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Math I notes Week of November 16, 2015 Identify-----Isolate-----Eliminate MONDAY Solving Quadratic Equations--- has to equal zero 1) 2) 3) 4) Graph Take square roots Factoring Formula Factoring 1) 2) 3) 4) GCF—greatest common factor (make a list) Difference of 2 squares Short cut what numbers can you use to multiple and add to get numbers in equation? “wrong” way what numbers can you use to multiple and add to get numbers in equation? Don’t forget to DIVIDE at the end Zero Product Property: If you know that (x)(y)=0, what can you tell me for sure about x or y? one of them HAS to be 0 When the product of two factors is zero, then one/both of the factors equal zero. Set each factor equal to zero and find each solution. What if I tell you that (x)(y)=6? Cannot determine numbers *Before you factor make sure the equation = 0. If it is x~2 you are looking for 2 answers. If you move the number to the opposite side of = then you need to change the charge from/to positive/negative. Once you get the factors (x-a)(y-b)=0 then separate (x-a)=0 (y-b)=0 Distribution: combine two products to create equation TUESDAY Area of triangle ½ base * height GCF---make a list (cut in half#/double#) Rectangle Area----length*height……………negative numbers don’t work in obtaining the area WEDNESDAY When taking the square root of numbers, always remember positive and negative If a two is the highest exponent it is a quadratic equation 4th technique to solve quadratic equations is FORMULA Solving quadratic through FORMULA See written formula (notes from 11/18/2015) Equation HAS to equal ZERO The entire formula is divided by 2a Opposite of –b Plus/minus lets us know there are two answers Be cautious of negative signs Don’t round off, give exact answer If the square root is not perfect, bust it up “are you sleeping” song will help you remember formula (Opposite of b, plus or minus square root, b squared minus 4ac, all over 2a) SONG: repeat each section If a=0 the formula is not quadratic b squared cannot be negative THURSDAY Review for test WEEK of November 23, 2015 MONDAY Math I Book I Combining FUNCTIONS (p.337)- add, subtract, multiply and divide linear and exponential functions. (p.337) bottom of page has Key Concepts and Function Operations -cannot divide by zero (undefined) -combine like terms -make a table -anytime you subtract a group, put () around whole group -Add & Subtract Functions -Multiply & Dividing Functions -have to divide entire group -if you add two numbers with the same base, base stays the same and exponents are added After Christmas----two practice EOC tests EOC test you get 4hrs (possible 1.5hrs for 1st part of EOC) HOMEWORK p.341 (7-10), p.342 (17-24) TUESDAY -homework review p.399 Correlation=relationship Trend lines and correlations Scatterplots Calculator to get “line of best fit” equal number of dots on each side Positive correlation: as one increases so does the other (if x increases then y increases) Negative correlation: as one increases then the other decreases (if x increases then y decreases) No correlation: increasing and decreasing have no meaning/relationship Strong/Weak Positive correlation: if the dots are not compact/tight together Strong/Weak Negative correlation: if the dots are not compact/tight together May be able to look at data and not graph, pending on your numbers CALCULATOR Turn on------2nd (+) 71 2 ----------------CLEAR RAM STAT-----EDIT (hi-lighted)-----enter L1 is group one of numbers L2 is group two of numbers Y= button---hi-light PLOT Y-----GRAPH ZOOM----- #9 ZOOM STAT----enter If you delete a list----GO TO----STAT----SetupEditor----enter-----go back to STAT---EDIT (list will be there) TO GET EQUATION STAT--- hi-light CALC---- #4 LinReg (linear regulation)----y=ax+b #5 QuadReg (quadratic) #0 ExpReg (exponential) 2nd----0 (catalog)-----list of