SAT Quick Challenge  Week 20
Saturday, December 12, 2020
Providence Baptist Church is concerned about you, your family, your friends, and the community. We are following the recommendations to limit the number of people in our inperson gatherings. For this reason, all SAT Preparation classes at the church are cancelled until further notice.
Although the current COVID19 pandemic abruptly ended the Spring 2020 SAT classes, technology offers an opportunity for students to refresh, retain, and/or acquire SAT knowledge and skills essential for answering various kinds of questions often found on the SAT. Therefore, the Providence Baptist Church SAT Preparation Program has provided this “SAT Quick Challenge” website. Each Saturday morning, this site will be updated with a set of activities that will help students acquire and maintain mastery of important skills that help lead to high SAT scores. Be sure to review the SAT Math Formulas and the SAT Math Operations each week. You can find them in the dropdowns at the bottom of this page.
So that we all can celebrate the birth of Christ and the arrival of a new year, no new lessons will be provided during the holiday season. The next set of lessons will be posted during the spring semester. As you plan your celebrations with others, please stay safe. Remember the three Ws: Wear, Wait, Wash. Have a Merry Christmas and a Happy New Year!
Verbal  This Week's Questions (December 12, 2020)
SAT QUICK CHALLENGE U
The Dangling Modifier
Saturday, December 12, 2020
Like the SAT misplaced modifier question, the SAT dangling modifier question also has a comma that divides a sentence into two parts in a very specific way. The first part of the sentence is an introductory phrase or clause that modifies a word or word group that is not (even) in the first part of the sentence. The comma comes next. Finally, the noun or pronoun modified by the introductory phrase or clause comes right after the comma when possible, but as close as possible to the comma otherwise. In the case of a dangling modifier, however, the word that should be modified is not in the sentence at all! Therefore, the modifier is just left hanging (or "dangling") with no word that it can modify. Note Examples AC below.
Example A.
When driving to work, a fallen tree blocking the road kept people from getting where they wanted to go.
Comments: The introductory phrase and the comma that divides the sentence into two parts are in place correctly. The word to be modified  the noun indicating who was driving to work  should follow the comma, but the word "tree" is in that position. Therefore, the sentence absurdly suggests that the tree was driving to work. The noun that tells who was driving to work is not in the sentence at all. Therefore, the introductory phrase is a dangling modifier. The writer needs to correct the sentence by placing after the comma a word that does make sense. (See Example B.) Another way to correct the sentence is to turn the introductory phrase into an introductory dependent clause that indicates who was driving to work. (See Example C.)
Example B.
When driving to work, my mom saw that a fallen tree blocking the road kept people from getting where they wanted to go.
Comments: The phrase my mom corrects the error. "My mom" could have been driving the car, and the introductory phrase logically modifies "my mom."
Example C.
When my mom was driving to work, a fallen tree blocking the road kept people from getting where they wanted to go.
Comments: The introductory dependent clause "When my mom was driving to work" corrects the error by telling who was driving the car. Moreover, that dependent clause modifies the noun phrase "fallen tree," which comes right after the comma.
Now, complete the exercise below, and use the answer key in the "SAT Verbal  Answers to This Week's Questions" dropdown to check your work.
SAT QUICK CHALLENGE U  The Dangling Modifier
Directions. On the line after each question, write the letter of the answer choice that corrects the underlined part of the question. If you think the underlined part is already correct, select choice A.  
1. During this long, cold snap, furnaces have been breaking down in homes all over town. Shivering in her freezing house, staying at her mom's home until the furnace repairs were complete was a really great idea.
A. NO CHANGE 

2. After getting his foot injured, a therapeutic boot was required for three months. A. NO CHANGE 
3. Exhausted from the long hike, a cool drink and a nice bed would be wonderful.

SAT Verbal  Answers to This Week's Questions (December 12, 2020)
SAT QUICK CHALLENGE U  The Dangling Modifier ANSWER KEY
Directions. On the line after each question, write the letter of the answer choice that corrects the underlined part of the question. If you think the underlined part is already correct, select choice A.  
1. During this long, cold snap, furnaces have been breaking down in homes all over town. Shivering in her freezing house, staying at her mom's home until the furnace repairs were complete was a really great idea. C A. NO CHANGE 

