New SAT MATH as your WORKBOOK

Graphs-Increasing Linear Functions: We learn about the difference between "strictly increasing" versus "increasing" linear functions. I share a tip on how to solve this question quickly.

Linear equation-Solution: I solve for the unknown coefficient in a linear equation, get the complete form, and evaluate the equation at another x-value.

Parallel Lines-Transversals-Angles: I first use this question as a setting to learn about the 8 angles formed when two parallel lines "l" and “m" are cut by a "transversal". We learn which of these 8 angles are equal and why; the ”corresponding angles","opposing angles" and "supplementary angles". Next, I solve the question using BOTH sets of parallel lines given in the question-both,"l" &”m" and "s" and "t".

Linear Equation-Word Problem: Key is to correctly translate an "English sentence" into a "linear equation” and then solve for the unknown variable "x".

Scatterplots: We are given 4 graphs (Scatterplots): I analyze each and look for the one that has a negative association between d-values and t-values.

UNIT conversion-Word Problem: I convert all the different units into to a common unit and solve for the number of 1-milligram doses in a 2 decagram container.

Bar Graph-Units: I discuss how to read a bar graph; construct a simple linear equation to find the unit of the y-axis variable.

Absolute Value-Linear Equation: I solve a simple linear equation where the unknown term is in "absolute value".

Linear Equation: I isolate the x-variable ( "t" ) in a linear equation. A tip is also shared for a quicker solution.

Linear Equation Word Problem: I solve the simple linear equation word problem defined in the previous question, by solving for the air temperature “t”, at which “a”, speed of the sound wave, is closest to 1000 feet per square.

Linear Inequality-Solution: I first solve for the solution set of the linear inequality we are given. Then, of the choices given, I find which is NOT a solution.

Histogram-Mean: We are given  "distribution" of 12 apples in the sample as a function of the number of seeds they contain; I find the average of number of seeds per apple in the sample.

Table Data-Survey: We need to find, of the 6 categories given, which one corresponds to 19% of all survey respondents. I first discuss how we read the table data in terms of "rows" and " columns”; then I find "all" survey responds in the table and solve.

Outlier-Mean-Median-Range: Based on the sample of 21 observations, I calculate the “mean, the “median" and the "range" for the full sample (21 observations) and see which of these 3 measures is impacted most when the outlier observation ("24-inch fish") is excluded from the full sample. I also use this question as a setting to discuss how we find the mean, median and the range of a sample of data. In particular, I discuss how we find the “median" when the number of observations in the sample is an "odd" number versus when it is an "even" number.

Linear Cost Function-Interpretation: I interpret the C-intercept in the h-C space (where h is the hours and C is total cost. Because it is instructive, I also discuss each of the choices. In addition, I introduce the important distinction between variable costs vs fixed costs.

Linear Function Word Problem: Follow up on Q-15. I find the linear function that defines the relation between "h" and “C” after finding the slope and the y-intercept.

Functions-Graphs: Given the graph of function f(x), I first discuss how we correctly interpret the graph of f(x): what it means for a point (x,y) to be on the graph of the f(x). I, then find the corresponding x-value for which the function f(x) is at its minimum.

System of Linear Inequalities-Solution: Exploiting the fact that (0,0) is a solution of the given system, I find the relation between the" y-intercepts" "a" and “b” and solve the relationship that must hold.

System of Linear Equation Word Problem: Based on the word problem (“the story”), I construct 2 linear equations in 2 unknowns (the number of salads and drinks). This problem is then equivalent to solving a "system of linear equations, which I solve by the “substitution” method. This a typical "system of linear equation SAT word problem", and should be understood and studied well.

Percent Word Problem: I discuss how to write percent equations when there is a discount on the price, and when there is a sales tax on the price, and solve the question. I also review, what a percent is, and how we convert it to a decimal number. This a typical percent SAT word problem, and should be understood and studied well.

Table Data-Probability: I discuss the logic of probability, and find the probability that the people come from Group Y, from all the people who recalled at least 1 dream.

Data-Average Rate of Change: Based on a table data of 6 categories (6 rows) over 4 years ( 4 columns), I calculate the "average rate of change" in the desired category: " agriculture budget" from year "2008 to 2010”. I also discuss time-saving rounding tips as we find not exact figures but best approximates in arriving at the correct answer.

Data-Ratios: Follow-up question to Q-22. I calculate the desired "ratios" from the data presented in the Table in Q-22.

Circle Equations Word Problem: I first introduce the “standard form equation for the circle” discussing the underlying logic behind it. I next discuss the "distance formula" and show how the two are related, and finally, solve the question at hand. This is standard mid-level challenging SAT Circle Equation question and should be understood and studied well.

Quadratic Equation-Parabola Word Problem:A ball is thrown vertically up; We are given the trajectory of height “h" as a quadratic function of time "t": h(t). I solve for the time when the ball hits the ground. This is a standard SAT Parabola word problem, and should be understood and studied well. I also note what piece of redundant info is given to "confuse" the student: a time saving warning!

Percent Word Problem: Setting up an equation, I solve for x, the number of pears Type B trees produce.

Data-Mean-Proportions: I extrapolate (that is, extend to a larger data size) what I learn in a small random sample on the sample mean. That is, I find the mean in a random sample of 10 observations, where each observation is the number of worms found by each one of the 10 students in his/her respective 1mx1m =1m2 field. Note, mean is the average number of worms found in each of the ten 1m2 fields. I extrapolate this small sample mean to the expected number of worms in the larger field 10mx10m= 100 m2

System of Linear Inequalities: We want to find out which Quadrant has no solution to the system of linear inequalities given. I start by discussing more generally, how a solution to a system of linear inequalities is found on the x-y space. I then solve for the solution to the system, and diagnose in which Quadrant there is no solution. I also give some tips on how to draw the implied lines in the system quickly.

