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Unit 5 Radicals and Equations of a Line

Unit 5 Radicals and Equations of a Line. This unit has a review section on solving/reducing radicals, and the laws of exponents. This is required to solve the Pythagorean Theorem.

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Unit 5 Radicals and Equations of a Line

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  1. Unit 5 Radicals and Equations of a Line • This unit has a review section on solving/reducing radicals, and the laws of exponents. • This is required to solve the Pythagorean Theorem. • This unit also covers the three standard equation of a line used to graph in the coordinate plane (Standard equation, slope-intercept, and point-slope). • The distance formula, midpoint, endpoint, and slope are all addressed in the coordinate plane, as well as parallel and perpendicular lines.

  2. Standards • SPI’s taught in Unit 5: • SPI 3108.1.4 Use definitions, basic postulates, and theorems about points, lines, angles, and planes to write/complete proofs and/or to solve problems. • SPI 3108.3.1 Use algebra and coordinate geometry to analyze and solve problems about geometric figures (including circles). • SPI 3108.3.2 Use coordinate geometry to prove characteristics of polygonal figures. • CLE (Course Level Expectations) found in Unit 5: • CLE 3108.1.4 Move flexibly between multiple representations (contextual, physical written, verbal, iconic/pictorial, graphical, tabular, and symbolic), to solve problems, to model mathematical ideas, and to communicate solution strategies. • CLE 3108.1.7 Use technologies appropriately to develop understanding of abstract mathematical ideas, to facilitate problem solving, and to produce accurate and reliable models. • CLE3108.2.3 Establish an ability to estimate, select appropriate units, evaluate accuracy of calculations and approximate error in measurement in geometric settings. • CLE 3108.3.1 Use analytic geometry tools to explore geometric problems involving parallel and perpendicular lines, circles, and special points of polygons. • CFU (Checks for Understanding) applied to Unit 5: • 3108.1.1 Check solutions after making reasonable estimates in appropriate units of quantities encountered in contextual situations. • 3108.1.2 Determine position using spatial sense with two and three-dimensional coordinate systems. • 3108.1.3 Comprehend the concept of length on the number line. • 3108.1.5 Use technology, hands-on activities, and manipulatives to develop the language and the concepts of geometry, including specialized vocabulary (e.g. graphing calculators, interactive geometry software such as Geometer’s Sketchpad and Cabri, algebra tiles, pattern blocks, tessellation tiles, MIRAs, mirrors, spinners, geoboards, conic section models, volume demonstration kits, Polydrons, measurement tools, compasses, PentaBlocks, pentominoes, cubes, tangrams). • 3108.1.7 Recognize the capabilities and the limitations of calculators and computers in solving problems. • 3108.2.6 Analyze precision, accuracy, and approximate error in measurement situations. • 3108.3.1 Prove two lines are parallel, perpendicular, or oblique using coordinate geometry. • 3108.3.2 Connect coordinate geometry to geometric figures in the plane (e.g. midpoints, distance formula, slope, and polygons). • 3108.3.4 Apply the midpoint and distance formulas to points and segments to find midpoints, distances, and missing information in two and three dimensions. • 3108.4.3 Solve problems involving betweeness of points and distance between points (including segment addition). • 3108.4.19 Use coordinate geometry to prove properties of plane figures.

  3. Reducing a Radical There are three conditions that must be met for a radical expression to be in its simplest form: • The number under the radical sign has no perfect square factors other than 1 • The number under the radical sign does not contain a fraction • A denominator does not contain a radical expression

  4. Reducing the Square Root of a Fraction Reduce this Square Root: √4/3 • We can rewrite this = √4 /√3 • The square root of 4 is 2, so we now have: 2/√3. • We cannot stop here, there is a radical in the denominator. • Therefore, we multiply the top and bottom by top and bottom by √3/√3 ( a form of 1) • So…( 2* √3) / ( √3* √3) = 2 √3 / 3

  5. Reducing a Radical with a Perfect Square • Reduce this radical: √24/ √3 • The square root of 24 can be rewritten as: (√4* √6)/ √3 • We can solve the square root of four (it is 2) and now have: (2* √6)/ √3 • The square root of 6 can be rewritten as the square root of 2 times the square root of 3, which gives us: ( 2* √2* √3)/ √3 • We can cancel out the square roots of 3, and now we have: 2* √2

  6. A Deeper Look at Reducing Radicals • Ultimately, to reduce radicals we look for perfect squares • For example, reduce this: √75 • This has one perfect square in it: 25 • Therefore, we have 25*3, which is 5*5*3 • Therefore we can reduce this to: 5 √3 • There are three ways to reduce a radical:

  7. How to Reduce a Radical • Recognize perfect squares. If you know your perfect squares up to about 13, you can often reduce a radical in your head (like reducing √75) • The Factor Tree. Whee  • Using a Calculator • What A calculator? How????