everything calculator can do-----hit GREEN D (x button)----DiagnosticON---enter-----enter again until it says done STAT---CALC---LinReg---scroll through list and calculate----r= is the correlation -1: 1 Is the answer close to -1:1? The closer to number -1:1 the better the correlation ~When dealing with Dates in a graphing situation…..first year in data =0 and then figure out the pattern ***watch for conversions (graph is in months and question is asking a year) ***Clear—clear it out ***Delete—getting rid of it ERROR message: DIM MISMATCH means that there are more numbers in one column than the other HOMEWORK p.407 #10-12 p.408 #18-21 #19* Week of December 4, 2015 MONDAY Homework answers p.407) #10 strong negative #11 y=-2.5x + 137 #12 12.6 degrees p.409) #18 7cm #19 Exponential; y= 8.4 x 1.075^x ; 0.9999943; yes; 1995 #20 y= 0.91x + 18; 154,178,000; no p.410) #21 CHAPTER 6—DATA ANALYSIS STATISTICS FREQUENCY TABLE displaying data and analyzing data p. 372 frequency table & histogram Frequency- how often it happens quickest time & slowest time RANGE= biggest number – smallest number To find INTERVALS: take Range and divide by 4 ** see written notes Relative Frequency= comparing current settings, how frequently it occurred within the group all of the relative frequency numbers added together should equal 1 HISTOGRAM p.372 (looks like a bar graph) Problem 2 (see written notes for picture of histogram) Subtract largest from smallest 105-80=25 Interval 25/4= 6 # of people relative frequency 80-86 4 4/13 87-93 4 4/13 94-100 3 3/13 101-107 2 2/13 p.373 UNIFORM, SYMMETRIC, SKEWED histograms in terms of their shape uniform- roughly bars are same height symmetric- close to mirror images if cut in half skewed- learns more one way than the other, peak is not in center a) 17-1= 16 16/4= 4----too small groups so change to 3.5 Intervals frequency 1-4.5 2 5-8.5 3 9-12.5 4 13-17 4 Answer: SKEWED TO THE RIGHT b) $9 x 5 days in a week Answer: $45/$50 CUMULATIVE FREQUENCY P.375 HOMEWORK: p.376 (9-13) p.377 (15-19) p.378 (21-27) #9 do a frequency and relative frequency table TUESDAY 6-2 Measures of Central Tendency and Dispersion (how is information spread out) MEAN, MEDIAN, MODE, RANGEnumbers can be decimals Mean= add all numbers and divide by # of numbers, AVERAGE Median= middle number, list from smallest to largest (vice versa); can be the average of the two middle numbers Mode= the number(s) that appears the most *****Which of those three best describe the set of data?? MEAN [p.379 Key Concept: When to Use] outlier= the number who is way smaller or larger than the other numbers if there is an outlier, the median would be best to describe set if there is NOT an outlier, the mean would be best to describe set CALCULATOR STAT- EDIT STAT-CALC (hi-lighted at top)- #1 1-Var Stats [to find median and mode] p.381 -see written notes line plot= how many times something appears (see chart on p. 381), use x- tallies to show how many of each number Problem 3: Finding the Range RANGE: biggest minus smallest C= range 6 mean 4.2 D= range 22 mean 11.2 *stock C is usually cheaper; stock C has a small range so it is more stable 5) E= 113 183 479outlier 120 117 F= 145 129 153 135 142 range 479-113= 366 mean: all numbers/5= 202.4 range 153-129= 24 mean: all numbers/5= 140.8 ***having a big range doesn’t mean that its better, it means that your numbers are spread out more HOMEWORK: P.385 #11-19, 23-25 WEDNESDAY p. 387 homework answers 12) mean 8.76 14) all describe central tendency of data, but describe different things—list definitions 18a) 5.8, 5.4, 1.2 25) mean 20x/3, median 6x, range 7x, mode 4x 6-3 Box and Whisker Plots p. 392 See chart at the bottom of p.392 Quartile= values that divide a data set into four equal parts FIRST QUARTILE (Q1) median of lower half of data THIRD QUARTILE (Q3) median of upper half of data Interquartile range= difference between