Interactive Notebook College Readiness Math 2017 Page 2 1 2 COVER PAGE TABLE OF CONTENTS 26 27 Unit 4 project continued U5 systems of equations and inequalities 3 TABLE OF CONTENTS 28 Solving using graphing 4 TABLE OF CONTENTS 29 Solving using substitution 5 TABLE OF CONTENTS 30 Solving using elimination 6 RULES AND EXPECTATIONS 31 Solving systems of inequalities 7 U1 Estimation of multiplication 32 Unit 6 Quadratic Functions 8 Calculating tips, scientific notation 33 Minimums and maximums 9 Rules for estimating 34 Transformations of graphs 10 Sample problems 35 Quadratic formula and factoring 11 Solving 1 step equations 36 Unit 7 Exponents - Exponent rules 12 Words to Math 37 Exponential growth and decay 13 Evaluate expressions 38 Simple and compound interest 14 U2 functions vs relations 39 15 Expression vs equation 40 16 word problems to equations 41 17 Different solutions 42 18 Graphing inequalities 43 19 U3 units of conversion 44 20 KHDuDCM 45 21 Terms to know 46 22 Trip project SUMMATIVE GRADE 47 23 U4 linear functions and slope 48 24 Graphing linear functions 49 25 Linear regression project 50
Page 3 51 76 52 77 53 78 54 79 55 80 56 81 57 82 58 83 59 84 60 85 61 86 62 87 63 88 64 89 65 90 66 91 67 92 68 93 69 94 70 95 71 96 72 97 73 98 74 99 75 100
Page 4 101 126 102 127 103 128 104 129 105 130 106 131 107 132 108 133 109 134 110 135 111 136 112 137 113 138 114 139 115 140 116 141 117 142 118 143 119 144 120 145 121 146 122 147 123 148 124 149 125 150
Page 5 151 176 152 177 153 178 154 179 155 180 156 181 157 182 158 183 159 184 160 185 161 186 162 187 163 188 164 189 165 190 166 191 167 192 168 193 169 194 170 195 171 196 172 197 173 198 174 199 175 200
Page 6 RULES AND EXPECTATIONS 8/7/17 AND 8/8/17 (Unit 1 Pages 6 13) 1. The notebook stays in the classroom. 2. You will write in this notebook daily. 3. The notebook will be graded once each semester and checked regularly. 4. You will benefit from writing in the notebook if you put an effort into it. 5. I will keep a master notebook. If you miss class, you are responsible for copying the material and the essential question. 6. A master will be available at http://tinyurl.com/zhd7zuw and will be updated weekly 7. Notebooks will be collected the day of each unit test and graded. This is an opportunity for extra credit. Grading: Tests, Projects 50% Quizzes 30% Final Exams 20% PUT YOUR NAME HERE AND SEVERAL OTHER PLACES IN THE BOOK Unit 1 algebraic expressions Unit 2 equations Unit 3 measurement and proportional reasoning Unit 4 linear functions Unit 5 systems of linear equations Unit 6 Quadratic functions Unit 7 exponential functions Unit 8 statistics School Rules 1) No bookbags 2) Follow dress code 3) Put your phones up unless asked to get them out 4) Come prepared each day with paper and pencil 8. If you miss a day, refer to the notebook at http://tinyurl.com/zhd7zuw
Page 7 Calculating tips http://lifehacker.com/5970143/who-should-i-tip-and-how-much 1) To find 10% of a number, move the decimal point one to the left. 2) To find 20%, double that number 3) 15% would be halfway between Scientific notation 3.5 x 10 3 is 3500 3.4 x 10-5 is.000035 Positive exponent, move decimal to the right Negative exponent, move decimal to the left If 37 x 42 is 1554 3.7 x 42 = 155.4 move decimal 1 to the left 420 x 370 = 155400 move decimal 2 to the right 1554 3.7 move decimal 1 to the right 15.54 42 = 0.37 move decimal 2 to the left Divide by smaller, answer should get begger Division decimal up front, move left (up) decimal in back, move right (back) warmups If 28 x 51 = 14280, find 1) 2.8 x 5.1 = this answer would have 2 more decimal places than the original 14.28 2) 280 x 5.1 = this answer would move 1 decimal place to the right, then 1 to the left or 14280 3) 14.28 5.1 = since 14.28 has 2 decimal places and 5.1 has 1, the answer would have 1, 2.8 4) 14280 51 = 14280 has 1 more decimal place and 51 is the same so the answer would have 1 more or 280 You can also get this by estimating 28 is almost 30 and 51 is like 50 so 30 x 50 = 1500
