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

Number Theory:types of numbersdivisibility rulesEx: a is divisible by b if there is a number c that meets the requirement2 x 5 = 105 is a factor of 102 is a factor of 105 is a divisor of 10 All True2 is a divisor of 1010 is divisible by 510 is divisible of 2Divisibility Rules:-ending: the last digitby 2: 0,2,4,6,8 Ex: 24 / 12,070by 5: 0,5by 10: 0-sum of the digits:by 3: if sum of the digits is divisible by 3 Ex: 24: 2 + 4 = 6by 9: if sum of the digits is divisible by 9by 6: if sum of digits is divisible by both 2 and 3-last digits:by 4: if last 2 digits are divisible by 4by 8: if last 3 digits are divisible by 8 (not very helpful)by 7: double last digitssubtract from remaining numberrepeat826/7= ?8262 x 6 = 1282-1270Yes!by 11: the "chop off" methodchop off last 2 digitsadd them to the remaining numberrepeat29,194 / 11 = ?29,194291+ 9438538503+ 8588 / 11Yes!The number 0 and 1 are neither Composite or Prime numbers.0:Additive Identity Element1: Multiplicative identity elementPrime #'s through 602,3,5,7, 11, 13, 17, 23, 29, 31, 37,41,43,47,59 * 57 51 * TestNumber Theory Video:https://youtu.be/3zHn5iIBJzM?si=krfvYcfzCy3GDiYZ

Week 8: TuesdayPrime Factorization24: 1,2,3,4,6,8,12,24Prims Factorization:24/ \4 6/ \ / \2 2 2 324= 2 x 2 x 2 x 3(Never Changes! DNA of a Number)Greatest Common Factor (GCF)Least Common Multiple (LCM)Multiples Ex: 3: 3,6,9,12…4: 4,8,12,16…(Never Ends)Factors Ex: 24: 1,2,3,4,6,8,12,2425 / 5 = 5 / 5 = 1100 / 5 = 20 / 5 = 4 (Need to know GCF)1 + 1 —> 3 +1 = 4 2 6 —> 6 6 = 12(When teaching use LCM)List Method GCF & LCM (24 & 36)GCF: 24: 1,2,3,4,6,8,12,2436: 1,2,3,4,6,9,12,18,36GCF (24,36) = 12LCM: 24: 24,48,72,96…36: 36,72,…LCM (24,36) = 72Prime Factorization (24 & 36)24/ \4 6/ \ / \2 2 2 336/ \6 6/ \ / \2 3 2 324 = 2 x 2 x 2 x 336 = 2 x 2 x 3 x 3GCF = 2 x 2 x 3 = 12LCM = GCF x 2 x 3 = 12 x 2 x 3 = 72Prime Factorization Video:https://youtu.be/XBnUWjo3TgM?si=70XPYbsLc61QkJRb(May be at a disadvantage using for LCM)

Week 8: FractionsWhat is a fraction?A symbol that is used to express the relationship of part of a wholeMeanings of Fractions:Part-WholeQuotient 3/6Ratio(just ratios)9 girls boys — 7 girls — 97 boys girls — 9 boys — 716 totalboys — 7 girls — 9students — 16 students — 16(part-whole)(ratios & fractions)Models:Area (pattern blocks and shading)Length (number line and folding paper)Sets (groups)1 Whole2 = 1 3 = 1 2 3 When the numerator and the denominator are the same, it equals 1 whole.4 = 1 a = 14 a Fractional parts are equivalent parts.8 = 1 xy = 18 xy16 = 1163 x 2 — 6 Why is it equal?5 x 2 — 10 Because you are not multiplying by 2, you are multiplying by 2/2 which is equal to 1. (Identity Property)Fractions Are Parts Video:https://youtu.be/CA9XLJpQp3c?si=C_N5IckDtVs6-Xgj

