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Honors Geometry Spring 2016 Mrs. Poppiti Day 1: February st 1 Objective: Form and meet study teams. THEN Build the biggest box possible from a single sheet of paper; decide how we will determine which box is the “biggest”. THEN Build an understanding of area and perimeter. Investigate how the perimeter and area of a shape change as the shape is enlarged proportionally. Seats and Fill out Index Card (questions on next slide) Introduction: Mrs. Poppiti, Books, Syllabus, Homework Record, Expectations Quick Pairs Introduction Activity: Building the Biggest Problems 1-19 and 1-24 Homework: Do Problems 1-3 to 1-7 AND 1-14 to 1-18; Supplies (Monday at the latest); Have parent/ guardian fill out last page of syllabus and sign; Extra credit tissues or lysol wipes Respond on Index Card: 1. When did you take Algebra 1? 2. Who was your Algebra 1 teacher? 3. What grade do you think you earned in Algebra 1? 4. What is one concept/topic from Algebra 1 that Ms. Poppiti could help you learn better? 5. What grade would you like to earn in Geometry? (Be realistic) 6. What sports/clubs/activities/jobs are you involved in this Spring? 7. E-mail address that you check regularly – PRINT CLEARLY! Support www.cpm.org Resources (including worksheets from class) Extra Support/Practice Parent Guide Free Homework Help Mrs. Poppiti’s webpage on the HHS website Classwork and Homework Assignments Worksheets Extra Resources Building the Biggest Your main task in this activity is to build the biggest box you can from a single sheet of construction paper. In this activity, “biggest” means “holding the most” and “box” means a container with four rectangular sides, a rectangular bottom, and no top. You can cut your construction paper and tape pieces together in any way you want as long as your final product is a box. If your first attempt does not satisfy you, try again. Keep working at it until you think you have built the biggest box possible. Once you have a container that you are satisfied with, it is time to test its capacity! Fill your paper container with puffed rice, and compare with other teams’ containers. Your teacher will instruct you as to how to make the comparison/measurement. The “winning” container will be the one that holds the largest amount of puffed rice. Day 2: February nd 2 Objective: Build an understanding of area and perimeter. Investigate how the perimeter and area of a shape change as the shape is enlarged proportionally. THEN Build an understanding of what an angle is and how it is measured. Introduce complicated shapes composed of triangles, and begin to use attributes of sides and angles to compare and describe those shapes. THEN Use spatial visualization skills to investigate reflection. HW Check/Correct in RED Finish Problems 1-19 and 1-24 Problems 1-37 to 1-40 Start Problems 1-47 to 1-52 Homework: Do Problems 1-25 to 1-29 AND 1-32 to 1-36; Supplies (Monday at the latest); Have parent/ guardian fill out last page of syllabus and sign; Extra credit tissues or lysol wipes Carpetmart • What is perimeter? • What does perimeter measure? • What are the units used to measure area? • Why are they different from the units used to measure length? • See Math Notes Box on Perimeter and Area • Look for patterns! • Help each other – everyone has strengths and weaknesses! Types of Angle Measures (pg 24) Acute: Between 0° and 90° Right: 90° Straight: 180° Obtuse: Between 90° and 180° Circular: 360° Day 3: February rd 3 Objective: Use spatial visualization skills to investigate reflection. THEN Understand the three rigid transformations (translations, reflections, and rotations) and learn some connections between them. Also, introduce notation for corresponding parts. THEN Begin to develop an understanding of reflection symmetry. Also, learn how to translate a geometric figure on a coordinate grid. Learn that reflection and reflection symmetry can help unlock relationships within a shape (isosceles triangle). HW Check/Correct in RED & Comlete Warm-Ups (on white board) LL Notes: “Complete Graph” – for HW tonight (and in the future) Finish Problems 1-50 to 1-52 Problems 1-59 to 1-62 Start Problems 1-69 to 1-72 Homework: Do Problems 1-42 to 1-46 AND 1-54 to 1-58; Get Supplies! (Supplies check Monday) Ch. 1 Team Test Monday? Individual Test Wed? Complete Graph When a problem says graph an equation or draw a graph: y On graph paper: Plot key points accurately Use a ruler! (-2,0) (3,0) x Scale your axes appropriately (0,-6) (.5,-6.25) Label the axes (with units if appropriate) Day 4: February th 4 Objective: Begin to develop an understanding of reflection symmetry. Also, learn