Friday, December 08, 2023

The Four Pillars

Successful math test prep rests on four pillars: skill, strategy, practice, and focus. To maximize scores, students must optimize performance in each of these domains. Nowhere is this more important than on the SAT/ACT. 

Of course, superior mathematical skill is required to achieve an excellent math score. This is obvious. Less well-known is the fact that mastery of key test taking strategies, implementation of a powerful practice regimen, and development of the ability to concentrate with an aggressive, razor-sharp, winning focus are often just as important.

A secure command of test material is the most fundamental prerequisite for success on the SAT/ACT. Unfortunately, improving mathematical skill is mainly a function of time. There are no shortcuts.

Years of diligent, dedicated effort inside and outside the classroom are necessary to develop “mathematical maturity,” a level of mastery at which reliable intuition starts to take hold, problem-solving abilities become potent and instinctual, and students experience math as a creative endeavor as well as an intellectual one. This is the level of competitive mathematics.

For a number of reasons, however, the vast, overwhelming majority of students never reach mathematical maturity. and must work on building strength in the three remaining pillars in order to reach impressive levels of performance on the SAT/ACT math test.

Strategy and tactics comprise the second pillar. Correct general approach and specialized test-taking strategies enable students to correctly answer difficult multiple-choice math questions with relative ease.

In 1946, Kaplan was the first company to offer SAT test prep courses. In the 1970s, Kaplan, Gruber, and others sold early test prep books alerting students to simple techniques they could use to quickly raise scores.

In the early 1980’s, the Princeton Review became famous for challenging the College Board’s claim that its SAT test could not be hacked, that test prep could only marginally improve scores. After a number of unsuccessful lawsuits targeting the Princeton Review, and the tremendous success and growth of the test prep industry in the 1990s, the College Board finally relented and began offering test prep strategy instruction in it’s own publications.

So, yes – learning and applying testing strategies can significantly improve results. Although compensatory modifications in recent years have made the SAT harder to game, mastery and consistent use of key test-taking strategies is still the best way to rapidly improve scores on both the SAT and ACT math tests.

Practice is the next pillar. An old story illustrates its importance.

A man was lost in downtown Manhattan, late for a concert. He stopped to ask a stranger for directions. The stranger happened to be a famous violinist. The man asked, “How can I get to Carnegie Hall?” The maestro answered, “Practice, practice!”

No athletic team competes for a championship without months of grueling daily practices. No good musician performs a recital until countless hours facing the same sheet of music are clocked before the big day. No professional dancer would think of performing without first mastering the choreography through a long series of arduous practice clinics and rehearsals.

We've all heard that “practice makes perfect.” Well, not really. Practice generally leads to improvement, however, and without lots and lots of it, most students won’t be able to significantly raise their SAT/ACT math scores. Disciplined, dedicated, dynamic practice is crucial in the quest for success.

But not all practice regimens are equally effective. At least 3-5 hours a week for several months must be devoted to practice testing. It will involve sacrifice. And there are rules:

Multi-tasking isn’t allowed. Realistic test conditions and use of official practice tests are musts. No music while studying (have you ever been allowed to listen to music while taking a test?). No late night practice work, and no test taking in the car or on the plane. The mantra: practice, critique, review, repeat!

Finally, an intense (even aggressive) yet controlled state of concentration is critical to performing at one’s best. This is true of any competitive activity, whether in athletic, performing arts, or academic contexts. Winning Focus enables students to make the most of what they already know, think quickly and creatively, and problem solve effectively. The ability to generate this state of mind at will is so important that, without it, almost nothing else matters. 

Most test prep coaches endeavor to raise scores solely by improving math skills. But this approach is misguided. Except in rare cases, a one-dimensional, skills-based approach cannot and does not take the place of simultaneous instruction in all four essential areas of preparation.

Working to bolster each of the four pillars is the best way to optimize SAT/ACT math scores.

-----


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

Friday, December 01, 2023

dSAT Reading and Writing Topics

This is the companion piece to a post I wrote earlier on dSAT Math Topics.

Questions in the Reading/Writing sections of the dSAT cover four broad topic areas: Craft and Structure, Information and Ideas, Standard English Conventions, and Expression of Ideas.

The entire Reading/Writing section is 64 minutes long with 54 multiple-choice questions.

-

Modules

Two Reading/Writing modules:

Each module is 32 minutes long, with 27 questions (1:11 per question)

Organized by question type, with similar skills grouped together

Questions within each group are presented in order of increasing difficulty

Each question has its own short passage or pair of passages

Passages are between 20–150 words

Wide range of topics, including literary excerpts and poetry

May contain informational tables and graphs 

-

Information and Ideas

12-14 questions, 26% of the section

Use information stated or implied in the passage (may include tables and graphs) to evaluate ideas in the text and draw conclusions.

Tasks:

Determine main idea and central details

Interpret details supporting the main idea

Use evidence from the text, graphs, and tables to support or respond to a claim

Draw reasonable inferences based on the passage

-

Craft and Structure

13-15 questions, 28% of the section

Focuses on vocabulary (words in context), text structure and purpose, and cross-text connections.

Tasks:

Determine the primary purpose of the text

Evaluate text structure rhetorically and how the author chose to organize information

Select the best word to complete the text (fill-in the blank)

Compare/contrast two related texts

Make supportable connections between two related texts

-

Expression of Ideas

8-12 questions, 20% of the section

Focuses on improving effectiveness of written expression. Test takers are asked to scrutinize topic development, accuracy, logic, cohesion, transitions, and language to improve, refine, and enhance the effectiveness and impact of a text and achieve specific rhetorical goals.