Page 8 rules for estimating
Page 9 sample problems 1) Elise is taking piano lessons. The first lesson is twice as expensive as each additional lesson. Her mom spends $270 for 8 lessons. How much was the first lesson? 2) Javier needs to get a tank of gas. Gas costs $3.79 per gallon. How much money does Javier need to fill up his 16 gallon tank? 3) you borrow $350 from you parents for a new Wii and games. They are not going to charge you interest but you want to pay them back as quickly as possible. If you pay them back $15 per week, how long will it take you to pay them back? 4) You want to buy five magazines that cost $1.95 each. When you go to buy them, the cost is $12.25. Is this right? (five times 1.95 is about five times two or $10) 5) You want to plant a row of flowers. The row is 58.3 cm long. The plants should be 6 cm apart. How many flowers do you need? (58.3 is nearly 60, divided by 6 is 10. So 10 plants should be enough.) 6) You are calculating 107 times 56 and the calculator shows 952.00. Is it right? 7) You are making invitation cards. It took yyou 3 minutes and 20 seconds to make one card, but you need to do 15 more. How long will it take?
Page 10 topics for test 1) Practice 1.1 estimations (if 13 x 17 = 221, what does 1.3 x 17 equal?) 2) Practice 1.2 (Felecia 70 mph for 1.5 hours, 30 mpg, how much gas does she need?) 3) Sidewalk patterns 4) Areas of rectangles (LW 2 x 3, x times x + 3) and the distributive property 5) Equivalent expressions are 2 expressions equivalent? 6) Solving one step equations 7) Words to math 8) Literal equations 9) Any of the warmups 10) Evaluate expressions 11) Terms of an expression
Page 11 Notes for solving one step equations. Addition x + 2 = 7-2 -2 you want to isolate the variable so subtract the 2 from both sides x = 5 you can check (5) + 2 = 7 Subtraction x - 3 = 7 +3 +3 you want to isolate the variable so add the 3 from both sides x = 10 you can check (10) - 3 = 7 multiplication 4x = 8 4 4 you want to isolate the variable so divide both sides by the coefficient (4) x = 2 you can check 4 (2) = 8 Division x 5 = 4 (which can also be written as follows) x 5 = 4 you want to isolate the variable so multiply both sides by the divisor or denominator (5). This eliminates the denominator x = 4 * 5 = 20 you can check 20/5 = 4 fractions 2 3 x = 4 to isolate the variable I need to multiply both sides by 3 and divide both sides by 2. It doesn t matter which order you do this.
Page 12 Words to math Add Sum increased by More Altogether Plus Total Combined Perimeter and Multiply Product Of Times Area Double/twice Triple 7n 7(6) Per/each Answer Solve Find express Evaluate Calculate Model joined both in all also solution simplify Subtract Difference Elapsed Deduct Used Less Minus Passed Remain Exceed Take away off Divide Quotient Division Average Distribute Half 12/2 Per/each Shared ratio equals Is Was Were Have Has Gives bigger smaller shorter taller longer loss least has left decreased reduce remainder split goes into totals
Page 13 evaluate expressions When you evaluate an expression, you are given values to plug into the expression. Example 2x + 3y where x = 5 and y = 7 2(5) + 3(7) = 10 + 21 = 31 Terms of an expression Terms are separated by operations (add, subtract, multiply, divide) An expression with one term is called monomial. Two terms is called binomial. Three terms is trinomial. More than that is polynomial. The leading coefficient is the NUMBER in front of the variable with the largest exponent. So in 3x 2 + 5x 3 2x + 3, 5x 3 ha s the largest exponent and 5 is the leading coefficient. The degree of an expression is the number of the largest exponent. And we usually put expressions in descending order. So 3x 2 + 5x 3 2x + 3 would be written 5x 3 + 3x 2 2x + 3 and would have a degree of 3 If I have multiple variables together (like 3x 2 y 3 z 5 ), you add the exponents together. So this has a degree of 10.
