Exponent Rules

After addition, subtraction, multiplication, and division, exponentiation serves as the 5th and final arithmetic operation.

Calculations involving exponents are crucial in algebra and are a major feature of SAT/ACT math. Seven basic rules and two additional corollaries govern exponentiation.

It’s important to understand these principles well and master their use through practice and application.

Below are the laws governing exponents you'll need to know and follow.


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Basic Exponent Rules (A ≠ 0, B ≠ 0)

Product of Equal-Base Powers: A^m*A^n = A^(m+n)
c: z^3*z^4 = z^7. 

Quotient of Equal-Base Powers: A^m/A^n = A^(m–n).
For example: x^-3/x^5 = x^-8.

Power of a Power: (A^m)^n = A^(mn)
For example: (y^3)^4 = y^12.

Power of a Product: (A*B)^n = (A^n)(B^n)
For example: (x^2*y)^3 = (x^6)(y^3).

Power of a Quotient: (A/B)^n = [(A)^n]/[(B)^n]
For example: (x^7.5/y^-2)^2 = [x^15]/[ y^-4].

Zero Powers: A^0 = 1 (A ≠ 0)
For example: (2z–1)^0 = 1 (z ≠ 1/2).

Negative Powers: A^-n = 1/(A^n)
For example: x^-3 = 1/(x^3).

Additional Corollaries

Quotient of Negative Powers: A^-m/B^-n = B^n/A^m
For example: y^-1/z^4 = z^-4/y^1.
(Changing positions of the lower and/or upper powers changes the signs on those exponents)

Negative Power of a Quotient: (A/B)^-n = (B/A)^n
For example: (x^-1/y^8)^-4 = (y^8/x^-1)^4.

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For practice, search Google for “exponent rules worksheet,” 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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General Functions – What You Need to Know

Much of high school algebra revolves around the study of input/output machines called functions, one of the most widely applicable concepts in all mathematics. Naturally, functions comprise a large fraction of questions found on the SAT/ACT. Fortunately, only knowledge of basic facts and processes is required.

Here’s what you need to know, generally, about functions. 

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[Note: “iff” means “if and only if.”]

Definition

A function is a relationship between two sets of numbers, one containing inputs and the other for outputs; these sets are called "the domain" and "the range," respectively. A function can can be understood as an input/output “machine” that takes a number in and returns a corresponding number out, such that no input is associated with more than one output. Normally, the input is called x and the output is called y. The function itself is named with a single letter, like f, in which case the output for general input x can be written “f(x),” pronounced “f of x.”

y = f(x) 
y and f(x) are interchangeable. 

Function values
The “value of a function” is an output value (y value).

Operations
The essential operation with functions is substitution.
“g(n)” means substitute n for x in function g.

Composition of Functions
Composite functions are “nested” functions. “f[g(x)]” means function g is nested inside function f.
For example: To find f[g(2)], first find g(2) and then substitute that value into f. 

Zeros of a function
Values of x (input values) that make y (output values) equal zero.
Zeros are found at x-intercepts.

When f(x) = 0, solutions are called “roots.”

Solutions iff roots iff zeros iff x-intercepts (“roots,” “zeros,” “solutions,” “x-intercepts” are essentially synonymous).

Intercepts of functions:
To find intercepts, let the other variable’s value be zero

For example: For the y-intercept, let x = 0).

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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.

Unit Conversions

Kilometers to centimeters? Gallons to tablespoons? Feet-per-second squared to miles-per-hour squared? 

Unit conversion is a pre-algebra topic that stops many students in their tracks. Questions about converting units pop up routinely on the SAT/ACT.

Basic conversions are easy to calculate by simple multiplication or division. More difficult problems require “Dimensional Analysis,” an easy and reliable way to perform conversion calculations.

The method is based on the following facts:

1. Equations relating units enable the creation of two fractions whose values are 1;

2. Multiplication by 1 never changes values and is therefore always allowed; 

3. "Per" implies division.  

To see how they enable the conversion of units using Dimensional Analysis, let's look at two questions.

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Simple example

1 mile = 5280 feet. Therefore, 1 mi / 5280 ft and 5280 ft / 1 mi are two fractions with values = 1. Let's convert 45 miles into feet. First write, as a fraction, the quantity to be converted: 45 miles / 1, and then multiply by 1 in the form of 5280 ft / 1 mi (we choose this fraction, with miles below, in order to cancel-out miles). Cancelling “mi” above and below leaves “ft” as the unit and 45 * 5280 as the calculation. So the answer is 237,600 ft.

Complex example

We'll convert 3500 meters per second squared to kilometers per hour squared. First write, as a fraction, the quantity to be converted: 3500 m/s^2. Since 1 kilometer = 1000 meters and 1 hour = 3600 seconds, multiply the initial fraction by 1 in the following forms: 1 km / 1000 m, 3600 s / 1 hr, and 3600 s / 1hr (to cancel s^2 below). Cancelling above and below leaves “m/ hr^2” as the unit and 3500 * 3600 * 3600 / 1000 as the calculation. So the answer is 45,360,000 km/hr^2.

