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Process Standards in the High School Mathematics Classroom Focus: Connections and Representations Michael Bolling TCTM – High School Breakout – 10.1.13 [email protected] Mathematical Connections Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study. Through the application of content and process skills, students will make connections between different areas of mathematics and between mathematics and other disciplines, especially science. Science and mathematics teachers and curriculum writers are encouraged to develop mathematics and science curricula that reinforce each other. 2 Mathematical Representations Students will represent and describe mathematical ideas, generalizations, and relationships with a variety of methods. Students will understand that representations of mathematical ideas are an essential part of learning, doing, and communicating mathematics. Students should move easily among different representations ⎯ graphical, numerical, algebraic, verbal, and physical ⎯ and recognize that representation is both a process and a product. 3 Multiplication and Area Concept of multiplication Connection to area 2 groups of 3 2x3 3 2 2x3=6 Area is 6 square units 4 Multiplication and Area Multiplying whole numbers – progression of complexity 8 12 10 23 8 x 10 8 groups of 10 5 5 Multiplication and Area Multiplying whole numbers 20 23 23 12 3 2x3=6 10 12 2 20 40 10 3 30 2 “Partial Products” 10 20 200 276 6 Multiplication and Area Connection to Algebra I x 3 ( x 3)( x 2) x · x = x2 x 2 · x = 2x 2 3 · x = 3x This will work for more than multiplying binomials! (unlike FOIL). This model is directly linked to use of algebra tiles. 2·3=6 x2 5x 6 7 Multiplication and Area x x 2 original warehouse 3 The sides of a square warehouse are increased by 2 meters and 3 meters as shown. The area of the extended warehouse is 156 m2. What was the side length of the original warehouse? New Zealand Level 1 Algebra 1 Asia-Pacific Economic Cooperation – Mathematics Assessment Database 8 Multiplication and Area 30 50 x original warehouse x The original warehouse measured 30 meters by 50 meters. The owner would like to know the smallest length by which she would need to extend each side in order to have a total area of 2500 m2. New Zealand Level 1 Algebra 1 (modified) Asia-Pacific Economic Cooperation – Mathematics Assessment Database 9 Multiple Representations • 7.12 – represent relationships with tables, graphs, rules, and words • 8.14 – make connections between any two representations (tables, graphs, rules, and words) • A.7f – make connections between and among multiple representations of functions (concrete, verbal, numeric, graphic, and algebraic) • AFDA.4 - transfer between and analyze multiple representations of functions (algebraic formulas, graphs, tables, and words) 10 Algebra I.7 AFDA.1 Algebra II.7 Relation or function? Continuity Domain/range (includes Domain/range Domain/range Zeros Zeros x- and y-intercepts x- and y-intercepts Function values for elements of the domain Function values for elements of the domain Connections among representations Connections among representations (AFDA.4) discontinuous domains/ranges) Zeros x- and y-intercepts Function values for elements of the domain Connections among representations Local/absolute max/min Local/absolute max/min Intervals of incr/decr Intervals of inc/dec End behaviors End behaviors Asymptotes Asymptotes Inverse functions Composition of functions 11 Rational Functions 12 Geometric Constructions Connections SOL 7.7 students learn properties of parallelograms, including that the diagonals of a rhombus bisect each other and are perpendicular. 13 Geometric Constructions Connections SOL 6.12 students learn to identify congruent polygons by their attributes. SOL 7.6 students demonstrate knowledge of congruent polygons when learning about similar polygons SOL G.6 students prove triangles congruent 14 Coordinate Geometry Connections SOL A.6 students determine the slope of a line SOL 8.10 students learn about the Pythagorean Theorem, a direct connection to the distance formula 15 Discussion • With which content could we do a better job of facilitating connections or using multiple representations? 16