Page 14 Unit 2 functions vs relations (Unit 2 pages 14 18) A relation just says that there is a relationship between the input (x) and the output (y). A function says that each input has one and only one output. This is not a function since 1 can be 0 or 15. This is also a function This is a function This is a function. Each input has one output. Function. There is no vertical line which will cross this twice. Relation. A vertical line (like the y axis) crosses this graph twice. This is a relation. There are 2 entries with the same x [(2.3) and (2, -2)] and different ys.
Page 15 Expression vs Equation An equation will have an equal sign in it (in inequality will have a less than or greater than sign). An expression is part of an equation. If you think of sentences, an expression is a phrase. Or, another way to say it is, the primary difference between the two is an equals sign. An 'equation' has a left side, a right side and an equals sign separating the sides. An 'expression,' by contrast, doesn't have any 'sides' and is simply what the name suggests an algebraic 'expression.'.
Page 16 Word problems to equations 331 students went on a field trip. Six buses were filled and 7 students traveled in cars. How many students were in each bus? 1. Define the variable (what do we know?) 6 buses filled with the same number of students x 2. Determine the constant (if there is one) 7 3. What is the rate (look for each, part, something that will repeat) 4. Write the equation. Solve and check. 5. 331 = 6x + 7 324 = 6x 54 = x the number of students per bus Check 6(54) + 7 = 331
Page 17 Different solutions Solving linear equations with one solution, no solution. Or an infinite number of solutions One solution x = a No solutions a = b Infinitely many solutions a = a or x = x equations 7x 3 = 5x + 5 7x 3 = 7x + 5 7x 3 = -3 + 7x Use 7x 3 = 5x + 5 subtract 5x 7x 3 = 7x + 5 subtract 7x 7x 3 = -3 + 7x5 subtract 7x properties 2x 3 = 5 add 3 3 = 5 not true 3 = -3 true for of equality 2x = 8 divide by 2 every x to solve X = 4 Check your solution End results 7(4) 3 = 5(4) + 5 28 3 = 20 + 5 25 = 25 true Work these out A) 6m 2 = m + 13 B) 4y + 9 = 4y - 7 C) 3c + 2 = 3c + 2
Page 18 graphing inequalities Inequality Symbols symbols < > Key Is less than Is fewer than phrases Is greater than Is more than Is less than OR equal to Is at most Is no more than Is greater than OR equal to Is at least Is no less than Graphing an Inequality The graph of an inequality shows all the solutions of the inequality on a number line. An open circle is used when a number is not a solution. A closed circle is used when a number is a solution. An arrow to the left or right shows that the graph continues in that direction. Hint: as long as the variable is on the left, the inequality will point in the direction the arrow will go. Real-life Application Each day at lunchtime, at least 53 people buy food from a food truck. Write an inequality that represents this situation x 53 Work these
Page 19 Unit 3 unit conversions (Unit 3 pages 19-22) Step 1: set up your conversion factors Step 2: cancel out the units Step 3: Multiply across the top and multiply across the bottom Step 4: Simplify Example: convert 30 miles per hour (mph) to feet per second 30 mi 1 hr 1760 yd 1 mi 3 ft 1 yd 1 hr 60 min 1 min 60 sec = 158400 ft 360 sec = 44 ft 1 sec Time Weight Distance 60 seconds 1 minute 60 minutes= I hour 24 hours = 1 day 7 days = 1 week 365 days= 1 year 52 weeks= 1 year 16oz = 1 lb 2.2 lbs = 1 kg 1000 kg = 1 ton 5ml = 1 teaspoon 3 teaspoon = 1 table spoon 1 gallon = 4 quarts 1 quart = 2 pints 1 pint = 2 cups 1000 m = 1km 100 cm = 1 m 10 mm = 1 cm 1000mm = 1 m 12in = 1 ft 3ft = 1yd 1760 yd = 1 mile Rule: Small to Big you divide ( ) Big to Small you multiply ( X )