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For practice, search Google for “converting units dimensional analysis worksheet,” 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.

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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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.

Essential Geometry

For more than 2500 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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Contact Chris

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

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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Contact Chris

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

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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Contact Chris

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

Orphaned Strategies

This post is intended as a home for useful but orphaned math strategies. Included are Triangles Within Triangles, Avoid Fractions on the SAT/ACT, and Two Reactions to Right Triangles.

Proceed to peruse these and other SAT/ACT math and general approach tips.

[Note: “iff” means “if and only if.”] 

Triangles Within Triangles 

On the SAT, whenever a triangle appears inside another triangle, all triangles will be similar, and setting up and solving a proportion will solve the given problem.

Avoid Fractions on the SAT/ACT

It’s often best to entirely avoid fractions on the SAT/ACT. Instead, fractions can be converted to decimals to make numbers more intuitive and calculations, estimates, and comparisons easier.

Two Reactions to Right Triangles

Whenever encountering a right triangle in a math problem, immediately ask two questions: Could I use the Pythagorean theorem? Could I use SOH CAH TOA? Nine times out of ten one of those approaches will lead to the solution.

No Solutions

Questions referencing equations or systems with “no solutions” are common on the SAT/ACT math tests. They mainly come in two flavors: quadratic equations, and linear/non-linear systems.

SAT/ACT questions involving quadratic equations with no solutions can be solved using the Discriminant: D = b^2–4ac. There are no real solutions to a quadratic equation iff D < 0, and this inequality will usually solve the given problem.

In general, linear or non-linear systems with no solution produce graphs with no points of intersection. Linear systems have no solutions iff the lines graphed are parallel iff slopes are equal. Desmos can be employed to draw the graphs of both equations, and see what needs to be done (e.g. with sliders) to ensure required conditions are met.

40 Top Tips for Taking Standardized Tests

Click here to read my collected musings and battle-tested general approach tips for improving scores on standardized assessments like the SAT and ACT.

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

Analytic Geometry – What You Need to Know

Analytic geometry is a core sub-topic in all algebra courses, blending geometry and algebra on a coordinate grid measured by x and y axes. 

As the story goes, famous French polymath René Descartes came up with the idea after watching a fly crawl up a wall.

These questions comprise a large fraction of SAT/ACT questions, and students must fully command key elements.

Following is what you need to know. 

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Equation of a line 
y = mx+b.
m = slope, b = y-intercept.

Equation of a parabola

Vertex form: y = a(x-h)^2 + k, vertex is (h,k).

Standard form: y = ax^2+bx+c, vertex is (h,k) with h = -b/2a, k = f(h).

Equation of a circle
(x-h)^2+(y-k)^2 = r^2.
Vertex is (h,k), radius = r.

To write the equation of a line containing 2 given points (x,y):
Use a Desmos table together with y1 ~ mx1+b to return m, b.

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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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Contact Chris

borlandeducational.com

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

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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Contact Chris

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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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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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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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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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borlandeducational.com

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

borlandeducational.com

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

U.S. News State Educational Rankings

U.S. News is famous for its annual college rankings, but it also ranks states on a number of educational benchmarks. Their recent College Readiness Rankings are a revelation.

Not surprisingly, Northeastern states occupy half of the first 10 spots. 

California? Number 49. Ouch.

When I was schooled as a boomer kid, California public schools were the best in the country – and, therefore, the best in the world. Number 49 is very hard to take.

Annual U.S. News college rankings (and others) are notorious as less-than-stellar indicators of college caliber. But they do, at least, provide a sense of relative quality. Similarly, one should probably take these overall educational rankings of the 50 states with a large grain of salt.

Nevertheless, if you'd like to see them, click here for the complete state rankings.

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Contact Chris

borlandeducational.com

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

GPS for Functions

Functions are of central importance in algebra, and problems based on functions form a major subset of questions appearing on the SAT/ACT.

Wouldn’t it be awesome if you had a easy, reliable way to navigate straight to the answer to any question involving functions? 

"GPS" will do it for you. 

The acronym outlines three options that will help solve any function-related question on the SAT/ACT. 

“G” stands for “graph.” (Would a graph be helpful?)

“P” stands for “points.” (Could you use coordinates of specific points?)

“S” stands for “substitute.” (Does a simple substitution solve the problem?)

Looking at virtually any function question on the SAT/ACT through these lenses will quickly and easily reveal the path to the answer.

Many times solutions can be found through simple visualization or basic pencil and paper techniques. 

Otherwise, Desmos can be used to carry out graphing, locating points, or doing substitution.

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For practice, search Google for worksheets covering “SAT ACT function problems,” 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 the GPS strategy.

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Contact Chris

borlandeducational.com

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

Teachers are Quitting

It seems our schools are always in trouble for this reason or that one. 