Polynomials-Factors-Remainder Theorem: Solution requires an understanding of how "polynomials" and "factors" are related to each other: I solve the question by elimination showing why choices A through C are NOT conditions that MUST hold; thus D is the answer by elimination. But, I also discuss choice D as it is related to the important Remainder Theorem. So we could alternatively solve this question directly using the remainder theorem.

Parabolas-Equivalent Forms: I use this question as a setting to study all of the 3 forms for the parabola: “standard”, “root" and the “vertex” forms. As the question asks for the "Vertex form” for the given parabola, I write the vertex form; I also evaluate each of the other choices and discuss why they would be eliminated.

Linear Inequality Rate Word Problem: I solve the question by figuring out the time Wyatt requires to eat 72 dozen of corn as a function of his "eating" rate. I also discuss the units.

Linear Inequality Word Problem: I write the linear inequality in terms of "x": the number of boxes, and solve for the max "x" that solves the inequality.

Graphs-Ratios: Solution is based on the correct reading of the graph that connects 6 data points on the x-y space. I calculate the number of players sold in 2008 as a fraction of players sold in 2011. Simple question not to be missed!

Linear Word Problem-Ratios: I calculate the number of half hour shows sold for 2 day broadcasting. Simple question not to be missed!

Volume Word Problem-Cylinder:  Right circular cylinder silo is given. I discuss what a right circular cylinder is and how we calculate its volume. I calculate the diameter of the silo given its volume, paying attention to the units involved.

Structure-Rational Equation: A rational equation h(x) is given; we need to find at what value of x, h(x) is undefined. “Seeing” the structure in the denominator of the fraction, I solve for the x-value, for which h(x) is undefined.  This requires knowledge of squares of sums formula.

Exponential Functions-Interpretation: Solution requires an understanding the "compound interest" logic and the associated "annual compound interest formula". I first discuss these crucial topics and then solve the question at hand easily.

Exponential Functions Word Problem: Follow-up compound interest formula question to the previous one, Q-37. I calculate compound return on J’s and T’s deposit of $100 for 10 years at an annual compound rate of 2% and 2.5% respectively. I conclude by finding the difference in the respective returns over 10 years between the two accounts.

Non-linear Expressions-Interpretation: I interpret the given expression n.K.l.h

Linear Equation-Solution: I solve a simple linear equation in two different ways: First I solve for r, and substitute; second, manipulate 6r+3 and solve.

Radicals/Roots: I find an equivalent form of the "fractional exponent" term given.

Linear equation word problem: I convert the verbal description into an equation.

Rational Equations-Solution: I solve for the unknown variable in the rational equation given. I also discuss cross multiplication as a short-cut method.

System of Linear Equations-Solution: I manipulate the given system such that I can use the "elimination" method to solve the system. I then calculate "x-y".

Polynomial factors-Graphs: A table that shows several (x,f(x)) points for function f(x) is given. I discuss what a "factor" for a polynomial is, and find a factor of f(x) among the choices given.

Linear Equations-Slope-Solution We are given equation of a line in its slope- intercept form, where the slope term is given as a letter "k". We are also told that the line passes through point (c,d). I solve for k as a function of c and d.

System of Linear Equations: A system of (two) linear equations is given. The coefficient of the x variable in the first equation is an unknown, denoted by letter “k”.  I solve for “k” if the system has no solutions.

Non-linear System-Solution: We are given a non-linear system where the first equation is a parabola and the second one is a horizontal line. I find the solution of this system by finding the two points where the line intersects the parabola-and by finding the horizontal distance between these two points.

Vertical Angles: We are given 3 statements and we want to find out which ones must be true. I show that 2 of these statements are true by using the equalities implied by vertical (or opposing) angles; I show that one of the statements is false by using the proof method of contradiction.

Quadratic Equations-Manipulation: We are given the equation of a parabola in its "root form” where the leading coefficient is an unknown "a". We are told that the Vertex of the parabola is (c,d). I find out what the unknown y-coordinate of the vertex, “d” is.

Rational Expressions-Matching Coefficients: We are given a rational equation, with the unknown "a" in the denominator. I solve for "a" using 2 different methods: 1) I manipulate the system in order to attain equal denominators in the fractions and use the method of "matching coefficients”; 2) I discuss a quicker method that works through evaluating the expression at a particular value, that may work better at time sensitive settings.

Quadratic Equation-Solution-Quadratic Formula: I introduce and discuss when it is optimal to use the quadratic formula to find the solutions of a quadratic equation. I solve for the solutions by using the quadratic formula.

Liner Equations-Interpretation: We are given a linear equation C as a function of F, and 3 statements linking C& F to each other; and we want to find out which ones are correct. This question asks interpreting the slope coefficient of a “slope-intercept" form of a linear equation. Very useful question to study to learn this critical topic.

Structure in Expressions: 4th degree equation is given: I simplify and solve "seeing" the structure in the equation after making a "change of variable".

Linear Equations-Solution: I solve a linear equation in 1  variable for the unknown variable x. Note each term is a fraction.

Similar Triangles-Isosceles Triangles: We are given a figure where 2 "isosceles" triangles are placed in between 2 lines. I solve for the unknown angle x.

System of Linear Equations Word Problem: I construct 2 linear equations as a function of the 2 unknowns. “calories in H” and “calories in F”; H for Hamburger; F for Fries. I use "substitution" method to solve for H.

Right Triangle Trigonometry-SOH-CAH-TOA: We are given 2 similar right triangles. The question asks for sin(F); one of the internal acute angles in the smaller triangle. I find sin (F) showing several time cutting short cuts on the way. I review and use SOH-COH-TOA , congruent triangles and Pythagorean theorem in the solution.

Graphs-Interpretation: I interpret a graph that relates time (x-variable) and distance (y-variable).

Table Data-Probability: Table data that shows distribution of Age (under 40 or above 40) and Gender (F or M) for 25 people is given. I find the probability that the winner selected at random is a F "under age 40" OR a M "age 40 or older".

Graphs-Trend I find/discuss the general trend of a graph that shows "Sales" (y-axis) against time in "Years since 1997 " (x-axis).