how to translate a geometric figure on a coordinate grid. Learn that reflection and reflection symmetry can help unlock relationships within a shape (isosceles triangle). THEN Learn about reflection, rotation, and translation symmetry. Identify which common shapes have each type of symmetry. HW Check/Correct in RED & Complete Warm-Ups (on white board) LL Notes Problems 1-69 to 1-72 Problems 1-87 to 1-91 Homework: Do Problems 1-63 to 1-67 AND 1-73 to 1-77; Get Supplies! (Supplies check Monday) Ch. 1 Team Test Monday? Individual Test Wed? Prime Notation (pg 34) When labeling a transformation, the new figure (called the image) is labeled with prime notation. – Example: If ΔABC is reflected across the vertical dashed line, its image can be labeled ΔA’B’C’ to show exactly how the new points correspond to the points in the original shape. B’ B C A C’ A’ Rigid Transformations (pg 34) Transformation: A movement that preserves size and shape Reflection: Mirror image over a line Translation: Slide in a direction Rotation: Turning about a point clockwise or counter clockwise Polygons (pg 42) Polygon: A closed figure made up of straight segments. Regular Polygon: The sides are all the same length and its angles have equal measure. Line: Slope-Intercept Form (pg 47) y = mx + b Slope Slope: Growth or rate of change. yintercept y m x y-intercept: Starting point on the y-axis. (0,b) Slope-Intercept Form 3 y x 3 2 You Next, use rise Firstcan plotgo the Now connect backwards if over run towith plot y-intercept on the points new thepoints y-axis anecessary! line! Parallel Lines (pg 47) Parallel lines do not intersect. Parallel lines have the same slope. For example: 5 y x 4 2 and 5 y x 1 2 Perpendicular Lines (pg 47) Perpendicular lines intersect at a right angle. Slopes of perpendicular lines are opposite reciprocals (opposite signs and flipped). For example: 2 3 y x 5 and y x 1 2 3 Everyday Life Situations Here are some situations that occur in everyday life. Each one involves one or more of the basic transformations: reflection, rotation, or translation. State the transformation(s) involved in each case. A. You look in a mirror as you comb your hair. B. While repairing your bicycle, you turn it upside down and spin the front tire to make sure it isn’t rubbing against the frame. C. You move a small statue from one end of a shelf to the other. D. You flip your scrumptious buckwheat pancakes as you cook them on the griddle. E. The bus tire spins as the bus moves down the road. F. You examine footprints made in the sand as you walked on the beach. Reflection across a side The two shapes MUST meet at a side that has the same length. 1-71: Reflections 1. Lines that connect corresponding points perpendicular are _____________ to the line of reflection. bisects each of the 2. The line of reflection ______ segments connecting a point and its image. Day 5: February th 5 Objective: Learn about reflection, rotation, and translation symmetry. Identify which common shapes have each type of symmetry. THEN Learn how to classify shapes by their attributes using Venn diagrams. Also, review geometric vocabulary and concepts, such as number of sides, number of angles, sides of same length, right angle, equilateral, perimeter, edge, and parallel. HW Check/Correct in RED & Complete Warm-Up (on white board) Explain Math Contest… Block 2: Finish Problems 1-71 to 1-72 Problems 1-87 to 1-91 Problems 1-97 to 1-98 AND 1-104 Homework: Do Problems 1-82 to 1-86 AND 1-92 to 1-96; Get Supplies! (Check on Monday) Extra Credit? Ch. 1 Team Test Tuesday Ch. 1 Individual Test Thursday 1-72 B A A’ Isosceles Triangle Sides: two sides of equal length Base Angles: Have the same measure Height: Perpendicular to the base AND splits the base in half Day 6: February th 8 Objective: Learn how to classify shapes by their attributes using Venn diagrams. Also, review geometric vocabulary and concepts, such as number of sides, number of angles, sides of same length, right angle, equilateral, perimeter, edge, and parallel. THEN Continue to study the attributes of shapes as vocabulary is formalized. Become familiar with how to mark diagrams to help communicate attributes of shapes. HW Check/Correct in RED & Complete Warm-Up (board & next slide) Supplies Check! Block 2: Finish Rotational Symmetry talk + Translation + Notes Problems 1-97 to 1-98 Problems 1-104 to 1-108 Homework: Do Problems 1-99 to 1-103 AND 1-110 to 1-114 Math Contest Tomorrow! Ch. 1 Team Test Tomorrow Ch. 1 Individual Test Thursday Day 6: Try This! Rotate Figure D 90° about the origin into each of the other 3 quadrants. Note any patterns that you see in the coordinates of the transformed images. 