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PUSD Math News – Mathematics 1 Module 8: Connecting Algebra to Geometry Module 8 Overview – Connecting Algebra to Geometry (Standards: F.IF.9, F.BF.1, F.BF.3, G.GPE.4, G.GPE.5, G.GPE.7) Student and Teacher materials can be found at Mathematics Vision Project http://www.mathematicsvisionproject.org/ (Curriculum>Secondary Mathematics One>Module 8: Connecting Algebra to Geometry) Module 8 makes explicit the bridge between algebra and geometry as it uses the tools and definitions of each to connect the two. This is done through the exploration of coordinate geometry and the development of compound functions that connect two or more existing functions to produce something new. Students now have all the tools to discuss the functions they have studied algebraically, graphically, numerically, in tables and verbally. The distance formula is introduced as it is used along with all they have learned in the previous units to perform coordinate proofs. Students continue to gain proficiency in the Standards for Mathematical Practices. The predominant SMP is MP7 (look for and make use of structure). The structure of mathematics is laid open for students through the connections they are making between all of the different representations of functions and the properties of each. They recognize the connections between algebra and geometry and shift their perspective in order to gain new understanding. They see complex algebraic expressions as being composed of simpler expressions, each of which contributes to the whole expression and determines its values. When this is expressed as a function they recognize that each part of the expression has an effect of the graph of the whole. Scan the QR code below to take you directly to the PUSD Secondary Math Resources webpage for Mathematics 1, Module 8: Connecting Algebra to Geometry You will find the student text, newsletter, standards for the module, homework help links and more! https://goo.gl/r8SFWI Vocabulary and Major Mathematical Concepts Sections designated with the letter H will appear in the Honors Mathematics 1 course only Prerequisite Concepts and Skills: Apply the Pythagorean Theorem Graph linear and exponential functions Write linear equations in standard, slope-intercept, and point-slope form Identify/solve for slope and x- and yintercepts of linear functions Solve multi-step equations Identify basic geometric shapes and characteristics Use function notation PUSD Math News – Mathematics 1 Module 8: Connecting Algebra to Geometry Additive identity – the value that when added to any number will result in the number itself, that value is zero. Component form of a vector – a notation for writing a vector that includes the horizontal and vertical components of movement. Component form is written as <a, b>, where a represents the horizontal distance and b represents the vertical distance. Example: Image- http://www.printable-math-worksheets.com/additiveidentity.html Additive inverse – the value that when added to a number will result in zero, opposites add to zero. Image- https://quizlet.com/9167697/mth-277-chapter-11-section-1flash-cards/ Image- http://www.knowmia.com/watch/lesson/20369 Associative property – states that when adding or multiplying, grouping can be done in any order. Determinant – a function which as an input accepts n x n matrix and whose output is a real or a complex number that is called the determinant of the input matrix. An example for a 2 x 2 matrix is given below. Example: Image- http://www.printable-math-worksheets.com/associativeproperty.html Commutative property – states numbers can be added or multiplied in any order. Imagehttp://www.statistica.com.au/finding_determinant_of_2_x_2_m.html Directed line segment – an arrow that is drawn to represent a vector. Direction – when examining vectors direction is in reference to the change in location from preimage to image (left, right up, down, left and down, right and up, etc). Image- http://www.mathsisfun.com/definitions/commutative-law.html PUSD Math News – Mathematics 1 Module 8: Connecting Algebra to Geometry Distance formula – given two points (x1, y1) and (x2, y2), the distance between the two points is given by the formula d = (x2 - x1 )2 + (y2 - y1 )2 . Magnitude – when examining vectors magnitude refers to the distance an object moves from preimage location to image location. Matrix multiplication – Imagehttp://www.teacherschoice.com.au/maths_library/analytical%20geomet ry/alg_15.htm Distributive property – the rule that multiplying a(b + c) gives the same result as ab + ac. Image- http://functionspace.com/topic/1007/Algorithm-formultiplication-of-two-square-matrix Multiplicative identity – the value that a number could multiply by and have the result be the number itself, that value is 1. Image- http://www.printable-math-worksheets.com/multiplicativeidentity.html Multiplicative inverse – the value that when multiplied by a number will result in 1. Reciprocals when multiplied