Linear Function Word Problem: We are given a table that shows the relation between variables “n" and “f(n)." I find what the relation is and write it as a linear function.

Percent Word Problem. I calculate the number of juniors and seniors who were inducted to Honors. Standard percent problem that should not be missed!

Polynomials-Addition: I add the two 2nd degree polynomials given by combining like terms

Linear Equation-Solution: I solve the linear equation given for the unknown variable "w". An easy question not to be missed!

Linear Model-Interpretation-Slope: I discuss how to interpret the slope coefficient in the slope-intercept form of the linear model given and pick up the correct choice.

Rate/Unit Word Problem: Given Nate walks 25 meters in 13.7 sec, I find the distance he walks in 4 min if he keeps the same rate walking.

Ratios Word Problem: I calculate the Weight of an object in Mercury that has a mass of 9 kg, using Weight= Mass x Acceleration formula with respective units Newton = kg* (m/sec2).

Ratios Word Problem: Follow up question to the setting introduced in the previous question: Q-10-T3. This time, we consider an object that has a weight of 150 newtons on Earth. I use the Weight(newtons) formula to figure out at which planet the SAME object is located at, if it has a weight of about 170 newtons.

Non-linear Equation Graphs-Roots: We are given graphs of 4 non-linear functions; we want to find out which of the graphs has 5 distinct zeros. I proceed by elimination, checking each of the graphs by the number of x-intercepts they have, to arrive at the correct answer.

Isolating quantities: Given a quadratic function h(.) in terms of variables h,t ,v and k, I solve for v(.) as a function of variables h,t and k. I also share a tip.

Linear Function Rate/Word Problem: We are given per min rate of phone call as $0.20: I calculate $ cost of a phone call for h hours .

Data Collection and Conclusions: I discuss what experimental research is. Note, for a cause-effect relation to be established: (1) population must be defined, (2) participants must be selected at random (3) participants must be assigned to Treatment group randomly. I discuss how these conditions eliminate choices B, C and D.

Quadratic Functions-Graphs: Graphs of quadratic functions (parabolas) f(x) and g(x) are given. The question tests knowledge of function notation: I look for the x value at which f(x) + g(x) =0.

Non-linear Expressions-Interpretation: Demand and Supply functions are given as functions of the price “p” in the market. I find how quantity supplied in the market changes, if the price of the product increases by $10.

System of Linear Equations-Market Equilibrium: I solve for the price at which market equilibrium takes place in the market we are introduced in the previous question, Q-17.

Ratio-Unit Word Problem: I find how many acres 48 ounces of Graphene cover, if 1 ounce of Graphene covers 7 Fields, by using conversions from field to acres in arriving at the solution.

Scatterplots-LBoF: A scatterplot and the corresponding line of best fit (LBoF) is given. I discuss what the scatter plot shows, and how the line of best fit is constructed in theory (not necessary for solution). I find the difference between the actual and the predicted values-corresponding heart beats per min (y-axis) for 34 min swim time (x -axis). 

Linear vs Exponential Growth: Of the 4 different types of saving plans given, I find which yields exponential growth. I first, discuss the "same amount" increase per year vs the "same factor" increase per year. This is a key difference which the SAT tests students on in every test! So be sure you are clear about it!  Also recall the exponential growth case we studied earlier: compound interest rate in Test 1 Part 4 Q37 and Q38.

Linear Equation Word Problem: I construct 2 linear equations as a function of the unknown three numbers, x, y and z, and solve for x.

Complementary Angle Property-Trigs: I easily solve this tough looking question by making use of a special trig relation we already learned in Test 1 P3 Q19, with regard to 2 acute angles x and y, whose sum is complementary. That is: "sin x= cos (90-x)", if x+y= 90 degrees". By using this special trig property (technically called: Complementary Angles Property), I easily solve for the unknown "k" in the question.

System of Linear Equation Word Problem: I algebraically represent the story in 2 equations as a function of the unknown “s": number of students, and solve for s.

Volume Word Problem-Cylinder-Cone: A silo is built from 2 right circular cones and a right circular cylinder. I solve for the volume of the silo using the volume formulas for a right circular cone and a right circular cylinder.

Lines-Slope: The question implies that the line passes through 3 points: 2 points are given in terms of k ; the origin is the 3rd point. I use definition of slope of a line to solve for k. Note, we can take any 2 points on the line to calculate the slope of the line.

Percent Word Problem-Rectangles: A rectangle's length is increased by 10%, and its width is decreased by p%. I solve for the unknown p.

Exponential Word Problem-Negative Growth: Population in a city decreases by 10% every 20 years. Hence, I modify the exponential function for time t (process elapse time) accordingly and find the estimate for population of the city t years from now. I also discuss negative exponential growth; how it differs from exponential growth cases we had seen earlier in savings questions.

System of Linear Equations and Probability: I find both the "left handed female"  and the "total number left handed" students through solving a system of linear equations. I find the probability that a left handed student selected at random is Female.

System of Linear Equations-Solution: We are given 2 linear equations as a function of x, y and constants b and c. I solve the system after  to translating the relation between constants b and c  into an equation. The final answer requires another translation from algebraic equation into words.

Linear Inequality Word Problem: I set up a linear inequality for the number of students x, the unknown, and solve for the possible solutions for x.

Mean-Calculation: Given a table data of 12 observations for the President’s age, I find the mean President age.

Polynomials-Matching Coefficients:  We are given an algebraic expression that is equal to the quadratic equation: ax^2+bx +c. I solve for the unknown coefficient “b” by "matching coefficients" after simplification. For a detailed early discussion of this method, see Test 1 Part 3 Q15.

Circles-Sector Area-Radians :A circle with center O with central angle AOB with a measure of 5pi/4 radians is given. I find the area of the sector formed by the central angle 5pi/4 as a fraction of the full circle. I first discuss radians as angle measure of central angles in circles as background and solve the question by applying the proportional equality rule for circles. This rule is defined by the measure of central angle relative to the full angle 360 degrees, or 2pi radians. This is a standard SAT circle question; and is always solved by the correct application of the proportional equality discussed.