Tasks:

Analyze bullet points and choose the one that best combines information to achieve a certain goal

Identify the flow and structure of a passage

-

Standard English Conventions

11-15 questions, 26% of the section

Use editing skills and knowledge to make texts conform to standard conventions of written English, emphasizing proper word usage, sentence structure and form, grammar, and punctuation.

Tasks:

Identify correct punctuation to join and separate sentences and clauses 

Find the best way to complete a sentence so that it follows conventions of standard written English (e.g. proper agreement, verb tense, etc.)

-----


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

Wednesday, November 08, 2023

Trigonometry You Need to Know

Trigonometry is one of the most useful and widely applied branches of mathematics. It comes in two flavors: triangle trigonometry and circular trigonometry. 

Triangle trigonometry, the kind featured on the SAT/ACT, is primarily concerned with solving triangles, finding unknown side lengths and angle measures.

A tiny number of basic trig facts and processes are tested in a relatively small number of SAT/ACT math questions. Nevertheless, students aiming for top scores must master each of these elements.

Following is what you need to know.

-

Three primary trig ratios
Definition: SOH CAH TOA (sin, cos, tan).

Solving right triangles using SOH CAH TOA

Converting radians to/from degrees 
1rad = 180/pi.
1deg = pi/180.

Sinusoidal curves
Amplitude: vertical distance between midline and upper/lower bound.
Period: horizontal distance between repeating points on the curve (e.g. peaks).

Cofunctions of compliments are equal
For example: sin(50 degrees) = cos(40 degrees).

-

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 important material.

-----


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

Wednesday, November 01, 2023

Look Inside the Digital SAT

The digital SAT has arrived, 

In the meantime, much has been written about all aspects of this latest incarnation of the SAT. All along, The College Board has offered limited information concerning what we who care should expect.

But the private analyses of those whose businesses and livelihoods hinge on gaining an early, accurate, and comprehensive view of this strange new beast are also well worth noting.

In addition to those featured on the SAT/dSAT resource page of my business site, below are several more such reports.

-




-----


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

Sunday, October 08, 2023

Handle the Stupid Stuff

No amount of test prep tutoring, practice, careful review, or lost Saturdays and Sundays can make up for lack of sleep the night before test day. 

Put simply, sleep deprivation is death to test scores.  Without sufficient sleep, all that effort and personal sacrifice go right out the door.

Dehydration, emotional upsets, or late arrival at the test site can also wreak havoc on results.

It’s vital that students give themselves the best possible chance of success on test day. “Stupid stuff” can sink math scores like nothing else, and must be assiduously avoided.

Don’t ruin you score over stupid stuff!

To perform at your best, you should:

1. Get eight to nine hours of sleep on each of the three nights immediately preceding test day.

2. Drink extra water on each of the three days immediately preceding test day.

3. Go to bed 30 minutes early, and get up in the morning 15 minutes early, on each of the three days immediately preceding test day, using the extra waking minutes to study your most important test prep notes.

4. “Night before, stuff at the door.” Before going to sleep on the night before the test, put all the things you’ll need at the front door, so you won’t forget anything (admission ticket, pens/pencils, snack, calculator, laptop, etc.)

5. Eat a normal breakfast on test day, with protein for extended energy. But don’t get too full (digestion can drain energy). Drink a glass of water, use the bathroom, and bring a small snack (e.g. a power bar) to eat during breaks.

6. Don’t sit during breaks! Get up and stretch, walk around, relax, and think about other things (to recharge your body and mind).

7. Leave in time to arrive at the test sight at least 15 minutes early.

8. To provide extra motivation and “Winning Focus,” write the name of your dream college atop your test page or scratch paper. Glance at this note when fatigued, and dive back into the test with renewed vigor.

9. If you’re preoccupied with anything upsetting, procrastinate dealing with it till after the test. Put the negative experience entirely out of your mind. You can deal with it later.

10. If you find yourself stressing out during the test, break the pattern of anxiety by taking three deep, slow, “belly breaths” and/or subtracting by sevens from 100; then re-focus on the test question right in front of you. 

-----


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

Sunday, October 01, 2023

dSAT Math Topics

After trying and failing to find a comprehensive online resource that thoroughly detailed math content covered on the dSAT, I finally decided to put together my own. I've also written a companion piece on dSAT Reading and Writing Topics.

Questions in the Math section of the dSAT cover four broad topic areas: Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry/Trigonometry.

The entire Math section is 70 minutes long with 44 multiple-choice questions.

-

Modules

Two math modules:

Each module is 35 minutes long, with 22 questions (1:35 per question)

75% multiple choice (33 questions), 25% student produced response (11 questions)

30% in-context word problems (≤ 50 words)

Progressive difficulty (easiest first, hardest last)

Calculators are allowed – but don't use a handheld – learn to use Desmos, instead

Questions from each topic area appear in each module
-

Algebra

35%: 13-15 questions total

Linear equations in one variable (analyze, solve, create)

Linear equations in two variables (analyze, test, solve, graph, create)

Linear inequalities in one variable (analyze, solve, create)

Linear inequalities in two variables (analyze, test, solve, graph, create)

Linear functions (create, analyze, interpret, graph)

Linear systems of equations and inequalities (create, analyze, solve)

-

Advanced Math

35%: 13-15 questions total

Equivalent expressions (including algebraic fractions)

Non-linear equations (solve): absolute value, quadratic, polynomial, exponential

Non-linear functions (create, analyze, interpret, graph): absolute value, quadratic, polynomial, exponential

