Elementary Mathematics^

Problem Solving is a process. The first step to problem-solving is understanding the problem and what it asks for. 4 steps to problem-solving:Understand Devise plan (strategies) Implement planLook back and check- is the answer reasonable?Use trial and error, blocks, and solving backwards as strategies to experiement. Example problems:Tag-of-war problem- Professor solved:R1: 4A= 5G 4 x 5= 5 x 4 20 = 20 R2: I= 2G+A R3: I+3G ? 4AA=5G=4 R2: I=2G+A 2(4)+5 I=13 3(4)R3: 13+12 ? 20 25>20 Ivan & the 3 grandmas will win the 3rd round. Discard the old books problem:

In 1945 George Polya wrote the "How to Solve it" book, that breaks down the 4 steps to problem-solving. The Cartesian product of two sets is the set of all possible ordered pairs that can be made from those sets. Combining items from one group to another- through multiplication.Example problem-solving questions:There are 7 people, if each person shakes hands with everyone at the table one time, how many handshakes are had? A: B, C, D, E, F, GB: C, D, E, F, GC: D, E, F, G D: E, F, GE: F, GF: GG:--------------------------------------21 handshakes This is because person A, cannot shake hands with themselves, and will therefore shake hands with 6 people. Similarily, with person B, which will shake hands with 5 people. To get the appropriate answer add up the numbers in sequence. ---------------------------------------------------------------------------------------There are four 3- cent stamps and three 7-cent stamps. How many postage amounts can you put together if you're sending letters? This is because multiplying the 4 groups of 3 cents ensures that you're accounting for all combinations. Then adding 4+3=7 ensures you have all the cent amounts. Adding 12+7=19 ensures all the cents and combinations are accounted for. 

Numeration Systems:Base-10: One-to-ten relationshipDigits Used:0,1,2,3,4,5,6,7,8,9, 10Example:: (1/10 (1/100) of the unit) hundreds tens ones tenths hundredths 3 7 5 . 2 5 ↝ 1/10 x 1/10= 1/100←x10 →(can be measured with money)Expanded Notation:375= 300+70+5 = (3x100)+ (7x10)+(5x1) = (3x10^2)+(7x10^1)+(5x10^0)(any number to the power of zero is one)Examples:1,078= 1000+70+8 = (1x1000)+(0x100)+(7x10)+(8x1) = (1x10^3)+(0x10^2)+(7x10^1)+(8x10^0)931= 900+30+1 = (9x100)+(3x10)+(1x1) = (9x10^2)+(3x10^1)+(1x10^0)-------------------------------------------------------Turning Base-5 numbers into Base-10 Example problems:1115 = (1x5^2)+(1x5^1)+(1x5^0) = (1x25)+(1x5)+(1x1) = 25+5+1 = 31 ←in Base-1010235 = (1x5^3)+(0x5^2)+(2x5^1)+(3x5^0) = (1x125)+(0)+10+3 = 125+13 = 138 ←in Base-10

The principles are the same for every base. When you change the base, the value of the number changes- but the principle is the same. Base 3-------------------ones - 3^0threes - 3^1nines - 3^227s - 3^3Digits: 0,1,2... Examples:Base 3---------------122.213 ↓↓ 1/3 1/912223=(1 x 3^3)+(2 x 3^2)+(2 x 3^1)+(2 x 3^0) = 27+18+6+2 = 53 Base 4--------------ones - 4^0fours - 4^116s - 4^264s - 4^3Example: 12224=(1 x 4^3)+(2 x 4^2)+(2 x 4^1)+(2 x 4^0) =64+32+8+2 =106Homework #2- 1/28/25Base 8-----------------ones - 8^0Eights - 8^164s - 8^2512s - 8^3a) 41.58=(4 x 8^1)+(1 x 8^0)+ (5 x 1/8) = 32+1+5/8 = 33 5/8b) 13415=(1 x 5^3)+(3 x 5^2)+(4 x 5^1)+(1 x 5^0) = 125+75+20+1 =221------------------------------------------a) Base 9 12= 13 9-----------------ones - 9^0nines -9^181s - 9^2729s - 9^3--------------------------------------------------------------b) Base 8 12= 14 8-------------------ones - 8^0Eights - 8^164s - 8^2512s - 8^3

Base 2-----------------------ones - 2^0twos - 2^1fours - 2^28s - 2^316s - 2^4Digits: 0,1...Expanded Notation Example Problem for the Base of 2:11112 = (1 x 2^3)+(1 x 2^2)+(1 x 2^1)+(1 x 2^0) = 8+4+2+111112 = 15-------------------------------------Solve and compare to find which is greater:20+3 12+5435 or 25623 > 1715+4 18+4345 or 34619 < 22Same Digits:45 = 461112 = 77 = 712+3 10+5236 or 23515 > 13Give the Base-10 numeral using expanded notation:a) 31.24 = (3 x 4^1)+ (1 x 4^0)+(2 x 1/4) = 12+1+ 2 2/4 31.24 = 13 2/4b) 221.23 = (2 x 3^2)+(2 x 3^1)+(1 x 3^1)+(1 x 3^0)+ (2 x 1/3) = 18+6+1+ 2/3 221.23 = 25 2/3c) 10002 = (1 x 2^3)+(0x 2^2)+(0 x 2^1)+ (0 x 2^0) = 8+0+0+010002 = 8-------------------------------------------------------Homework #3- 1/30/251.a) 29 in base 3= 29-27= 02 29=10023ones ← 2threes ←0nines ←027s ←1 group b) 69 in base 2= 69-64= 0569=10001012ones ←1twos ←0fours ←1 group8s ←016s ←032s ←064s ←1 groupc) 115 in base 5= 115=4305ones ←05s ←3 groups25s ←4 groups125s ←02.a) 10= 243 b) 13 3/4= 25.34 In both equations, the numbers have digits that are greater than the base; that is the mistake.