6, 3 7, 8 1.87-88 6 Ref. 60 rot. 0 Ref. No Rot. 1 Ref. No rot. 3 Ref. 120 Rot 2 Ref. 0 Ref. 180 rot. No rot. 4 Ref. 90 rot. 1 Ref. No rot. 1.87-88 2 Ref. 180 Rot Inf. Ref. Inf. Rot 0 Ref. No Rot 1 Ref. No Rot 0 Ref. No Rot 1 Ref. No Rot 5 Ref. 72 Rot 0 Ref. 180 Rot Symmetry Symmetry: Refers to the ability to perform a transformation without changing the orientation or position of an object Reflection Symmetry: If a shape has reflection symmetry, then it remains unchanged when it is reflected across a line of symmetry. (i.e. “M” or “Y” with a vertical line of reflection) Rotation Symmetry: If a shape has rotation symmetry, then it can be rotated a certain number of degrees (less than 360°) about a point and remain unchanged. Translation Symmetry: If a shape has translation symmetry, then it can be translated and remain unchanged. (i.e. a line) Venn Diagram #1: Has two or more siblings #2: Speaks at least two languages Venn Diagrams (pg 42) Condition #1 Condition #2 Satisfies condition 2 only Satisfies condition 1 only A B C Satisfies neither condition Satisfies both conditions D Problem 1-98 (a) #1: Has at least one pair of parallel sides #2: Has at least two sides of equal length Problem 1-98 (a) Has at least one pair of parallel sides Both Has at least two sides of equal length Neither Problem 1-98 (b) Has only three sides Both Has a right angle Neither Problem 1-98 (c) Has reflection symmetry Both Has 180° rotation symmetry Neither 1.98 a 1. Has at least 1 pair of Parallel sides 2. Has at least 2 sides of equal length 1.98 b 1. Has only 3 sides 2. Has a right angle 1.98 c 1. Has reflection symmetry 2. Has 180 rotation symmetry 1.104 1. Quadrilateral 2. Equilateral Describing a Shape Day 7: February th 9 Objective: Assess Chapter 1 in a team setting. THEN Continue to study the attributes of shapes as vocabulary is formalized. Become familiar with how to mark diagrams to help communicate attributes of shapes. HW Check/Correct in RED & Complete Warm-Up (on the board) Chapter 1 Team Test (≤ 50 minutes) Finish Problems 1-107 to 1-108 Homework: Problems 1-121 to 1-129 AND Review Worksheet Ch. 1 Individual Test Thursday Day 7: Try This! Algebraically, solve for x: 1 2x 1 x 5 1 1 2 3 3 3 6 x 5 2x 1 6 6 2 x 1 x 5 12 x 6 x 5 6 12 x 6 x 5 11x 6 5 11x 11 x 1 Shape Toolkit Shape Toolkit Day 8: February th 10 Objective: Develop an intuitive understanding of probability, and apply simple probability using the shapes in the Shape Bucket. HW Check/Correct in RED Review Chapter 1 Team Test Problem 1-108 Problems 1-115 to 1-119 Extra Review (Practice) – 6 Questions on Slides Homework: Problems CL1-130 to CL1-134 & STUDY!!! Ch. 1 Individual Test Tomorrow Day 8: Try This! 1. Solve the following equation for x: 2 1 3 x x 7 3 3 5 3 5 2 x x 21 5 5 10 x 3 x 105 7 x 105 x 15 2. Rotate the point (-6, 2) 90 degrees clockwise about the point (-1, 4). Where is the image located? Probability (pg 60) Probability: a measure of the likelihood that an event will occur at random. Number of Desired Outcomes P event Total Possible Outcomes Example: What is the probability of selecting a heart from a deck of cards? Number of Hearts 13 1 P select a heart 0.25 25% Total Number of Cards 52 4 Shape Bucket Question 0 1. The area of the following rectangle is 2 x 2 54. Write an equation involving all of the information given. x 3 Solve your equation. 2x 2 2. Find the equation of the line perpendicular to y x 5 7 that passes through the point 2, 7 . Show the algebraic process. Question 1 If the length of a rectangle is 5 less than 4 times the width, and the perimeter is 40 inches, what is the area of the rectangle? [A] 15 in2 [B] 75 in2 [C] 30 in2 [D] 5 in2 Question 2 Solve for x: [A] x = -4 [B] x = -3 [C] x = 10 [D] x = 3 Question 3 Which of the following represents the system below? Question 4 If the point (6, -3) is rotated 270 degrees clockwise about the origin, where would the new image be located? [A] (3, -6) [B] (-6, -3) [C] (-3, -6) [D] (3, 6) Day 9: February th 11 Objective: Assess Chapter 1 in an individual setting. • Silence your cell phone and put it in your school bag (not your pocket) • Get a ruler, pencil/eraser, and calculator out • First: Take the test • Second: Check your work • Third: Hand the test & formula sheet to Mrs. Poppiti when finished • Fourth: Check last night’s homework • If time remains, we’ll start Chapter 2 together! Homework: Problems 2-8 to 2-12 Day 10: February th 12 Objective: Learn how to name angles, and learn the three main relationships for angle measures, namely supplementary, complementary, and congruent. Also, discover a property of vertical angles. HW Check/Correct in RED Problems 