Page 20 1 Gallon 1 Quart 1 Pint 1 Cup ¼ cup 1 Tbsp 4 quarts 2 pints 2 cups 8 ounces 4 Tbsp 3 tsp 8 pints 4 cups 16 ounces 240 ml 12 tsp ½ ounce 16 cups 32 ounces 480 ml 2 ounces 15 ml 129 ounces.95 liters 60 ml 3.8 liters
Page 21 Note: for unit rate, the denominator is 1, so you have divided the numerator by the denominator numerator denominator Reasonable distances (for the trip project) Distance around the equator 40,075 kilometers (24,901 miles). Distance to the moon 384,400 km (238,900 mi) Distance from Washington DC to San Francisco 4529.9 km (2814.7 miles) Distance from Bangor, Maine to San Diego 5108.6 km (3174.4 miles)
Page 22 (when you copy the notes, you do not need to put the map in your notes) For the project, measure the distance from 1 city to the next, using whichever side of the ruler you prefer. Then convert the distance using the key in the bottom right corner of the map. Remember you cannot drive in a straight line from Miami to Houston as you would be driving over water. Unit 3 project available at http://lithiasprings.dch.schoolinsites.com/?pagename=teacherpage&page=% 27DocumentsCategory%27&CategoryID=259739&SubCategoryID=31252&iSe ction=teachers&correspondingid=289435
Page 23 Unit 4 Linear functions (unit 4 pages 23 26) A linear function is a function that forms a line. The function will have 2 variables, normally x and y, where x is the independent variable and y is the dependent variable (y is dependent on the value of x and whatever we are doing with the x.) We talk about a vertical line test: can we run a vertical line across the function and only have the function touch the vertical line once). SLOPE is defined as the rate of change, rise over run, or Δy Δx or m = (y 1 y 2 ) (x 1 x 2 ), where (x 1, y 1 ) and (x 2, y 2 ) are points on the line. If 2 lines have the same slope, they are parallel if the y intercepts are different. If 2 lines have the same slope, they are the same line if the y intercepts are the same. If 2 lines have slopes that are negative reciprocals of each other, they are perpendicular. (negative reciprocal of 2 is 1 2 ) Any other scenario, the lines will intercept Slope intercept form is y = mx + b or f(x) = mx + b (where m is the slope and b is the y intercept.) Standard form is ax + by = c Point Slope intercept form is (y - y 1 ) = m (x - x 1 ) A negative slope descends from left to right. A positive slope ascends. Negative slope positive slope zero slope ( y =) undefined slope (x =)
Page 24 graphing linear functions In order to graph a linear function in the format y = mx + b, first find the y intercept, then count the slope. For y = 3x 3, the y-intercept is (0, -3) and the slope is 3 up and 1 to the right (1, 0) and (2, 3) Graph these problems 1) y 2x 1 2) y 3x 8 3) y = 4 To graph using point slope, graph the intercepts (the y intercept is where x is 0; the x intercept is where y is zero) and connect the points. 4) 2(x 2) = (y + 3) 5) 2(x + 5) = (y + 0) 6) (x 5) = (y + 1)
Page 25 Linear Regression project. The unit 4 project is available at http://lithiasprings.dch.schoolinsites.com/?pagename=teacherpage&page=%27documentscategory%27&ca tegoryid=259739&subcategoryid=31253&isection=teachers&correspondingid=289435 Please follow the rubric. You can find samples of the work expected in other files on that same site. Meanings of terms x xbar the average of the x-values x sum of the x values x 2 sum of the x 2 values N - the total number of items. For the project, this should be 9 y y bar the average of the y-values y = the sum of the y values y 2 sum of the x 2 values xy sum of the x y values (note: -1 r 1 and 0 r 2 1) [When you copy the notes, you do not need to copy what is in boxes] This is an example of the tables filled out
Page 26 project The rubric tells you how the project will be graded. Please make sure you do all of the steps (listed below the rubrics) Write an equation each time you are asked for one. Also answer the analysis questions.
