Thursday, May 01, 2025

How to Fail a Test with Dignity

Sometimes, you just can't win.

In that case, there's no harm or shame in surrender. So why not have a sense of humor about it?

These students tried and failed, but succeeded in turning loss into laughs.



























































































































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

Tuesday, April 08, 2025

The Two-Fisted Approach

If you were a boxer, would you rather fight with two hands, or with one tied behind your back? Of course, two hands would be best.

Likewise, a two-fisted strategy in attacking SAT/ACT math questions is superior to a one-dimensional, purely analytical approach. 

Secure command of several key multiple-choice math strategies enables students to correctly answer questions that would otherwise be difficult, time-consuming, or impossible to answer. 

When answering multiple-choice math questions, students always have two options.

Plan A: Do the Math

If math is easy, just do the math. Make drawings. Show your work. Write out all the steps. Internally verbalize your work to avoid careless mistakes.

Plan B: Use a Strategy

If the math isn’t easy, use a strategy! Make it real, and keep it simple.

Dozens of useful strategies can make correctly answering difficult SAT/ACT math questions much easier. Fortunately, it’s not necessary to master them all. A small number of tricks confer most of the benefit:

Sample Values, Try the Answers, Seeing Is Believing, Boil-Down Questions, Choose Your Battles, Two-Pass Approach, GPS for Functions, Joe Average, Thought Experiments, Guessing Machines.

The best way to acquire these powerful test taking tools is to work with an expert coach knowledgeable and accomplished in the strategic approach to test-taking.

I'm happy to help.

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

Tuesday, April 01, 2025

Advanced Math Tricks

SAT/ACT math sections often contain questions requiring advanced techniques, and students are expected to creatively problem-solve their way out of these unfamiliar boxes (this is especially true of the difficult second module of the SAT).

Although these questions are challenging and unpredictable by nature, two solution methods have emerged that can be studied.

Compound Expressions

These are simple expressions with more than one part. Such questions might ask students to find 4x + y, for example, rather than simply x or y.

When encountering compound expressions questions, students are usually tempted to first find the values of the unknowns, one at a time, and then substitute them into the compound expression itself. But on the SAT/ACT, this is never the best way to solve the problem. It’s faster and easier to deduce a clever way to create the compound expression directly from what’s given.

For example, suppose you’re given that 15n – 5m = 70, and asked to find the value of 3n – m. Noticing the similarity between the left side of the given equation and the compound expression, one sees it’s easy to derive the expression itself directly simply by dividing both sides of the given equation by 5, yielding 3n–m = 14.

Matching Forms

The trick here is to match the form of expression on the right side of a given equation with that on the left, and then equate corresponding terms. The forms typically required on the SAT/ACT are standard quadratic and linear forms.

For instance, suppose we’re given i^2 = -1, a and b are real numbers, and a + b + 5i = 9 + ai, and asked to find b. Both sides can be written in the same linear form: (a + b) +5i = 9 + ai. Since the expressions on either side of the equation showing matching forms are equal, corresponding terms must likewise be equal. Thus a + b = 9 and 5i = ai, which implies a = 5 and b = 4.

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

Saturday, March 08, 2025

Y1Y2

Inexpensive handheld math computers like the Ti-84 series of graphing calculators came of age in the early 1980s and revolutionized the teaching and learning of mathematics. The youngest child in the family, the Ti-84 Plus CE, is still a standard classroom tool, and a powerful one.

While Desmos is the only graphing calculator tool allowed for use on the SAT, the Ti-84 Plus CE is the best of many calculator option for the ACT. “Y1Y2” is one the most helpful “cool calculator tricks” the Ti-84 Plus CE can perform, leveraging raw computing power to solve equations by graphing.

To solve any equation on the ACT using the Ti-84 Plus CE:

1. Set Y1 equal to the left side of the given equation to be solved, and Y2 equal to the right side. 

2. Hit the graph button, and make sure both graphs appear in the viewing window. 

3. Note the coordinates of the points of intersection. The x-coordinate(s) will be the given equation's real solution(s).

It can sometimes be hard to finagle the graphs of Y1 and Y2 so that both show up in the same viewing window, however.

In that case, carry out the following steps to make viewing both graphs easier,:

1. Remember that since Y1= Y2, Y1–Y2 = 0. 

2. Set Y3 equal to Y1–Y2, turn off Y1 and Y2, and graph Y3 alone. 

4. Real solutions to the original equation will be zeros of Y3. These solutions should now conveniently appear as x-intercepts (adjustment of window variables Xmin and Xmax may be necessary to show both zeros).

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

Saturday, March 01, 2025

How to Read SAT/ACT Math Questions

Success in mathematics is rooted in efficient reading. Most math mistakes on the SAT/ACT are, in fact, reading mistakes! Learning to avoid them is an opportunity to quickly and significantly boost scores without learning any new math.

Four simple strategies help ensure students aren’t misreading math questions or failing to catch important information.

The Middle Lane

The most effective way to prevent reading mistakes is to slow down a little. It’s counter-intuitive, but slowing down can actually help one move more quickly through the test, since doing so leads to better comprehension, less confusion and stress, and far less re-reading. All this contributes to stronger SAT/ACT math scores. A comfortable, moderate reading pace is the goal. The middle lane of the freeway. Not too fast, not too slow. Students should no faster than they can fully understand what they're reading. 