This was not at all the case when I was a boy in the 1960’s. California public schools, K-College, were unquestionably the best in the world, and all were levels were free or very low cost. So it’s especially hard for boomers like myself to see the kite hit the ground like this.

As a private academic coach for more that 45 years, I’ve tried to figure out what went wrong. I still don’t know, after all these years.

And now, a sobering new trend is emerging. Teachers are leaving K-12 schools in droves.

The reasons are many and various. But what I hear over and over again is that they boil down to several core issues: low salary, poor mental health, toxic work cultures, unsupportive administrations, lack of respect inside and outside school, lack of authority in the classroom, and out of control students and parents.

I have no answer, but I do have suggestions. At a minimum:

Pay teachers like doctors, require the same achievement, give them the same respect inside and outside schools, and weed out poor teachers (the Finnish approach); give teachers back-up they need and deserve within schools hierarchies; return classroom authority to teachers; allow sensible grading and disciplinary procedures, including compassionately but unapologetically holding back students who don’t meet grade level standards; a teacher-aid in every classroom; effective protection against parent-zillas think their little Johnny can do no wrong; make it unnecessary for teachers to buy learning supplies their students need; and provide enough public funding to pay for all this, realizing that doing so is less expensive than not.

For further information, I recommended the following:

I quit teaching for better mental health: former teachers share the jobs they got after teaching

Why Teachers Quit: Lack of respect, abominable working conditions, and more.

After Teaching For 11 Years, I Quit My Job. Here's Why Your Child's Teacher Might Be Next.

I quit my job as a teacher after 6 years to work in tech sales. I make $20,000 more ...and am so much happier now.

Teachers Who Quit Are Sharing The Moment They Realized It Wasn't For Them

Why Teachers Quit + Top Signs Quitting Teaching Is The Right Move

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Contact Chris

borlandeducational.com

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

Solving Literal Equations

To beginning algebra students, literal equations, those involving more than one letter, often seem inherently more difficult to solve than simpler, univariate ones. But this isn’t so. 

The trick is to treat the extra letters like simple numerals using the same steps you would ordinarily.

Solving equations is a matter of “undoing” what’s been done to the variable, using inverse operations, starting as far from the variable as possible.

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Example 1

To solve 3(x+4)–8 = 19, we first undo subtracting 8 by adding 8, then divide by 3 to undo multiplying by 3, and finally subtract 4 to undo adding 4. The result is x = 5. [Note: we could first simplify by distributing 3 across (x+4) and adding like terms, but this would take four steps, not three.]

If the letters a, b, and c were to replace 3, 4, and 19 in the same equation, we’d carry out exactly the same series of steps. This time we’d start with a(x+b)–8 = c. We’d then add 8, divide by a, then subtract b to get x = (c+8)/a – b.

When solving univariate equations, it’s important to add like variable terms as soon as possible. Sometimes this isn’t possible when solving literal equations. In such a case, factoring out the variable leads to a simple solution.

Example 2

To solve (nx–mn)/q = x+p for x, we first multiply by q and then collect and isolate variable terms on the same side by subtracting qx and adding mn to yield nx–qx = qp+mn. Since nx and x are not like terms, we factor out the variable x to produce x(n–q) = qp+mn, and divide by n–q. The result is x = (qp+mn)/(n–q).

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For practice, search Google for “ solving literal equations 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 important topic.

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Contact Chris

borlandeducational.com

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

Desmos – the New Standard

The online Desmos graphing calculator is fast taking over from the venerable Ti-84 series of handheld calculators as the default calculator tool in secondary education. Desmos is now included as an integral part of the digital SAT, and acquiring intermediate-level Desmos skills is fundamental to maximizing math scores.

[Familiarity with the Ti-84 Plus CE handheld graphing calculator is still crucial to optimizing math scores on the ACT.]

I'm not aware of any succinct, comprehensive exposition of Desmos skills required for use on the dSAT (I'm working on it).

At this point, the best one can do is to peruse the various official materials linked in the "Desmos First Steps" and "Desmos Graphing Calculator" sections below. 

Check out each link, read the information provided, and do the sample exercises until you've covered all topics presented (search Google for additional help with particular topics).

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Desmos First Steps

User Guide

Quick Start Guide

Getting Started: Desmos Graphing Calculator

Getting Started: Creating Your First Graph

Getting Started Articles

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Desmos Graphing Calculator

Graphing Calculator

Graphing Calculator: Essential Skills

Graphing

FAQ: Graph

FAQ: Student Graphing

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Desmos Geometry

Geometry

Geometry Tool

Transformations

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Other Desmos Calculators

Scientific Calculator

Matrix Calculator

3-D Calculator

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Desmos Miscellaneous

Digital SAT Math Training – DESMOS

1600.io Mastering Desmos Course

Tutorials

Desmos YouTube Channel

Keyboard Shortcuts

Help

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Contact Chris

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