Lesson #3 The Essentials of Algebra Math as a language for determining relationships to assist problem solving in the real world and on the SAT

The 6-Tiered Approach to achieving success on the SAT: 1 – get to know the test √ 2 – learn B.T.S. (basic test-taking strategy) √ 3 – learn format-specific methodology √ 4 – learn the content 5 – learn advanced strategy √ 6 – gain consistency through practice Lesson 3

PUT AWAY YOUR BOOKS—IT IS TIME FOR THE… P.O.T.D!!! Lesson 3

Sit back and enjoy the mathemagic show… Lesson 3

Believe it or not, a little algebra is all we need to reveal the trick! 1) If the trick works for any 3-digit number, there must be a formula behind it. So.. instead of saying 123, or 476, or 885, etc., we can say ABC. 2) Assign A to hundreds place, B to tens, and C to ones. Re-write the equation. (100A + 10B + C ) - (100C + 10B + A ) = 99A - 99C = 99(A - C), Voila! This reveals the fact that our solution to the palindrome will always be a multiple of 99. Why is this significant? 3) 99(A - C) can also be written as ( )(A - C). Which means: we increase the hundreds digit 1 and decrease the ones digit 1. Now we have an arithmetic sequence where the hundreds and the ones digit will always add up to 9s and because the tens digit of a multiple of 9 will always be 9, we get , the 18 is like , so that 1 carries over into the hundreds place, and finally we arrive at Math is Awesome!!!! Lesson 3

CONCEPT #1 – FUNDAMENTALS OF ALGEBRA (115 pts.): Now that we have been witness to the incredible power of algebra, let’s make sure we know how to weald it like a sword to cut down those tough SAT algebra problems. Lesson 3

CREATING EXPRESSIONS: TRANSLATING WORDS TO SYMBOLS: here’s what our equation should look like… Lesson 3

TRANSLATING: Let’s take a look inside the words-math translation guide, so to speak… Lesson 3

SOLVING EQUATIONS PRACTICE #1: Remember Alma? We all agree that she and her stupid laptop is annoying, but let’s see if she can help us out just one more time.

Prowess’ s SAT Math Method: Our approach on the SAT math is called Q-C-T. Lesson 3 Q = Question (ask yourself, “What exactly is my question asking me to figure out?) -underline or take notes C = Concept (ask yourself, “What type of math is involved here?” -organize the mental closet) T = Terrain (ask yourself, “Is there a unique format to how this question is written?) - see if you can use a shortcut should you need one

SOLVING EQUATIONS PRACTICE #2: Try this one. It is another real SAT question taken straight from your diagnostic. Wyatt can husk at least 12 dozen ears of corn per hour and at most 18 dozen ears of corn per hour. Based on this information, what is a possible amount of time, in hours, that it could take Wyatt to husk 72 dozen ears of corn? Step 1. assign your symbols… Step 2. re-write the problem in your own words… Step 3. Translate from those words to math… Step 4. Isolate your unknown… Lesson 3

TRANSLATING: We are going to stop a second to add a few more words to our dictionary… Lesson 3

CREATING EQUATIONS AND INEQUALITIES: Let’s go behind the scenes of an equation to get a closer look at what it is we’re actually trying to do… Lesson 3

SOLVING EQUATIONS PRACTICE #3: Now we understand how to translate from English to math. Can we do it in reverse? y = M + (W + K)x -What does y represent? -What does x represent? Lesson 3 store Materials’ Cost, M (dollars) Rental cost of wheelbarrow, W (dollars per day) Rental cost of concrete mixer, K (dollars per day) A B C

SOLVING EQUATIONS PRACTICE #4: How about this one? Kathy is a repair technician for a phone company. Each week, she receives a batch of phones that need repairs. The number of phones that she has left to fix at the end of each day can be estimated with the equation P = 108 – 23d, where P is the number of phones left and d is the number of days she has worked that week. What is the meaning of the value 108 in this equation? A) Kathy will complete the repairs within 108 days. B) Kathy starts each week with 108 phones to fix. C) Kathy repairs phones at a rate of 108 per hour. D) Kathy repairs phones at a rate of 108 per day. Lesson 3

SIMPLIFYING: The first step to solving any equation involves simplifying it. This means reducing the amount of terms to as few as possible. Lesson 3

SOLVING EQUATIONS PRACTICE #5: Here is a simple question, no pun intended, for you to practice your simplifying skills. Lesson 3 Q- C- T-

ISOLATING: The Golden Rule says, “Do unto one side as you have done to the other.” Let’s try to understand what this means visually. Lesson 3 24

ISOLATING: You can do anything you want to one side, so long as you do it to the other too. Look what happens if I pull 6 off the left plate but not the right. Lesson 3 24 Could we still say these two terms are equal??

