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PUSD Math News – Mathematics 2 Module 6: Similarity and Right Triangle Trigonometry Module 6 Overview – Similarity and Right Triangle Trigonometry (Standards: G.SRT.1, G.SRT.2, G.SRT.3, G.SRT.4, G.SRT.5, G.SRT.6, G.SRT.7, G.SRT.8, G.CO.9, G.CO.10, G.GPE.5, G.GPE.6, F.TF.8) Student and Teacher materials can be found at Mathematics Vision Project http://www.mathematicsvisionproject.org/ (Curriculum>Secondary Mathematics Two>Module 6: Similarity and Right Triangle Trigonometry) In Module 6 students develop and formalize the definition of dilations and proportional reasoning. With dilations and proportions in mind, a formal development of similarity follows. They then identify whether figures are similar and use the proportions within similar figures to solve problems. Students analyze relationships formed by transversals and parallel lines by identifying and using the similar triangles. Then students narrow their focus to derive the location of a point between two points at a given proportion and to use proportions of similar triangles to prove the Pythagorean Theorem. Finally, similarity is used to develop an understanding of sine, cosine, and tangent leading to a study of trigonometry. They learn how to find unknown angles and sides, and how to solve a right triangle. Students continue to gain proficiency in the Standards for Mathematical Practice. Module 6 includes an extensive use of MP8, look for and express regularity in repeated reasoning, as student repeatedly go back to the concepts of similarity learned at the beginning of the unit and use them to develop each new concept. Students also focus on MP7, look for and make use of structure, as they continually use proportions throughout the unit. MP3, construct viable arguments and critique the reasoning of others, is used in several of the tasks as students are required to make or examine conjectures and then verify them. The trigonometry at the end requires MP4, Model with mathematics, as students use trigonometry to find measures in real world contexts. Scan the QR code below to take you directly to the PUSD Secondary Math Resources webpage for Mathematics 2, Module 6: Similarity and Right Triangle Trigonometry You will find the student text, newsletter, standards for the module, homework help video links and more! https://goo.gl/1D5T4e Vocabulary and Major Mathematical Concepts Note: Section numbers followed by an H will be addressed in the Honors Mathematics 2 course. Prerequisite Concepts and Skills: Angle relationships with parallel lines Transformations (Translations, rotations, and reflections) Pythagorean Theorem Slope Sum of the angles of a triangle Solving equations Congruent triangles Arithmetic mean Geometric mean PUSD Math News – Mathematics 2 Module 6: Similarity and Right Triangle Trigonometry AA Similarity postulate (6.3) – if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Example: Image A- http://www.mathwords.com/a/aa_similarity.htm Angle of depression (6.10) – from a horizontal, look down, the angle from the horizontal to your line of sight is called the angle of depression (see Image B). Angle of elevation (6.10) – from a horizontal, look up, the angle from the horizontal to your line of sight is called the angle of elevation (see Image B). Example: Image Bhttp://bscstudent.buffalostate.edu/kajfkr79/web/Masters_Project/Angle s_of_Elevaation_and_Depression.html Arithmetic Mean (6.7) – the arithmetic average of a set of numerical values, calculated by adding them together and dividing by the number of terms in the set. In section 6.7 this process may be used to find missing terms in a sequence. Center of dilation (6.1) – a fixed point in the plane about which all points are expanded or contracted. A dilation stretches or shrinks the original figure according to a scale factor and the center of dilation (see point O in Image D). Congruent (6.3) – two figures are congruent if they have exactly the same size and shape. Angles and line segments are congruent if they have the exact same measurement. Cosine (6.9) – the trigonometric function that is equal to the ratio of the length of a side adjacent to an acute angle (in a right triangle) to the length of the hypotenuse (see Image J). Dilation (6.1-6.3) – a transformation of the plane that produces an image that is the same shape as the original, but is a different size. If O is the center of the dilation and a non-zero number k is the scale factor, then A’ is the image of point A if O, A and A’ are collinear and Example: 𝐴′ 𝐴 =k Angle of reference (6.8) – the angle of a right triangle used to determine the trigonometric ratios sine, cosine and tangent. The example below uses angle A as the angle of reference. If Angle C is the angle of reference opposite and adjacent sides would be reversed. Example: Image Dhttp://www.regentsprep.org/regents/math/geometry/gt3/ldilate2.htm Image C PUSD Math News – Mathematics 2 Module 6: Similarity and Right Triangle Trigonometry Geometric mean (6.7) – the nth root of the product of n numbers. For example, the geometric mean of 2, 4 and 8 is the cube root of 64 or 4. In section 6.7 this process may be used to find missing terms in a sequence. Hypotenuse (6.8) – the longest side of a right triangle, it is the side opposite the right angle (see Image C). Proportional (6.1) – when two ratios are equivalent. In this module corresponding sides of similar shapes are proportional, which means each set of corresponding sides has the same ratio. Example: Image (6.1) – the final shape and position of an object after a transformation is performed on a pre-image (see triangle A’B’C’ in Image D). Inverse trigonometric functions (6.10) – in this module, inverse trigonometric functions (sin-1, cos-1 and tan-1) are used to find a missing angle when two sides of a right triangle are given (see video link Inverse trigonometric functions on a graphing calculator at the end of this newsletter). Example: Image F- http://math-problems.math4teaching.com/theorems-onproportional-sides-of-triangles/ Proportionality statement (6.2) – an equation that equates two ratios. Example: In the diagram, ∆BTW ~ ∆ ETC, Image E- https://www.mathsisfun.com/algebra/trig-inverse-sin-costan.html Pre-image (6.1) – the original shape and position of an object prior to a transformation (see triangle ABC in Image D). Write the statement of proportionality. 