# math 221 # fall 1999 Monday, September 6 Note: Labor Day Note: No Class Tuesday, September 7 Topic: Introduction Act: A1: 3-Digit Sums @ handouts/activities-a.txt Read: Syllabus @ home.shtml Note: Note: Class Meets 10:55-11:20 Group formation & introductions Introduction to course Look at 3-digit sums. Tell students to begin when they arive next time Thursday, September 9 Topic: Teaching Math, Topic: NCTM Standards Act: A1: Sums, Triangle @ handouts/activities-a.txt Read: VdW 1 @ handouts/reading-quest-a.txt Read: Standards handout @ http://standards-e.nctm.org/1.0/print/standards/stdsK12int.html Read: LDT 1.2 (HW 1) @ hw01.shtml Introduce Quizzes and HW (do projects tomorrow) Quiz 1 NCTM MCTM meetings in GR NCTM trial membership Standards & NSF projects Standards 2000 5th theme standard reaction to standards, VdW1, etc. introduction to first month of course (handout calendars) discussed 3-digit sums problem guess and check pattersns how to generate more solutions introduce V1 (2-digit sums response) Friday, September 10 Topic: Doing Math Read: VdW 2, Figure 3.9 @ handouts/reading-quest-a.txt Read: LDT 1.3 (HW 2) @ hw01.shtml Act: V1: Double-Column Addition @ handouts/videos.txt introduce Projects watch video -- brief wrap-up discussion increased/decreased emphasis group work on V1 and A1 (triangle 1-9) 1-9 triangle puzzle -- brief discussion, digit sums and corners == Monday, September 13 Topic: Kinds of Understanding Activity: A2: Procedural vs. Conceptual @ handouts/activities-a.txt Read: VdW 3@ handouts/reading-quest-a.txt Read: LDT 1.4 (HW 2) # Due: A1: Triangle Puzzle (G) @ handouts/activities-a.txt Due: HW 1 @ hw01.shtml handouts kinds of understanding (talsma handout) Flanders article R1-R3: effects of problem solving reminder about HW1 scope and sequence charts for Cur. Project 1 group work on A2 discussion of procedural vs conceptual knowledge discussion of why division by zero is undefined group work on V1, if necessary answer questions regarding types of understanding procedural vs. conceptual relational vs. instrumental connections (words, symbols, models, pictures, "real world") Tuesday, September 14 Topic: Problem Solving Read: VdW 4 @ handouts/reading-quest-a.txt Read: LDT 1.5 (HW 2) @ hw01.shtml Act: A3: Some Problems @ handouts/activities-a.txt Due: V1: 2-Digit Sums 1-3,5 (G) @ /handouts/videos.txt Due: A1: Triangle Puzzle (G) @ handouts/activities-a.txt Polya's 4 steps: understand the problem devise a plan carry out the plan look back (correctness, generalize) Problem Solving ingredients (Schoenfield) knowledge heuristics control (meta-cognition) belief Heuristics make list with students Thursday, September 16 Topic: Assessing Problem Solving Read: VdW 5 @ handouts/reading-quest-b.txt Act: A4: More Problems @ handouts/activities-a.txt Due: HW 2 @ hw01.shtml review problem solving heuristics, etc rubrics (rubric for HW 3) Friday, September 17 Topic: Patterns Read: VdW 18 @ handouts/reading-quest-b.txt Act: A5: Find the Pattern @ handouts/activities-a.txt presentation of solutions to A3 highlight strategies, multiple soutions, connections patterns -> formulas in A5 connnections in problem 2 == Monday, September 20 Topic: Classification Read: VdW 18@ handouts/reading-quest-b.txt Act: B1-4: Attribute Blocks Due: HW 3 @ hw01.shtml Due: Cur. Project 1 @ projects/ schedule today: B1-4, (A5, if time) look at A5 (patterns) problem 2 before tomorrow hand out a blocks (B1-4), set ops (A6) and reding questions discuss attribute sets (attributes, values) get cur. project data on overhead from each pair wrap-up people pieces and sizes of attribute sets cur. project 1 wrap-up (graph results and look) Tuesday, September 21 Topic: Sets and Set Operations Read: LDT 2.1 (HW 4) @ hw01.shtml Act: A5: Find the Pattern @ handouts/activities-a.txt work on A5 (finish B1-4 if necessary) repeating vs. growing patterns Thursday, September 23 Topic: Effects of Problem Solving Read: Wheatley (R1) @ handouts/reading-quest-b.txt Read: Frank (R2) @ handouts/reading-quest-b.txt Read: Vergano (R3) @ handouts/reading-quest-b.txt Act: A6: Set Operations@handouts/activities-a.txt Due: A5: Find Pattern (G) @ handouts/activities-a.txt Due: B1-4: Attribute Blocks (G) quiz 3 A6 go over pattern formulas discuss effects Friday, September 24 Topic: Miscellaneous and Catch-Up Due: HW 4 @ hw01.shtml quiz 3 wrap-up assessment Van de Walle's purposes evaluate student achievement (give grades) evaluate programs (did this teaching method work?) monitor student progress (how is an individual student doing?) make instructional decisions other purposes focus attention on what is important teach (students learn while being evaluated) affective issues (boost egos, temper) guiding principle: assess what you value example: tests in this class shoenfield's components knowledge heuristics list (save for patterns stuff) control belief list of bad beliefs from Frank article) math is computation math can be done quickly goal = right answers students get info and show that they have it teachers give info and check that students get it other bad beliefs you either get it or you don't i'm no good at math, so i can't ever do it patterns (A5) problem 1 repeating vs non-repeating (core) notation for patterns repeated addition = multiplication repeated multiplication = exponentiation repeating pattern in d problem 2 how to attack understand problem plan: organized tables, pictures, etc do it: look back: connections between these problems point out connections before solving solve them set operations (A6) == Monday, September 27 Topic: Number Concepts Read: VdW 6 Read: LDT 2.2 #Due: A6: Set Ops (G) @handouts/activities-a.txt # decided not to collect -- running a bit behind shedule handout solutions to set ops Counting procedural - sequence of words - one-to-one assignment conceptual - cardinality principle - more/less/same [war] counting to twofy (..., fen, alf, bet, thirfen, fourfen, twofy) Number Procedures - symbolism - counting on, counting back Number Relations spatial [dice, dominoes, family] 1 or 2 more/less part-part-whole 5 and 10 benchmarks Tuesday, September 28 Topic: Numeration Systems Read: LDT 3.1, 3.2 (HW 5, HW 6) Ask about MCTM Number systems (tally, additive, multiplicative/place value) Roman numbers and modifications intro to positional system mention representations models - words - symbols - pictures - real world situations test questions? did venn diagram stuff Due: Cur. Project 2 @ projects/ Thursday, September 30 Topic: Test 1 @ test-info.shtml Note: covers thru 9/24 Friday, October 1 Topic: Place Value Read: VdW 9 Act: Video: stars, Jonathan Act: A12-13: Base 5 blocks @ handouts/activities-b.txt Due: HW 5 @ hw02.shtml save this for a quiz? models for place value: groupable, pre-grouped proportional, nonproportional == Monday, October 4 Topic: Place value Topic: Mental Arithmetic, Estimation Read: VdW 10 Read: LDT 3.5 Act: A14-15: FUNN @ handouts/activities-b.txt Intro to mental math when to teach spread throughout curriculum as it arrises before pencil-and-paper algorithms are learned standard algorithms are "unnatural" avoid "mental blackboard" methods how to teach let it arrise naturally encourage multiple answers/methods explain reasoning provide some examples/suggestions prerequisites number sense operator sense example Extending Basic facts to tens, hundreds, etc. fen + threefy = fourfy threefy + threefy = alf fen = figgle fen (110_5) estimation: threefy-one + threefy-three is approx threefy+fourfy = bet fen = figgle twofy = 120 Tuesday, October 5 Topic: Meaning of Addition/Subtraction Act: A16: Add/Sub Word Problems@ activities-b.txt Read: VdW 7 (thru 126) Read: LDT 2.3 quiz 4 other models for place value not proportional (money, washers, chips, abacus) groupable, proportional (counters & cups, bundles of sticks, unifix cubes) pregrouped, proportional (beans on sticks/rafts) Finish FUNN, if necessary Meaning of addition/subtraction categories: situation & unknown info dynamic join (init, change, result) separate (init, change, result) static part-part-whole (part, whole) compare (large, small, difference) Thursday, October 7 Topic: Meaning of Multiplicaiton/Division #Act: A17: Meaning of Mult/Div @ activities-b.txt Read: VdW 7 Read: LDT 2.4 Due: HW6 @ hw02.shtml Due: A12-15: FUNN @ handouts/activities-b.txt Models for addition/subtraction Meaning of multiplication/division categories equal groups (whole, group size, number of groups) multiplicative comparison (product, set size, multiplier) cartesian product (product, size of a set) product of measures (product, one of the measures) models any model for addition/subtraction counters (in piles), etc. numberline (with hops) multibase blocks (for place-value algorithms) array equal groups (repeated subtration vs. fair share) for division models and word problems discussion of remainders Friday, October 8 Topic: Basic Facts Read: VdW 8 Act: A18: Basic Facts@ activities-b.txt review from thursday quiz 5 Basic Facts def'n approach to teaching 0. understand operations and number relationships 1. develop strategy ("discover" strategy) 2. practice strategy 3. practice strategy selection (strategies for addition facts) 0 1 more/2 more doubles near doubles make 10 (elevator strategy) 6 (pairs of) remaining facts (strategies for subtraction) think addition benchmark 10 (strategies for multiplication facts) 0,1 2: doubles 5: fives 9: nifty nines remaining 25 facts (10 pairs plus 5 squares) strategies for division facts (why not on worksheet?) think multiplication near multiplication facts operation tables (in vdw) show which facts can be done with which thinking strategies Mult/Div Division ideas missing factor fair share repeated subtraction Important properties (inverse relation) order (0,1) Postpone this until division algorithm distributive law ====================================================================== Monday, October 11 Topic: Addition Algorithms Read: VdW 11 (thru 221) Act: A19: Addition Algorithms a few arithmetic properties why did add/sub and mult/div go in pairs? -- inverses commutativity 0, 1 Mental Math - Extending Basic facts to higher decades algorithms for arithmetic operations def: step by step procedure, recording scheme emphasize algorithms as recording schemes use of models do then write problem solving with models available recording schemes partial sums crowded house (overloading columns, then exchanging) "standard algorithm" prerequisites meaning of operations place value (exchanging) basic facts Tuesday, October 12 Topic: Subtraction Algorithms Act: A20: Subtraction Algorithms Read: LDT 3.3 Thursday, October 14 Topic: Test 2 @ test-info.shtml Notes: covers thru 10/8 Notes: Friday, October 15 Topic: Multiplication Algorithms Act: A21: Multiplication Algorithms Notes: Notes: MCTM in GR Due: HW7 @ hw02.shtml Non-standard subtraction algorithms elevator instead of borrowing 9's complement adjust up/down (arrows) uses higher decade and place value mental math ideas Comparing algorithms efficiency accuracy closeness to conceptual basis [correctness] Lattice Algorithm how does it work? why does it work? comparison to standard adn "all pairs" algorithms Models base 10 products (1*10, 10*10, 10*100, etc) modeling base 10 products area model Multiplication Algorithms All pairs Lattice Standard (role of zeroes) == Monday, October 18 Topic: Division Algorithms Read: LDT 3.4 Read: VdW 11 (221-236) Act: A22: Division Algorithms Division Algorithms Standard (role of zeroes) Scaffolding (Big seven) Column-wise recording of exchanges and distributions Tuesday, October 19 Topic: Algorithm Wrap-Up Act: A23-24: Algorithm Review Due: MCTM Project @ projects/ Thursday, October 21 Topic: Calculators & Mental Math Read: Calculator handouts Due: HW8 @ hw02.shtml quiz 6: pros and cons of calculators 432 - 234 in base 6 (two different algorithms) 43 * 21 in base 6 (sketch model, lattice, all pairs) Options - Calculator/Computer, Mental Math, pencil and paper How to teach