  8. Recognize the Square • Reduce this: √60 • This is 4 x 15. Therefore it reduces to 2 √15 • Reduce this: √45 • This is 9 x 5. Therefore it reduces to 3 √5 • Reduce this: √90 • This is 9 x 10. Therefore it reduces to 3 √10 • This is a good method for simple radicals

  9. Factor Tree • Reduce this: √80 • The factor tree requires you to reduce a number to it’s prime numbers • 80 is 8 x 10, which is 2 x 2 x 2 x 2 x 5 • Therefore, I can pull out the 2’s and leave the five • This gives me 4 √5 • Most of you remember how to do this

  10. Calculator • Let’s look at this one again: √80 • Again, I’ll look for perfect squares, but this time using the calculator • Go to the “y=“ button (top left) • Enter this: 80/x2 (use the “x,t,Ø,n” button for X, and then use the “squared button”) • This forces the calculator to divide 80 by every single rational number from 1 to 80

  11. Calculating the Square • The calculator first assigns X the value of 1. Therefore Y is 80 (80/12) • It then continues up in infinite increments basically forever • What we want to do is identify the largest value of X, and whatever is left over will still be under the radical sign

  12. The Table • After we enter 80/x2, we need to enter this: • Press the “2nd” button, then “table F5” (top right button) • Here you will see an X / Y chart • What we focus on is the “Y” column • We need to find the smallest value of Y, where Y is a whole number greater than 1 (1 can’t be a perfect square, right?)

  13. Analyzing the Table • As we look at the table, we see that the smallest whole number in the Y table is 5 • We see the X value is 4 • We can therefore write this: 4 √5 • Remember, the value of Y is what is left under the radical sign • If we cannot find a whole number for Y, we cannot reduce the fraction

  14. Try Another • Reduce this using the Calculator: √135 • Go to the “y =“ button (you may have to hit “clear” to remove your last problem) and enter: 135/X2 • Now hit “2nd” and “Table” • We look for the smallest whole value of Y, and find 15. The matching X value is 3 • So we write: 3 √15 • To confirm this is right, we would multiply 3 x 3 x 15, and we get 135.

  15. Example • You try: • √175 • Answer: 5 √7 • √294 • Answer: 7 √6 • √168 • Answer: 2 √42 • √270 • Answer: 3 √30

  16. Assignment • Page 399 1-15 • 3 Radicals Worksheets

  17. Unit 5 Quiz 1Reduce all radicals to their simplest form Answers 2 √39 4 √15 2 √85 2 √19 2 √10 5 √6 14 3 √10 2 √55 6 √15 • √156 • √240 • √340 • √76 • √40 • √150 • √196 • √90 • √220 • √540

  18. Unit 5 Quiz 2Reduce all radicals to their simplest form Answers 2√39 - 6√2 5√15 √85 2 20√2 3√6 8 (√10) / 2 2√6 10√15 • (√156) – (√72) • (√240) + (√15) • (√340) / 2 • (√76) / (√19) • (√40) * (√20) • (√150) – (√24) • (√196) – (√36) • (√90) / 6 • √(240/10) • (√540) + (4√15)

  19. Multiplying with Exponents • A quick look at exponents • When multiplying the same base number with exponents, add the exponent • For example: 53 x 52 = 55 • This is the same as writing this: • 51 x 51 x 51 x 51 x 51 which equals 55 53 52

  20. Multiplying with Exponents • What is y3 x y3 • it would be y6 • What is ya x yb • it would be ya+b • What is m6 x m-4 • it would be m2 • What is n2 x n3 x n • it would be n6

  21. Parenthesis and Exponents • When we have parenthesis, we multiply the outer exponent times the inner exponent • For example (x4)3 = x12 • This is because we are really doing this: • (x4) x (x4) x (x4) • And we know that when multiplying the same base number, we add exponents • Thus we add the exponents 4 + 4 + 4 = 12 • Which is how we get x12

  22. Parenthesis and Exponents • What is (x5)3 • It would be x15 • What is (5x2)2 • It would be 52x4 • What is (52x3)-2 • It would be 5-4x-6 • What is (3-3y-4)-2 • It would be 36y8

  23. Dividing with Exponents • Dividing base numbers with exponents • When we divide the same base number with exponents, we take the top exponent minus the bottom exponent • For example: • X5/X3 = X2 • An example with real numbers: 53/52 = 5 • This is 125/25 = 5, so we have proved this to be true

