Thursday, February 08, 2024

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

Thursday, February 01, 2024

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


Monday, January 08, 2024

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

Monday, January 01, 2024

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

Friday, December 08, 2023

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.

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