2-1 to 2-7 Circle Activity (10 minutes) Homework: Finish Problems 2-2 to 2-6 AND Do 2-18 to 2-22 2-2 A C’ a. B B’ C b. c. 6 Day 11: February th 16 Objective: Use our understanding of translation to determine that when a transversal intersects parallel lines, a relationship exists between corresponding angles. Also, continue to practice using angle relationships to solve for unknown angles. THEN Continue to apply knowledge of corresponding angles, and develop conjectures about alternate interior and same-side interior angles. Also, learn about how light reflects off a mirror. HW Check/Correct in RED Review Problems 2-4 to 2-7 as necessary LL Notes: Notation for Angles & Angle Types Problems 2-13 to 2-17 Start Problems 2-23 to 2-28 Homework: Problems 2-29 to 2-33 Notation for Angles F E D Name or Measure Correct: If there is only one angle at the vertex, you can also name the angle using the vertex: Y W X Z ? Incorrect: ? Incorrect: Angle Types (pg 76) Complementary Angles: Two angles that have measures that add up to 90°. 30° 60° x° y° x° + y° = 90° Supplementary Angles: Two angles that have measures that add up to 180°. Example: Straight angle Congruent Angles: Two angles that have measures that are equal. Example: Vertical angles 70° 110° x° y° x° + y° = 180° 85° 85° x° y° x° = y° Marcos’ Tile Pattern How can you create a tile pattern with a single parallelogram? Marcos’ Tile Pattern a. Are opposite angles of a parallelogram congruent? Pick one parallelogram on your paper. Use color to show which angles have equal measure. If two measures are not equal, make sure they are different colors. Marcos’ Tile Pattern b. What does this mean in terms of the angles in our pattern? Color all angles that must be equal the same color. Marcos’ Tile Pattern c. Are any lines parallel in the pattern? Mark all lines on your diagram with the same number of arrows to show which lines are parallel. Marcos’ Tile Pattern J a L c w N y b M d x P z K Use the following diagram to help answer question 2-15. Day 12: February th 17 Objective: Continue to apply knowledge of corresponding angles, and develop conjectures about alternate interior and same-side interior angles. Also, learn about how light reflects off a mirror. THEN Discover the triangle angle sum theorem, and practice finding angles in complex diagrams that use multiple relationships. HW Check/Correct in RED Finish Problems 2-15 to 2-17 Problems 2-23 to 2-28 Start Problems 2-34 to 2-37 Homework: Problems 2-38 to 2-42 2-16 X X Why Parallel Lines? 53° x 2-23 (a) a b More angles formed by transversals 132° 48° 48° 132° 132° 48° 48° 132° > > a. Alternateb.Interior(1) Same Side (2) Interior (3) Angles formed by parallel lines and a transversal Corresponding - Congruent b a > > > 100° a=b Alternate Interior - Congruent b a > > a=b 100° > 22° 22° > > Same-Side Interior - Supplementary b a > > a + b = 180° 60° 120° > > Day 13: February th 18 Objective: Discover the triangle angle sum theorem, and practice finding angles in complex diagrams that use multiple relationships. THEN Learn the converses of some angle conjectures. Also, apply knowledge of angle relationships to analyze the hinged mirror trick from Lesson 2.1.1. HW Check/Correct in RED & Complete Warm-Up (Next Slide) Problems 2-34 to 2-37 Problems 2-43 to 2-50 Homework: Problems 2-51 to 2-55 AND 2-61 to 2-65 Day 13: Try This! Name the relationship between these pairs of angles: 1. 2. 3. 4. 5. b and d a and x d and w c and w x and y b c a d Possible Choices: w x z y Vertical Angles Straight Angle Alternate-Interior Angles Corresponding Angles Same-side Interior Angles Triangle Angle Sum Theorem The measures of the angles in a triangle add up to 180°. Example: B 45° A 65° 70° C 2-37: Challenge! f g h k m p m 57° 123° h k57° 123° 99° p 81° q g 99° 81° f q r s u v 42° s r 81° 57° v 57° u 123° 2-43 and 2-44 > x y > 2-43 and 2-44 A 100° C B E 80° D 2-43 and 2-44 > 112° 68° > 2-45 80° > 100° 80° > 80 100° ° 80° > > If Same-Side Interior angles are supplementary, then the lines must be parallel. If Corresponding angles are congruent, then the lines must be parallel. If Alternate Interior angles are congruent, then the lines must be parallel. Day 14: February th 19 Objective: Learn how to find the area of a triangle, and develop multiple methods to find the area of composite shapes formed by rectangles and triangles. THEN Use rectangles and triangles to develop algorithms to find the area of new shapes, including parallelograms and trapezoids. HW Check/Correct in RED & Complete Warm-Up (Handout) Review Chapter 1 Individual Test Problems 2-66 to 2-68 Problems 2-75 to 2-79 Homework: Problems 2-70 to 2-74 AND …? Area