HOW TO SOLVE ALMOST ANYTHING: (10 PRINCIPLES) Lesson 3 1. ??? 2. >Try to understand things visually 3. ??? 4. Throw things out and see what sticks 5. ??? 6. ??? 7. ??? 8. ??? 9. ??? 10. Use known values to procure a reasonable range

ISOLATING: Free “X” (or y or z or q) of its attachments. Or, put another way, we could say “remove the disguise of x.” Lesson 3

SOLVING EQUATIONS PRACTICE #6: See if you can remove the disguise from “x” in this equation. Lesson 3 Q- C- T-

MULTIPLE VARIABLE: Sometimes more than one variable is in disguise. The first step stays the same: simplify… then isolate and substitute or multiply and combine. 3x + 4y = -23 2y – x = -19 -Substitution involves figuring out what one variable is in terms of the other, then re-writing the equation using just the one variable. -Combination involves creating a third equation to get rid of one variable. Lastly, we plug our undisguised variable (the value) back in to find the other. Lesson 3 x = 3 y = -8

SOLVING EQUATIONS PRACTICE #7: Isolate and substitute or multiply and combine to solve the problem below. b = 2.35y x = c = 1.75y x = In the equations above, b and c represent the total price (including packaging) per pound, in dollars, of beef and chicken, respectively. The buyer wanted to compare the total costs of y boxes and x pounds of each to determine which would cost more given the number of pounds he needs. After determining this, he recorded his results and found that chicken would cost 0.90 more. What was the cost per box of the beef? Good luck! This one is a challenge! Lesson 3

MULTIPLE VARIABLE: Let’s run through it again… Step 1: re-write your equations. Step 2: make sure you have it in simplest form Step 3: Isolate or Combine - figure out what x is in terms of y and substitute - multiply one equation until you can subtract or add a term out of their sum or difference. Step 4: Plug the newly-known value back in to replace x or y. In general, we need #of variables equations to solve for # of variables variables. Lesson 3

FINDING y “IN TERMS OF” x: If you get stuck, choosing numbers is often a great shortcut for these tough questions… Lesson 3

FINDING x USING INEQUALITIES: Just like an equation, but here we are solving for a range rather than just one or two possible values. Lesson 3 Q- C- T-

FINDING x USING INEQUALITIES: Let’s try to understand this visually. Lesson 3

FINDING x USING ABSOLUTE VALUE #8: Just like an equation, but here we are solving for both the positive and the negative on the other side of the equal sign. The cardinal rule of solving questions with absolute value is “solve it two ways.” …. but why? If | 2x + 5 | = 11, what is/are the possible value/s of x ? A) -8 B) -3 C) {-3, 8} D) {3, -8} Lesson 3 Q- C- T-

HOW TO SOLVE ALMOST ANYTHING: (10 PRINCIPLES) Lesson 3 1. ??? 2. Try to understand things visually 3. ??? 4. Throw things out and see what sticks 5. ??? 6. ??? 7. ??? 8. ??? 9. >Understand simple ideas deeply 10. Use known values to procure a reasonable range

A CLOSER LOOK #9: Remember this one. Did you recognize that formula we used? y = M + (W + K)x If the relationship between the total cost of buying the materials and renting the tools at Store C and the number of days for which the tools are rented is graphed in the xy-plane, what does the slope of the line represent? A) The total cost of the project B) The total cost of the materials C) The total daily cost of the project D) The total daily rental cost of the tools Lesson 3 store Materials’ Cost, M (dollars) Rental cost of wheelbarrow, W (dollars per day) Rental cost of concrete mixer, K (dollars per day) A B C

CONCEPT #2 – LINEAR GRAPHS (55 PTS.): To be able to accurately “read” linear graphs we have to be fluent in our understanding of the coordinate plane and how it works. Simply a way to divide up a flat surface so that we can more readily see the locations of objects placed upon it…more or less, here’s where it came from: Linear graphs became tools for us to visually understand trends. Linear means that the rate of change produces “a straight line.” … Before we get to all that, though, let’s talk about the surface that these lines will be drawn upon. Lesson 3

THE COORDINATE PLANE: A great way to determine the location of objects in 2- dimensional space. Lesson 3

THE FORMULA FOR EQUATION OF A LINE: y = mx + b. Lesson 3

A FEW LINE RULES: Here, we will take a closer look at what all this means for helping us determine certain trends. Lesson 3

LINEAR GRAPHS PRACTICE #1: Can you match the line on the graph with its correct equation? A B C D E E C A B