  24. Dividing with Exponents • What is 105/102 • It would be 103 • What is x7/x3 • It would be x4 • What is 53x5/5x2 • It would be 52x3 • What is 43y2/45y7 • It would be 4-2y-5

  25. Creating Positive Exponents • Negative exponents are not considered “simplified” • A proper answer in math will only have positive exponents • To turn a negative exponent into a positive exponent, put it under a dividing line • For example: 10-2 = 1/102 • This can be rewritten as 1/100 • Here is how it works…

  26. There is a Pattern • 103 = 1000 • 102 = 100 • 101 = 10 • 100 = 1 • 10-1 = 1/101 • 10-2 = 1/102 or 1/100 • 10-3 = 1/103 or 1/1000 • So we simply put the base number under the divider line and turn the negative exponent into a positive exponent

  27. Converting Negative Exponents • What if the exponent is already under the dividing bar, and is negative? • For example: 5/x-2 • In this case, we move the number above the dividing bar and make it positive: 5x2 • So our rule will be this: If we have a negative exponent, go to the opposite side of the dividing bar, and rewrite the exponent as a positive exponent

  28. Converting Negative Exponents • What is x-3 • It would be 1/x3 • What is 5-2y-6 • It would be 1/52y6 • What is 4-3y4/x2z-3 • It would be y4z3/43x2 –Here we moved the 43 under the dividing line, and the z3 above the dividing line, and left the other two variables where they were • What is 4-3/4-5 • It would be 45/43, or 45-3 or 42

  29. The Zero Exponent • ANYTHING TO THE ZERO POWER IS ONE. • 1 • ONE • 1! • Get it?!!! • For example 50 = 1, or x0 = 1

  30. Solving the Zero Exponent • What is 5a0 • It would be 5 (because 5 x 1 = 5) • What is 5ab0 • It would be 5a (because 5 x a x 1 = 5a) • What is (45810x9y2)0 • It would be 1 (that whole expression is to the zero power, so the whole thing is 1) • What is (2343y8)0 x (1210y8z9)0 • It would be 1 x 1, which is of course 1.

  31. Summary • Multiplying base numbers: Add exponent • Dividing base numbers: Subtract bottom exponent from top exponent • Exponents in Parenthesis: Multiply outside exponent times inside exponent • Negative Exponents: Move expression to opposite side of dividing line, convert exponent from negative to positive • Exponent = 0, the entire expression which is raised to the zero power = 1

  32. Unit 5 Quiz 2aReduce all radicals to their simplest form Answers 2√39 - 6√2 5√15 √85 2 20√2 3√6 8 (√10) / 2 2√6 10√15 • (√156) – (√72) • (√240) + (√15) • (√340) / 2 • (√76) / (√19) • (√40) * (√20) • (√150) – (√24) • (√196) – (√36) • (√90) / 6 • √(240/10) • (√540) + (4√15)

  33. Unit 5 Quiz 3 • When multiplying numbers with the same base, you ________ exponents • When dividing numbers with the same base, you _______ the bottom exponent from the top exponent • When solving exponents in parenthesis, you ________ the outer exponent times the inner exponent • When solving negative exponents, you _______ the base number, and make the exponent positive • Any number with an exponent of zero = __________ • Reduce (leave in exponent form): m6 x m-4 = ________ • Reduce (leave in exponent form): (5x2)2 = ____________ • Reduce (leave in exponent form): x7/x3 = _____________ • Reduce (leave in exponent form): 5-2y-6 = _____________ • Reduce: (2343y8)0 x (1210y8z9)0 = _________________

  34. Reducing Radicals with Exponents Example Answer √13 13 13 √a a a a √a a2 a2 √a a3 • √13 • √13 x √13 • (√13)2 • √a • (√a)2 • √a2 • √a3 = √ a2 x a • √a4 = √ a2 x a2 • √a5 = √ a4 x a • √a6 = √ a3 x a3

  35. Assignment • Page 829 1-20 • Exponents Handout • Another Exponents Handout

  36. Pythagorean Theorem (500 BC) • It was believed that Pythagoras discovered this theorem when waiting for the tyrannical ruler, Polycrates. • While looking at the floor’s square tiling of the palace of Polycrates, Pythagoras thought of this interesting idea: A diagonal line may be used to cut or divide the square, and two right triangles would be produced from the cut sides. • Other stories tell us that he then spent time on the sand at a beach drawing diagrams until he came up with the final equation

  37. The Distance Formula • Background: The distance formula is based upon the Pythagorean Theorem • The Pythagorean Theorem states that in a right triangle, where you have side A, side B, and side C, you can calculate a side thusly: a2 + b2 = c2, where c is the hypotenuse, or the longest side of the right triangle