pencil-and-paper algorithms Be sure prereqs are in place Begin with models Emphasize algorithms as recording schemes first Move toward more efficient algorithms and more abstract understanding of them gradually Give students choices let them see multiple algorithms discuss advantages/disadvantages emphasize them as means toward end Why to teach pencil-and-paper algorithms Some applicability to daily life (but less than in past) Reinforce number sense, etc Algorithms are important work on A17-A18: Algorithm Review Friday, October 22 Topic: Lockers Act: A25: Locker Problem Due: Cur. Project 3 @ projects/ ====================================================================== Monday, October 25 Topic: Number Theory Act: A26-27: Sieve of Eratosthenes Read: VdW 20 (436-439) Lockers Wrap-Up and intro to Factors, Primes, etc Tuesday, October 26 Note: Reading Recess Note: No Class review session at 1pm Thursday, October 28 Topic: Test 3 @ test-info.shtml Note: covers thru Algorithms Friday, October 29 Topic: Number Theory Read: LDT 4.2, 4.3 Act: A36: Missing Digits Act: A28-29: Factor Game Due: A26-27: Sieve look through vdw calculator stuff in groups find most important claims he makes Factor game Primes, composites and factoring trial division 210=2*3*5*7; 221=13*17; 715=11*13*5; 3465=5*7*9*11; 667 = 23*29; 3335=5*23*29; 3035 = 5 * 607 30030= 2*3*5*7*11*13; 5400= 8 * 27 * 25 primes: 547, 607, 1033, 2203, 3217, 4253, 4423, 9689 divisibility tests (did 2,5,3,9,6) prime divisors largest necessary divisor factor trees finding prime numbers sieve == Monday, November 1 Topic: GCD and LCM Act: A30: Number Theory Problems Act: A31: Common Divisors/Multiples Due: HW 9 @ hw03.shtml GCD definition from prime factorization from factor list (did not do Euclid's algorithm) LCM definition from multiple list from prime factorization relation to gcd Tuesday, November 2 Topic: Fraction Concepts Read: VdW 12 Act: A32: Cuisenaire Rods Act: A33: Pattern Blocks Act: Set Models for Fractions Due: Reading Projects 1 & 2 @ projects/ meaning of fractions models for fractions cuisenaire rods pattern block activity set models for fractions Thursday, November 4 Topic: Fraction Concepts Act: A34: Comparing Fractions Act: Division with Pattern Blocks Read: VdW 12 Due: HW 10 @ hw03.shtml meaning/models of ops with fractions addition & subtraction pattern block operations with fractions new groups Friday, November 5 Topic: Fraction Operations Read: VdW 13 Read: LDT 6.2 Due: HW 11 @ hw03.shtml quiz 7 work on pattern block division sheet ====================================================================== Monday, November 8 Topic: Fraction Operations Read: VdW 13 Read: LDT 6.2 Due: HW 12 @ hw03.shtml Due: A34: Comparing Fractions quiz 8 algorithms for fraction ops (chalk talk) connection to whole number operations connecction to estimation and ad hoc methods connections to models addition/subtraction, common denominator equivalent fractions set models (quiz) area models (length models) multiplicaiton what does multiplication of fractions mean? models area model especially connection to old area model for mult both factors less than one one factor less than one both factors greater than one Tuesday, November 9 Topic: More Fraction Operations Due: HW 13 (not collected) @ hw03.shtml division algorithms for fractions common denominators (repeated subtraction) invert and multiply (missing factor) mental math: front-end addition Thursday, November 11 Topic: Test 4 @ test-info.shtml Note: covers through fractions Friday, November 12 Topic: Decimals Read: VdW 14 Read: LDT 7.1 teaching a new concept the big triangle (plus 2) (manipulative) models symbols words (real world situations) (pictures) connections to other mathematics topics to "real world", other disciplines, etc procedural/conceptual knowledge application of above to decimals connections: fractions place value operations on whole numbers, fractions understanding the decimal place value system what does the decimal point mark? modeling