Inequality Word Problem: A store receives ratings between 1 and 100 inclusive. First 10 ratings with an average 75 is given; We want to find out the least value the store can receive for the11th rating and still have an average of at least 85 for the first 20 ratings. Correct solution relies on an understanding of the calculation of the mean and ability to write and solve linear inequalities.

System of Linear Inequality Word Problem: A system of linear inequalities is given. Point (a,b) lies in the solution system of the 2 linear inequalities given, and we want to find the maximum possible value of “b”-,the y-coordinate of point (a, b). No graph is given in the question; but I make a graph that shows how 4 different areas are created when the system of inequalities is drawn; note, solution points lie only in area 4 of the graph. I find point with the max y-coordinate.  I also discus how to make a quicker to do sketch that still gives us the correct solution without spending time on drawing the lines on the x-y space precisely on paper or on a graphical calculator.

Application of the "Little's Law”-Units: During business hours, 84 shoppers make a purchase per hour and they spend an average of 5 min on the check out line. Converting hour rate to min rate, I apply the Little's Law, and find the average number of shoppers on the check out line.

Percents-Little's Law: Follow up question to Q-37. In the new store, shoppers enter into the store 90 min per hour, and spend on average 12 min in the store. Applying the Little's Law, I find the average number of shoppers. Compared to the old store, this implies a 60% reduction.

Linear Equations-Graphs: Given the graph of a line, we want to find its equation. I discuss 2 different methods to find the equation of the line. I also discuss all the other answers, so that we learn how to write the equation for the "horizontal line” , the "vertical line" and "line from the origin".

Circles-Arc Length-Central Angles: I discuss what arcs are and how central angles and arc lengths are related through a proportional equality relationship.

Quadratic Equations-Solution: I solve for the roots (i.e zeros) of the quadratic equation given.

Polynomial Factors-Graphs: Of the 4 choices given, I look for the function with no x-intercept. That is, no zeros/roots. I move by elimination; discussing the relation between roots of a quadratic function and its discriminant (choices B and C); and graph of a line (choice A); and a cubic polynomial with at least one real zero. Critical test question where we discuss/learn about roots/zeros of a first, second and third degree function and their respective graphs.

Radical Equations-Solution: I solve a radical equation as a function of x. Note the function has a constant k which is unknown. We want to solve for "k" when x takes value 9. Key to solve the question is to start by substituting 9 for x and solve for k.

Polynomials-Addition: I add 2 polynomials through combining like terms. Very simple questions never to be missed!

Systems of Linear Inequalities Word ProblemsThe question simply asks us to write the 2 inequality equations that define the system. One is a constraint on Earnings; the other one is a constraint on Time. The logic we learn here forms the basis for economic optimization problems we learn in economics courses later in college.

Linear Functions-Interpretation: I interpret the intercept term in a linear function. A time saving tip with regard to the y-intercept is also shared.

Non-linear System-Solution: I solve the non-linear system of equations given: the first equation is a quadratic equation; the second, a linear equation.

Structure in Expressions: A question about seeing the structure in the equation we get after making the necessary substitutions. I make use of the “square of a sum" formula in the solution: (a+b)^2 = a^2 +2ab +b^2.

Volume Word Problem-Cylinder: We are given a right circular cylinder with volume equal to 22. I find the volume of the same cylinder after 2 adjustments: increase the radius of the cylinder twice; and halve its height.

Exponents-Roots: We are given a fractional exponent: 9^(3/4). I find its equivalent through applying rules of exponents: power rule; multiplication rule, power of a product rule and the fractional exponent rule. Be sure to learn about the rules themselves discussed in the Appendix as well.

Linear Functions-Interpretation: We are told that n cups of tea are made by adding t tea bags to hot water. If t=n+2, I find the number of additional number of tea bags needed to make each additional cup of tea. Interesting problem!

Graphs-Transformation-Exponential: We are given f(x) = 2x +1. I find y=-f(x) from the four choices A through D we are given using method of elimination. In particular, I analyze the behavior of the function y=-f(x) as x increases, and the y-intercept, when eliminating the potential choices. Note; we could just focus on the y-int and find the answer much quicker!

Linear Equation Word Problem: This is a gas mileage type of problem. Alan drives 100 miles per week on average; his r can travel 25 miles per a gallon of gas; and a gallon of gas costs $4. Alan decides to decrease his travel time “m” average miles per week such that he saves $5 on his weekly gas expenditures. I solve for “m” such that Alan’s weekly gas expenditure decreases by $5.

Linear Inequality Word Problem: This is a basic linear cost function problem: rental costs for a boat are $60 per hour and safety course is $10. I discuss and distinguish between the fixed versus variable costs. I solve for the max number of hours Maria can rent the boat if she does not want to spend more than $280, and if she can rent the boat for only a whole number of hours.

Linear Equation-Solution: I solve the linear equation for the unknown variable "p". Simple question not to be missed!

System of Linear Equations-Solution: I solve the system for the solution point (x,y) in the x-y space, and find the x-coordinate of the solution point as desired.

Rational Equations-Matching Coefficients: I solve for the unknown constant "a" in the equivalent expression through the method of matching coefficients. We have already seen several such questions; as in Test 3, Part 3 Q13, for example. A good idea is to try the quicker method suggested there and see that you obtain the same solution.

Triangles-Angles: We have a figure where 3 lines, s, t and r intersect such that they form a triangle. I find the unknown angle “x” using 2 properties:(1) sum of supplementary angles equals 180; (2) sums of interior angles in a triangle equals 180. A quicker method would be to use the property that sum of 2 interior angles equals the exterior angle of the remaining third angle.

Graphs-Slope: Time (years) against "Number of 3D movies" is plotted. I discuss the graph and how to read it, and find at which 2 consecutive years, change in 3D movies released was the greatest. I find it both visually; and algebraically.