Read Party-By-Part

When reading gets tough, the best thing to do is slow down, read each sentence part by part, and make sure you understand each part before going on to the next. Yes, this will take some time. But what’s the alternative? Not fully comprehending the question is the best way to answer it incorrectly. Taking the extra time to read slowly and carefully enough to fully understand the problem, students actually save more time than the strategy costs, even as it greatly improves the likelihood of choosing correct answers. 

Boil-Down Questions

This is another great way to boost comprehension and improve the odds on difficult SAT/ACT math questions. The acronym “RCU” outlines the steps: 

“R” stands for read the question all the way through. 

“C” means circle the main words in the question (i.e. the last sentence, with the question mark). 

“U” reminds us to underline the clues. 

Taking these notes on scratch paper increases awareness of the meat and bones of the question and leads naturally to higher scores.

Point to Get the Point

“Tracking” is a tried and true method for improving concentration and comprehension. 

All you do is your pencil or index finger under the text as you read, and pay close attention to both the meaning of the text and your pointer. 

While this technique precludes reaching maximum reading speed, who cares? As we’ve already pointed out, comprehension is more important than speed on the SAT/ACT math section. Point to get the point!

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

Saturday, February 08, 2025

Mastering Desmos for the SAT

A special version of the
Desmos online graphing calculator is included as part of the BlueBook app students use to take the SAT.

Clever use of Desmos on the SAT math section can give a significant boost to scores, but Desmos is as complicated as it is powerful, and many students are unaware of its most helpful features. 

Which Desmos skills are most important to master for use on the SAT? Narrowing the list is critical.

Click here to read my outline of Desmos essentials.

The calculator is no cure-all, however, and the boundaries of its usefulness on the SAT must be kept firmly in mind. Desmos is helpful in solving the following types of problems: basic calculation, and simple data analysis, solving equations and systems, finding intercepts/zeros/max-min values, graphing equations and functions, and finding points along these curve. During the test, students should keep the calculator open but minimized, expanding it when in use.

Desmos is growing in importance in most high school and college math courses. Facility with the calculator beyond beginning levels confers multiple benefits, and taking the time to upgrade one's Desmos skills is highly encouraged. 

Students can progress to the next stage through independent study using targeted Google searches to learn and practice any underdeveloped skills listed in the linked document above.

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

Saturday, February 01, 2025

SAT Interpretation Questions

For many students, SAT problems requiring interpretation of details found in equations or graphs can be some of the most difficult. 

These questions are simply unanswerable without a basic understanding of the particular mathematical models tested on the SAT.

Fortunately, firm grounding in those aspects of linear, quadratic, and exponential functions tested on the SAT isn't hard to obtain.

Following are key facts about linear, quadratic, and exponential models students must understand to answer SAT interpretation questions.

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Linear functions
y = mx+b: y-intercept b is the “initial value” (y value when x = 0).
Slope is the “rate of change” (y compared to x, “y per x”).

Quadratic functions
y = ax^2+bx+c: y-intercept c is the “initial value” (y value when x = 0).
max/min value is the y coordinate at the vertex.

Exponential functions
y = A*B^x.
x is usually time, t.
A is the initial value (y value when x or t = 0).
B is the “multiplier” (number repeatedly multiplied in the problem.
B = 1 ± r, where r is the rate of increase or decrease (respectively)..
For example: If the rate of decrease is 15%, B = 85% = .85.

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For practice, search Google for “SAT interpretation questions 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 question type.

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

Wednesday, January 08, 2025

Exponential Functions – What You Need to Know

Exponential models are useful in a number of real-world scenarios, from predicting declines in population to forecasting growth in asset values. This can all get rather complicated, and both the SAT and ACT require some familiarity with exponential functions. 

Fortunately, only knowledge of bare basics is required. 

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

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Simple definition
Exponential functions have an unknown in the exponent.

Exponential functions – general form
y = A*B^x.
x is usually time, t.

Constants
A is the “initial value” (y when x=0).
B is the “multiplier.”

Growth factor
B = (1±r), where r is rate of growth/decay.
(Time and rate units must match).

Parent Graph
Rising curve through (0,A).
Horizontal asymptote y = 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.

Wednesday, January 01, 2025

Essential Ti-84 Plus CE Skills

Texas Instrument’s handheld Ti-84 Plus CE graphing calculator is a powerful math computer, the best available calculator for use on the ACT (a different  graphing calculator, Desmos, is featured in the BlueBook app used to take the SAT).

Clever use of the Ti-84 Plus CE can make significantly improve scores on the ACT math test. Unfortunately, the calculator is as complicated as it is powerful, with hundreds of features and functions. Shortening the list is critical.

Which calculator skills are most important?

Click here for my outline of essential Ti-84 Plus CE skills. Students are encouraged to do independent study using targeted Google searches to learn and practice any underdeveloped skills listed in the linked document above.

Graphing calculators like the Ti-84 Plus CE have played an important role in teaching and learning mathematics for decades, and as of this writing, this calculator is still standard technology in high and college mathematics courses.

Mastering the Ti-84 Plus CE beyond beginning levels confers multiple benefits, and is highly recommended.

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