Week 10: Fractions ContinuedWhen adding fractions with the same denominator, only add the numerators together. (Same with subtraction)1 + 1 —> 2 + 3 = 56 4 12 12 12(equivalent fractions)2 + 4 —> 10 + 12 = 22 improper fraction3 5 15 15 151 7 Mixed Number53 1 = 7 (Multiply 3 and 2 and add the 1)5 2Multiplying Fractions:Fractions get smaller when multiplying because you are using part of a partDividing Fractions:2 / 4 —> 2 x 5 = 103 5 3 4 12Adding and Subtracting Fractions Video:https://youtu.be/5juto2ze8Lg?si=8JZ58VBodADFdl3o

It’s important that even in first grade, the students are using the correct vocabulary of fractions. For example, when talking about fourths, they should also understand that they can use the terminology “a quarter of” or “a fourth of.” In third grade, students begin to see fractions written as a symbol. It’s important for us as teachers to place emphasis on the visual representation of fractions. An example of this is using a length model, a set model or an area model.Students need to understand that when comparing fractions, we have to be talking about the same size whole. 1/4 can be bigger than 1/2, if the whole that the 1/4 came from is larger than the whole that the 1/2 came from. The Progression of Fractions Video:https://youtu.be/f1nApsDKOdY?si=KDwptOiOL9EG6VVh

Problem Solving Fractions Practice: Question 2:Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took ¼ of the bars. Then Ken took 1/3 of the remaining bars. Next, Len took 1/3 of the remaining bars, and Max took the remaining 8 bars. How many bars were in the bag originally? How many bars did Jim, Ken, and Len each get? How is this problem (with regards to fractions) different from Problem 1?| 6 | 6 | 6 | 6 | - 1/4 Jim (6 bars)| 6 | 6 | 6 | - 1/3 Ken (got 6 bars)| 4 | 4 | 4 | - 1/3 Len (got 4 bars)| 8 bars |- MaxQuestion 3:There was ¾ of a pie in the refrigerator. John ate 2/3 of the left-over pie. How much pie did he eat?| | | | | - 1/4 gone| | | | - John ate 2/3 John ate 2/4 or 1/2 of the pie.Problem Solving Fractions Video:https://youtu.be/WrvDWD9HvOs?si=rrjwhcStrilgHbg8

Million Ten Thousands Hundreds Ones Hundredth Ten Thousandths Millionths 9, 6 0 5, 8 7 2 . 1 4 5 6 7 3 Hundred Thousands Tens Tenths Thousandths Hundred Thousands Thousandths$111 . 11-- Penny (1/10 of a dime) | Dime (1/10 of a dollar)375 . 35 |Decimal Point (A symbol that shows the relationship between a part and the whole) (*Sits to the right of the unit)$375 . 35 --> $3,753.5 --> $37,535 Dollars Dimes PenniesAn Introduction to Decimals Video:https://youtu.be/KrAQneGhyuE?si=3-FL5ht0NU5MLMVE

Multiplication: 2.2 ~ 2 1.23x 3.4 ~ 3 x 2.1 8 8 1 2 36 6 0 2 4 6 07.48 2. 583 (inference because of reason and logic) 4 x 2 = 8 1 x 3 = 310 10 100 10 100 1000Division: _____27/3= 3| 27 Exactly the Same 1.12 _____ 3| 3.36 3.1| 3.36 --> 31| 33.6 -3 03 - 3 06 - 6 0Fraction --> Decimal 0.833 <---- Repeating 5 = 5/6 --> 6| 5.000 6 - 48 5 = 0.83 (only over the 3) 20 6 -18 20 -18 20 If: 1.272727 ---> 1.27 0.1251 --> 0.125 <--- Terminating 8| 1.0008 - 8 20 -16 40 - 40 __| | = 3.14 <--- irrational number (Pi)Pi is just like love! It is irrational and never ending! <3How to Convert Fractions to Decimals:https://youtu.be/guBVW5PiHLs?si=uD5ZBXcryjl9cnlv