Linear Function Word Problem: A table of x and f(x) is given for the linear function f(x): I find the equation for the line using slope-intercept formula. Note, it is enough to find the slope to find the correct choice here. As it may not be enough in other settings, I also show the method to find the y-intercept as well.

Units-Word Problem: One muffin is made with 2.5 ounces of chocolate. I find how many pounds of chocolate is needed to make 38 muffins. I solve it by setting up the "proportional ratios" equality and making the necessary unit conversion from ounces to pounds. (1 pound = 16 ounces)

Linear Equations-Solution: I solve a standard linear equation in 2 unknown constants: c and d.

Ratio Word Problem : Given the weight of the object on Earth, I calculate its weight on Jupiter and Venus according to the given ratios, and find the difference between the two.

Systems of Linear Equations Word Problems: Store sells novels and magazines; each n costs $4, and each m costs $1 -("n" for novels; "m" for magazines). If S purchased a total of 11 n and m that cost $20, I solve for the number of n she bought.

Linear vs Exponential Model Word Problem: Store DBA increases membership by a total of n businesses per yer. I make a chart that shows how membership behaves over time. Note that there are 2 linear and 2 exponential models in choices A through D. I make the distinction between “same amount" versus “same factor” change from year to year; and discuss and choose the right model.

Operations with Polynomials: I expand and simplify the algebraic expression given with coefficient terms with decimals. Watch out and do not make a mistake with the decimals!

Unit Word Problem: Marathon extended from 40 km to 42 km: I find the increase in distance in terms of miles, by making use of the "proportional ratios”.

Isolating quantities: I algebraically represent an equation in “words” and solve for the desired variable m, as a function of d and V.

Perpendicular Lines: We are given a linear equation in "standard form". I solve for the equation that is perpendicular to this line, by using the slope property for perpendicular lines (Note Typo: question # written as Q-13; should be Q-11.)

System of Linear Equations-Solution: I solve the system by the method of elimination and find the desired x-coordinate of the solution. I also use the substitution method as an alternative, as it is equally easy to use it in this question.

System of Inequalities-Solution: We want to find out, of the 4 choices given the point that satisfies the system. The point that satisfies the system of inequalities should satisfy both of the equations in the system. I solve by method of elimination, first checking if the point satisfies inequality (2). This process leaves only one choice; I also check and show that, this point also satisfies inequality (1) as well in the system.

Data Table-Probability: A 2x2 table of survey data for 607 surgeons (S) is given. An S is either O-Surgeon or G-Surgeon; AND an S either does T or R. This creates 4 categories: G-T, G-R, O-T, O-R. I find the prob that a randomly selected surgeon is O-R.Note: O for orthopedic, G for G General; T for teaching, R for research.

Data Collection and Conclusions (ccl): We are given a survey design and 3 statements. The question asks us which statement HAS to be True. In each case we are changing the sample and asking if the ccl of the survey (the exact percentage result) would hold. Discussing each case individually, I show why NONE has to be true.

Table-Data Word Problem: A data table is given that specifies "Growth Factor" for 8 different trees species. We are told: Approximate age, in years, of a tree of a particular species is found by multiplying the diameter of the tree, in inches, by a constant called the growth factor unique to that that species. We find app. age for "American Elm tree" with a diameter of 12 inches.

Data Collection-Scatterplot-Slope: Given a scatterplot of 26 observations of Tree Age against its Diameter for a SINGLE species, I estimate the implied Growth Factor, and find to which tree it belongs, from the growth factor data table.

Table-Data Word Problem: If a White Birch and Pin Oak tree each now have a diameter of 1ft, we find the difference in their diameters in 10 years from now, in inches. (1 ft= 12 inches)

Right Triangle Trigonometry: I start by finding all the angles in the triangle ABC. Finding 2 congruent right triangles , I find the size of AC. This is a classic 30-60-90 right triangle problem.

Graphs-Interpretation: The question is about seeing what the figure is charting as the wheel turns rightwards: Is it drawing the behavior of the wheel or the mark on the rim of the wheel? I discuss why it is the mark on the wheel. I also discuss the other choices and eliminate them. a) implies a horizontal line; b) implies line y=x; c) implies a horizontal line; y=radius, a constant.

Structure in Expressions-Rational equation: I manipulate the left side to obtain c-1>1; where c=a/b- and conclude c>1.

Data-Survey-Best Estimator: Based on the mean of a representative sample for the “number of siblings" each Grade 8 Student has in State X, I calculate the mean of the population for the  number of siblings. Note the former is observed and measured while the latter is estimated.

Linear Function Word Problem: We are given a table data that relates "p=purchase price" (2nd column) to "r=rental price" (3rd column) for 5 properties-and also told there is a linear relationship "r(p)" between r and p. Also note that there is a twist in units:while "r" is defined in dollars, "p" is defined in thousands of dollars. I write the equation r(p) in the slope intercept form. I solve for the linear relationship r(p).

Percent Word Problem: A follow-up problem on Q-23. We are given the purchase price of Glenview property after 2 sequential discounts: a 40% discount on the original price, followed up by a further 20% discount on the discount price. I solve for the original price.

Linear Function Inequality Word Problem: I put into an algebraic representation an inequality relation given in words.

Non-Linear Relation-Interpretation: We are given surface area of a cube as a function of “a”. I find out the perimeter of one face of the cube as a function of “a”.

Mean Word Problem: We are given the mean score of 8 players. When the highest score is removed, a new mean score for the remaining 7 players is calculated. I solve for the highest score that was removed by setting up an equation with one unknown.

Linear Equations-Graphs We are given the graph of the linear function f(x) on squared paper-so that we know the points it passes through. We are also told that, there is another linear function g(x), whose graph not given; instead we are told that its slope is 4 times the slope of f(x); and that g(x) passes through point (0,-4). I first solve the slope-intercept form for g(x); and then find g(9).