Non-linear systems (solve)

-

Problem Solving and Data Analysis

15%: 5-7 questions total

Ratios, rates, proportions, units

Percentages

Analyze and interpret one-variable data: distributions and measures of center and spread (mean, median, mode, range, standard deviation)

Analyze and interpret two-variable data: models and scatter plots

Simple probability

Conditional probability (from a data table)

Inference from sample stats and margin of error

Evaluating statistical claims: observational studies and experiments

-

Geometry and Trigonometry

15%: 5-7 questions total

Perimeter, area, volume

Lines, angles, triangles

Right triangles

Trigonometry (SOH CAH TOA)

Circles

-----


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

Friday, September 08, 2023

The Backward-Forward Process

Unfortunately, little or no time is spent in most math classrooms discussing "heuristics," the art of problem solving. 

This often leaves students grasping at straws, struggling even to know where to begin when staring down an unfriendly, unfamiliar math question.

George Pólya's How to Solve It is a classic on this subject, required reading for all serious math students.

In another classic, How to Read and Do Proofs, author Daniel Solow advances a powerful problem solving approach he calls the “Forward-Backward Method.” 

I’ve found it helpful in my own teaching and mathematical work to reverse the method, first thinking backward from the ultimate goal to various subgoals which, if achieved, would enable direct progress to the original objective.

Whether writing complex proofs or tackling simple algebra problems, this “Backward-Forward” process provides students with a simple yet powerful structure for solving problems.

Just as an archer would prefer to move the target closer, so can a math student make a problem easier by finding a nearer target to shoot at. The next step would be to think further backward, to find another even closer target tied directly to the first, and so on. These subgoals are set by repeatedly asking the same question: “What would I need to know to find that?” And then “What would I need to know to find that?” Subgoals should be written down, to keep the trail clear.

After looking backward as far as possible, it’s time to reason forward from each given fact, with the last subgoal in mind. A different question governs the forward process: “What can I imply from that fact?” And then “What could I imply from that?”

Eventually, forward progress enables us to hit the target. All that’s left is to follow the string of subgoals up the ladder to the desired result.

-----


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

Friday, September 01, 2023

Bring a Mouse to the SAT

Most people find use of a mouse to be a time saver, especially if you’re accustomed to employing one.

On the digital SAT, every second counts, and use of a computer mouse can save time on this important assessment. 

The College Board allows use of a "USB A" mouse (wired or wireless) with mouse pad on the digital SAT. Practice taking digital SATs at home using your favorite mouse. Then bring the same mouse with you on test day, plug it in, and go. Don't forget to bring a small mouse pad, as well.

This seems like a minor detail, but there’s almost no such thing when it comes to high-stakes testing.

-----


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

Tuesday, August 08, 2023

How to Handle Super-Hard Questions

For high scorers, only the hardest problems near the end really matter. That’s where top math students get lost, waste time, feel stress, and lose points.

How to avoid freaking out over crazy-hard questions?

1. Handle intimidation

It’s normal to feel intimidated by a long, wordy, or complex problem. Most of the time, however, you’ll be able to answer the question correctly in a reasonable amount of time, if you just give yourself a chance. Always approach such a problem with the mentality that you can solve it. Boil down the question. Reread and make sure you understand each sentence, part by part. You probably can, in fact, do this. If you slow down a bit, read carefully, take notes, and focus intently, you’ll be successful most of the time.

2. Dodge nightmare questions

Instead of getting hung up on an unsolvable problem, it’s better to surrender, as quickly as possible, and live to fight another day. Prevent the wasted time, loss of energy, frustration, and stress caused by battling impossibly difficult “nightmare questions.” The fight isn’t worth it. Avoid the bottomless pit. Earn points somewhere else. Once you realize a question is over your head, just guess and move on. Do not come back later, even if you have time.

3. Employ Skip-Guessing

Naturally, top students want top scores, and often assume they’ll have to work hard to crack every single question. But this is not true. Students can generally afford to skip-guess lots questions without worry (to confirm this, study the scoring tool for any official practice test). When faced with an uncomfortably difficult question, don't get stuck, whatever you do. Quickly make your best guess, skip the question for now, flag it for later review, and move merrily along. 

What matters is making the utmost of what you’re able to do on test day. To optimize your score, spend your limited time where it can be maximally effective, not on crazy-difficult problems you’ll just get wrong, anyway.

-----


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

Tuesday, August 01, 2023

The Official Digital SAT Study Guide

The Official Digital SAT Study Guide by The College Board has been the sine qua non of SAT work for decades. 

But with the SAT's switch to a new digital format, just how useful is this latest incarnation of the venerable test prep tome? Why buy a book printed on paper when the test is now taken on-screen?

These are good questions.

The new edition contains four non-adaptive paper versions of the digital SAT – the very same "linear" tests made available as free downloads to students everywhere – which are roughly 70% identical to the four official adaptive on-screen tests contained in the College Board's BlueBook app (the only official tests available in on-screen adaptive form).

So that's even less reason to buy the book, right?

Right. Except for one thing.

Currently, there's a severe scarcity of official SAT practice materials (this happens each time the College Board decides to overhaul the test). Every new official SAT question made available to the public for practice gives valuable clues as to what to expect on the test, and is therefore worth gold.

As it happens, the new College Board Official Guide to the SAT contains 192 printed practice questions different from those provided anywhere else. That's nearly the equivalent of two additional full-length digital SAT tests.

