Friday, November 01, 2024

Seeing is Believing

On the SAT/ACT, test takers are warned figures aren’t necessarily drawn to scale. In recent years, however, questions with misshapen diagrams have become vanishingly rare.

Nowadays, unless a drawing is clearly distorted, students can assume all figures to be scale drawings. And from this can be inferred a tremendously helpful geometry strategy.

Based on the realism of figures drawn to scale, the notion that “seeing is believing” can be used to make good estimates helpful in answering even the most irksome questions.

For example: Angles that seem equal probably are equal. Lines look parallel? Call it true. If one segment appears to be slightly less than half the length of another, that can be assumed.

Known information in geometric figures can thus be used to “ball park” reasonable guesses about unknown information in the same figure, and this is often all it takes to find the correct answer or at least eliminate wildly incorrect ones.

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

Tuesday, October 08, 2024

Transforming Functions

Questions involving reflecting or shifting graphs stump a great many students.

But this needn’t be! 

Four simple rules govern all transformation questions encountered on the SAT/ACT. Master these laws, and all such questions suddenly become easy ones.

Listed below is what you need to know.

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Horizontal Reflection (across the y-axis)
Replace x with -x. 
For example: If f(x) = x^2– x+1, the horizontal reflection is f(x) = (-x)^2–(-x)+1 = x^2+x+1.

Vertical Reflection (across the x-axis)
Replace y with -y.
For example: If g(x) = 3x–2 i.e. y = 3x–2, the vertical reflection is (-y) = 3x–2 and y = -3x+2. Therefore, g(x) = -3x+2.

Horizontal Shift, h units
Replace x with x–h.
For example: If f(x) = x^2–x is shifted 4 units left, h = -4, h–k = h–(-4) = h+4, and the shifted function is f(x) = (x+4)^2–(x+4) = x^2+8x+16–x–4 = x^2+7x+12.

Vertical Shift, k units
Replace y with y-k (or simply add k to the function).
For example: If y = |6x–1| is shifted 3 units up, k = 3, y–k = y–3, and the shifted function is y–3 = |6x–1|. Therefore, y |6x–1|+3.

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For practice, search Google for worksheets covering any or all topics listed above, 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.

Tuesday, October 01, 2024

Essential Geometry

For well over 2000 years, since the time of Euclid, geometry has occupied a central place in the study of mathematics, and these problems form an important subset of questions encountered on the SAT/ACT. 

Luckily, the particular facts and concepts you need to know are few and easy to review. 

A comprehensive list of these elements follows. Make sure you’ve mastered each one.

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Perimeter (with radius r and diameter d)
Polygons: Add all sides.
Circumference of a circle = 2πr or πd

Area formulas (with base b and height h)
Triangle = bh/2.
Parallelogram: bh (includes rectangles and squares)
Trapezoid = h(b1+b2)/2.
Circle = πr^2.

Volume formulas
Rectangular prisms = Bh, where B = rectangular base area and h = height of the object (includes boxes, including cubes), 
Right cylinders = Bh, where B = circular base area and h = height of the object.
Right cones = (1/3)Bh, where B = circular base area and h = height of the object.

Famous figures
See study sheet here.

Triangle inequality theorem
The length of any side in a triangle must be between the sum and difference of the other two sides.
For example: The lengths 8, 10, and 2 could not form a triangle since 2 is not between 2 and 18.

Pythagorean Theorem: a^2+b^2 = c^2 (where, for any right triangle, a and b are legs and c is the hypotenuse
Apply the Pythagorean Theorem to find missing sides in a right triangle.
Use key right triangle “triples” (3x : 4x : 5x, 5x : 12x : 13x) to find missing sides in a right triangle.
Use ratios of sides in special right triangles (30-60-90 = x : x√3 : 2x, 45-45-90 = x : x : x√2) to find missing sides in a right triangle.

Parallel lines are cut by a transversal
Know how to use "big angles" and "small angles" formed to find measures of unknown angles in figures.

Regular hexagon
A regular hexagon can be divided into six equilateral triangles by drawing segments between opposite vertices. Each equilateral triangle can then split into two 30-60-90 triangles, from which various lengths can be inferred.

Questions involving circles and radii
Circle problems can often be solved by drawing radii to indicated points on the circle and noting that all radii have the same length.

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For practice, search Google for worksheets covering any or all of the above, 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.

Sunday, September 08, 2024

Scratch Paper Strategies

There are obvious uses for scratch paper on the SAT and ACT. There are other more creative uses for it, as well. 