the third and first quartiles p.393 1a) 60 65 75 75 75 85 85 85 95 100 105 (put in order first) Min=60 max=105 Q1= 75 (1st 75) median=85 Q3= 95 60 65 70 75 80 85 90 95 100 105 I------------I--------I-------I-----------I 1b) 5 7 11 19 21 53 Min=5 0 max=53 10 20 median=15 30 40 50 Q1= 7 Q3=21 60 I---------I------------I----------------------------------I Practice #3 & #4 p.394 *Put in order first 3) 206 374 421 469 489 505 531 573 702 min=206 Q1=397.5 max=702 Q3=552 median=489 *Put into a more uniform way 200 250 300 350 400 450 500 550 600 650 700 I-----------------------I---------I----------I-----------------I 4) 30 70 90 120 220 224 280 410 Q1= 77.5 Q3=252 min=30 max=410 median= 179.25 30 60 90 120 150 180 210 240 270 300 330 360 390 420 I-----------------------I---------------------I------------------------I CALCULATOR (when listing in L1 can be in any order, just need to make sure you put all numbers) STAT—(hi-light) EDIT----enter info in L1----STAT (hi-light) CALC---#1 Var Stats----enter, enter, enter(where it says calculate) 2nd---Y= (above is STAT PLOT)----enter choose Plot 1-----hit enter to turn plot on-----Type: (hi-light) <right arrow key to> whisker plot-----STAT----#9 zoom STAT-----TRACE <will give you all info> p.396 6) 46%tile To find a percentile rank: percentage of data values that are less than or equal to that value, separate into 100 equal parts HOMEWORK p.396 (8-12) (16, 18, 19, 20, 21) THURSDAY p. 396 homework answers 10) Class B 11) the box 12) 88 18) unlimited answers Test next Tuesday 6-5 Two-Way Frequency Tables p. 414 Frequency (how many times) **Relative frequency (decimal/fraction---frequency over total number) can turn into percentages Joint frequency= has to have two conditions (ie. girls and own a bicycle) Marginal frequency= entry in the total row/column See written notes for example ~ two way frequency table---Great way to display data p.416----adding the relative frequency category turning decimals into percentages joint relative frequency===== what percent of?? “given”= if you know p.418 problem 3 READ WORDS CAREFULLY TO SEE WHAT THE PROBLEM IS ASKING HOMEWORK p.421 (7-10, 15-19, 21) NOTEBOOK CHECK (14.9 #1, 15.3 G, 15.4 #1, 14.7 #1---workbooks), p341 #7, p341 #17, p.407 #11, p.407 #18, p. 377 #315b, p.385 #11 Week of December 11, 2015 MONDAY p. 425 Chapter 6 Review, test tomorrow CALCULATOR if /r/ does not show up go to 2nd Catalog----diagnostic (turn on) Review sheet TUESDAY Review Test WEDNESDAY 7-2 Points, Lines, and Planes p.441 (name it with capital letter) Point= indicates a location and has no size (name it with lower case line l, AB, BA) Line= straight path that extends in two opposite directions w/o end and has no thickness, contains infinitely many points (use capital letter such as plane P or plane ABC) Plane= flat surface that extends w/o end and has no thickness, contains infinitely many lines (goes on forever---but not in all directions) Collinear points= points that lie on the same line Coplanar= points and lines that lie in the same plane Geometric figure= set of points Space= set of all points in three dimensions (AB) Segment= part of line that consists of two endpoints and all points between them, piece of a line (AB) Ray= part of line that consists of one endpoint and all the points of the line on one side of the endpoint Opposite rays= two rays that share the same endpoint and form a line ANSWERS: p.442 Problem 1 a) b) c) RQS, TQN d) N, T Practice A 1) EBF 2) G, E, F, B p. 443 Problem 2 Practice A 3) RW, ST, RT, TW, SW, RS 4) RW, TW, ST (go down) WR, TR, SR (go up) P.444 Postulate/axiom= basic building blocks of the logical system in geometry Postulate 1= through any two points there is exactly one line *Intersection: set of points the figures have in common Postulate 2= if two distinct lines intersect, then they intersect in exactly one point Postulate 3= if