2. After getting his foot injured, a therapeutic boot was required for three months. D A. NO CHANGE 
3. Exhausted from the long hike, a cool drink and a nice bed would be wonderful. B 
SAT Math  Answers to Last Week's Questions (December 5, 2020 )
 │ 2 │ = 2
 │7 │ = 7
 │ 46 │ = 46
 │25│ = 25
 │ 4 │ = 4
 │ 9 │ = 9
Absolue Value Equations:
Some equations contain absolute value expressions. The definition of absolute value is used in solving these equations. Remember, │ 4 │ = 4 and │ 4 │ = 4; thus, what is between the absolute value bars can be positive and negative. Thus when solving absolute value equations, there will be two equations and usually two values of x.
│x  3│ = 7 also means x – 3 = 7 and x – 3 = 7
│x  18│ = 5 also means x – 18 = 5 and x – 18 = 5
Solve each equation. Check your work by substituting your answers into each original equation.
 │x  3│ = 7
x  3 = 7 x  3 = 7
x = 10 x = 4
There are two values of x that solve this equation, 10 and 4. Now we need to check our answers by substituting into the original equation,
│x  3│ = 7.
Check for x = 10: Check for x = 4:
│x  3│ = 7 │x  3│ = 7
│10  3│ = 7 │4  3│ = 7
│7│ = 7 │ 7│ = 7
7 = 7 7 = 7
Our check shows that both values of x solve the equation.  │x  18│ = 5
x  18 = 5 x  18 = 5
x = 23 x = 13
There are two values of x that solve this equation, 23 and 13. Now we need to check our answers by substituting into the original equation,
│x  18│ = 5.
Check for x = 23: Check for x = 13:
│x  18│ = 5 │x  18│ = 5
│23  18│ = 5 │13  18│ = 5
│5│ = 5 │ 5│ = 5
5 = 5 5 = 5
Our check shows that both values of x solve the equation.
For most absolute value equations there will be two solutions. However, there are some equations where there is no solution, and in some cases there is only one solution.
Consider the following two examples.
Example 1:
│4x  2│ + 7 = 0
│4x  2│ = 7
This sentence is never true; there is no value of x that solves the equation. Thus there is no solution for this equation.
Example 2:
│x + 6│ = 3x  2
x + 6 = 3x  2 x + 6 =  ( 3x – 2)
6 = 2x  2 x + 6 =  ( 3x – 2)
8 = 2x x + 6 = 3x + 2
4 = x 4x = 4
x = 1
There are two calculated values of x that solve this equation, 4 and 1. Now we need to check our answers by substituting into the original equation, │x + 6│ = 3x  2
Check for x = 4 Check for x = 1
│x + 6│ = 3x – 2 │x + 6│ = 3x  2
│4 + 6│ = 3(4) – 2 │1 + 6│ = 3(1)  2
│10│ = 12 – 2 │5│ = 3 – 2
10 = 10 5 = 5 NO
Since 5 does not equal 5, the only solution is x = 4; x = 1 is not a solution.
It is important, then, to check each answer when solving absolute value equations. Even if the equation is solved correctly, one of the answers may not be an actual solution to the equation.
SAT Math  This Week's Questions (December 12, 2020)
ABSOLUTE VALUE
The absolute value of a number is the size of the number without regard to the sign.
 4  = 4 The absolute value of 4 is 4.
  4  = 4 The absolute value of 4 is 4.
The symbol │ x │ is used to represent the absolute value of a number x.
Another way of looking at absolute value is to consider the number line.
.
┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼┼
5 4 3 2 1 0 1 2 3 4 5
The absolute value of a number is the distance from 0 on the number line. Since distance is nonnegative, the absolute value of a number is always nonnegative. The distance from 0 to 4 is 4 units, and the distance from 0 to 4 is 4 units. Again, the absolute value of a number is the size of the number without regard to the sign.
SAT Math Formulas
Formulas Given in the Test Booklet
At the beginning of each math section these formulas are given in the test booklet. If you haven’t memorized them, you should be familiar with what they mean.
 Area of a circle: A = πr^{2 }
 Circumference of a circle: c = 2πr
 Area of a rectangle: A = lw
 Area of a triangle: A = ½ bh
 Pythagorean theorem: c^{2} = a^{2} + b^{2}
 30° – 60° right triangle:
 the length of the hypotenuse = twice the length of the side opposite the 30° angle.
 the length of the side opposite the 60° angle = the length of the side opposite the 30° angle times √3
 the length of the side opposite the 30° angle = ½ the length of the hypotenuse
 45° – 45° right triangle:
 the two legs are equal
 the length of the hypotenuse = the length of the either leg times √2
 The volume of a rectangular solid: V = lwh
 The volume of a cylinder: V = πr^{2}h
 The volume of a sphere: V = (4/3) πr^{3}
 The volume of a cone: V = (1/3)πr^{2}h
 The volume of a pyramid: V = (1/3)lwh
 The number of degrees in a circle = 360
 The number of degrees in a triangle = 180
 The number of radians in a circle = 2π