Addition:Meaning: putting together, joining.Properties of Addition:Identity:a+0=aex. 2+0= 2 (in addition the answer is the sum)If you add a 0 to a number in addition, the identity of the number doesn't change. Commutative: (order property)a+b= b+aex. 3+7=7+3The order in which you add numbers doesn't matter in addition. Associative: (grouping)(a+b)+c=a+(b+c)ex. (7+3)+5=7+(3+5)The outcome is the same, no matter how the numbers in the problem are grouped. Subtraction:Meaning: a) Take Awayex. 4-3=1 (in subtraction the answer is the difference)b) Comparision ? (as in word problem comparisions)c) Missing Addend. ?ex. 3+ ?=7 Emily had 3 cookies, Addison gave her some cookies, Emily now has 7 cookies. How many cookies did Addison give her?The action we take as adults to solve this problem is subtraction, but to a 1st grader they will likely try to add, not subtract. Therefore, if a 1st grader reads this problem and asks "why are we taking away"- it is valid because the problem is sort of an addition problem, so telling our students to "take away" doesn't make sense, quite yet. There are no subtraction properties because of negative numbers. Multiplication:Meaning: Groups of the same quantity. Multiplication is repeated addition. ex. 3 x 4= 12 (in multiplication the answer is the product) 3 multiplied by 4, means there are 3 groups of 4. Key Note: Don't teach students isolated skills- make connections. Properties of multipl.Identity: a x 1= aThe identity property says that when you multiply by 1, the property doesn't change. Commutative: (order property)a x b= b x a ex. 3 x 4= 4 x 3 The order in which you multiply doesn't matter. Array example:Associative: (grouping)(2 x 5) x 5= 2 x (5 x 5)Zero: a x 0=0 Multiply any number by 0, and the answer will be 0. Examples:0.25 x 0= 0 3/5 x 0= 0 -4 x 0=0 Grouping problem example:3 x 7= 3 x (5+2)= (3 x 5)+(3 x 2) (partial products)Distributive Property: When you multiply a number by the sum of two other numbers the outcome is the same. You must multiply each of the numbers by the addends, and add the partial products to get your answer. Example:3 x 9= 3 x (5+4)= (3 x 5)+(3 x 4)= 15+ 12= 27 (you could group in various other ways as well to get the same answers, but you have to follow the property).

Divison: Meaning: Sharing There are no properties for division. In divison the answer is the quotient. Different signs to show division:8 ÷ 2 ÷8/2 - (fraction bar)2 ⟌8 ⟌Consider these problems: a) John has 15 cookies. He puts 3 cookies in each bag. How many bags can he fill?b) John has 15 cookies. He puts the cookies into 5 bags with the same number of cookies in each bag. How many cookies are in each bag?a) 15 ÷ 3= ?To solve students will make groups of 3 with the manipulative materials, till no materials or "cookies" are left. b) 15 ÷ 5=? To solve students will have 5 groups and spread out the 15 pieces of the manipulative materials or "cookies" till none are left. Explaination: Both of these problems are the same. However, to students problem B might seems more challenging because the action taken to solve the problem might be longer and different. Students will more than likely use trial and error when solving division problems such as this. ------------------------------------------------------------In 4th grade the long division algorithim will be taught:Standard American Algorithimex. The Standard American Algorithim should be the last way of solving taught to students when it comes to divison. Because students should be able to explain why these steps work to solve and why. ---------------------------------------------------------Place Value Algorithim:This place value heavily talks about place value- which makes it easier for students to understand. --------------------------------------------------------------Alternative Algorithim:Looking at divison as repeated subtraction. Example problem: I have 197 cookies, I want to put them into boxes, making sure that each box will have 16 cookies. How many boxes will I need?

Test 1: Study GuideProblem Solving: G. Polya's Steps- name + describeBases: Write #'s in different bases -Use expanded notation to give b-10 number-Compare numbersOperations: Properties: Names, meaning, applications --------------------------------------------------------------------An algorithm is a mathematical process.With addition, subtraction, and multiplication, the American Standard algorithm should be the last step. Students should first be able to explain their work and understanding through expanded notation, and then transition to the American Standard. Addition algorithms: (Example Questions in HW #4)a) American Standard R→ L algorithm b) Partial Sums R→ L algorithm c) Partial Sums with Place ValueR→ L algorithm (adding with place value in mind)d) Left-to-rightL→ R algorithm (Uses place value)e) Expanded Notationf) Lattice Method--------------------------------------------------------------------Subtraction Algorithms: (Example Questions in HW #4)a) American Standard R→ L algorithm b) Reverse Indian L→ R algorithm c) Left-to-rightL→ R algorithm (Uses place value)d) Expanded Notatione) Integral Subtraction Algorithm(+,-)---------------------------------------------------------Multiplication Algorithms: a) American Standardb) Place Valuec) Expanded Notationd) Lattice Method Example problems for each multiplication algorithm:Homework #4- 2/11/25