Circles-Expanded Form-Center: We are given the equation of a circle in expanded form; we want to find the coordinates of the center of the circle. I convert the expanded form to the standard form for the equation of a circle through the method of completing the squares and find the center.

Quadratic Equations-Manipulation: I write the equivalent form of the y=x2-4, making use of the critical "difference in squares" formula. Be sure to know and be able to use this formula, as it will definitely be on your test at least once!

Proportions-Unit Conversion:  We are told that 5 Horse Power is equal t 3730 Watts. We  set up a proportional ratio equality to find out the number of Watts equal to 2 Horse Power.

Proportions-Word Problem:   A copy of a rectangle painting is reproduced in inches, where each dimension is 1/3 of the original. We find the height of the reproduction, in inches again. As there is no change in units, a ratio question easy to solve! I also use a second method to solve for the height of reproduction setting up proportional ratio.

Linear Equation Word Problem: A line segment is divided into 3 parts; length of each segment is given as a function of x: I use the relation between equal segments PQ=RS to solve for x. Having found x, I easily find the length segment PS.

Function Notation-Quadratic Equation: We are given a 2nd degree function where the constant term is an unknown "k". We are also told that point (2,5) is on this function. Plugging this value into this function, I solve for k.

Quadratic Word Problem: I let width of the rectangle garden be "w"; then, length of the rectangle is "w+5".  We are also that the area of rectangle is 104. Obtaining area as a function of "w", I solve for w, given area equals 104. Once I know what "w" is,  easy to get "l" as well, as l=w+5.

Circle Geometry Problem-Radius-Angles: I draw line AP equal to radius r (this is the key trick to solve the problem); obtain 2 isosceles triangles: triangles APB and  ABC. As AB=PB, I find measure of < APB; as AB=PC, I find measure of <APC. Given the measures of these angles, it is easy to solve for x, as sum of x and  < APB and <APC equals 360.

Speed-Distance-Unit Word Problem: There are 3 segments for drive from Home to Work. We are given data for distance (miles) and average speed(mph) for each segment. We are also told that the drive (All 3 segments) takes 24 min. I solve for the average speed used, in terms of miles per hour of the full drive.

Speed-Distance Word Problem: Variation on the previous question. Now a slight change is introduced: if the driver leaves half an hour late at 7:00 am from Home, the time to complete the second segment of the drive increases by 33% due to traffic, but all else remains the same. I find how many more minutes it takes to complete the whole trip if the driver leaves late at 7:00am relative to the regular trip with no traffic.

Linear Function-Interpretation: We are given a standard linear cost problem: I interpret the y-intercept term. (Learned linear cost functions, earlier; Test 5 Part 3 Q16.)

Linear Function Word Problem: Gardener buys two different kinds of fertilizers: F-A, and F-B. F-A contains 60% filler by weight; F-B contains 40%. Total weight of filler is 240 lb. I set up the linear equation, where x is pounds of F-A; and y is pounds of F-A . We also need to use the percentages correctly when setting up the equation correctly.

Complex numbers-Addition: I add two complex numbers as usual! Learned how to do this in Test 1 Part 3 Q2. Same logic applies!I add two complex numbers as usual! Learned how to do this in Test 1 Part 3 Q2. Same logic applies!

Quadratic Function-Matching Coefficients:-We are given a quadratic function equal to the product of 2 binomial terms with unknown coefficients. I solve for the unknown coefficients through the use of matching coefficients as usual. A quicker solution applies if you are able to see the structure on the right side where the key formula of "difference in squares" applies, that is a^2-b^2= (a+b)(a-b).

Linear Equation-Graph: We are given a linear equation; we want to find which of the 4 choices given shows its graph. Rather than trying to graph the line we are given, I proceed by elimination. I first eliminate 2 choices based on the y-intercept. Then based on the sign of slope term, I eliminate one of the remaining the 2 choices and arrive at the correct answer. Note I never need to use the magnitude of the slope term to solve the question if I solve by elimination.

System of Linear Equations-Solution: I solve the system algebraically through the method of substitution. Once I know what "x" is, I can easily find the value of 2x-3.

Isolating Quantities: We are given= 7*l*h. I solve for l.

Function Notation: We are given a table of 5 rows that gives values of w(x) and t(x) at x=1,2,3,4,5. I find for which x value, w(x)+t(x)=x

Radicals: I solve a radical equation.

Linear Inequality Word Problem: We are told J prepares for a bike race; and that he has already biked for 240, 310 and 320 miles per week for the 1st, 2nd and 3rd weeks consecutively. His goal is to have an average of at least 280 miles per week for the first 4 weeks. I write the inequality that represents the miles he needs to bike in the 4th week, such that he gets an average of at least 280 miles per week for the first 4 weeks.

Parabola-Standard/Vertex Form: We are given the graph of a parabola and its equation with coefficients as variables; and also another quadratic equation whose coefficients are related to coefficients of the first parabola. By elimination, I find which of the choices A through D is the graph of the 2nd quadratic equation given.

Rational Expression-Long division: A second degree function/polynomial is divided by a linear function: I find its equivalent by long division, which yields remainder “-2”.

Quadratic Equation-Discriminant: We are given a quadratic equation where the constant term is an unknown: “t”. We are told that the quadratic has no solution. I find of the given choices A through D, which is a possible value for “t”. Very important question where I show how the discriminant (D) of a quadratic equation shows the numbers of solutions a quadratic equation has. (D>0-> 2 real; D=0-> 1 real; D<0->no real solutions/roots.)

System of Linear Inequality Word Problem: I put the verbal story into algebraic representation, writing 2 linear inequalities: (1) Constraint on weight; (2) constraint on number of detergents relative to the number of softeners.

Exponents: We have a binomial term raised to 2nd power. I start by expanding and simplifying, adding the like terms. Alternatively, for a quicker solution , we could make use of “square of sums" formula in raising the binomial term to 2nd power: (A+B)^2= A^2+2AB+B^2.