So yes, at this point, you should buy the new edition – if only to gain additional practice with those 192 precious questions.

-----


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

Saturday, July 08, 2023

Well Begun is Half Done

Aristotle’s admonition to make a good start on any journey counts doubly on difficult SAT/ACT math problems. 

Beginning is often half the battle, and almost anything you can do to get yourself going will probably be helpful.
 
This is sometimes easier said than done, but there are several things you can generally try. It's good to keep in mind a few tips to help grease the wheels when stuck at the beginning of a tough question.

Primary among these is to reread the question slowly and carefully, at half-speed. Many times, you’ll find you simply missed something, and can now solve the problem. Easy as that.

To get a feeling for what’s going on, experiment with simple, realistic numbers in place of unknown quantities. Let the cost of the sweatshirt be $20, for instance. Use that number in the problem, and see what happens. Based on what you learn, the solution may reveal itself.

You can try making up a “simpler similar problem.” Solve that simpler problem, and apply the same approach to the more complex one you’re tackling.

For multi-part questions, pick the easiest part, and work that out first. With such a “jump start,” you may find you’re able to make progress and find your way to the answer.

-----


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

Saturday, July 01, 2023

Commercial Test Prep

Needless to say, when I graduated high school in 1975, it was an entirely different world.

With plenty of well-paying blue collar jobs available in the U.S., a college education was seen as an optional luxury, not at all a requirement to live a good middle-class life. I remember befriending a Golden Gate Bridge worker in the late-1980’s who was paid an annual salary of nearly $60,000 – $150,000 today – taking tolls!

Not many students used any kind of prep, though. I took the PSAT in high school, cold, no prep or pre-test studying at all, as a lark (and hit 98th percentile). But I never told the SAT (didn’t feel like wasting a Saturday morning). Most of my friends acted similarly. College just wasn’t a must-do, at the time.

Stanley Kaplan invented the modern test prep industry in 1939, and between 1940 and 1980 his company’s courses and books were essentially the only ones available to help interested students maximize scores on the standardized tests like the SAT.

Then along came the Princeton Review in the mid-1980’s, upending the entire educational testing scene. Despite protestations from the College Board and others, PR showed everyone just how easy it was to game these tests and quickly raise scores without doing much to improve nominal academic ability.

When I began tutoring professionally in the late 1970’s, test coaching wasn’t yet a thing. Following the huge success of PR, the test prep industry as we know it today was born. 

The test prep universe is vast. Companies old and new seem infinite in number. With the advent of distance learning on a mass scale during the Covid epidemic, this number has grown further.

It’s not easy to make a choice, nowadays, given the multitude of options. To aid in your search, listed below are my current favorites, based on my own long experience and most recent research on the subject:


Thursday, June 08, 2023

Polynomial Functions – What You Need to Know

Providing students familiarity and facility with general polynomial functions is one of the most important goals of every advanced algebra course. 

There’s a lot to learn, and it’s common to devote several weeks to covering key aspects of these functions and their graphs. Fortunately, only a small number of key concepts and processes must be mastered to ace the SAT/ACT.

Following is a list of the things you need to know about polynomial functions.

-

[Note: “iff” means “if and only if.”]

Definition
Polynomial functions: functions of the form p(x) = ax^n+bx^(n–1)+cx^(n–2)+…+z.

Polynomial Standard form
y = ax^n+bx^(n–1)+cx^(n–2)+…+z.

Factored form
y = a(x–s1)(x–s2)
Let y = 0. s1, s1 … are solutions (a.k.a. “roots”) iff s1, s2, … are x-intercepts (a.k.a “zeros”).

Zeros are values of x for which y equals zero (i.e x-intercepts).

Factors of p(x) iff zeros of p(x) iff x-intercepts iff roots iff solutions of p(x) = 0. That is, “factors,” “zeros,” "x-intercepts," “roots,” and “solutions” always go together (any one implies all the rest).

Factor theorem
Given polynomial function p(x), c is zero iff x–c is a factor.

-

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 important material.

-----


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

Thursday, June 01, 2023

Mock dSAT Practice Tests

It’s always the same.

Whenever the College Board trots out a new version of the SAT, years elapse before we have enough official practice material to adequately prepare students for the test.

True to form, to date, the College Board has only made released four official computer-based adaptive SAT tests to the public. To do a good job preparing for the SAT, students need 3-4 times that number.

As always, we’re left to evaluate the various mock SAT practice test offerings currently available. Luckily, most companies offer a free sample test.

In fact, you could pay nothing (or almost nothing) for multiple mock dSAT practice tests simply by signing up for free trials from the list of providers below.

-

Monday, May 08, 2023

Pacing and Time Management

Maintenance of a steady, productive pace and effective management of limited time are keys to unlocking optimal performance on the SAT/ACT. Getting stuck, spending far too much time on a small number of crazy-hard problems, can destroy test results. So can mindless rushing. These twin evils, once discovered, must be quickly and thoroughly exorcised. 

Fortunately, bad testing habits like these can be uprooted and replaced with good ones.

Getting hung up on annoying, difficult questions for long periods of time makes several things happen, all of which are bad for SAT/ACT math scores. Having wasted valuable minutes, you’re now behind the clock and feel an even greater need to rush. The added time pressure also increases stress. Realizing those sticky questions were too hard, and that you probably answered them all incorrectly despite the extra time and effort spent on them, you’re in a bad mood. All this is quite distracting and makes it hard to be energetic, creative, and attentive. Things spiral downward, and you feel you have no choice but to rush even more and nail every question. Which only makes things worse.