Page 27 Unit 5 Systems of linear Equations and Inequalities (Unit 5 pages 27-31) A system of linear equations is 2 or more linear equations. The solution is the value that makes all the equations true. You can solve using graphing, substitution or elimination or using matrices or calculators. No solution: the lines are parallel;, solutions means the two lines are the same; one solution, they intersect and the intersection point makes both equations true. Page 28 Solve by Graphing Graph each equations. If they intersect, the answer is a point (2,1). This method is the most subject to error but is one way to check your work. Another way is to lug the found values into the equations. 2 + 1 = 3 2 2(1) = 0
Page 29 Solve Substitution Generally, one or more equation is set up as y = or x =. Simply take what the variable is equal to and substitute it for the variable in question. In the example below, y = -3x -14. Put (-3x 14) for x in the next equation and you get 4x + 3(-3x 14) = -22 and solve for x. Once you have x, plug that value in and solve for y. Then check the correctness of the answer 3(-4) + (-2) = -12-2 = -14 4(-4) + 3(-2) = -16 6 = -22
Page 30 Solve by Elimination Find variables where the coefficients are the same but opposite signs and add the two lines, eliminating that variable. (You may have to multiply one or more lines by a constant or 1 in order to make this so). Once you know what one variable is, you can plug it in to determine the other In the example above, you have -2y on the left and y on the right. Multiplying the right equation by 2 gives you 4x 2y = 4 and 4x + 2y = 12, allowing you to add them together and eliminate the y. Then solve for x. Once you have the x value, put that in for x (in one of the original equations) and solve for y. Then check, like on page 29. 4(2) 2(2) 8 4 = 4 2(2) + (2) 4 + 2 = 6
Page 31 Graphing of systems of inequalities y > -x 1 and y x + 4 > or < is a dotted line, or is a solid line To determine where to shade, pick a point, like (0,0) not on a line. If it makes it true, shade the portion with the point. If it is false, shade the other side. 0 is not greater than 0 1, so it is shaded above that line REMEMBER TO CHECK by plugging your answer into both of the original equations to make sure it is true. Solve these problems Solve by substitution or elimination. 1. 2x y 5 4x 6y 12 2. 2x 3y 1 10x y 11 3. 4x + 2y = 3 6x 3y 8 4. Solve the system of linear equations by graphing. Graph the solution set to the system of inequalities State the solution! on the axes below to check your answer. 3y 3 6x 2x 2y 14 5 y x 2 2 1 y x 2 2
Page 32 Unit 6 Quadratic Functions (pages 32 35) Exploring Properties of Parabolas In vertex form, a quadratic is f(x) = a(x-h) 2 + k, where the vertex (or point where the parabola turns is (h, k) Quadratic functions can also be written in standard form, f(x) = ax 2 + bx + c, where a 0. You can derive standard form by expanding vertex form. An axis of symmetry is a line that divides a parabola into mirror images and passes through the vertex. Because the vertex of f(x) = a(x h)2 + k is (h, k), the axis of symmetry is the vertical line x = h. Page 33
Page 34 effect of transformations F(x) = x 2 + 2 up 2 F(x) = x 2-2 down 2 F(x) = (x + 2) 2 left 2 F(x) = (x - 2) 2 right 2 F(x) = 2 1 x 2 F(x) = 2x 2 F(x) =-x 2 fatter, wider, shorter longer, thinner, skinnier, stretched flipped over x axis Page 35 Quadratic formula If a quadratic is written in standard form, it is f(x) = ax 2 + bx + c Those coefficients can be used to solve the quadratic using the quadratic formula x = b ± b2 4ac 2a Factoring x 2 + 5x + 6 x 2-5x + 6 both positive sum both negative sum (x + 2)(x + 3) (x - 2)(x - 3) x 2 + x - 6 x 2 - x 6 bigger positive difference bigger negative difference (x + 3)(x - 2) (x + 2)(x 3)
Page 36 Exponent Rules(unit 7 exponents, pages 36 38) Add 2x 2 + 3x 2 = 5x 2 Add coefficient. Do not change exponent. Multiply 2x 2 3x 3 = 6x 5 Multiply coefficients. Add exponents. 5 Divide 6 x Subtract exponents. 3 2x 2 3x Divide coefficients. Raise to a Power (2x 2 ) 3 = 8x 6 Raise coefficient to power. Multiply exponents. Root 4x 2 = (4x 2 ) 1 2 = 2x Find root of coefficient. Divide exponent by 2. Zero 2 0 = 1 Anything to the 0 power is equal to 1
Page 37 Exponential growth and decay y = abx, where a 0,b > 1 (for growth) and 0 < b < 1 for decay Exponential growth is such things as population growth or investment income. Exponential decay is such things as radioactive decay. Page 38 Simple and Compound interest Your investments grow faster with compound interest as you are earning income from the principal (investment) as well as from interest on the interest.