𝐸𝑇 𝑇𝐶 𝐶𝐸 = = 𝐵𝑇 𝑇𝑊 𝑊𝐵 Image G Ratio (6.1) – the quotient or comparison of two quantities (for example a and b) measured in the 𝑎 same units. In this module it is written as , but a ratio can also be written as a : b. 𝑏 PUSD Math News – Mathematics 2 Module 6: Similarity and Right Triangle Trigonometry Scale factor (6.1) – the ratio of lengths of two corresponding sides of two similar polygons. Similar figures (6.1) – Two figures are similar if one is the image of the other under a transformation from the plane into itself that multiplies all distances by the same positive scale factor, k. That is to say, one figure is a dilation of the other. Trigonometric ratios (6.8) – the ratios defined below as sine, cosine and tangent. Example: Similar polygons (6.3) – two polygons are similar if all corresponding angles are congruent and all corresponding pairs of sides are proportional. Example: Image J- http://precalc0.weebly.com/41.html Facts about Dilation Image H- http://www.dummies.com/how-to/content/how-to-identifyand-name-similar-polygons.html Sine (6.9) – the trigonometric function that is equal to the ratio of the side opposite an acute angle (in a right triangle) to the hypotenuse (see Image J). Tangent (6.9) – the trigonometric function that is equal to the ratio of the side opposite an acute angle (in a right triangle) to the side adjacent to the same acute angle see Image J). Transversal (6.4) – a line that intersects two or more lines in the same plane at different points. Example: Image Ihttps://gcps.desire2learn.com/d2l/lor/viewer/viewFile.d2lfile/6605/11 031/Angle_Pairs_print.html Lines are taken to lines, and line segments to line segments of proportional length in the ratio given by the scale factor. Angles are taken to angles of the same measure. A line not passing through the center of a dilation is taken to a parallel line, and lines passing through the center of dilation are unchanged. To describe a dilation, specify a center of dilation and a scale factor. The center of dilation is a fixed point in the plane about which all points are expanded or contracted. It is the only invariant (unchanging) point under a dilation. Dilations create similar figures – the image and pre-image are the same shape, but different sizes (unless the scale factor is 1, then the image and pre-image are congruent). PUSD Math News – Mathematics 2 Module 6: Similarity and Right Triangle Trigonometry Theorems used in this module Vertical angles are congruent. Measures of interior angles of a triangle sum to 180 degrees. When transversals cross parallel lines, alternate interior angles are congruent and corresponding angles are congruent. A line parallel to one side of a triangle divides the other two sides proportionally. If an altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the lengths of the two segments formed on the hypotenuse. If an altitude is drawn to the hypotenuse of a right triangle, the length of each leg of the right triangle is the geometric mean between the length of the hypotenuse and the length of the segment on the hypotenuse adjacent to the leg. Main Topics Note: Section numbers followed by an H will be addressed in the Honors Mathematics 2 course. Section in student text – Task done in class Related Homework Help Videos 6.1 – Describing the essential features of a dilation Comparing side lengths after dilation https://goo.gl/mavaKs 6.2 – Examining proportionality relationships in triangles that are known to be similar to each other based on dilations How to perform a dilation https://goo.gl/c6wmtD 6.3 – Comparing definitions of similarity based on dilations and relationships between corresponding sides and angles Solving proportions https://goo.gl/FrPAat 6.4 – Examining proportional relationships of segments when two transversals intersect sets of parallel lines How to find a missing measure with similar triangles https://goo.gl/WGzoTH Angles formed by parallel lines and transversals https://goo.gl/iyJm5p Figuring out angles between transversals and parallel lines https://goo.gl/SbsYag 6.5 – Applying theorems about lines, angles and proportional relationships when parallel lines are crossed by multiple transversals See video suggestion for sections 6.3 and 6.4 6.6 – Applying understanding of similar and congruent triangles to find midpoint or any point on a line segment that partitions the segment in a given ratio Finding the mean https://goo.gl/DWuXf Finding a point part way between two points https://goo.gl/EoE1Ld PUSD Math News – Mathematics 2 Module 6: Similarity and Right Triangle Trigonometry 6.7 – Using similar triangles to prove the Pythagorean theorem and theorems about geometric means in right triangles 6.10 – Solving for unknown values in right triangles using trigonometric ratios Pythagorean theorem proof using similarity https://goo.gl/ylShwW Tutorial Activity: Angles of elevation and depression https://goo.gl/hP2dCD Finding arithmetic means (missing terms) https://goo.gl/1b3RMq Using trigonometry to solve a right triangle https://goo.gl/N2jqEz Finding geometric means (missing terms) https://goo.gl/Yslbqy Inverse Trigonometric functions on a graphing calculator https://www.youtube.com/watch?v=FOKJn1wU krI 6.8 – Developing an understanding of right triangle trigonometric relationships based on similar triangles Triangle similarity and the trigonometric ratios https://goo.gl/72dCZK Finding the trigonometric ratios in a right triangle https://goo.gl/YJrUVG How to solve a quadratic by factoring (to find xintercepts) https://goo.gl/5G9a5i 6.9 – Finding relationships between the sine and cosine ratios for right triangles, including the Pythagorean identity Match trigonometric values and side ratios https://goo.gl/S2K5e6 Introduction to the Pythagorean trigonometric identity https://goo.gl/rqBE6O 6.11 – Practicing setting up and solving right triangles to model real world context How to solve a right triangle word problem with a missing angle https://goo.gl/wVbgRl