two distinct planes intersect, then they intersect in exactly one line Postulate 4= through any three noncollinear points there is exactly one plane Problem 3 a) front & right / plane ABFE, BCGF (F&B are in both planes) b) two points make a line and the two planes intersect at a line Practice A 5. bottom & right / RS 6. front & left / QUR, XUQ Problem 4 p.446 a) P b) JM, ML HOMEWORK (P.447 # 9-25, 27, 29-35, 42, 44, 46) THURSDAY Review homework 9) line XR 10) ray RY, ray RX 11) RS 12) segment RS, SR 13) no 14) because the line goes in both directions 15) –diff: one has one arrow and the other two arrows, one side of both of them goes forever, both have 2 points 16) yes 17) yes 18) no 19) yes 20) no 21) no 22) can draw 23) cannot draw 30) sometimes 31) always 24) can draw 25) cannot draw 29) always 32) sometimes 33) always 34) never 35) triangular prism b. (1) left 42) yes a. (1) bottom 44) no c. (1) diagonal d. e. yes 46) yes p. 451 7-3 Measuring Segments Postulate 5- Ruler postulate= every point on a line can be paired with a real number Postulate 6- Segment Addition postulate= if three points are collinear ABC and B is between A & C, then AB+BC=AC Coordinate= the real number that corresponds to a point Distance= absolute value of the difference of the coordinates **cannot have negative distances Congruent Segments= if two segments have the same length Midpoint= point that divides the segment into two congruent segments Segment Bisector= a point, line, ray or other segment that intersects a segment at its midpoint (cuts a segment in half) Problem 1 UV=(4) SV=(18) 1) 9 Problem 2 JK 2) 6 make an equation 4x+6+7x+15=120 4(9)+6= 42 KL 11x+ 21= 120 11x=99/11 x=9 7(9)+15= 78 3) 24 4) 25 a. No b. 5-(-2)=? 7, wrong (-2)-5=? b. TU UV Problem 3 Problem 4 35 35 5) TV 70 Since you know they have to equal the same in length, put them equal to each other, find x and fill it in to find the answer (3 step process) Practice 7 a. XA=9 3x=5x-6 b. 9, 18 Lesson Check p. 455 8) B 9) A,G 10) D 11) BD, CE, DF, EG HOMEWORK: worksheets Measuring Segments Week of December 18, 2015 MONDAY p.458 7-4 Measuring Angles) angle= two rays with the same endpoint (for ways to name an angle see bottom p.458) acute 0<x<90 right x=90 (needs the symbol) obtuse 90<x<180 straight x=180 congruent angles= angles with the same measure, could be length or size rays= sides of angle vertex= endpoint of an angle (the point at which you change directions) interior of angle= region containing all of the points between the two sides of the angle exterior of angle= region containing all of the points outside of the angle circle= all points in a plane that are a given distance from a given point, 360 degrees WHEN YOU NAME ANGLES USING 3 POINTS, THE VERTEX MUST GO IN THE MIDDLE One way to measure the size of an angle is in degrees To indicate the measure of an angle, write a lowercase m in front of the angle symbol (m<A=) Problem 1 a) <2, <LMK b) no, because you don’t know what angle is being identified Practice 1) <1, <ABC, <CAB, <B 2) <JKM, <MKJ, <2 POSTULATE 7= protractor postulate: allows you to find the measure of an angle Practice A 2) Right 4) acute 5) acute p. 461 (bottom) shows how to write congruent angles p.462 Practice A 6. 75 7. 130 POSTULATE 8= angle addition postulate: angle addition postulate is similar to segment addition postulate (see middle of p.462) Problem 4 (11x-12) + (2x+10)= 180 get x= 14 and THEN PLUG IT IN!!!! to get answers 8) (5x+4) + (8x-3) = 79 get x and then PLUG IT INTO EQUATIONS TO GET ANSWERS 9) (6x+20) + (2x+4)= 180 get x and then PLUG INTO EQUATIONS TO GET ANSWERS HOMEWORK P.463 #10-12, 14, 20-22, 24-29 P. 7-5 Exploring Angle Pairs Adjacent angles= two coplanar angles with a common side, common vertex, and no common interior points (sit side by side) Vertical angles= two angles whose sides are opposite rays (sit across from each