You are given these 12 formulas and three geometry laws on the test itself. It can be helpful and save you time and effort to memorize the given formulas, but it is ultimately unnecessary, as they are given on every SAT math section.
Formulas NOT Given in the Test Booklet
The following formulas are not printed on the test booklet; you will have to memorize them.
 The area of a square: A = s^{2}
 The perimeter of figure = the sum of all of the sides
 Area of a parallelogram: A = lw
 Area of a trapezoid: A = ½ h(b_{1} + b_{2})
 Given a radius and a degree measure of an arc from the center of a circle, find the area of the sector that is defined by the angle and the arc:
Area of a sector of a circle: A = (t/360) πr^{2} when t = the number of degrees in the central angle  Given a radius and a degree measure of an arc from the center, find the length of the arc:
Length of an arc: L = (t/360) (2πr) when t = the number of degrees in the central angle  When the angles of triangle A are equal to the angles of triangle B, the sides of triangle A are proportional to the sides of triangle B.
 x^{2} – y^{2} = (x + y)(x – y)
 (x + y)^{2} = x^{2} + 2xy + y^{2}
 (x  y)^{2} = x^{2}  2xy + y^{2}
 A function in the form of f(x) = 3x + 12 is the same as y = 3x + 12.
 The equation of the line in the slope/intercept form:
y = mx + b, where the slope = m, and the yintercept = b.  The equation of the line in standard form:
Ax + By = C, where the slope = A/B and the yintercept = C/B.  Slope – four ways to determine the slope:
 Slope = rise (vertical change)/run (horizontal change)
 Given two points on a line, (x_{1}, y_{1}) and (x_{2}, y_{2}), the slope = (y_{2} – y_{1})/(x_{2} – x_{1}).
 If the equation of the line is in the slope/intercept form, y = mx + b, the slope = m.
 If the equation of the line is in standard form, Ax + By = C, the slope = A/B
 The standard form of a parabola equation: y = ax^{2} + bx + c
 Vertex form of the parabola equation:
y = a(x – h)^{2} + k, where the vertex is the point (h,k).  Equation of a circle? (x – h)^{2} + (y – k)^{2} = r^{2} where the center of the circle is the point (h,k)
and the radius of the circle is r.  The quadratic formula:
For ax^{2} + bx + c = 0, the value of x is given by:
x = (−b ± √ b2 − 4ac ) / 2a  The key to solving average problems is to find the total of the items before doing anything else.
There are two ways to find the total:  Total= sum of the items
 Total = the average times the number of items. This method is usually required on SAT problems.
 Average speed = total distance / total time; Distance = (speed) x (time)
 SOHCAHTOA (applies to a right triangle)
 sine of an angle = side opposite the angle over the hypotenuse (SOH)
 cosine of an angle = side adjacent to the angle over the hypotenuse (CAH)
 tangent of an angle = side opposite the angle over the side adjacent to the angle (TOA)
 180 degrees = π radians
 Imaginary numbers
 i = √1
 i^{2} = 1
 i^{3} = i
 i^{4} = 1
 i^{5} = i
 i^{6} = 1
 i^{7} = i
 i^{8} = 1
etc.
 A present amount P increases at an annual rate r for t years. The future amount F in t years is:
F = P(1 + r)^{t}  A present amount P decreases at an annual rate r for t years. The future amount F in t years is:
F = P(1  r)^{t}  Item sold at discount: discount amount = original price x discount percent
 Item sold at discount: reduced price = original price x (1discount percent)
 Given two points, A(x_{1},y_{1}), B(x_{2},y_{2}), find the midpoint of the line that connects them:
Midpoint = the average of the x coordinates and the y coordinates:
(x_{1} + x_{2}) / 2, (y_{1}+y_{2}) / 2  Given two points, A(x_{1},y_{1}), B(x_{2},y_{2}), find the distance between them:
Distance = √[ (x_{2  }x_{1})^{2 }+ (y_{2  }y_{1})^{2}^{ }]
Actually, this is one formula you do not need to memorize, since you can simply graph your
points and then create a right triangle from them. The distance will be the hypotenuse, which
you can find by using the Pythagorean Theorem.  Probability of x = (number of outcomes that are x)/(total number of possible outcomes)
SAT Math Operations
Operations You Need to be Able to Perform
 Substitute values for a variable and simplify.
 Add fractions with different denominators, where the denominators are numbers.
 Add fractions with different denominators, where the denominators are variables.
 Know how to simplify complex fractions.
 In a fraction, the denominator cannot equal zero. If an equation is solved and the value of the variable makes the denominator = zero, then that value cannot be a solution to the problem.