of a Triangle What is the area of the right triangle below? Why? 4 cm 10 cm What about non-right triangles? Height Height Where is the height? Base? Height Base Base Base Height Obtuse Triangle Extra Base Area of Obtuse Triangle = Area of Right Triangle = ½ (Base)(Height) Area of a Triangle The area of a triangle is one half the base times the height. Base Base Height Height Height 1 A bh 2 Base Can we find the area? YES! YES! YES! YES! YES! YES! YES! YES! Day 15: February nd 22 Objective: Use rectangles and triangles to develop algorithms to find the area of new shapes, including parallelograms and trapezoids. THEN Explore how to find the height of a triangle given that one side has been specified as the base. Also, find the areas of composite shapes using what has been learned about the areas of triangles, parallelograms, and trapezoids. HW Check/Correct in RED & Complete Warm-Up (Handout) Finish Problems 2-75 to 2-79 Problems 2-86 to 2-89 Homework: Problems 2-81 to 2-85 AND 2-90 to 2-94 Area of a Parallelogram h h Height Base b h h Area Rectangle! = b.h Area of a Parallelogram h b . Area = b h Area of a Parallelogram The area of a parallelogram is the base times the height. . Area = b h h b 20 Ex: 13 5 20 13 A = 20.5 = 100 Area of a Trapezoid b21 b1 h Base One h Height b2 b2 h b21 Base Two b1 Parallelogram! Duplicate Translate Reflect Area = (b1 + b2) h Area of a Trapezoid b1 h b2 1 Area = b1 b2 h 2 Area of a Trapezoid The area of a trapezoid is half of the sum of the bases times the height. b1 1 Area = b1 b2 h 2 h b2 Ex: 9 5 5 4 15 A = ½ (9+15) 4 = ½ . 24 . 4 = 48 Answers to 2-79 a. 0.5(16)9 = 72 sq. un b. 26(14) = 364 sq. un c. 11(11) = 121 sq. un d. 0.5(6+21)8 = 108 sq. un Notecard = Height Locator Base “Weight” Day 16: February rd 23 Objective: Review the meaning of square root. Recognize how a square can help find the length of a hypotenuse of a right triangle. THEN Learn how to determine whether or not three given lengths can make a triangle. Also, understand how to find the maximum and minimum lengths of a third side given the lengths of the other two sides. HW Check/Correct in RED FIX 2-88 and Finish! Problems 2-95 to 2-99 Problems 2-105 to 2-108 Homework: Problems 2-100 to 2-104 AND 2-109 to 2-113 Ch. 2 Team Test Tomorrow Ch. 2 Individual Test Friday Triangle Inequality Theorem Longest Side: Slightly less than the sum of the two shorter sides Shortest Side: Slightly more than the difference of the two longer sides Triangle Inequality Theorem Each side must be shorter than the sum of the lengths of the other two sides and longer than the difference of the other two sides. b a c |a – b| < c < a + b |a – c| < b < a + c |b – c|< a < b + c Day 17: February th 24 Objective: Develop and prove the Pythagorean Theorem. THEN Assess Chapter 2 in a team setting. HW Check/Correct in RED Problems 2-114 to 2-117 If time: Mini Review Lesson on Simplifying Radicals Chapter 2 Team Test Homework: Problems 2-118 to 2-122 AND Review WS Ch. 2 Individual Test Friday The Pythagorean Theorem a a c b a c c a b 2 a +b =c 2 2 b c b c a b 2 c a c b b c 2 a a b c a b 2 Pythagorean Theorem Leg B a C 2 b 2 2 a +b =c c A Leg When to use it: • If you have a right triangle • You need to solve for a side length • If two side lengths are known Day 18: February th 25 Objective: Review concepts from Chapters 1-2 in preparation for the individual test. HW Check/Correct in RED Review Chapter 2 Team Test Mini Lesson/Review: Simplifying Radicals + Practice Calculate EXACT perimeters of shapes in Problem 2-90 Pythagorean Theorem slide (tough type) Extra Practice Options: Areas, Mixed Review, etc? Homework: Problems CL2-123 to CL2-131 AND STUDY!!!!!! Don’t know how to study? See me for ideas! Ch. 2 Individual Test Tomorrow Pythagorean Theorem Example Solve for the length of the missing side: 3 2 a 2 b2 c2 x 6 5 3 2 2 x 6 5 2 2 18 x 2 180 x 2 162 x 162 x 81 2 x 9 2 un. Day 19: February th 26 Objective: Assess Chapters 1-2 in an individual setting. • Silence your cell phone and put it in your school bag (not your pocket) • Get a ruler, pencil/eraser, and calculator out • First: Take the test • Second: Check your work • Third: Hand the test & formula sheet to Mrs. Poppiti when finished • Fourth: Check last night’s homework • If time remains, we’ll start Chapter 3 together! Homework: Worksheet – Pythagorean Theorem & Its Converse Day 20: February th 29 Objective: Learn the concept of similarity and investigate the characteristics that figures share if they have the same shape. Determine that two geometric figures must have equal angles to have the same shape. Additionally, introduce the idea that similar shapes have proportional