Elementary Mathematics

Polya's Four Steps to Problem Solving Understanding the ProblemDevise a Planfind a strategy that works best for youCarry Out the PlanBe patient when solving your problemLok Back (Reflect) Does my answer make sense?

Problem Solving:Every problem has a different strategy to find the solution.Find the strategy the works best for you"Think for simple"Teachers need to use different modalities for students who learn differently.Need to remember that problem solving is a process. A problem is a problem because it is:challenginghas a solutionExample:There are 7 people in a room. Each person shakes hands with everyone only once. How many handshakes took place?1 --> 2, 3, 4, 5, 6, 72 --> 3, 4, 5, 6, 73 --> 4, 5, 6, 74 --> 5, 6, 75 --> 6, 76 --> 7 21 handshakes altogether

Expanded Notation: Understanding place value and expanded notation is important. In Base 10 we have: ones, tens, hundreds, thousands...In Base 10 we have: 100, 101, 102, 103...In Base 5 we have: ones, fives, 25s, 125s, 625s...In Base 5 we have: 50, 51, 52, 53, 54...In Base-3 we have: ones, threes, nines, 27s, 81s, 243s…In Base-3 we have: 30, 31, 32, 33, 34…Example:723 = 700 + 20 + 3 723 = (7 x 100) + (2 x 10) + (3 x1)723 = (7 x 102) + (2 x 101) + (3 x 100)

Base 10: The number system used in school and society. *Any number you see without a number as a base will always be base 10* Example: hundreds ->2 7 5<- ones . 3 2<- hundredths ^tens ^tenthsBase 10/DecimalOne - Ten Relationship10 ones = 1 ten --> 10 tens = 1 hundred, etc. 10 dimes = $1 dollar --> 10 pennies = 1 dime, etc. Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Base 5:Digits Used: 0, 1, 2, 3, 4Example: 5 (fives) ->105<--ones 115 = (1 x 5) + (1 x 1) = (1 x 51) + (1 x 10) = 6 115 = 6 2. 1005 = (1 x 52) + (0 x 51) + (0 x 50) = (1 x 25) + (0 x 5) + (0 x 1) = 25 + 0 + 0 = 25 1005 = 25Base 3:Digits used: 0, 1, 2Example: 2223 = (2 x 32) + (2 x 31) + (2 x 30) = (2 x 9) + (2 x 3) + (2 x 1) = 18 + 6 + 2 = 26 2223 = 26

Base 2:*Most commonly seen and known as the binary system (language computers use)*Digits used: 0, 1In base 2 we have, ones, twos, fours, 8s, 16s, 32s, 64s...In Base 2 we have, 20, 21, 22, 23, 24, 25, 26When a number is written in Base 2, they can look like: 10110112 Example: 10110112 = (1 x 26) + (0 x 25) + (1 x 24) + (1 x 23) + (0 X 22) + (1 X 21) + (1 + 20) = (1 X 64) + (0) + (1 X 16) + (1 X 8) + (0) + (1 X 2) + (1 X 1) = 64 + 0 + 16 + 8 + 0 + 2 +1 = 91 10110112 = 91 

Addition:The process of combining group "A" to group "B"Also considered to be "join" problemsThere are three properties of Addition:Identity Property of Addition:When you add a number to zero (0), the identity of that number does not changeExamples: 7 + 0 = 7, -3 + 0 = -3, .32 + 0 = .32, 3/4 + 0 = 3/4Communtative Property of Addition:Also known as the ORDER propertyOrder of the numbers does not matterExample: a + b = b + aAssociative Property of Addition:Also known as the Grouping PropertyHow you group the nu does not matter.Example: (a + b) + c = a + (b + c) --> (3 + 4) + 5 = 3+ (4 + 5) --> 7 + 5 = 3 + 9 --> 12 = 12Addition AlgorithmsStandard American AlgorithmRight to left Not the best algorithmThis should be the last step we show our studentsPartial SumsLeft to right algorithmPartial Sums with Emphasis on Place ValueEmphasis on place valueVery similar to the partial sums algorithm, but you are adding the value of the number placementExample: It is NOT 7 + 7 = 14 ---> It is actually 70 + 70 = 140Left to Right AlgorithmEmphasis on the place valueExpanded NotationThis gives the visual to the "carry the one" in the standard algorithmLattice Algorithm

