1Unit 1: Teaching Number and Operation SenseRational NumbersConnected Math: Bits and Pieces IIUsing Rational Numbers Advanced Topics from Math Session Field of Rational Numbers Multiplicative Inverse Transition from Number to AlgebraInvestigation 7: Dividing Fractions Connections with advanced topics Concept behind dividing fractions (Prob. 7.1, Part 2, pg 154; 7.2, Part 2, pg 155; and 7.3 Follow-Up, Part 1, 3, pg 157) Application – division in context (Applications 26, pg 159) Inverse Operation – reciprocal (Extensions, Prob. 45, 46, pg 161)2Research PSSM Companion Ch. 7: Fractions and Multiplicative Reasoning Misconceptions: a/b denotes only a part-whole relationship (3/7 vs. 7/3) Fraction versus rational number Understanding – concept image versus concept definition Conceptual Scheme of operations that constitutes a person’s understandingResearch PSSM Companion Ch. 7: Fractions and Multiplicative Reasoning Measurement Schemes Ratio is at heart of measurement Multiplicative comparison where measuring unit imagined apart from thing measured Magnitude vs. Measure – magnitude independent of unit of measure Numerical equations vs. quantity equations Proportional reasoning central to understanding measurement – single vs. multiple proportionsResearch PSSM Companion Ch. 7: Fractions and Multiplicative Reasoning Multiplication Scheme Repeated addition conception supports evaluation of products but not conceptualization Thinking of 1/n as one out of n parts Leads to sense of inclusion Conceptualized Multiplication Entailing multiple proportion Multiplicative reasoning3Research PSSM Companion Ch. 7: Fractions and Multiplicative Reasoning Division Scheme Two settings Sharing or partitioning – distributing something among participants Segmenting or measuring – putting an amount into parts Operative imagery Envision result of acting prior to acting Suppress attention to the processResearch PSSM Companion Ch. 7: Fractions and Multiplicative Reasoning Fraction Scheme Collection of Objects misconception Comparison vs. Inclusion Invert and Multiply conception Rule a/b x c/d = ac/bd Generalizations must be rooted in meaning-making activities, not algorithms Symbolic rules should not be provided in place of generalization from conceptualizationTeacher Resources Fraction activity-coloring parts of shapes and ordering: Fraction Sorter http://www.shodor.org/interactivate/activities/fracsorter/index.html Fraction activity-using geometric shapes to introduce the concept of fractions http://math.rice.edu/~lanius/Patterns/ Fraction game-card game “Make One” http://mathforum.org/paths/fractions/one.game.html4Teacher Resources Adding Fractions activity http://matti.usu.edu/nlvm/nav/frames_asid_106_g_3_t_1.html Ordering and comparing fractions activity http://matti.usu.edu/nlvm/nav/frames_asid_159_g_3_t_1.html Discussion and example problems on ordering fractions http://mason.gmu.edu/~mmankus/talks/bars/disc.htmTeacher Resources Fraction Multiplication Activity: From Musser website http://www.wiley.com/college/musser/CL_0471263796_S/emanipulatives/nav/frames_asid_230_g_0_t_0.html Fraction Division Activity: From Musser website http://www.wiley.com/college/musser/CL_0471263796_S/emanipulatives/nav/frames_asid_229_g_0_t_0.html Lesson Plan - students break words into fractions to decode a message: Fraction Hunt http://www.lessonplanspage.com/MathFractionHuntPuzzleIdea46.htmTeacher Resources Lesson Plan linking fractions with jazz rhythms  http://www.pbs.org/jazz/classroom/rhythmicinnovations.htm Lesson Plan on fraction sense, equivalent fractions, comparing fractions http://www.pbs.org/teachersource/mathline/xyencounter/tg_part4_intro.shtm Short Lesson on Dividing fractions by whole numbers—includes short definition of reciprocal