corresponding side lengths. THEN Determine that multiplying (and dividing) lengths of shapes by a common number (zoom factor) produces a similar shape. Use the equivalent ratios to find missing lengths in similar figures and learn about congruent shapes. CIRCLE THEN *** NEW SEATS *** HW Check/Correct in RED Problems 3-1 to 3-5 Problems 3-10 to 3-14 Homework: Problems 3-6 to 3-9 AND 3-17 to 3-21 Dilation A transformation that shrinks or stretches a shape proportionally in all directions. Enlarging 3-10 Similar Figures Exactly same shape but not necessarily same size • Corresponding angles are congruent • The ratios between corresponding sides are equal 21 127° 7 5 127° 90° 15 90° 12 4 53° 90° 10 53° 90° 30 Day 21: March st 1 Objective: Examine the ratio of the perimeters of similar figures, and practice setting up and solving equations to solve proportional problems. THEN Apply proportional reasoning and learn how to write similarity statements. HW Check/Correct in RED & Complete Warm-Up (Next Slide) Check/Review Problems 3-10 to 3-14 Problems 3-22 to 3-26 Problems 3-32 to 3-37 Review Chapter 2 Individual Test Homework: Problems 3-27 to 3-31 AND 3-38 to 3-42 Day 21: Try This! 1. If Rob has three straws of different lengths: 4 cm, 9 cm, and 6 cm. Will he be able to make a triangular picture frame out of the straws? Why or why not? 2. Find the area of the following shapes: 20 ft 28 ft 40 ft 7 ft 10 ft 3 ft 10 ft Zoom Factor The number each side is multiplied by to enlarge or reduce the figure x2 x2 Example: 18 4 9 12 x2 24 Zoom Factor = 2 8 George Washington’s Nose 720 in 60 ft in ? ft ? ft in ft ? in Day 22: March nd 2 Objective: Apply proportional reasoning and learn how to write similarity statements. THEN Learn the SSS~ and AA~ conjectures for determining triangle similarity. THEN Learn how to use flowcharts to organize arguments for triangle similarity, and continue to practice applying the AA~ and SSS~ conjectures. HW Check/Correct in RED & Finish 3-34 to 3-37 Check/Review Problems 3-34 to 3-37 LL Notes to summarize Problems 3-43 to 3-47 Start Problems 3-53 to 3-58 Homework: Problems 3-48 to 3-52 & Radicals WS 1-11 ODDS Chapter 3 Team Test Tuesday Chapter 3 Individual Test Thursday Notation Angle ABC Line Segment XY The Measure of Angle ABC The Length of line segment XY Notation Acceptable KT GB Not Acceptable KT GB Writing a Similarity Statement Example: ΔDEF~ΔRST The order of the letters determines which sides and angles correspond. B Z C Y A ΔABC ABC ~ ΔZ XY X Writing a Proportion B s C W 13 A 25 X 10 D Z AB ABCD WXYZ BC ~ WX XY WX AB = XY BC 25 13 = s 10 Y First Two Similarity Conjectures SSS Triangle Similarity (SSS~) If all three pairs of corresponding side lengths share a common ratio, then the triangles are similar. AA Triangle Similarity (AA~) If two pairs of corresponding angles have equal measure, then the triangles are similar. Day 23: March rd 3 Objective: Learn how to use flowcharts to organize arguments for triangle similarity, and continue to practice applying the AA~ and SSS~ conjectures. THEN Practice making and using flowcharts in more challenging reasoning contexts. Also, determine the relationship between two triangles if the common ratio between the lengths of their corresponding sides is 1. x 3 = HW Check/Correct in RED & Solve: 4 x +1 Problems 3-53 to 3-58 Problems 3-64 to 3-67 Homework: Problems 3-59 to 3-63 & 3-68 to 3-72 Chapter 3 Team Test Tuesday Chapter 3 Individual Test Thursday 3-54 T D 3 C 16 4 2 F 12 Q 8 R Conjecture will we use: SSS~ Facts 12 4 3 8 4 2 16 4 4 Conclusion ΔCDF ~ ΔRTQ SSS~ Another Example Y B 100° 100° A 60° C 60° X Z What Conjecture will we use: AA~ Facts mB mY mA mZ Conclusion ΔABC ~ ΔZYX AA~ Similarity and Sides The following is not acceptable notation: AB ~ CD OR AB CD Acceptable: AB CD OR AB CD Day 24: March th 4 Objective: Practice making and using flowcharts in more challenging reasoning contexts. Also, determine the relationship between two triangles if the common ratio between the lengths of their corresponding sides is 1. THEN Complete the list of triangle similarity conjectures involving sides and angles, learning about the SAS~ Conjecture in the process. HW Check/Correct in RED QUICKLY! Finish Problems 3-65 to 3-67 Problems 3-73 to 3-77 Homework: Problems 3-78 to 3-82; Finish ODDS on Add/Subtract Radicals WS Chapter 3 Team Test Tuesday Chapter 3 Individual Test Thursday Day 25: March th 7 Objective: Complete the list of triangle similarity conjectures involving sides and angles, learning about the SAS~ Conjecture in the process. THEN Practice using the three triangle similarity conjectures and organizing reasoning in a flowchart. HW Check/Correct in RED QUICKLY & Complete Warm-Ups Problems 3-73 to 3-77 Problems 3-83 to 3-86 Homework: Problems 3-88 to 3-92 AND Pages 1-2 in Ch. 3 Review Packet Chapter 3 Team Test Tomorrow Chapter 3 Individual Test Thursday Conditions for Triangle Similarity If you are testing for similarity, you can use the following conjectures: SSS~ All three corresponding side lengths have the same zoom factor AA~ Two pairs of corresponding angles have equal measures. 