Getting stuck and rushing originate in the false belief that one must answer every SAT/ACT math question correctly to earn a high score (“finish the test” at all costs), and that slowing down when necessary is an unaffordable luxury.

This may well be true for tests in class. But on the SAT/ACT, nothing could be further from the truth. Students taking the SAT/ACT math test can safely make many more mistakes than would ordinarily be the case.

The scoring tool on a recent ACT showed students could make 20 errors out of 60 questions to earn an ACT Math 27 (90th percentile), 15 errors for an ACT Math 29 (95th percentile), 10 errors for ACT Math 32 (97th percentile), or seven errors for an ACT Math 34 (99th percentile). In most classes, a score of 90, 10% wrong answers, is an A-, a very respectable grade. On the ACT, however, an identical “grade” is granted even when answering 33% of the questions incorrectly! The same is roughly true on the SAT.

[Note: Since each question has four answer choices, blind guesses are right 1/4 of the time. Guessing on 20 out of 60 questions on the ACT math test would likely earn 5 extra points, which should cancel out any mistakes made on the other 40 (easier) questions. Impossible as it sounds, a student only needs to work on the 2/3 of the questions on the ACT math section, avoiding all the hardest questions, to earn a 90th percentile score.]

In the end, effective time management on the SAT/ACT boils down to not getting stuck. To avoid falling victim, “skip-guess” any difficult question on your first pass through the test, answer it with a random guess, flag the question for later review, and quickly move on to the next problem.

By skip-guessing all the hard problems and working only on relatively easy ones, you’re sure to reach the end of the test super early. With loads of extra time and energy, you can then evaluate how best to spend your remaining minutes. You can calmly and rationally “choose your battles,” deciding which of the harder, flagged questions to come back to on your second pass and the order in which to work on them, slowing down and spending more time on each question to help avoid careless mistakes and ensure success. Knowing you can safely lose a large number of points and still earn a great score, impossibly difficult “nightmare” questions can be safely ignored during your second pass.

Much more time. Much less stress. Far fewer careless errors. No discouragement, headache, or heartache from trying (and failing) to answer crazy-hard questions in a rush. Do you think your score would likely improve under such circumstances?

By employing these three simple strategies – Skip-Guessing, the Two-Pass Approach, and Choosing Your Battles – students routinely cut errors in half. Virtually overnight, an ACT Math 27 (20 errors, 90th percentile) can turn into an ACT Math 32 (10 errors, 97th percentile). That’s an entirely different college! No new math to learn. No additional practice work. All that’s required is the adoption of a new, more comfortable, vastly improved general approach to the SAT/ACT math test.

Following are two additional pacing strategies offering concrete help in the fight to avoid getting stuck.

The 10-Second Skip

It’s crucial to decide quickly whether to work on a given problem during your first pass through the test or skip-guess the question for possible review later on. This is where getting stuck often happens. The rule is: give yourself only 10 seconds to decide. When in doubt, skip-guess the problem.

The 20-Second Skip

Getting stuck can also happen in the middle of a difficult question as one bargains with oneself for “just a little more time.” A “little more time” can easily turn into several wasted minutes. This is bad. Here’s the rule: Anytime you stop making progress on a problem for 20 seconds, skip-guess the question.

WARNING: If you find yourself finishing early without skip-guessing a large number of questions, you’re rushing. To improve your score, you must skip-guess a lot more problems, and slow down a bit. As you see your practice test scores dramatically improve, you’ll be convinced and converted to this new approach.

Going too fast can be just as bad as going too slow. Slowing down slightly doesn’t cost you points! It actually helps you move faster and more successfully than before, since you won’t waste loads of time and energy to needless confusion and stress, nor will you lose nearly as many points to carelessness. Don’t go slow as a snail, of course! It’s best to drive in the middle lane of the freeway. Not too fast, not too slow.

Whatever you do, don’t get stuck! 

Remember that, working carefully to avoid pitfalls, there’s really no need to rush. Let go of false assumptions and unnecessary, uninformed expectations about having to perform perfectly. Your test-taking experience will improve, and so will your SAT/ACT math score.

-----


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

Monday, May 01, 2023

Cracking BlueBook Second Modules

With only six available official digital adaptive SATs and PSATs, it’s critical that students get access to all questions contained in the “second modules” of each section of each test in the BlueBook app. 

There’s no way to do so directly, but an easy workaround gets the job done: 

-

1. After completing your practice test, save the answer page.  

2. Retake the same practice test. 

3. To access the harder second module, enter only correct answers when you retake the test; to access the easier module, enter only incorrect answers. 

-

Click here to view and download (File > Download) correct answers to first modules from BlueBook SAT Tests 1, 2, 3, and 4.
 
You may also want to screenshot important questions as you practice (hardest problems, any you’ve answered wrongly, etc.) and keep these in a folder for further study.

These few official tests are best used sparingly to assess progress in your prep work or as “dress rehearsals” before test dates.

They're the only ones you’ve got. Don’t waste them!

-----


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

Saturday, April 08, 2023

Quadratic Functions – What You Need to Know

Most intermediate-advanced algebra courses cover second degree polynomial functions, called “quadratic functions,” long before introducing general polynomial functions.
 
Because quadratic functions are simpler versions of general polynomial functions, graph as parabolic curves, and are featured in many real-world situations, this subtopic holds a prominent place in the syllabus.

Following is a list of concepts you should have firmly under your belt before taking the SAT/ACT.

-

[Note: “iff” means “if and only if.”]