MORE LINE RULES: midpoint and distance. Lesson 3

A SECOND LOOK: Let’s answer this one, now. y = M + (W + K)x If the relationship between the total cost of buying the materials and renting the tools at Store C and the number of days for which the tools are rented is graphed in the xy-plane, what does the slope of the line represent? A) The total cost of the project B) The total cost of the materials C) The total daily cost of the project D) The total daily rental cost of the tools Lesson 3 store Materials’ Cost, M (dollars) Rental cost of wheelbarrow, W (dollars per day) Rental cost of concrete mixer, K (dollars per day) A B C

LINEAR GRAPHS PRACTICE #2: When 2 lines intersect, this means that their separate equations yield the same values for precisely one ordered pair. Same means equal. The lines y = - x - 9 and 2y = - x - 25 meet at exactly one location on the standard coordinate plane. What is that location? A) (-9, -25) B) (9, -25) C) (7, -16) D) (-7, -16) Lesson 3 Q- C- T-

LINEAR GRAPHS PRACTICE #3: This one is straight out of your blue book. Which of the following equations represent a line that is parallel to the line with equation y = -3x + 4 ? A) 6x + 2y = 15 B) 3x – y = 7 C) 2x – 3y = 6 D) x + 3y = 1 Lesson 3 Q- C- T-

LINEAR GRAPHS PRACTICE #4: This one is also straight out of your blue book. x = 2y + 5 y = (2x – 3)(x + 9) How many ordered pairs (x, y) satisfy the system of equations shown above? A) 0 B) 1 C) 2 D) Infinitely many Lesson 3 Q- C- T-

LINEAR GRAPHS PRACTICE #5: Guess where this one came from?? Yep…blue book. Line l in the xy-plane contains points from each of Quadrants II, III, and VI, but no points from Quadrant I. Which of the following must be true? A) The slope of the line is undefined. B) The slope of the line is zero. C) The slope of the line is positive. D) The slope of the line is negative. Lesson 3 Q- C- T-

CONCEPT #3 – FUNCTIONS (20 PTS.): f(x), g(x), and other function models. Why do functions look so scary?? Hypothesis: we don’t understand them, visually. function- cadabra!

f(x) = y: It’s all about transformation! “f” means “function”  This is the magical incantation. “( )” means “of”  This is the hat “x” is the “input value” or “domain”  This is the trinket “y” is the “output value” or “range”  This is the rabbit x f(x) OR y

FUNCTIONS PRACTICE: A Challenge Question (not from the blue book). Q- C- T- Lesson 3

FUNCTION MODELS #1: Pretty much the same. Just substitute. h = 3a A pediatrician uses the model above to estimate the height h of a boy, in inches, in terms of the boy’s age a, in years, between the ages of 2 and 5. Based on the model, what is the estimated increase, in inches, from ages 2 to 5 ? A) 3 B) 6 C) 9 D) 15 Lesson 3 Q- C- T-

FUNCTIONS PRACTICE #2: Let’s try this one together. Q- C- T- Lesson 3

FUNCTIONS PRACTICE #3: Q- C- T- Lesson 3

FUNCTIONS PRACTICE #4: This one is an SPR (student-produced response) Q- C- T- Lesson 3

The 6-Tiered Approach to achieving success on the SAT: 1 – get to know the test √ 2 – learn B.T.S. (basic test-taking strategy) √ 3 – learn format-specific methodology √ 4 – learn the content √ 5 – learn advanced strategy √ 6 – gain consistency through practice Lesson 3

LET’S SUMMARIZE: 1.The Heart of Algebra is composed of 3 major concepts: solving equations, linear graphs, and functions. Algebra is a language of mathematics; this requires us to be able to translate from words to symbols and vice-versa. It is also a tool for determining relationships and creating formulae. 2.Solving equations is all about doing to what side exactly what you did to the other to keep things equal. Inequalities solve for a range, which means we are not looking for one value but for some segment/s of the number line. Absolute value is a way of determining distance, and distance is never negative, at least not in the fields we are concerned with. 3.Linear graphs use the formula y = mx + b and are represented on the coordinate plane. M is the slope, which means the amount the line rises divided by the amount it runs. b is the y-intercept, or the value of y when x = 0. Often, the best way to scrutinize a line is to simply sketch it out using a rough coordinate plane. 4.Functions are used to model some relationship between a collection of variables and constants. They are used almost on a daily basis, often without our noticing. The best way to think of them is to imagine y as a transformation of x via the function f(x). In class, we used the trick of pulling a rabbit out of a hat for this.

LESSON ANSWER KEY