14 6 7 3 10 5 55° 40° 40° 55° SAS~ Two pairs of corresponding lengths have the same zoom factor and the angles between the sides have equal measure. 70° NO CONJECTURE FOR ASS~ 3 0 40 20 70° 15 Day 26: March th 8 Objective: Assess Chapter 3 in a team setting. THEN Apply knowledge of similar triangles to multiple contexts. HW Check/Correct in RED QUICKLY & Complete Warm-Up Chapter 3 Team Test Problems 3-94 to 3-95 Homework: Problems 3-96 to 3-100 AND Finish Ch. 3 Review Packet Chapter 3 Individual Test Thursday Day 27: March th 9 Objective: Apply knowledge of similar triangles to multiple contexts. THEN Review concepts in preparation for tomorrow’s assessment. HW Check/Correct in RED QUICKLY & Complete Warm-Ups Review Chapter 3 Team Test Tough Trapezoids Practice Hallway Relay Problems 3-94 to 3-95 Carousel Cards: Proof/Flowchart Practice? Homework: Problems CL3-102 to 3-110 (Skip 106) Chapter 3 Individual Test Tomorrow – STUDY!!! You’re Getting Sleepy… Eye Height Eye Height x cm 200 cm Lessons from Abroad x 316 ft 12 + 930 = 942 6–2=4 12 Day 28: March th 10 Objective: Assess Chapters 1-3 in an individual setting. • Silence your cell phone and put it in your school bag (not your pocket) • Get a ruler, pencil/eraser, and calculator out • First: Take the test • Second: Check your work • Third: Hand the test & formula sheet to Mrs. Poppiti when finished • Fourth: Check last night’s homework Homework: Problems 4-6 to 4-10 Day 29: March th 11 Objective: Recognize that all the slope triangles on a given line are similar to each other, and begin to connect a specific slope to a specific angle measurement and ratio. THEN Connect specific slope ratios to their related angles and use this information to find missing sides or angles of right triangles with 11°, 22°, 18°, or 45° angles (and their complements). ***BRIEF CIRCLE*** HW Check/Correct in RED Problems 4-1 to 4-5 Problems 4-11 to 4-15 Homework: Problems 4-16 to 4-20 AND 4-25 to 4-29 Day 30: March th 14 Objective: Use technology to generate slope ratios for new angles in order to solve for missing side lengths on triangles. THEN Practice using slope ratios to find the length of a leg of a right triangle and learn that this ratio is called tangent. Also, practice re-orienting a triangle and learn new ways to identify which leg is Δx and which is Δy. Learn how to find the slope ratio using a scientific calculator. ***BRIEF CIRCLE*** HW Check/Correct in RED & Warm-Up (Next Slide) Lesson 4.1.3: Problems 4-21 to 4-24 Problems 4-30 to 4-35 Homework: Problems 4-36 to 4-40 AND Tangent Worksheet Try This! Solve for x: x 68° 25 un Day 31: March th 15 Objective: Practice using slope ratios to find the length of a leg of a right triangle and learn that this ratio is called tangent. Also, practice reorienting a triangle and learn new ways to identify which leg is Δx and which is Δy. Learn how to find the slope ratio using a scientific calculator. THEN Apply knowledge of tangent ratios to find measurements about the classroom. THEN Learn how to list outcomes systematically and organize outcomes in a tree diagram. HW Check/Correct in RED Finish Problems 4-33 to 4-35, 4-36 Problems 4-41 to 4-42 Problems 4-48 to 4-53 Homework: Problems 4-43 to 4-47 AND 4-54 to 4-58 AND Front side of Tangent Worksheet Trigonometry Theta ( ) is always an acute angle Opposite (across from the known angle) Δy h Hypotenuse (across from the 90° angle) Δx Adjacent (forms the known angle) Trigonometry (LL) Opposite Theta ( ) is always an acute angle h Adjacent Trigonometry (LL) Adjacent Theta ( ) is always an acute angle h Opposite When to use Trigonometry 1. You have a right triangle and… 2. You need to solve for a side and… 3. A side and an acute angle are known Use Trigonometry Day 32: March th 16 Objective: Learn how to list outcomes systematically and organize outcomes in a tree diagram. THEN Continue to use tree diagrams and also introduce a table to analyze probability problems. Also, investigate the difference between theoretical and experimental probability. THEN Learn how to use an area model (and a generic area model) to represent a situation of chance. HW Check/Correct in RED Review Ch. 3 Individual Test Finish Problems 