Subtraction:*There are no properties of subtraction*Subtraction can become difficult when the number on the bottom of the problem is larger than the number on the top. As some children will not have grasped the concept of place value. There are three concepts of subtraction:Take Away: There are 7 cookies and I take away 3, how many cookies are left?Comparison: I have 4 books and Nicole has 3 books, how many more do I have than Nicole?Students will put the two groups together and count on. They will compare the two groups.Missing Addent: Anthony has 4 cookies, his mom gives him some more, now he has 7. How many did Anthony's mom give him?Children will see this problem as addition in their mind.Students will count on from 4 to get to the answer of 7.Even though the action looks like addition, this concept is still subtraction. Subtraction Algorithms:American Standard AlgorithmStarts from right to leftHas no emphasis on place valueWill be the last step we show our students.European - MexicanStarts from right to leftHas no emphasis on place valueReverse - IndianStarts from left to rightNo emphasis on place valueLeft to RightStarts left to rightHas an emphasis on place valueExpanded NotationHas an emphasis on place valueInteger SubtractionEmphasis on place value

Multiplication:Is the process on "repeated addition"Another way of thinking of multiplication as "groups of"There are three properties of MultiplicationIdentity Property of MultiplicationAny number multiplied by 1, the identity of that number does not changeExample: a x 1 = aZero Property of MultiplicationAny number multiplied by zer0 (0), than answer/product will always be zero (0). Example: a x 0 = 0Commutative Property of MultiplicationAlso known as the "order" propertyThe order of the numbers do not matterExample: a x b = b x a --> 9 x 3 = 3 x 9 --> 27 = 27Associative PropertyAlso known as the "grouping" propertyThe way the numbers are grouped, the answer will remain the sameThe order of the numbers does not change, but the parenthesis changes how the numbers are groupedExample: (a x b) x c = a x (b x c) --> (2 x 3) x 4 = 2 x (3 x 4) --> 6 x 4 = 2 x 12 = 24 = 24Distributive Property of MultiplicationIf you multiply a number by a sum, it is the same as multiplying that number by each partial sum, and then adding the partial products together. Example: a x (b +c) = (a x b) + (a x c) 3 x 9 = 3 x (4 + 5) = (3 x 4) + (3 x 5) = 12 + 15 = 27Multiplication AlgorithmsStandard AlgorithmNo emphasis on place valueThis should be the last step Place ValueEmphasis on place valueWrite everything on the sideExpanded NotationExpanded notation is where we can show our students the concepts that are applied with the standard algorithmLattice: No emphasis on place value

Division: Is the same as repeated division'Division symbolFraction form VinculumDivision AlgorithmsStandard Algorithm"How many times does 3 go into 4?""Bring down the 5""Subtract 3 from 4"Alternate (The OMG Algorithm)Dividing into "pockets"The Stolen Algorithm*Refer to Notes*

Mental Strategies are used for solving problems in your headThey give flexibility for solving problemsAddition Mental Strategies 1) Left to Right: Starting at the front of the number and working towards the back. Example: 347 + 129 = 300 + 100 = 400 40 + 20 = 60 7 + 9 = + 16 476 2) Compensation Example: 67 + (29) --> change to 30 67 + 30 = (97) --> need to subtract you added 97 - 1 = 96 3) Compatible Numbers: pick the numbers that work best together to solve the problem Example: 130 + 50 + 70 + 50 + 20 130 + 70 = 200 50 + 50 = 100 + 20 320*Gives you the flexibility to solve the problems based on what is best for you 4) Breaking up and Bridging: Example: 67 + (36) = 30 + 6 = 67 + 30 = 97 97 + 6 = 103

