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.

Advanced Math Strats

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 specific solution methods are unpredictable here, two question types 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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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 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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How to Read SAT/ACT Math Questions

Successful 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.

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.

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 unanswerable without secure grasp of the mathematical models tested on the SAT

Fortunately, secure grounding in particular concepts of linear, quadratic, and exponential functions tested on the SAT isn't hard to attain.

Following are the key facts about linear, quadratic, and exponential models students need to know 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 subtopic.

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Contact Chris

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

Exponential Functions – What You Need to Know

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

Fortunately, only knowledge of bare basics is required.

Below 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 important subtopic.

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Contact Chris

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

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. Narrowing 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 thus confers multiple benefits, and is highly recommended.

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

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 is a complete list.


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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 important subtopic.

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Contact Chris

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

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 about general 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 important subtopic.

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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.

“Dimensional Analysis” is a simple method to handle these basic math problems quickly and easily, and it’s important to understand and master this technique.

Premises: Equations relating units enable the creation of fractions whose values are always 1, multiplication by 1 never changes values and is therefore always allowed, and the word “per” can be translated “divided by.”

To see how these concepts enable the conversion of units, it’s helpful to study examples.

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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 important subtopic.

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Contact Chris

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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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Contact Chris

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

Transforming Functions

Questions involving reflecting or shifting graphs stump many students, but this needn’t be so! 

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

Following 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 important subtopic.

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Contact Chris

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

Geometry You Need to Know

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 important subtopic.

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Contact Chris

borlandeducational.com

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 helpful tips 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

If 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.

Read on 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 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
y = a(x-h)^2 + k.
Vertex is (h,k).

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 important subtopic.

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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 the prohibited plastic or metal ones. But you can easily and legally improvise each of these tools with materials you’re allowed to use on the SAT/ACT. 

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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Contact Chris

borlandeducational.com

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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Contact Chris

borlandeducational.com

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