4-50 to 4-53 Problems 4-59 to 4-62 Start Problems 4-68 to 4-70 Homework: Problems 4-63 to 4-67 AND 4-72 to 4-76 My Tree Diagram Read Write S T A R T #41 #28 #55 #81 Listen Read Write Listen Read Write Listen Read Write Listen One Possibility: Take Bus #41 and Listen to an MP3 player 4-60: Tree Diagram S T A R T $100 $300 Keep $100 Double $200 Keep $300 Double $600 Keep $1500 Double $3000 $1500 Day 33: March th 17 Objective: Learn how to use an area model (and a generic area model) to represent a situation of chance. HW Check/Correct in RED Warm-Up – Complete individually and hand-in when done Problems 4-68 to 4-70 Problems 4-77 to 4-81 Homework: Problems 4-82 to 4-86 AND 4-91 to 4-95 4-77: Area Model Spinner #1 Spinner #2 I T F U A 1 2 1 6 3 4 IT UT AT 1 1 24 1 12 4 IF UF AF 3 3 3 12 1 3 8 8 24 1 Day 34: March th 18 Objective: Learn how to use an area model (and a generic area model) to represent a situation of chance. THEN Learn about the sine and cosine ratios. Also, start a Triangle Toolkit. HW Check/Correct in RED & Calculate EXACT area and perimeter for each shape drawn on the board Problem 4-80 Problems 5-1 to 5-6 Homework: Chapter 4 Review Sheet Optional: Extra study items (2 packets) Chapter 4 Team Test Wednesday Chapter 4 Individual Test Friday Day 35: March th 29 Objective: Learn about the sine and cosine ratios. Also, start a Triangle Toolkit. THEN Develop strategies to recognize which trigonometric ratio to use based on the relative position of the reference angle and the given sides involved. HW Check/Correct in RED Answer questions from Chapter 4 Review efforts Problem 5-6 (Toolkit Entry) Problems 5-12 to 5-15 Homework: Problems 5-7 to 5-12 AND 5-16 to 5-20 Chapter 4 Team Test Wednesday Chapter 4 Individual Test Friday Day 36: March th 30 Objective: Assess Chapter 4 in a team setting. THEN Understand how to use trigonometric ratios to find the unknown angle measures of a right triangle. Also, introduce the concept of “inverse.” HW Check/Correct in RED Chapter 4 Team Test Problems 5-21 to 5-25 Homework: Problems 5-26 to 5-30 Chapter 4 Individual Test Friday Chapter 5 Team Test Next Thursday (?) Day 37: March st 31 Objective: Understand how to use trigonometric ratios to find the unknown angle measures of a right triangle. Also, introduce the concept of “inverse.” THEN Use sine, cosine, and tangent ratios to solve real world application problems. HW Check/Correct in RED Review Chapter 4 Team Test Problems 5-21 to 5-25 Problems 5-31 to 5-35 Homework: Problems 5-36 to 5-40 AND STUDY! Chapter 4 Individual Test Tomorrow Chapter 5 Team Test Next Thursday (?) Day 38: April st 1 Objective: Assess Chapters 1-4 in an individual setting. • Silence your cell phone and put it in your school bag (not your pocket) • Get a ruler, pencil/eraser, and calculator out • First: Take the test • Second: Check your work • Third: Hand the test & formula sheet to Mrs. Poppiti when finished • Fourth: Check last night’s homework Homework: Problems 5-33(c) to 5-35 AND Trig/Inv Trig WS Chapter 5 Team Test Next Thursday (?) Ch. 5 Individual Test IS the midterm… Day 39: April th 4 Objective: Use sine, cosine, and tangent ratios to solve real world application problems. THEN Recognize the similarity ratios in 30°60°-90° and 45°-45°-90° triangles and begin to apply those ratios as a shortcut to finding missing side lengths. NEW SEATS!! HW Check/Correct in RED & Complete Warm-Ups (On Board) Review Problems 5-33 to 5-35 LL Entry: Trig Memory Tool Review Ch. 4 Individual Test Problems 5-41 to 5-44 Notes (Toolkit and Learning Log Entries) Homework: Problems 5-46 to 5-50 Chapter 5 Team Test Thursday (?) Trigonometry h o a SohCahToa opposite o sin( ) hypotenuse h adjacent a cos( ) hypotenuse h opposite o tan( ) adjacent a 30° - 60° - 90° A 30° – 60° – 90° is half of an equilateral (three equal sides) triangle. 30° s 60° 0.5s s You can use this whenever a problem has an equilateral triangle! Long Leg (LL) 30° - 60° - 90° 30° 60° Short Leg (SL) 30° - 60° - 90° Remember √3 because there are 3 different angles You MUST know SL first! √3 30° ÷2 1 ÷√3 SL LL x√3 x2 Hyp 2 60° Isosceles Right Triangle 45° - 45° - 90° Remember √2 because 2 angles are the same 45° √2 1 45° ÷√2 Leg(s) 1 Hypotenuse x√2 Isosceles Right Triangle 45° - 45° - 90° A 45° – 45° – 90° triangle is half of a square. 45° s You can use this whenever a problem has a square with its diagonal! d 45° s Pythagorean Triple A Pythagorean triple consists of three positive integers a, b, and c (where c is the greatest) such that: a2 + b2 = c2 Common examples are: 3, 4, 5 ; 5, 12, 13 ; and 7, 24, 25 Multiples of those examples work too: 3, 4, 5 ; 6, 8, 10 ; and 9, 12, 15