those with the greatest number of common features), and eliminate outliers.

The Last Letter

Answers near the end of the math section tend to be near the end of the alphabet (test makers deliberately design tests this way, knowing most students will examine the first answer first). Since you’ll be guessing mainly on harder questions coming near the end, unless you have a good reason not to do so, you should pick an answer at or near the end.

The “Last Letter Strategy” is a “guessing machine” that quickly and easily provides the best guess on any question. After eliminating wrong answers, always pick the last available answer choice. For example, if both “B” and “C” have been eliminated, pick “D.” If only “D” has been eliminated, pick “C.”

When blind guessing, it’s best to pick randomly and quickly move on without further thought. Save time and energy. Just obey the guessing machine!

Don’t Second Guess

Never change a first guess to a second guess. The intuition used to make your initial pick may give you an advantage. If later on you realize with certainty that your guess isn’t correct, you should, of course, change it.

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Mischievous Engineers

Easter Eggs aren’t just for kids, and they aren’t found only on Easter – but they’re always hard to find and never fail to spark joy. 

Hiding “Easter Eggs” in software began in earnest in the 1970’s and continued through the Atari era into the modern age of computing. Engineers with too much time on their hands would deliberately program all kinds of surprises (little games, silly graphics and animations, text info, etc.) into their software projects.

Although Easter Egg grinches like Steve Jobs and Bill Gates banned the practice within their own companies, comedic Google engineers have managed to continue the tradition.

A Business Insider article gives a partial rundown of hidden tricks and treats to be found within the Google search bar. Give some of these a try! 

Easter Eggs provide a window into the minds of bored code monkeys, and furnish fatigued students and professionals a way to punctuate their day with diverting amusement.

A Wikipedia entry provides historical context and further info.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Percents

Many students don't have a secure understanding of percentages. This is a problem, for several reasons.

Percentages pervade our lives, and so it's important to have a good "feel" for them. On-the-fly estimates involving money, medicine, politics, and the like often require their calculation. Percent questions also frequently appear on the SAT/ACT.

In a nutshell, percents are fractions with denominator 100. The “50-25-10” rule enables use of simple unit fractions as guides in estimating percentages: 50% = 1/2. 25% = 1/4. 10% = 1/10.

Example 1

To estimate 62% of 48,300, first, round 62% to 60%, and 48,300 to 48,000, for convenience. 60% is 10% more than half. Half of $48,000 is 24,000, and 10% of 48,000 is 4,800. Altogether, this makes $28,800. Since we rounded down, adjust the answer up a little, to perhaps 30,000. The correct answer is 29,760.

To work out tricky SAT/ACT percent problems, it’s sometime best to pick a sample value to work with, and see what happens. In such a case, 100 is a good default choice.

Example 2 

On your test you’re asked to find the percent of change when a number is first increased by 10% and then decreased by 10%. The trap answer is to assume this is a wash, that there’s no change at all. But using 100 as a sample value enables us to find the surprising answer quite easily. Increasing 100 by 10% yields 110. 10% of 110 is 11, and decreasing 110 by 11 produces 99, which is 1% less than 100. The percent of change is 1%, not 0%.

Problems involving “percent of increase or decrease” would seem to require two calculations, but in practice these questions can easily be answered in a single step. First, simply add or subtract the percent of increase/decrease to/from 100 percent. A 70% decrease equates to direct calculation of 30% of the number. For a 60% increase, take 160% of the number.

Example 3

Suppose your dentist gives a 5% discount to patients who pay at the time of service. Your dental work will cost $420. You could first find 5% of 420, and then subtract this number from 420. But that requires two steps. Instead, remember that 5% off is the same as 95% on! So, simply calculate 95% of 420, and you’ve got your answer: $399.

Example 4

You’re having dinner out and it’s time to pay. The cost of the meal is $74, total, and a 15% tip is standard. You could first calculate 18% of 74 and add that back, but again that’s two calculations. Instead, realizing that a 15% increase produces 115% of the original number, you could simply multiply 74 by 1.15, to arrive at the amount to pay: $85.10. 

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For practice, search Google for “SAT ACT percent problem worksheets,” pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Trivium and Quadrivium

The reason I've always been captivated by the Trivium and Quadrivium is almost certainly that these ancient western educational models happen to coincide with six main interests of mine: math, music, astronomy/cosmology, logic, writing, and debate. 

Moreover, philosophy, another one of my main interests, was considered such an obvious part of classic liberal arts training that it wasn't included in the list of subjects for either the Quadrivium or Trivium.

From the Wikipedia article on Quadrivium:

"From the time of Plato through the Middle Ages, the quadrivium (plural: quadrivia[1]) was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the trivium, consisting of grammar, logic, and rhetoric. Together, the trivium and the quadrivium comprised the seven liberal arts,[2] and formed the basis of a liberal arts education in Western society until gradually displaced as a curricular structure by the studia humanitatis and its later offshoots, beginning with Petrarch in the 14th century. The seven classical arts were considered "thinking skills" and were distinguished from practical arts, such as medicine and architecture."

One has to wonder what our society would look like if schools prioritized these essential subjects in grades K-12.

St. Ann Classic Academy is a school trying to implement such a curriculum.

For an excellent book on the Quadrivium, try Quadrivium: The Four Classical Liberal Arts of Number, Geometry, Music, & Cosmology. 

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Hybrid Notes

Word problems make most math students a little nervous.

Generally, it’s the translation from English into algebra that poses the problem. Instead of getting stuck, consider taking “hybrid notes,” written partly in English and partly in math, at least initially. Once you gain more clarity, you can shift completely into algebraic sentences (i.e. equations).

Do translation in stages, in baby-steps, rather than a single leap. First write notes that mix English and math (e.g. Expense = Burgers * Price, or Total Time = Time Running + Time Walking), then translate fully into mathematics as you gain more understanding.

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Example

You own a boat rental business. Boats rent for $100 per hour plus a $150 security deposit. On average, your expenses amount to 30% of the hourly fee for each rental. Write an equation relating profit and rental hours for a typical boat trip.

It’s hard to translate that into algebra in a single step. Using hybrid notes will help.

Profit = Revenue – Expenses.

Revenue: 100*hours+150
Expenses: .3*(hourly fee)

Hours: h
Hourly fee: 100h

Profit = (100h+150)–.3(100h) = 100h+150–30h = 70h+150

Answer: P = 70h+150

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If you’re dealing with a particularly puzzling word problem, don’t let yourself get tripped up over language. Direct translation from English to math may be too much to ask.

Instead, use hybrid notes to get things going.



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Contact Chris

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.