MATHEMATICS BY CHRISTINA ROBBINS

Polya's 4 steps to problem solvingUnderstanding the problemDevising a plan Carry out the planLooking back (reflect)Understand the problem 1.What are you asked to find out or show? 2.Can you restate the problem in your own words? 3.Can you draw a picture or diagram to help you understand the problem?Devise a plan 1.What problem solving strategy are you going to use 2.Guess and check/Trial and error 3.Draw a picture or a diagram 4.Make a table 5.Act it outCarry out the plan 1.Carrying out the plan is usually easier than devising the plan 2.Be patient ~ most problems are not solved quickly nor on the first attempt 3.If a plan does not work immediately, be persistentLook back (reflect) 1.Does your answer make sense? 2.Is it a reasonable one? 3.Did you answer all of the questions? 4.Could you have solved this problem in a different way?

There are seven people in a room. If each person shakes every other person’s hand, how many handshakes will take place? One way to solve this problem is to make a chart.Person 1 shakes hands with: 2, 3, 4, 5, 6, and 7. 6 handshakesPerson 2 shakes hands with: 3, 4, 5, 6, and 7.5 handshakesPerson 3 shakes hands with: 4, 5, 6, and 7.4 handshakesPerson 4 shakes hands with: 5, 6, and 7. 3 handshakes Person 5 shakes hands with: 6, and 7.2 handshakes Person 6 shakes hands with: 7.         1 handshakePerson 7 has already shaken hands with everyone.0 handshakes               21 handshakes in total

Solve both of the following problems. Show enough details, draw pictures, and give written explanation, so I can tell how you solved the problem.1. There are 12 basketball teams in a league. If each of the teams plays each of the other teams once and only once, how many games take place?2. I have four 3-cent stamps and three 7-cent stamps. Using one or more of these stamps, how many different amounts of postage can I make?

Numerations systems have been developed over the centuries as a way of recording quantity.In essence a numeration system is a system for representing quantity or a number in a consistent manner.For example, the symbol “11” may represent a variety of numbers depending on the numeration system being used.Some number systems are positional which means that the placement of digits (or numerals) in the number specifies the value of the number. An example of such a number system is the decimal system that we use (Base-10)

The number system we use in our schools and society is the Base-10 system. There is a consistent one-to-ten relationship between the digits of any number in the Base-10 system.•One unit•Ten units make a ten•Ten tens make a hundred•Ten hundreds make a thousand, etc.•A tenth of a unit is a tenth•A tenth of a tenth is a hundredth•A tenth of a hundredth is a thousandth, etc.Digits usedIn Base-10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9Expanded Notation723=700+20+3723=(7x100)+(2x10)+(3x1)723=(7x10^2)+(2x10^1)+(3x10^0)Decimal Point23.14•A tenth of a unit is a tenth•A tenth of a tenth is a hundredth•A tenth of a hundredth is a thousandth, etc.The same idea applies to the other bases!

Base-5 ones, fives, 25s, ectThe first digit after the point is a fifth of the unitThe second digit after the point is a 25th of the unit etc.(Example)ones, fives, 25s, 125s, 625s5^0, 5^1, 5^2, 5^3, 5^4..Expanded Notation:23 in base 5 = 13 in base 10(2x5^1)+(3x5^0)(10)+(3) = 13Digits Used:0,1,2,3,4The same concept is used in all of the other bases

#21. Give the Base-10 numeral for each given number. Use expanded notation to explain your answer:a) 41.58=b) 13415=2. Write the number 12 in each given base:a) Base 9b) Base 8c) Base 7

Answer all questions and show your work.1. Write each of these numbers:a) 29 in base 3=b) 69 in base 2=c) 115 in base 5=2. How do you know there is an error in each statement?a) 10=243                   and b) 13 3/4=25.34*2 the answer: YOU CAN NOT HAVE A NUMBER BIGGER THAN THE BASE

Standard American Algorithm (last step)(R-L) (no reference to place value)Partial Sums (R-L)Problem Solving (emphasis on place value) (R-L)Left to Right (L-R)Expanded notation Lattice Algorithm

Add: 478+394

American Standard (R-L) (Last Step) (no emphasis on place value)European/Mexican (R-L)(no emphasis on place value)Reverse Indian (L-R)(no emphasis on place value)Left to Right (L-R)*(place value)*Expanded Notation *(place value)*Integer Subtraction

Subtract: 645-279

A number A is divisible by a second number B, if there is a third number C that meets this requirement: CxB=A - 2x5=10 (2&5 are factors) - 10/5=2

10 is divisible by 5 or 5 divides 105 is a divisor of 105 is a factor of 1010 is a multiple of 5

(GCF)Definition: The greatest number that is a factor of two (or more) other numbers. When we find all the factors of two or more numbers, and some factors are the same ("common"), then the largest of those common factors is the Greatest Common Factor.How do you calculate the greatest common factor?To find the greatest common factor (GCF) between numbers, take each number and write it's prime factorization. Then, identify the factors common to each number and multiply those common factors together.GCF:24: 1,2,3,4,6,8,12,2436: 1,2,3,4,6,9,12,18,36 (The greatest common factors of 24 & 36 is 12)

(LCM)Definition: The least common multiple is the first common multiple for the given numbers. For 36 and 48, the number 144 is the LCM. The dimension of using LCM of two numbers starts with basic math operations such as addition and subtraction on fractional numbers.How do you calculate least common multiple?One way to find the least common multiple of two numbers is to first list the prime factors of each number. Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs.LCM:(24) (36)2: 24 362: 12 182: 6 93: 3 93: 1 3 1 1LCM: 2x2x2x3x3=72LCM=72

Numbers List:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59Definition: A prime number is a number greater than 1 with only two factors themselves and 1

#3 - Correct answerNo...Multiples are not factors and multiples of numbers continue forever

Division - 3/5 - 3 divided by 5 - quotientPart/Whole relationship - symbolRatio - example: 20 students12 girls8 boysboys/all students= 8/20girls/ all students = 12/20

Region - surface areaLength - the number lineSets - groups of things

Objection: Adding fraction with the same denominator (easy to add and subtract)1/4 + 2/4 = 3/43/12 + 4/12 = 7/121/6 + 4/6 = 5/6Least Common Multiple: 1/4 + 1/6 - 3/12 + 2/12 = 5/121/2 + 1/6 - 3/6 + 1/6 = 4/63/12 + 1/4 - 3/12 + 3/12 = 6/12 - 1/2

When multiplying fractions the part will get smaller When multiplying fractions - multiply across3 groups of 2 = 3 x 21/2 of 1/2 = 1/2 x 1/2 = 1/4 1/2 x 1/4 = 1/81/3 x 1/8 = 1/24