When picking numbers, consider positive numbers, negative numbers, zero, decimals, and extreme numbers.

Understand the definitions of the terms digit, integer, number, prime number, factor, multiple, divisible, reciprocal of a number, absolute value of a number.

Know the absolute value sign.

Know the common fractiondecimalpercent equivalents.

Know how to change a fraction to a decimal or to a percent.

Know how to change a decimal to a percent.

Know how to change a percent to a decimal.

Understand the factorial concept.

Know how to compute permutations and combinations: n items taken x at a time.

Know when to use Venn diagrams.

When angles are formed when a line crosses parallel lines, several equal angles are created.

When a diagram is given in a geometry problem, consider adding one or more lines to create another figure.

Geometric figures are not necessarily drawn to scale; lines that look equal may not be equal; angles that look equal may not be equal.

In a triangle, the length of sides opposite equal angles are equal.

In a triangle, the length of a side opposite a larger angle is greater than the side opposite a smaller angle.

Know the third side rule for triangles: the length of any one side of a triangle must be less than the sum of the other two sides, and greater than the difference between the other two sides.

Two triangles are congruent if the sides of one triangle are equal to the corresponding sides of the other triangle and the angles of one triangle are equal to the corresponding angles of the other triangle.

Two triangles are similar if the angles of one triangle are equal to the corresponding angles of the other triangle and the sides of one triangle are not equal to the corresponding sides of the other triangle.

If two triangles are similar, their corresponding sides are proportional.

The measure of an angle inscribed in a circle is half the measure of the central angle that intercepts the same arc.

The length of an arc is a fraction of the circumference of a circle.

A line tangent to a circle produces a right angle at the point of tangency between the line and another line that connects the point of tangency to the center of the circle.

Know the exponent rules.

Know how to express a number with alternative bases using appropriate exponents; the most common problems involve changing a number to a base of 2 or a base of 3.

Know how to determine the three averages: mean, median, and mode.

Know how the normal curve, mean, and standard deviation interact.
 Read ratio problems carefully,
 A ratio can express a part to part relationship.
For example, a ratio of 1 to 2 = 1:2 = ½.  A ratio can express a part to whole relationship.
For example, a ratio of 1 to 2 has two parts and a whole (1 + 2 = 3). One part is ⅓, the other part is ⅔.  Solve linear equations when the answer is a number (one equation and one unknown.)
 Solve linear equations when one variable is in terms of other variables (one equation with all variables.)
 Solve simultaneous equations (two equations in 2 unknowns.)
 Solve quadratic equations by factoring, by using the quadratic equation, and by completing the square.
 Find the radius of a circle from the formula of a circle.
 Know how to write the formula of a circle in standard form.
 Factor an expression.
 Type 1: 3xy + 7x = x(3y + 7)
 Type 2: 2x^{2} + 13x + 15 = (2x + 3)(x + 5)
 Type 3: 2^{13} – 2^{11} = 2^{11}(2^{2} – 1) = 2^{11}(41) = 2^{11}(3)
 Solve inequalities.
 Find the price of an item after a sales tax is added.
 Find the price of an item after a percent increase.
 Find the price of an item after a percent decrease.
 Find the percent of a number.
 When one number is greater than another, find the percent greater.
 When an amount changes, find the percent change.
 Know the three averages: mean, median, and mode.
 In an average problem, the first thing to do is to find the total; there are two ways to find the total.
 Find the average of a set of numbers.
 Find the missing number in a set of numbers when the mean is known.
 From the equation of a line, determine the y intercept, x intercept, and slope.
 Understand positive slopes, negative slopes, and slope = zero.
 Equation of a parabola.
 Coordinate geometry: locate points in the xy plane.
 Know the I, II, III, and IV quadrants.
 Evaluate information in a chart.
 Word problems: write down each detail; proceed in a step by step fashion.
 Trigonometry: find the sine of an angle; find the cosine of an angle; find the tangent of an angle (remember SOHCAHTOA.)
The sine of an angle = the side opposite the angle divided by the hypotenuse (SOH)
The cosine of an angle = the side adjacent to the angle divided by the hypotenuse (CAH)
The tangent of an angle = the side opposite the angle divided by the side adjacent to the angle (TOA)  In right triangle ABC, if angle B is the right angle, then the sine of angle A = the cosine of angle C.