  38. Distance Formula • You can draw any line on a graph, and then create a right triangle with that line as the hypotenuse And So On

  39. Distance Formula • Remember, to find the length of the hypotenuse (the long side C) we need to know the length of the two short sides (side A and side B) • We can easily calculate that –it would be the big number minus the small number –just like on a ruler • So the length of side A is the big x number minus the small x number = 4 – (-4) = 8 • The length of side B is the big Y number minus the small Y number = 5 – (-2) = 7 (4,5) C B (-4,-2) (4,-2) A

  40. Distance Formula • From the previous slide, we know that side A is 8 units long, and side B is 7 units long • We also know that A2 + B2 = C2 • So we have 82 + 72 = C2 • Therefore to find out how long C is, we have to take the square root of both sides: • √C2 = √(82 + 72) • Or: C = √(82 + 72) (4,5) C B (-4,-2) (4,-2) A So C = 10.63

  41. Distance Formula-I will ask this equation on a test • Using the previous example, we can come up with a formula which will work to find the distance of any line: • Instead of C =, we will D = (for distance) • d= √(x2-x1)2 + (y2-y1)2 • Remember, this is big x minus small x, and then square it, and big y minus small y, and then square it, add those two squares, then take the square root of that sum

  42. Examples • Given point a (5,2) and point b (-4,-1) find the distance between the two points • D = √(5-(-4))2 + (2 – (-1))2 • D = √(9)2 + (3)2 • D = √(81 + 9) • D = √90 • D = 9.48 Would it matter if you subtracted the numbers the other way around? Hmm… D = √(-4 – 5)2 + (-1 -2)2 D = √(-9)2 + (-3)2 D = √(81 + 9 D = √(90) no, it does not D = 9.48 matter

  43. Midpoint • If you wanted to know the halfway point between two numbers, you would add them together, and divide by two. • For example, what is halfway between 1 and 9? • 1 + 9 = 10, then divide by 2, or 10/2 = 5 • Therefore, 5 is the midpoint of 1 and 9

  44. Midpoint Formula • If we have segment AB, with the coordinates of A = (x1, y1) and the coordinates of B = (x2, y2) all we do is add the “x”s and divide by two, and add the “y”s and divide by two. • We will call the midpoint M. • Therefore, to find Mx = {[(x1+ x2)/2], My = {[(y1+ y2)/2]}

  45. Example • Segment QS has endpoints Q (3,5) and S (7,-9) • What is the midpoint M? • X coordinate of M = (3+7)/2 = 5 • Y coordinate of M = (5 + -9)/2 = -2 • The coordinates for midpoint M = (5,-2) • By the way. I have this loaded on my calculator. You’re welcome to have it. 

  46. Finding an Endpoint • What if you have an endpoint, and a midpoint, and want the other endpoint? • The midpoint of Segment AB = (3,4) • Endpoint A = (-3,-2) • What is Endpoint B? • Just substitute into the midpoint formula • Solving for X2, we have 3 = (-3 +X2)/2 • Therefore, -3 + X2 = 6, and X2 = 9 • Solving for Y2, we have 4 = (-2 + Y2)/2 • Therefore, -2 + Y2 = 8, and Y2 = 10 • The coordinates for B = (9,10)

  47. Assignment • Page 54 6-44 • Worksheet 1-6 • Pythagorean Theorem Word Problems Handout • Pythagorean Word Problems #2

  48. Unit 5 Quiz 4 • Problems 1 through 5: Graph points A,B,C,D,E (Hand draw one graph with five points –label each) • Problems 6 through 10: Use the same points: Calculate the midpoint between the two points in each set – show answers below graph • Problems 11 through 15: Use the same points: Calculate the distance between the two points in each set – show answers below graph #6, #11. A (1,4) B (3,-5) #6 (midpoint) #11 (distance) #7, #12. C (0,1) D (2, 6) #7 (midpoint) #12 (distance) #8, #13. E (0,0) F (4,6) #8 (midpoint) #13 (distance) #9, #14. G (-3,-4) H (-2,7) #9 (midpoint) #14 (distance) #10, #15. J (-1,6) K (4,9) #10 (midpoint) #15 (distance)

  49. Quiz Graph Answers .D Midpoint: :__________ :__________ :__________ :__________ :__________ Distance: :__________ :__________ :__________ :__________ :__________ .A .C .E .B

  50. Slope • Slope • The slope of a line is a measure of how “tilted” the line is. A highway sign might say something like “6% grade ahead.” What does this mean, other than that you hope your brakes work? What it means is that the ratio of your drop in altitude to your horizontal distance is 6%, or 6/100. In other words, if you move 100 feet forward, you will drop 6 feet; if you move 200 feet forward, you will drop 12 feet, and so on.

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