Definition
Quadratic functions: functions involving one first-degree variable and one second degree variable (typically y^1 and x^2).

Quadratic Function Standard Form
y = ax^2+bx+c.

Parabola
Vertex = (h,k) with h = -b/2a, k = f(h).
Axis: x = h.
y-intercept = c.

Vertex Form
y = a(x–h)^2+k.
Vertex = (h,k).
Radius = r.

Discriminant
D = sqrt(b^2–4ac).
D > 0 iff 2 real solutions.
D = 0 iff 1 real solution.
D < 0 iff no solutions.

Sum and Product Formula
x^2–(sum)x+(prod) = 0.

-

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 important material.

-----


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

Saturday, April 01, 2023

Factfulness

Despite rumors to the contrary, things aren't so bad. They're probably quite a bit better than you think. That's the thesis of this book by Swedish author and public health expert Hans Rosling.

From the book's description on amazon:

"It turns out that the world, for all its imperfections, is in a much better state than we might think. That doesn’t mean there aren’t real concerns. But when we worry about everything all the time instead of embracing a worldview based on facts, we can lose our ability to focus on the things that threaten us most.

"Inspiring and revelatory, filled with lively anecdotes and moving stories, Factfulness is an urgent and essential book that will change the way you see the world and empower you to respond to the crises and opportunities of the future."

Here's what Bill Gates had to say about Factfulness:

“One of the most important books I’ve ever read―an indispensable guide to thinking clearly about the world.”

And Barack Obama:

"Factfulness by Hans Rosling, an outstanding international public health expert, is a hopeful book about the potential for human progress when we work off facts rather than our inherent biases."

Rosling isn't a Pollyanna who refuses to acknowledge the very real problems and challenges humanity still faces. His point is that, if we actually follow the facts – instead of persistent, unconscious biases – there's much more reason to be proud and hopeful than we might ordinarily believe.

I enjoyed Factfulness, and highly recommend it to all with an interest in the state of the world.

-----


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

Wednesday, March 08, 2023

Winning Focus

Test taking is a competitive activity.

As is athletics or the performing arts, outstanding achievement on the SAT/ACT depends on maintenance of powerfully sharp, energetic, mindful concentration and “winning focus.”

In athletics, the problem is the opposing team. In the performing arts, it’s stage fright. In test taking, your adversaries are the test questions. Test takers, like athletes and stage performers, need to confront and conquer pressure, stress, fatigue, distractions, and other challenges with sustained enthusiasm, relentless drive, and full commitment to success.

An intense, aggressive, yet controlled state of concentration is critical to performing at one’s best in any competitive activity. “Winning Focus” is so important, in fact, that without it, almost nothing else matters.

By definition, intensity always involves some amount of tension. Undisciplined, raw intensity can lead to varying degrees of frustration, anxiety, and even exhaustion. Athletes, performers, and test takers must learn to thread the needle between two extremes, remaining fully responsive and focused on winning without getting rattled or falling victim to tenseness, tiredness, or negative emotion.

When intelligence is inhibited by inattention, distraction, tiredness, or emotional upset, reduced cognitive bandwidth causes a mental traffic jam, disrupting the efficient flow of ideas.

On the other hand, Winning Focus lights up the brain like a Christmas tree. All kinds of connections, insights, and creative solutions come to mind that would otherwise go unnoticed. Most important of all, careless mistakes are minimized.

Fully half the errors on a typical student's SAT/ACT math test are due to carelessness. If these careless errors were reduced by half, a student scoring 650 on an SAT math section would instead score 720, and one scoring 27 on the ACT would score 31!

These dramatic improvements require neither new math smarts, nor improved testing acumen, nor additional practice – just a 50% cutback in careless mistakes. In fact, math test prep could be restricted entirely to development of Winning Focus, and most students would be well served.

The vast majority of test coaches pay no significant attention to the mental game of test taking. In my work, Winning Focus takes center stage. Using proprietary methods and materials, a natural knack for inspiring and igniting motivation, and a unique ability to train all aspects of effective test prep that only 45+ years in business can bring, I teach my students to track errors, avoid unnecessary mistakes, and generate and sustain powerful, energetic, productive concentration.

The best way to train Winning Focus is to build strength and endurance through gradual exposure to practice tests of increasing length and difficulty, bolstered by professional performance critiquing, feedback, error analysis, and personal support. It boils down to habituating a determined, enthusiastic, mindful concentration on test questions, and training higher levels of focus for longer and longer periods.

Fortunately, most high-performing students already know how to do this. Such students are typically athletes, actors, dancers, and/or musicians, and already know how to generate the state of mind necessary to excel at competitive activities. It’s just a matter of transferring this well-developed ability to the domain of test taking. Although the particulars are more complex, it’s really that simple.

-----


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

Wednesday, March 01, 2023

lizardpoint

Lizardpoint is a wonderful all-around early learning site for people of all ages. Students can use lizardpoint to learn geography, flags, world leaders (current and past), art, and math.

The site is great for adults as well as for children.

Sections on geography help students master finding countries and provinces, naming their capitals, topographical features, airport codes, and more.

The section on art features quizzes on major western artists and their most well-known works, fine art terminology, and important movements.

Math offerings include on-screen and downloadable/printable worksheets (with answer sheets) for arithmetic operations, decimals, percents, and fractions.

Quizzes on trivia, important definitions, and more round out learning activities presented.

New stuff is added regularly, collected in What's New.

Like Freerice.com, lizardpoint can be addicting, but in a good way. These two educational sites keep interested students learning and growing while having fun.

