Geometry Hacks

Did you know you can use a ruler, protractor, and straight-edge on the SAT/ACT? 

Actually you can’t, at least not prohibited plastic or metal ones. 

"Too bad," you might be thinking. It would make answering most geometry questions a whole lot easier.

But you can easily and legally improvise each of these tools with materials you're allowed to use on the SAT/ACT. 

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Ruler

Along the edge of your scratch paper, mark the endpoints of a segment of known length taken from the given figure. Add a mark in the middle, to indicate the half-way point. Knowing the overall length of this “ruler,” and the length of the half-way mark, you can lay the ruler over the segment you need to find in the figure (on screen or on paper) to closely guess its length. Now, eliminate answers, and choose the best one remaining.

Protractor

You can create a “protractor” by using the right angle at any corner of your scratch paper. Carefully fold your scratch paper edge-to-edge at the corner. This forms a perfect 45 degree angle. Fold it again, like you’re making a paper airplane. Unfold and flatten the page. The angles formed are 22.5 degrees each. You’ve now created a “protractor” with angles 22.5 degrees, 45 degrees, 67.5 degrees and 90 degrees.

Straight-Edge

The edge of a sharply-folded scratch page makes an excellent edge for quickly drawing perfectly straight lines. In mathematics, neatness matters, nowhere more so than in the realm of geometry.

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Note: 

To form nice, sharp edges, run your pen or pencil firmly across the folds as you create each crease.

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

The New Online ACT

It had to happen. Following the lead of the SAT, the ACT has been streamlined and is now offered online.

In its new format, the ACT takes less time to complete (only 125 minutes), there are fewer questions in each section (with more time per question), math questions have only four answer choices instead of five, English and Reading passages are shorter, and the science section is optional.

Beginning in April 2025, students can choose to take the ACT on paper or online. The new format will be introduced in April 2025 for online ACT tests only. From September 2025 forward, all ACTs will be given in the new format.

Click here for complete official information about recent changes to the ACT.

ACT now provides students with practice in the new digital format. Unfortunately, only a single official full-length online ACT practice test is offered, separated into section tests (English, Mathematics, Reading, Science).

To find the sample test, go to ACT Official Online Practice Tests, and click the button near the bottom marked "Launch ACT Free Online Tests" (you'll first need to create a MyACT account). 

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Contact Chris

borlandeducational.com

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

The Backdoor

To enter, if the front door's open, walk right in. If it’s not, go ‘round back. This common sense advice applies to the SAT/ACT math section, as well.

While “doing the math” should always be plan A, plan B is to use a strategy, a “backdoor” approach to selecting the right answer. If the math is easy, do the math. If not, use a strategy!

Following are two classic backdoor strategies for answering questions on multiple-choice math questions. Each of them takes practice to master, but the advantage gained is well worth the effort. 

Try the answers

Math problems are typically solved by working forward from given information to the answer. This approach is the only option on typical “fill in the blank” tests taken in school, and on the SAT and ACT, students are expected to employ the same approach. But on multiple-choice tests, students have another option: it might be better to work backwards, from the answers to the problem. If answer are all numerals, this a good strategy to consider. 

Here's how it works:

Pick an answer choice. Take that answer back into the question and see if it’s consistent with all the given information. If there’s any inconsistency, eliminate that one, and try another. Only the correct answer will work. 

This strategy works best when answer choices are all numerals.

[Note: Since answers are arranged numerically, starting your search in the middle of the list will enable faster elimination and likely save time.]

Make up numbers

If variables are confusing you, don’t use variables. Use simple, real numbers instead! Here's how it works:

Pick simple sample values for the unknowns (don’t use 1, 0, or any number already appearing in the question), and answer the question using those numbers. Plug the same values into the answers, and pick the one that gives you the same result.

This approach works best when solving algebra problems and you're tempted to write some kind of difficult equation and/or when the answers all contain variables.

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borlandeducational.com

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

Best Local Stargazing and Dark Sky Sites

Nothing says applied mathematics like physics, and nothing says physics stars and cosmology.

Following is a list of best places in the SF Bay Area and around California to find dark, clear skies ideal for star gazing.

A good pair of 7x50mm binoculars makes stargazing even more amazing. Click here for a great deal on an excellent set for beginners.

For those who'd like to pursue the hobby further, click here for info and tips on how to get started as a "backyard astronomer."

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Santa Rosa Clear Sky Chart

California Clear Sky Charts

Where to catch a clear view of the stars (Press Democrat)

The best places to stargaze in the Bay Area (SF Gate)

Star Gazing in the Bay Area

10 Twinkling Places to Stargaze in the Bay Area

Top 10 Places To Stargaze In The Bay Area

San Francisco Amateur Astronomers (SFAA): Dark Sky Sites

SFAA – Star Viewing Nights on Mount Tam (Members Only)

Top California RV Parks for Stargazing

These Are California's 10 Darkest Places For Stargazing

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Contact Chris

borlandeducational.com

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

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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Contact Chris

borlandeducational.com

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