% = per cent = per 100 = out of 10030% OFF | 3 = 0.3 = 0.30 = 30 %$300.00 | 10 $210.00 | 3 = 0.03= 3% | 100Practice:21 = 42 --> 0.42 --> 42%50 100 7 --> 28 --> 0.28 --> 28%25 100 0.272 3 --> 0.27 --> 27% 11| 3.00011 -22 80 -77 30 -22 8 If: 0.5555 (repeating) --> 0.5 --> 56%a. 8 is what percent of 22? | is: = | what: nb. 8% of 22 is what number? | of: x | 8%: 0.08c. 8% of what number is 22? |a. 8 = n x 22 --> n = 8 --> 0.36 = 36% 22b. 0.08 x 22 = n --> n = 1.76c. 0.08 x n = 22 --> n = 22 --> n = 275 0.08Converting Fractions to Percentages:https://youtu.be/tZMj0OX5WJ8?si=4Sg_rlvpWpqD-Ia6

A student takes a test with 46 questions and gets 37 questions right. What is their percent on the test?37 is what % of 45?37 = n x 45 --> n = 37 45How We Teach It: Q | % 45 | 100 37 | n? (Cross Multiply)45n = 37 x 100n = 37 x 100 (Immediately a percentage! Don't have to convert it!) 45 Sandals | Blue 100 | 32 820 | n?100 x n = 820 x 32n = 820 x 32 = 82 x 16 = 262 100 5Percent Word Problem Video:https://youtu.be/YXpZhN6iwXI?si=F4lnUC3DGJSuIbF_

What were some eye-opening experiences in this class this semester?One of the most eye-opening experiences for me was realizing that I actually know more math than I thought. I’ve always struggled with math, so I came into this class expecting it to be really difficult and overwhelming. But as the semester went on, I started to notice that things were clicking in ways they hadn’t before. It surprises me how much I actually understand and that I’m able to help out the people in my group when they are confused. What were some of your success stories and some of your frustrating ones?A big success for me has been passing both exams so far. That felt like a huge accomplishment, especially given how nervous I was about this class in the beginning. Also, I  feel like I’ve gained a much better understanding of certain topics that used to confuse me, like fractions and converting to decimals, and that’s helped me build confidence. On the frustrating side, since I’ve been doing well so far, I’ve been putting pressure on myself to continue succeeding. I get more anxiety about doing well as the semester goes on. Do you feel better about your knowledge and ability to explain math concepts now than at the beginning of the semester?Yes, definitely. At the beginning of the semester, I didn’t feel very confident at all, but now I feel much better about both my understanding and my ability to explain math concepts clearly. I still have room to grow, but I’ve come a long way. I feel way better about teaching these math concepts in the future!

We want to represent positive and negative integers on a vertical number line because research shows:Students can identify place value easierStudents can understand negative numbers through real world examples"Chip Method"- negative + positiveAddition: (+++++) + (+) = (+5) + (+1) = +6(-----) + (-) = (-5) + (-1) = -6(+++++) + (-) = (+5) + (-1) = (+4) (+ - ) = zero pair (-----) + (+) = (-5) + (+1) = (-4)Subtraction:(+++++) - (++) = (+5) - (+2) = (+3)(-----) - (--) = (-5) - (-2) = (-3)(-----) + (+-) + (+ -) - (+) - (+) = (-5) - (+2) = (-7)(+++++) + (+ -) + (+ -) - (--) = (+5) - (-2) = (+7)Multiplication:2 groups of 3(+++) (+++) = 2 x (+3) = +62 groups of (-3)(---) (---) = 2 x (-3) = +6-3 groups of 2 --> difficult to demonstrate(-3) x (+2) = (+2) x (-3) <-- Communative Property(---) (---) = (-6)" the opposite of" -3 x (-2) = (+6)(--) (--) (--) = (-6) --> the opposite ofHow to Subtract Integers With Counters Video:https://youtu.be/DfnhVAnxMVI?si=34_zQaSO_mBk3nYM