with base 10 pieces (decimals vs. fractions: which comes first?) review of test 4 questions and hand back tests ====================================================================== Monday, November 15 Topic: Percents Act: A36: Some Percent Problems Read: VdW 14 Read: LDT 5.1 Due: Cur. Projet 4 @ projects/ Tuesday, November 16 Topic: Decimals, Percents, Fractions Act: A37: Decimals, Percents, Fractions Due: HW 14 @ hw04.shtml percents quiz Thrusday, November 18 Topic: Decimal Operations Read: VdW 14 Read: LDT 7.2 decimal algorithms : addition, subtraction, multiplication, division Friday, November 19 Topic: Ratio and Proportion Act: A38: Some Proportion Problems Read: VdW 15 Read: LDT 7.3 Act: Orange Juice Due: HW 15 @ hw04.shtml converting repeating decimals into fractions proportions worksheet cups of orange juice problem trolls ====================================================================== Monday, November 22 Topic: Integer Concepts Read: VdW 20 (444-450) Read: LDT 5.1 Due: HW 16 @ hw04.shtml fractions, ratio, and proportion integers (negatives) situations models (length, set) inverse of addition 5 + __ = 0 addition/subtraction from counter model human marcher debts/paychecks mailman/bills integer operations from models from patterns from properties Tuesday, November 23 Topic: Integer Operations Read: LDT 5.2, 5.3 Due: Reading Projects 3 & 4 @ projects/ number line model for integers rules for signs of products and quotients (based on patterns) fractions and division Thursday, November 25 Note: Thanksgiving Note: No Class Friday, November 26 Note: Thanksgiving Break Note: No Class ====================================================================== Monday, November 29 Topic: Rationals and Reals Read: LDT 6.3 Act: A39: Number Systems Due: HW 17 @ hw04.shtml rationals and irrationals (root 2, pi, growing pattern of digits) Tuesday, November 30 Topic: Rationals and Reals Read: VdW 20 (450-456) Act: A40-41: Irrationals on Geoboards Pythagorean Thm. (statement and application) geoboards Thursday, December 2 Topic: Test 5 @ test-info.shtml Friday, December 3 Topic: Rationals and Reals more geoboard stuff root(2) is irrational tests back =========== month break =============================== Monday, Dec 6 Topic: Variables and Equations Act: A43: Algebraic Sentences Read: VdW 21 Tuesday, Dec 7 Topic: Functions and Graphs Read: LDT 2.5 Act: Graphs and Functions Act: Graphs from Information base five "pentimals" problem from test 5 Thursday, Dec 9 Topic: Algebra Act: A43: Algebraic Sentences Due: Calc. Project @ projects/ look at graphs from information discrete vs. continuous linear vs. non-linear intercepts, negative values setting up equations specifying varibles carefully expressing relationship between quantities solving systems of equations tit-for-tat: do same thing to both sides substitution: substitute equals for equals equations (mathematical expressions with an equals sign in them) express relationships between quantities based on operations allow us to succinctly record relationships by manipulating equations, we can discover new relationships from old ones types: arithmetic fact (1 + 1 = 2) formula (computational form) definition (let tan x = sin x / cos x) identity (sin x = cos x tan x) property (1 = n * 1/n) relationship (AB = 2) (inequalities) (variables) specific unknown quantities (to solve for) place holder for any of a set of numbers arguments, paramters (constants: known number like pi, e, physical constants) models for algebra counters and cups scales box/dot diagrams: (x+2)(x+3) area: (a+b)(c+d) broken decimal point problem from test 5 ? Friday, Dec 10 Topic: Course Wrap-Up ==End Cal Wednesday, Dec 15 Final Exam Review number systems definitions of rational/real/irrational closure under division how to demonstrate closure densisty -- how to prove for rationals terminating and repeating decimals coversion to fractions (use algebra) which fractions have term.decimal representations geoboard lengths and area