How about making an improvised ruler and protractor for use in geometry problems? This trick is little known, but perfectly legal. 

Proper use of scratch paper is critical in tackling SAT/ACT math problems. Following is a list of things to keep in mind.

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When “doing the math,” write out all the steps.

Boil-down tough questions by jotting down notes about the clues you're given and what you’re trying to find (circle the main question words, underline the clues).

Write hybrid notes, half math, half English, to help make sense of difficult word problems. 

Keep scratch work neat and organized (mark notes with problem numbers, etc.).

Re-draw on-screen figures for convenience and for illustrating known information.

Ask for more scratch pages, if you need them. 

The SAT allows pens or pencils. Pencils can be mechanical pencils or wooden number 2 pencils. Don’t use mechanical pencils with .5mm lead (best to use .7mm or unbreakable .9mm lead). Bring at least two pens or pencils, in case one breaks.

The ACT only allows wooden number 2 pencils. Pens are prohibited.

Sharp pencils are best for scratch work. Slightly dull ones are better for filling-in bubbles quickly. Bring two of each, in case one breaks.

You'll need a good eraser, one that works and won't dig a hole into your paper.

If you're planning to use your pencils' erasers, first test each one by erasing fresh scribbling on paper. 

However, tiny erasers on pencils can easily break off. It's best to bring a new rectangular eraser or "click" eraser. Make sure to "break in" the one you'll be using by erasing fresh scribbling on paper.

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

Sunday, September 01, 2024

Concentration Hacks

Done correctly, preparation for the SAT/ACT cultivates essential skills not fostered in class yet vital to success in academia and beyond. One of these is the ability to generate robust, energetic concentration and deliberate, laser-like "winning focus.”

Highly intentional attitude lights up the brain like a Christmas tree, enabling students to think quickly and cleverly, solve problems creatively, and make the utmost of what they already know about mathematics. Maintenance of a sharp, energetic. mindful “winning focus” throughout the test is critical. This is so important that, without such attentiveness, almost nothing else matters.

Creating and sustaining optimal energy is vital to maximal success. Test taking is a competitive activity, and just as is the case in athletics or the performing arts, lagging attention and lackluster commitment won’t cut it.

Following are six “concentration hacks” that, in my work with students on SAT/ACT test prep over nearly five decades, have proven to be effective in developing students’ focusing skills.

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Good Questions

Asking a good question automatically initiates an internal quest, pointing and propelling the mind in a productive direction. “Good questions” lead us toward the goal; “bad questions” lead away from it. “What am I trying to find in this problem?” That’s a good question! “Why do I always mess up?” That’s a bad question. Asking and answering good questions is the best way I know get on track and stay there.

Verbalizing

By this I mean the process of discussing math internally, deliberately talking-out every step and calculation, mentally conversing with oneself about what one is doing and why at every moment. Verbalizing makes thinking conscious, draws out and connects ideas, exposes errors, and keeps the mind precisely attuned. For students not already in the habit, verbalizing can be trained.

Point and Trace

A visualization technique just as useful as verbalizing, “pointing and tracing” refers to pointing at and tracing each object mentioned in a geometry problem as one reads or thinks about the question. This makes key features of figures and diagrams stand out, allowing thoughts and ideas to gel and creativity to flow freely.

Tracking

The SAT and ACT are long hauls, and one of the first shoes to fall is reading comprehension. “Tracking" (physically pointing a finger or pencil at text as one reads) revives awareness and makes thinking “louder” and less likely to ebb. Point to get the point!

Thought Experiments

In these self-created multi-sensory imaginative experiences, students fully immerse themselves in the scene of an SAT/ACT word problem, mentally play it out, and closely observe what happens. Believing the question at hand to be an urgent matter (not just some arbitrary, boring word problem), the mind is compelled by the realness of the simulation to quickly find the right answer.

Get into it!

Enthusiastic engagement fuels concentration and creativity. The mind has a hard time telling the difference between a good act and factual reality, and, done convincingly, artificial excitement can generate the real thing. Fake it till you make it. Get psyched up. “This is great! I love this! What’s next!” Odd as it sounds, this actually does work.

Dream School

Write and underline the name of your “dream school” in large capital letters at the top of your scratch page, and return to this note whenever your energy starts to flag. Remember the reason you’re taking the test in the first place. This will automatically stimulate inspiration, motivation, and stronger focus. For extra effect, add an “!” point each time you do so. Employing this strategy repeatedly during practice testing has a cumulative effect, maximizing its impact on test day.

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