-----


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

Wednesday, February 08, 2023

Linear Functions – What You Need to Know

Linear functions is one of the most frequently tested subtopics on the SAT/ACT, and students continually employ these basic functions as they progress through more and more rigorous math courses. Indeed, there’s nothing more important to master in high school mathematics than linear models and methods.

But which ideas should students fluently understand and keep in mind to score well on the SAT/ACT?

Here’s the list.

-

Definition
Linear Functions: functions involving two first-degree variables (typically y^1 and x^1).

Vertical line equations
x = #. 
For example: x = 4 graphs as a vertical line through 4 on the x axis.

Horizontal line equations
y = #. 
For example: y = -4/5 graphs as a horizontal line through -4/5 on the y axis.

Oblique lines

Linear equation standard form: Ax+By = C, slope = -A/B. 
For example: 3x – 4y = 12, slope = -(3/-4) = 3/4.

Slope-intercept form: y = mx + b, slope = m = "rise/run," y-intercept = b. 
For example: y = -x, slope = -1 = -1/1, y-intercept = 0.

Parallel Lines
m1 = m2 (equal slopes).

Perpendicular Lines
m1 * m2 = -1 (opposite reciprocal slopes).

-

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 important material.

-----


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

Wednesday, February 01, 2023

Call the Capitol Switchboard!

It's hard to not be cynical about politics, today. 

Citizenship, however, isn't politics. 

Rather, citizenship is responsibility we all share. Citizen participation is a requirement in a functioning democracy, a duty we can each discharge in multiple ways. One of the simplest is to call the Capitol Switchboard on a regular basis to leave messages for your congressional and senatorial representatives.

Here's the phone number: (202) 224-3121.

I recommend storing the number in your phone, so that when you have a few minutes you can make that time count by calling the Switchboard and leaving a message or two. A great time to do this is during the morning/evening commute, while waiting in line somewhere, or everyday just after breakfast (to make it a habit).

"The problem with democracy is that people get the government they deserve."

Do we Americans still deserve a strong, healthy democracy? Time will tell.

It’s easy to feel there’s nothing one can do. Which only furthers the degradation of the democracy we still have.

I invite you to do what you can. Calling the Capitol Switchboard only takes a few short minutes. Do so on a regular basis.

You and yours will know you did what you could to keep democracy alive in America, which is all that's required of any of us.

-----


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

Sunday, January 01, 2023

How Young Adults Can Build Credit Fast

Credit allows adults to borrow funds to make important purchases too costly to buy with their own cash: a house, an education, a car, among other things. "Credit” is borrowed money in various forms: credit cards, retail store credit, car loans, personal loans, college loans, and a mortgage (a large loan used to buy a house).

Next to earning a good living, an excellent credit score is the most important financial goal to pursue, and young adults can begin to do so as soon as they turn 18. Credit scores range from 300 to 850, and your goal should be to keep yours at or above 760, the point at which you’ll get the most favorable terms (lowest interest rates, down payments, etc.) when borrowing money.

The way you create and maintain a high credit score is simple: pay all your bills in full, on time, always.

Below is a rough outline of the steps I followed to help our daughter to establish and grow her credit score once she graduated from high school.

-

Before Age 18: 

Work and save

This project will take $1500 or so to get started, so you’ll need to earn that seed money before you can begin the process outlined below.

-

Age 18:

Regular income

You’ll need a source of regular monthly income greater than you expect to spend each month. To be safe, it’s preferable to earn at least twice what you’ll be using credit to pay for.

** Note: Each month, on a regular basis, you must deposit earned income into your checking account. Without that regular deposit of income, in large enough amounts, this project will not work. 

Checking account – backed by an overdraft savings account

Put $500 into a fee-free checking account, and $500 into a linked overdraft savings account that will automatically pay your bills if you overspend your checking account. 

Set up your accounts at a bank or credit union near you. I recommend credit unions over big banks. Credit unions support the local area economy and generally providing better service to customers.

Credit Card 1

First, apply for a “secured credit card” or "student card." This is a special kind of credit card for first-time borrowers with no credit history.

With a secured credit card account, you pay a deposit to the bank issuing the card (perhaps $500) to guarantee your purchases will be paid, as promised. Once you’ve shown a six-month history of responsible payments (paying all charges each month in full and on time), the deposit will be returned to you. Your “secured” card then becomes an ordinary “unsecured” card.

Use you credit card sporadically to pay for restaurants and other low cost items. Be sure to keep your spending well under the spending limit for that card (known as your “credit limit”).

* Note: If at any time you have questions about your card, you can call the customer service phone number listed on the back of the card to get answers.

Set Up Autopay on Credit Card 1

From the beginning, you should enable “auto-pay” through your bank’s online banking system to automatically pay credit card charges and other regular bills and payments directly from your checking account. Autopay ensures that all your bills are paid in full, on time, each and every month.

With autopay, all you have to do is check online at the beginning of each month to make sure your checking account balance (the amount in your account) is large enough to easily cover the automatic payments that will be withdrawn from your account that month (if possible, maintain a checking account balance at least double the amount withdrawn by autopay each month to pay bills).

In this way, you won't have to write paper checks to pay your monthly bills, add stamps to payment envelopes, and actually drop them in a local mailbox on time. Making manual payments like this is not just a hassle; it raises the possibility that you might make a payment late, or forget to do so altogether. With autopay, that never happens – as long as you keep your checking account balance at the proper level.

-

Age 19: 

Become an “Authorized User”

Now that you have some clean credit history, you can apply to become an “authorized user” on a credit card owned by someone else who trusts you, like your mom or dad.

