MTE 280

Problem Solving Week 1:Steps to Problem Solving:Understand the ProblemDevise a Plan (Strategies)Implement PlanLook Back (Reasonable Answer?)Tag of War: Acrobats, Grandmas, and IvanR1: 4A = 5GR2: I = 2G + AR3: I + 3G ____ 4A <----- (2G + A) + 3G _____ 4AAnswer: I + 3G > 4A <------ 5G + A > 4AGive: Time and Manipulatives (Hands on Tools)Problem Solving Steps Video:https://youtu.be/kn8frIzQupA?si=oMIOF-95AUMSO2b7

Week 2: Tuesday"How to Solve It" - 1945 George PolyaProblem: How many handshakes? (Without repeating handshake)x x x x x x Solve: 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakesStrategies: Whatever makes the most sense to the studentBigger Number: SimplifyProblem: 4 3 cent stamps 3 7 cent stampsHow many different postage amounts can we put together?Pattern: Only 3 cents: Only 7 cents:x xx x x xx x x x x xx x x x 4 x 3 = 12 Simplify: 5 11 centsx 6 9 cents 305 + 6 + 30 = 41Cartesian Product: x x 4 x 2 = 8 y y y y Polyas 4 Steps & Describe: Test

Week 2 Thursday Numeration SystemsBase 10 SystemPositional systemNumbers get their value based on its place3 3 3| | || | ones| tenshundredsDecimal system because it is a 1-10 relationship.hundreds| tens| | ones| | | tenths3 7 5 . 2 5 ----- hundredths (1/10 x 1/10 = 1/100)| |1/10 |1/100Related to Money:3 $100 bills7 $10 bills5 $1 bills2 dimes5 pennies (1/10 of a dime)Digits used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 910 11x xx xx xx xx xx xx xx xx xx x xExpanded Notation:375 = 300 + 70 + 5375 = (3 x 100) + (7 x10) + (5 x 1)375 = (3 x 10^2) + (7 x 10^1) + (5 x 10^0)Example:1078 = 1000 + 0 + 70 + 81078 = (1 x 1000) + (0 x 100) + (7 x 10) + (8 x 1)1078= (1 x 10^3) + (0 x 10^2) + (7 x 10^1) + (8 x 10^0)Different Bases:Base 5 Digits Used: 0, 1, 2, 3, 4 Expanded:ones 5^0 1 1 1 base 5fives 5^1 | | ones25s 5^2 | fives125s 5^3 25s111 base 5: (1x 5^2) + (1 x 5^1) + (1 x 5^0)111 base 5: (1 x 25) + (1 x 5) + (1 x 1)111 base 5: 25 + 5 + 1 = 311023 base 5: (1 x 5^3) + (0 x 5^2) + (2 x 5^1) + (3 x 5^0)1023 base 5: (1 x 125) + (0 x 25) + (2 x 5) + (3 x 1)1023 base 5: 125 + 0 + 10 + 3 = 138Khan Academy Video:https://youtu.be/ku4KOFQ-bB4?si=PNILbWj735gYMFI3

Week 3: TuesdayBase 3: Digits Used: 0, 1, 2 Example:ones 3^0 x x x = 10 base 33s 3^1 x x x|x x = 12 base 39s 3^227s 3^3Examples:1222 base 3: (1 x 3^3) + (2 x 3^2) + (2 x 3^1) + (2 x 3^0)1222 base 3: (1 x 27) + (2 x 9) + (2 x 3) + (2 x 1)1222 base 3: 27 + 18 + 6 + 2 = 531222 base 4: (1 x 4^3) + (2 x 4^2) + (2 x 4^1) + (2 x 4^0)1222 base 4: (1 x 64) + (2 x 16) + (2 x 4) + (2 x 1)1222 base 4: 64 + 32 + 8 + 2 = 106Base 10 Base 5 Base 3122.21 122.21 122.21|| || ||1/10| 1/5| 1/3|1/100 1/25 1/9Base 3 Number System Video:https://youtu.be/X7HLvBzB9-I?si=0RDnQT48a4QpNoeV

Week 3 Thursday:Base 2 Digits Used: 0-1ones 2^0twos 2^1fours 2^28s 2^316s 2^41111 base 2: (1 x 2^3) + (1 x 2^2) + (1 x 2^1) + (1 x 2^0)1111 base 2: (1 x 8) + (1 x 4) + (1 x 2) + (1 x 1)1111 base 2: 8 + 4 + 2 + 1 = 15Which number is bigger?43 base 5 __>___ 25 base 6 -----> 23 __>__ 1743 base 5: (4 x 5^1) + (3 x 5^0) | 25 base 6: (2 x 6^1) + (5 x 6^0)43 base 5: 20 + 3 | 25 base 6: 12 + 543 base 5: 23 | 25 base 6: 17Compare:23 base 5 __<___ 23 base 6 *Same digits but second base is larger