Rational Exponents: Given a^(b/4) = 16, “a” and “b” positive integers, we want to find one possible value for b. I look at 2 cases: Case 1) I let 16=2^4 and obtain b=16; and Case 2) let 16=4^2 and obtain b=8. Other potential answers for b are 1,2 or 4.

Linear Equation-Solution: I solve a linear equation for the unknown variable "t".

Right Triangle Geometry: We are given a large triangle and a smaller right triangle inscribed within it. I show that there are two similar to triangles; using side proportionality rule, I solve for CE, the hypotenuse of triangle CAE.

Linear Equation Word Problem: 2 types of saline solutions with 25% and 10% salt percentages are mixed .While 3 liters of the second solution (10% salt saline) is used in the mix, the amount of the first solution (25% salt saline) used is unknown. I solve for the amount of the second solution used such that a 15% salt concentration saline is obtained in the mix. This is a classic mixture type word problem.

Circle Arc Length: Points A, and B lie on a circle with radius 1, and arc(AB) has length pi/3. No figure is given, but I make a basic sketch to help us visualize to think. I solve for the length of arc AB as a fraction of the circumference of the circle, using the “central angle-arc length” proportionality rule. “Another example for arc length question is Test 5 Part- 3 Q2, where central angle measure is given in degrees, but the arc length is found using the same proportionality rule.”

Operations with Polynomials: I simplify the algebraic expression given adding like terms to find its equivalent form.

Graphs-Linear Equation-Interpretation-y-intercept: Since both runners run at a constant rate, we know their positions follow a linear path. Mark who was given a head start implies his linear trajectory had a positive y-intercept. Note, each unit of increase on the y-axis corresponds 6 yards-visually seen on squared paper.

Graphs-Linear-Interpretation: Piecewise linear function describing the process for snow accumulation. Assuming no melting, key is to see that, if snow falling stopped, there would be a flat segment; and if it started falling at a faster rate, the slope would be larger, corresponding to a steeper slope; and because snow does not melt, there would be no segment with a negative slope. I proceed discussing each choice and proceed by elimination.

Linear Equation Word problem-Slope: A standard linear cost problem-I solve for the slope term.

Linear Inequality: I simplify by factoring out the common factor "3" on both sides to find the equivalent form.

Percents-Survey: Given a table summarizing 1200 responses to a survey question, I find the number of respondents who got most of their medical info from Doctors OR Internet.

Data Collection and CCL: Council wants assess opinions of all city residents about converting an open field into a dog park: So a sample of 500 city residents who owned dogs were surveyed. The survey showed that a majority of those who surveyed were in favor of the dog park. Of the 4 choices given A through D, I discuss each and determine which choice is True about the survey.

Table Data-Probability: A 2x2 table shows the Flavors of ice cream (Vanilla and Chocolate) and Toppings (Hot Fudge and Caramel) chosen by the people at a party.I discuss the setting more generally and find, of the people who chose Vanilla ice cream, what fraction chose Hot Fudge as a topping.

Units-Word Problem: We are given Total Land Area that also covers Water, and the Population of the region. I solve for Population Density in the region. Easy question not to be missed!

System of Linear Equations Word Problem: I set up 2 linear equations based on Vespucci Voyages story as a function of V-1 and V-2, the number of days each trip took respectively. I solve the system of linear equations by substitution for V-2 .

System of Linear Equations-Solution: I use the method of elimination to solve the given system; and find, “x-y” using the solution point.

Graphs:Average Growth Rate: We look for the time interval (days, x- axis) when average growth (height, y-axis) is the least. I find the exact average growth rate for each choice by applying the "average growth rate formula". I also discuss an alternative quicker method.

Linear Functions-Interpretation-Slope: We look for the interval where sunflower growth is app. linear and interpret the coefficient "a" in the l slope-intercept equation: h(t) = a(t) +b. I also discuss through representing algebraically what the other choices B through D show, and eliminate them.

Linear Equations-Modeling: We look for the linear equation representation of sunflower growth h(t) from Day 14 through Day 35. I choose points (14, 36.36 ) and (35,131) from data table in Q12 to write the slope-intercept form, and determine which choice best models the sunflower height "h" as a function of "t" for the interval Day 14 through Day 35. For a quicker solution, sufficient to find only the slope term and eliminate the wrong choices.

Linear-vs-Exponential Functions: We are given a data table that relates x to y values for 5 points. I analyze the data in the table and determine whether it s a linear or exponential process. I get the full form of the function also in arriving at my answer, though not necessary.

Right Triangle Geometry-Similar Triangles: We are given 2 right triangles. I show that these are similar triangles; set up the implied side proportionality ratio and find out the equivalent ratio in the second triangle to the specified ratio ("BC/AB") in the first triangle.

Isolating Quantities: We are given "riser-thread formula": 2h+d=25; I solve for h(d). I also use this question to introduce the architect's problem of building a stairway; not necessary for this basic linear algebra question, but necessary for the following 2 questions.

System of Liner Inequality Word Problem: Follow up question architect's staircase design problem. Buildings codes require: “d" to be at least 9 inches; “h" to be at least 5 inches. I find all the possible values of "h" that meet buildings code requirement.

Linear Inequality Word Problem: Final question on the architect's stairway design problem. New design has to have a "total rise of 9 ft; and “h" to be between 7 and 8 inches; and, an ODD number of steps. I find the unique "d" value that satisfies these constraints. Challenging problem, where the solution requires full understanding of architect’s design and any additional constraints.

Quadratic Equations-Solution: We are given the product of 2 binomial terms that equal  0. Note, this is the factored form of a quadratic equation; as such, we can easily find the solutions of this quadratic equation. We find the sum of the solutions.

Data Inferences: A random sample is taken from a pond of fish that contains 150 LM-Bass and other fish. LM-Bass in the sample is weighed, and it is seen that 30% of LM-Bass in the sample weighs more than 2 lb. Based on this random sample, 4 conclusions are drawn in choices A through D. I discuss each and show why only choice D is valid.