You’ll be able to use the card to make purchases, which helps strengthen your credit history, but you won't be responsible for any payments. This is where “trust” comes in.

To make the most of this score-building opportunity, be sure to remain as an authorized user on this account, making sporadic purchases with it, for at least one year.

Credit Card 2

Now that you can show a longer history of good credit habits involving two credit accounts, you can apply for another unsecured credit card under your own name. Having more open credit accounts improves your credit score – as long as you always pay the full amount due on every one of your accounts, every month, on time. Search online for info on the best credit cards to get.

Use this card sporadically. As usual, have payments for your second credit card made through your checking account autopay service.

* Note: Once you’ve established at least one year of clean credit history, you can remove yourself as an authorized user on your parents’ card. Time spent as an authorized user has served its purpose. Now that you’ve got your own cards, you don’t need to have your name on theirs any longer.

-

Age 20:

Retail Store Credit Card

A “mix” of different types of credit also helps build a higher credit score. In addition to bank-issued credit cards, retailers ofter issue credit cards to customers.

Go to a favorite retail store, and sign up for a store credit card to make a some additional purchases. As usual, pay the balance on that card in full each month using autopay.

Credit Card 3 – Rewards Card

Now that you’ve had two years' clean credit history on three different credit accounts, you’re ready to apply for your “go-to” credit card: a Rewards Card. “Rewards Cards" offer users extra advantages.

The best card to get is a “Cash-Back” card, one that offers at least 2% back on each purchase. Your rewards card now becomes your “go-to” card that you'll use for nearly all purchases.

If you shop on amazon, you’ll want to get an amazon Prime credit card, which gives users 5% off on all amazon Prime purchases. Costco members can get a Costco membership credit card that gives 4% back on all purchases of gas, which could save a lot of money for those who do a lot of driving.

* Note: Don’t keep too many cards in your wallet. I keep only two, my rewards card, and my Costco card. I keep the amazon card at home, since I only use it for online amazon purchases. Still, it’s good to have more than one credit card with you, just in case one doesn’t work for some reason. It's unnecessary to have more than three or four credit card accounts, total.

-

Age 21:

Consumer Tech Loan

The Apple Store and other tech retailers will give loans to customers to pay for major purchases of tech gear: laptops, iPads, desktop computers, etc. Buy a good product (something you need and were going to purchase anyway) new or refurbished from one of these major retailers, and have the loan payments made automatically through autopay.

Keep the term (length) of all loans under three years (less is better, but never pay off a loan in less than a year, to give your credit score the best boost).

-

Age 22:

Personal Loan

With a fairly long credit history, you can now apply to a bank for a “Personal Loan.” This is a loan you take for something personal, like a vacation. Apply at the bank or credit union holding your checking and savings accounts.

Take a short trip with a friend, and pay for it with funds provided by your personal loan. Then, as usual, put the loan payments on autopay.

Don't borrow more than you can easily afford to pay back. But a small personal loan of $1000-$2000 paid over a two year period will help you credit score a lot.

-

Age 23: 

Pay Bills

Get a cell phone account, put your name on your house utility bill, and sign up for a streaming service or two; then, put all these bills on autopay though your checking account or pay them automatically using your go-to credit card (call your credit card company to set this up).

Signing up for special credit services can give scores a boost for those who pay basic bills on time (especially phone and utility bills). Renting an apartment may require a such a history.

-

Age 24:

Car Loan

At this point, you’ve established excellent credit over a period of several years, and have a score at or above 760, proof of top-tier “credit worthiness.” 

With such an excellent credit record, you can make your first major purchase using credit – a car. Either arrange a car loan in advance with your bank or credit union, or buy the car through a dealership and use their financing department to arrange the loan.

To save money (and get a better deal), it’s best to buy a car that’s used but only a few years old. Going through a dealer is super-convenient and makes a lot of sense for first-time buyers. Be sure to research best models and prices, before making your purchase.

-

Age 25:

Home Mortgage

With three or four credit cards, a store card, a consumer loan, a personal loan, and a car loan on your credit history, each account over one year old and with a flawless repayment history, you’ll be able to realize the American Dream: home ownership.

The largest purchase most people will ever make is buying a home. A home loan, called a "mortgage," a huge step, but one you’re ready to make, now that you’ve learned how to earn and deposit money and handle spending and payment responsibly though auto payment.

The only other requirement is a regular income from work large enough to handle the mortgage (monthly income will need to be at least four times the monthly mortgage payment).

-

Search online for other suggestions relating to building and maintaining excellent credit. The more you know, the better. Next to your career, and your home, your credit rating is the most valuable asset you possess.

Following the approach above, in five years, by age 23, one year out of college, your credit score is now well over 800. You're in rarified air with one of the best of credit scores, with a credit history strong enough to qualify to buy your first home under the best possible mortgage terms.

* Never close a credit card account, once opened, even if you owe nothing on the account. Generally speaking, it’s best to keep all credit card accounts open forever, since the older your credit history, the higher your credit score.

Track your credit score regularly using a site like creditkarma.com.

At this point, further credit isn't necessary, and you should avoid opening any additional accounts, unless truly necessary. Just maintain your excellent credit history, use the few credit cards you have, always pay all your bills in full and on time each month using autopay, and make sure your checking account balance is higher than you'll need to cover auto payments for at least two months (with a similar sum in your linked overdraft savings account, for extra safety and as an emergency cash fund).

You have arrived. Good job.

Enjoy!

For another look at this important topic, click here.

-----


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