Week 4: TuesdayAddition Meaning and Properties:Putting together; joiningIdentity: a + 0 = a *When I add 0 to any number, the identity of the number does not changeCommutative (order): a + b = b + a *The order that you add does not matterAssociative (grouping): (a + b) + c = a + (b + c) *The way you group your numbers does not matterSubtraction:Take away: 4 - 3 = 1Comparison: How many more equationMissing Addend 3 + ___ = 7*As adults we subtract, a first grader would use addition*Answer: This would be confusing to a first grader because the equation doesn't tell them to take away. It looks like an addition problem.Multiplication:3 x 4: 3 groups of 4 3 x 4 = 12factors productxxxx xxxx xxxxRepeated Addition Cartesian Product: Combining GroupsTelling Time a a x b b bSkip Counting 2 x 3 = 6 combinationsProperties:Identity: a x 1 = a *When I multiply by 1 the identity of the number does not changeCommutative (order): a x b = b x a *The order in which I multiply do not matterAssociative (grouping): (a x b) x c = a x (b x c) *The way we group our problem does not matterZero: a x 0 = 0 *Any number multiplied by zero equals zero3 x 7 = 7 + 7 + 75 + 2x x x x x x xx x x x x x xx x x x x x x(3 x 5) (3 x 2)a x (b + c) = (a x b) + (a x c)3 x 7 = 3 x (5 + 2) = (3 x 5) + (3 x 2)partial productsDistributive Property: When I multiply a number by the sum of two other numbers. It is the same as multiplying the number by each addend.3 x 9 = 3 x (5 + 4) = (3 x 5) + (3 x 4) = 15 + 12 = 273 x 9 = 3 x (7 + 2) = (3 x 7) + (3 x 2) = 21 + 6 + 273 x 9 = 3 x (6 + 3) = (3 x 6) + (3 x 3) = 18 + 9 = 27Addition Properties:https://youtu.be/a0deCn5QNFI?si=WKsH9-tYEBDXmPHH

Week 4 DivisionDivision is about sharing or splittingJohn has 15 cookies. He puts 3 cookies in each bag. How many bags can he fill?John has 15 bags. He puts the cookies into 5 bags with the same number of cookies in each bag. How many cookies in each bag?*They are the same but completely different to our first graders. Long Division: Usually taught Standard American AlgorithmAlternative Algorithm: How many boxes? ___12____16| 197 -160 -----> 10 boxes 37 + -32 -----> 2 boxes 5 12

Week 5 - Tuesday AlgorithmsAddition:1. American Standard 576 + 279  855 (Right to Left)(Not the same way we read)(No place value)2. Partial Sums 576+279 1 514+ 7  855 (Right to Left)(No place value)3. Partial Sums with Place Value576+ 279 1 51 4 0 + 7 0 08 5 5 (Right to Left)(With place value)4. Left-to-Right 576 + 279 700 140+ 15 855 (Left to Right)(With place value)5. Expanded Notation 576 = 500 + 70 + 6 + 279 = 200 + 70 + 9 855 = 800 + 50 + 5 Use this in lesson planTeaches students place value and the reason why it’s important when transitioning to standard algorithm6. Lattice Method 576 + 279(Diagram with diagonal boxes)(Adding each “channel”)Subtraction Algorithms1.American Standard 476 - 289 287 (Right to Left)(No place value)(Can’t move on until they explain place value)2.Reverse Indian 576 - 289(Left to Right)5 - 2 = 37 - 8 (borrow from 3, make 17)6 - 9 (borrow from 9, make 16)Final Answer: 2873.Left-to-Right  576 - 289300 (500-200)(200)90 (170 - 80) borrowing from 300  (80) (16 - 9) borrowing from 90   (7)(Left to Right)(Borrowing explained step by step)4.Expanded Notation:576 = 500 + 70 + 6-289= 200 + 80 + 9287 = 200 + 80 + 75.Integer Subtraction Algorithm576-289-3 290 - 3 = 287-10+300Not for elementaryIt’s fun for kids (middle school)Work from bottom upMultiplication Algorithms1.American Standard12 3x 1 41 9 2+2 3 03 2 2No place value2.Place Value2 3x 1 43224 x 3=124 x 20 = 8010 x 3 = 3010 x 20 = 200322Seen a lot in elementary3.Expanded Notation(Take 23 14 times)1023 = 20 + 3x 14 = x 10 + 4100 90 + 2 <—— by taking my 23 4 times 200 + 30 + 0 <—— no ones (it takes that place since we are multiplying by 10s now)300 + 20 + 24.Lattice Methoddraw “/“ in each boxmultiply each numberdrag out the "channels"If there is a number to carry over put it in the next columnLattice Method for Addition:https://youtu.be/tYmF2GW0wwQ?si=3N3K6khH90-T4ko8