other) Complementary angles= two angles whose measures have a sum of 90 (C comes 1st and 90 comes first) Supplementary angles= two angles whose measures have a sum of 180 Problem 1 a) Yes b) no c) yes Practice A 1. <DOC, <AOB 2. <AOB, <DOC p.468 CAN find from diagram CANNOT find from diagram Angles are adjacent angles/segments are congruent Angles are adjacent and supplementary an angle is a right angle Angles are vertical angles angles are complementary HOMEWORK p.472 # 8-11, 13-17, 20-23, 29-32 TUESDAY Homework review p.464 20) x=8, <AOB=30, <BOC=50, <COD=30 21) x=18, <BOC=52, <AOD=108 22) A 24)180 25) 150 26) 30 27) 120 28) 30 29) 30 p.469 7-5 Exploring angle pairs, con’t c) don’t know for sure, would have to have right angle symbol d) don’t know for sure, PQ bisects TV, but unknown if TV bisects PQ Practice A 3) Yes, diagram tells me that 4) yes, but may need more information POSTULATE 9= linear pair postulate: two angles form a linear pair(has to share a side), supplementary = 180 Problem 3 a) b) add (3x+14) + (5x-2)= 180…………….then find x=21……………then plug in answers Practice A 5)<QMN, <LMQ <LMP, <PMN 6) add (4n+15) + (2n+21)=180………………..find x…………………..then plug in answers Angle bisector= ray that divides an angle into two congruent angles, endpoint is at the angle vertex, the ray/segment is cutting the angle in half Problem 4 36 degrees Practice A a) (3x-3)=(4x-14) because they are congruent measures of angles b) <HGI= 30 c) <FGI= 60 HOMEWORK p.472 # 8-11, 13-17, 20-23, 29-32 x=11……………plug in answers to find p. 475 7-6 Midpoint and Distance in the coordinate plane ***On the number line: the coordinate of the midpoint is AVERAGE/MEAN of the coordinates of the endpoints ***In the coordinate plane: the coordinates of the midpoint are the average of the x-coordinates and the average of the y-coordinates of the endpoints FORMULA--- the coordinate of the midpoint M of AB is a+b/2 Do not use Coordinate Plane Formula on p. 475 Problem 1: finding midpoint b) Average up the x #s and average up the y #s but it needs to equal a point on graph (4, -2) Problem 2: finding endpoint (-3,-5)-----------(4,-9) --------------- (x,y) (-3+x/2)=4 Multiply both sides by 2 to get rid of fractions -3x+x=8, x=11 (-5+y/2)=-9 -5+y=-18, y=-13………plug in answers HOMEWORK: finish all of p.472, p.479 #9-10, p.481 #14-22----only do part (b) WEDNESDAY Homework answers p. 472 8) <AFE, <CFD 9) <AEF, <DEF 10) <BCE, <ECD 14) x=5, 50 16) x=11, <56 17) x=12, 112 20) 90 22) 155 23) 115 p.474 29) <KML, <PMR, <MQP, <KMQ, <MNR 32) <PMR, <KMQ, <KML, <MQP 30) <KML p.473 20) 90 23) 105 21) 25 p.479 9) ½, 11/2 (.5, 5.5) 17) -2.5, 3 18) 1, -4 22) 155 10) 7, -8 19) -4, -4 11) 20 21) 25 31) <LMN, <PMQ 14) 4.5, 4 15) 1.5,0.5 16) -1.5, .5 20) 21) 3, .5 22) 3.5, 1 problems worked out 14) P(3, 2) Q(6, 6) 3+6/2 2+6/2 (9/2, 8/2 or 4.5, 4) 10) BC(5, -2) B(3,4) C(x,y) 3+x/2=5 3+x=5(2) 4+y=-2(2) C= (7, -8) 4+y/2=-2 multiply both sides by 2 p.478 Problem 3 a. SR S(-2,14) R(3,-1) [looking for a distance not a point] how do you find a diagonal distance? 10^2 + 2^2 = h^2 100+4=h^2 104=h^2 take square root of both h=10.2 Pythagorean Theorem Not!!! leg^2 + leg^2 = hypotenuse^2 a^2+ b^2 = c^2 p.477 DISTANCE FORMULA (x,y) (x,y)------(x-x) (y-y)---------- d^2=x^2 + y^2 Practice A 5. R(0,5) S(12,3) d^2=x^2+y^2 d^2= -12^2 + 2^2 If you square a negative number in the calculator, you must put them into parenthesis d^2= 144 + 4 [d^2=148 square root both] d= 12.2 may use longer formula on p.477 if you have fractions or large numbers HOMEWORK p. 479 #9-11, 13-22, 26-28 review tomorrow and possible test or review all tomorrow and test Friday THURSDAY Homework review p. 480: 11) 8.5 13) he didn’t do x-x or y-y, he did x-y 14) 5 15) 5.8 16) 7.1 17) 5.4 18) 10 19) 20) 6.7 21) 5.4 22) 2.2 26) midpoint -1.5, 0 distance 19.2 27) distance 10.8 midpoint 3,-4 28) distance 5.4 midpoint Notebook Check 2b: p.341 #7, p.407 #11, p.377 #16, p.385 # 13, p.391 #10, p.421 #7, p.447 #10, p.456 #15, p.463 #10, p.472 # 14 ****Math I Geometry Review packet FRIDAY Finish reviewing Geometry Review packet & TEST Week of January 8th, 2016 REVIEW FOR EOC Monday Review of Test from before break North Carolina READY EOC Assessment, Math I (last year’s 2015 test) for practice and timing *careful on how you bubble in sheet, watch the columns *cannot answer in mixed numbers, so has to be decimal or improper fraction *anything you can come up with that is equivalent to the answer, you can enter it Tuesday Review and complete last years practice test Wednesday *count A, B, C & D answers, see how many of each you have *questions really don’t know ?s ----leave blank, when you finish the test you are going to average how many of each letter you have, then use the letter that was used the least--number for all the unknown answers *do all first 15 questions (inactive) and cannot go back, but keep track of letter answers still *Factor, square root, make a table, GCF *if given equation, it says graph it……make a table *draw pictures, scrap paper *read the English and write it as Math *Distribute, combine like terms *System of equations: eliminate, multiple top equation by (-2) *do not need to reduce to lowest form ANSWERS from pretest 1) B 8) 5 9) 6 10) 10 2) C 3) B 11) -5 4) D 5) D 12) 16 6) 120 7) .75 or ¾ or 45/60 13) 9 14) 4 15) 1 THURSDAY *Quadratic: factoring, song, calculator Factoring-----divide by like terms to lower numbers Make a list if needed----what two numbers multiplied together equal the same when added together (ie. t^2-4t-12=0) so [ -6x2= -12 and -6+2= -4 ] *CALCULATOR----2nd ---Calculate----ZOOM find where item hit ground *When an exponent is negative, have to change its location, so 2^-2 would = ½^2 (1/4) *Combine like terms when they are on the same side *don’t forget that x= 6+ - 12/2 equals 6+12/2 AND 6-12/2------two problems to solve *a length being negative doesn’t make sense *read questions carefully to really understand what the question is asking *make sure you watch what the numbers labeled as….. aka miles, yards, minutes, hours and make sure you convert them correctly, changing one unit to another *try to cancel out as much as possible when using fractions ANSWERS to pretest: 16) B 26) D 27) A 28) B 17) C 29) D 18) D 30) B 19) C 31) D 20) B 21) D 22) C 23) B 24) D 25) B *Cube root of 3, 4, 5------have to change to (8x^2y^3z^4)^1/3 = 3|8x^2y^3z^4 (cube root) *Parabola (U-shaped) Calculator Zoom Fit will fit onto screen to show part of it. Zoom Out if needed. 2nd calculate-----Minimum Calculator-----Stat-----Edit-----clear out columns, enter info------STAT----Calculate----#4 LinReg FRIDAY *Continue review of last year EOC *adding every time is linear and multiplying every time is exponential when creating a list of numbers *slope equation: m= delta y/delta x * y=mx+b don’t plug in for b equation of a line you need slope & y intercept * parameter of a triangle: add up all lengths of the sides---use pathagrium theorem *make sure you put in order, make sure you have all numbers,, make sure you are using the same unit type *standard deviation: how far the numbers are away from a central number * read the question to know what it is asking!!! ANSWERS to the last part of the test 32) C 33) B 45) B 34) A 46) A 35) C 47) C 36) C 48) C 37) D 49) B 38) C 50) A 39) A 40) A 41) A 42) C 43) C *16-end of test: leave ones you don’t know blank….. at end, count how many of each letter you have……use the least used letter for all unanswered answers * STAT---CALC----Linear Equa * 2nd---Catalog---Diagnostic---On----enter, enter, enter *remember to change decimals to fractions as needed *see a TABLE, automatically think STAT on calculator *when seeing (AT LEAST) use less than/equal to *when seeing (NO MORE THAN) use less than/equal to * 44) C