together will result in 1. Image- http://nc5thgrademath.weebly.com/distributive-property-ofmultiplication.html Hypotenuse – the longest side of a right triangle, it is the side opposite the right angle. Kite – a quadrilateral that has two pair of congruent adjacent sides. Examples: Image- http://slideplayer.com/slide/2557575/ Parallel – a description of two or more coplanar lines or line segments that have the same slope. Example: Image- http://www.ck12.org/book/CK-12-GeometryConcepts/r2/section/6.7/Kites---Intermediate/ Image- https://new.edu/resources/parallel-and-perpendicular-lines PUSD Math News – Mathematics 1 Module 8: Connecting Algebra to Geometry Perpendicular – a description of two lines or line segments that intersect at a 90-degree angle. Their slopes are opposite (negative) reciprocals of another. Example: Scalar multiplication – multiplying a vector by a scale factor. This changes the magnitude of the vector, but not the direction. For example multiplying a vector by 3 represents a vector that has the same direction as the original, but whose magnitude is three times the original. Slope triangle – a visual tool that helps to find the slope of a line by drawing in the vertical and horizontal distance between two points. Image- https://new.edu/resources/parallel-and-perpendicular-lines Reciprocal – a pair of numbers whose product is one. A number and its multiplicative inverse are reciprocals of one another. Image- http://www.math.com/school/subject1/lessons/S1U4L1DP.html Resultant vector – the vector that results when two or more vectors are added together. Example: Image- http://study.com/academy/lesson/what-is-slope-intercept-formdefinition-equation-examples.html Square matrix – a matrix with the same number of rows and columns. An n-by-n matrix is a square matrix of order n. Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations. Translation form equation – an equation written in a form so that the vertical and/or horizontal translation(s) from pre-image to image can be seen in the equation. For example if f(x) is the pre-image and g(x) is the image that has been translated up 3 units, then g(x)=f(x)+3 is the translation form equation for that transformation. Vector – a quantity that has both magnitude and direction. Usually represented by an arrow or directed line segment. Image- http://calculator.tutorvista.com/resultant-vectorcalculator.html# PUSD Math News – Mathematics 1 Module 8: Connecting Algebra to Geometry Main Topics Sections designated with the letter H will appear in the Honors Mathematics 1 course only Section in student text – Task done in class Related Homework Help Videos 8.4 – Write the equation f(t)=m(t)+k by comparing parallel lines and finding k Function transformations https://goo.gl/B41xKq 8.1 – Use coordinates to find distances and determine the perimeter of geometric shapes 8.5 – Determine the transformation from one function to another Slope triangles https://goo.gl/DTmfTb Describing statistical distributions https://goo.gl/tkpfnO Distance between points https://goo.gl/pAzo73 Spread of distribution https://goo.gl/raLZ48 8.2 – Prove slope criteria for parallel and perpendicular lines 8.6 – Translating linear and exponential functions using multiple representations Writing equations of lines in point-slope form https://goo.gl/Uiv0u1 Graphing exponential functions https://goo.gl/tLIzdW Writing equations of lines in standard form https://goo.gl/CEus7k Proving relationships using a coordinate grid https://goo.gl/9rnJbx Slopes of parallel and perpendicular lines https://goo.gl/CnPRJa 8.3 – Use coordinates to algebraically prove geometric theorems Perimeters of geometric shapes https://goo.gl/kfsJaF Properties of quadrilaterals https://goo.gl/9kkclQ 8.7H – Defining and operating with vectors as quantities with magnitude and direction Arithmetic of vectors https://goo.gl/PGOJMg Solving equations https://goo.gl/WKQ63B 8.8H – Examining properties of matrix addition and multiplication, including inverse properties Properties of matrix arithmetic https://goo.gl/2IMIaQ Solving systems of equations https://goo.gl/9J3EwK PUSD Math News – Mathematics 1 Module 8: Connecting Algebra to Geometry 8.9H – Finding the determinant of a matrix and relating it to the area of a parallelogram Determinant of a matrix https://goo.gl/RXDRnS Solving systems with matrices http://goo.gl/fgjDQr 8.10H – Solving a system of linear equations using the multiplicative inverse matrix Solving systems with inverse matrices https://goo.gl/iqnzNi Properties of arithmetic https://goo.gl/BDL2kC Finding inverse matrices of 3x3 matrices https://goo.gl/L1ZVFm Solving 3x3 systems using inverse matrices https://goo.gl/yJhgJM 8.11H – Using matrix multiplication to reflect and rotate vectors and images Transformation using matrices https://goo.gl/iYHlU7 Function transformations with tables https://goo.gl/yUugcY 8.12H – Solving problems involving quantities that can be represented by vectors Scatterplots and trend lines https://goo.gl/AM97s2