Data-Table-Median: We are given a data table for the "Number of Electoral Votes in 2008” for 21 states that have “More than 10 Electoral Votes”. Among these 21 states, I find the median "number of electoral votes". I also make a dot plot that visually depicts the data and complete the discussion by finding the median observation.

Graphs-Elapsed Time: A ball is dropped and is allowed to bounce repeatedly off the ground until it comes to rest. The graph represents the relationship between the time elapsed (x-axis) after the ball is dropped and the height of the ball (y-axis) above the ground. I find the number of times the ball was at a height of 2ft after the it was dropped. I give a tip to find it easily as well.

Percent-Increase-Word-Problem: Bill is given as $75.74-the old bill. After a rate increase, the new bill comes as $79.86. I find the percentage increase in the bill discussing the main percentage increase formula.

Linear Function Word Problem:  Three (x, f(x) ) values are given; we want to find f(3). I find the slope and write the linear equation for f(x) in its slope-intercept form; and, evaluate it at x=3.

Ratios/Proportions-Word Problem: A gear ratio r:s is defined as the ratio of the number of teeth of two connected gears. The ratio of the number of revolutions per minute (rpm) of two gear wheels is s:r. In the diagram of the 3 interconnected gears A, B, and C, Gear A is turned by a motor. The turning of Gear A causes Gears B and C to turn as well. If Gear A is rotated by the motor at a rate of 100 rpm, I find rpm per minute for Gear C. I discuss in detail how I arrive at the solution through using the definitions of both gear ratio and rpm.

Circle Equations-Expanded form: We are given the expanded form for a circle and need to find its radius. To do so, I convert the expanded form of a circle to the standard form-through the method of completing the squares. We have seen how to do this in detail earlier at Test 5, Part 4 Q-29.

Absolute Value Word Problem: Two different points on a number line are both 3 units from the point with coordinate −4. I discuss how we could represent this with absolute value.

Interpretation: Non-Linear Expressions: A car travels "s" inches in "t" seconds, where s=16.t.root(t). I find the average speed of the car, in inches per second, over the first t seconds after the car starts.

Scatterplots-Quadratic Function: We are given an inverse U shaped scatterplot of 10 data points. I discuss how we choose which of the 4 quadratic functions best models the data in the scatterplot using the signs of the quadratic functions in A through D.

Quadratic Function-Word Problem: A group of friends decide to divide the $800 cost of a trip equally among themselves. When two of the friends decide not to go on the trip, those remaining still divide the $800 cost equally, but each friend’s share of the cost increases by $20. I solve for how many friends were there in the group originally. This requires solving a quadratic equation as a function of the unknown x, number of friends in the group.

Linear Equations-Solution: I simplify the linear equation and solve for the unknown "x". Easy question not to be missed!

Volume-Word-Problem: Cylinders, with a radius of 222 inches and a height between 7.75 inches and 8 inches are produced. I find one possible volume of a cylinder produced. (Rounded to the nearest cubic inch)

Linear-Quadratic Systems: On the x-y plane, y=3x^2-14x intersects y=x at points (0,0) and (a,a). I find the value of "a" by solving this system of equations through the method of "substitution".

Linear-Equations-y-Intercept We are given a linear equation; I find the x-coordinate of the y-intercept.

Mean-Word Problem: Andrew and Maria, each collected 6 rocks whose masses are shown in the table. Mean of the masses of the rocks Maria collected is 0.1 kilogram greater than the mean of the masses of the rocks Andrew collected. I solve for x, the unknown mass for the first rock Maria collected. Note this is another example of the standard "mean" question SAT asks over and over again: It is critical to be able to correctly represent the algebraic relation between the respective means of the rocks Maria and Andrew collected.

Exponential-Word-Problem: J deposits "x" dollars in his account on January 1, 2001. Amount of money in the account doubles each year until J had 480 dollars in the account on January 1, 2005. I solve for "x" in 2 different ways. A caveat is given as to how SAT may try to confuse you, if you use the exponential function method.

Percent-Word-Problem: People are invited to a committee: Of those invited 15% are Parents (P); 45% are Teachers (T); 25% are Administrators (A); and the remaining 6 individuals are Students (S). I find how many more T were invited than A: key to the solution is to correctly define the unknown.

I conclude the class by Congratulating  the Students for their hard work and asking for their help in reviews that will guide me in the follow up book I work on, such that it would be of most use to them.

This section covers 10 Basic Math Facts. The Full 10 facts set is attached as a pdf file: Basics-10. I give a video record of only Facts 1 through 4 here, given the technical difficulties to display all.  A video record for Facts 5-10 are covered in Lecture 234, below.

This section covers Basic Facts 6 through 10 as a video record;  a pdf file is also attached for your ease depending on how you'd prefer to learn.

This section covers the "eight laws of exponents" given on the Table: Laws of Exponents- both as a video record and as a pdf file attached. On the table, first column gives the  the general laws/rules; second column gives a very simple application of the rule with small numbers. The 8th law is the fractional exponent law : Note, this is equivalent to the definition of a root- as a root is simply a fractional exponent!

The section also has practice questions for both exponents and roots. There are 15 questions for the exponents: 9 basic, 6 harder. Questions are chosen with the goal of clarifying how the rules are applied. The harder questions are the ones where the test makers may trick students if they are not sufficiently clear about what the rules say.

There are 5 questions for the roots section: 5 basic; 2 harder. This section also has a file that gives the properties of roots. This involves rules for multiplication and division with roots, and warns against a potential mistake against addition with roots.

I prepared the practice questions such that you will have space to work on the problem, if you choose to print the questions.  I also attached the solutions for all the questions for you to check your results against.

SAT makes ample use of exponents and roots when they prepare their questions: They may either ask out right an operation with exponents and/roots asking for equivalent terms; or these may be fed into an algebra or a word problem. So it is very important to be clear about the definition of exponents and roots and how to apply the laws of exponents.