Mental Strategies are used for solving problems in your headThey give flexibility for solving problemsSubtraction Mental Strategies 1) Left to Right: Starting at the front of the number and working towards the back. Example: 47 - 32 = 40 - 30 = 10 7 - 2 = + 5 15 2) Compensation Example: 47 - (29) --> change to 30 47 + 30 = (17) --> need to remember to add the extra you took away 17 + 1 = 18 3) Compatible Numbers: pick the numbers that work best together to solve the problem Example: 170 - 50 - 30 - 50 50 + 50 = 100 170 - 100 = 70 - 30 = 40*Gives you the flexibility to solve the problems based on what is best for you 4) Breaking up and Bridging: Example: 67 - (36) = 30 + 6 = 67 - 30 = 37 37 - 6 = 31*The number you usually break up is the number you are taking away*

Multiplication Mental StrategiesThere are 2 mental strategies for multiplication 1) Left to Right Example: 3 x 123 = 3 x (100 x 20 x 3) --> Distribution Property Multiplication 300 + 60 + 9 = 369 2) Compatible Numbers: order of the numbers does not matter Example: 2 x 9 x 5 x 20 x 5 = 2 x 5 = 10 20 x 5 = 100 100 x 9 = 900 900 x 10 = 9000Division Mental StrategiesThere is only mental strategy for division 1) Compatible numbers Example: 105 / 3 (90 + 15) / 3 ---> 90 / 3 = 30 15 / 3 = + 5 35

On Thursday we were given our work papers and went over all the questions on the exams.

pempleDivisibility: A number (a) is divisible b a second number (b) if there is a third number (c) that meets this requirements. Example: c / b = a 10 / 5 = 2*10 is divisible by 5, to get to the number of 2*Important Terminologya x b = c --> product \ / Factors10 is divisible b 5 or 5 divides 105 is a divisor of 10 (2 is also a factor of 10)10 is a multiple of 5Divisibility Rules1) Ending - the last digit of the number By 2: 0, 2, 4, 6, 8... Example: 734 is divisible by 2By 5: 0, 5 Example: 75 is divisible by 5 300 is divisible by 5By 10: 0 Example: 1000 is divisible by 102) Sum of Digits - add all the digits to get the sum. Id the sum is divisible, so is the numberBy 3: Example: 543 = 5 + 4 + 3 = 12 12 is divisible by 3BY 9: Example: 543 = 5 + 4 + 3 + = 12 12 is not divisible by 93) OtherBy 6: If the number is divisible by 2 and 3 Example: 3702 By 2: ends in 2 \ / By 3: 3 + 7 + 0 + 2 = 15 --> 15 is divisible by 3 Divisible by 64) Last Digits - To see if a number is divisible by 4 or 8, you need to look at the last 2 - 3 digitsBy 4: The last 2 digits Example: 7948 --> 48 is divisible by 4By 8: The last 3 digits (Not a really helpful rule at all!) Example: 7948 --> 948 is divisible by 85) Special Numbers By 7: Look at the last digit and double it. Subtract that sum by the main number (may continue this process if necessary) Example: 826 --> 82|6 -> double it = 12 --> Subtract from the rest of the main number 82 - 12 = 70 --> if this number is divisible by 7, then so is the main number. Example: 92739 --> 9273|9 --> 9 x 2 = 18 9273 - 18 9255 925|5 --> 5 x 2 = 10 925 - 10 915 91|5 --> 5 x 2 = 10 91 - 10 81 8|1 --> 1 x 2 = 2 8 - 2 6 --> not divisible by 7By 11: "Chop - off" the last 2 digits and add the main number. Continue this until you get a number you can compare Example: 29,194 --> 291|94 --> 291 + 94 385 --> 3|85 --> 85 + 3 88 --> 88 is divisible by 11

Composite Numbers: have a lot of factorsPrime Numbers: has two factors - 1 and itselfList all Prime numbers thru 60:2, 3, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59*0 and 1 are neither prime nor composite*Number 1: multiplicative identity elementNumber 0: additive identity element

Greatest Common Factor (GCF) - SimplifyLeast Common Multiple (LCM) - Add/Subtract FractionsThere are 2 ways to find GCF & LCM1) List MethodGCF: 24: 1, 2, 3, 4, 6, 8, 12, 24 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 GCF: 12LCM: 24: 24, 36, 72 36: 36, 72 LCM: 72Where to use GCF The Simple Way | The GCF Way ** Ex: 25/100 = 5/20 = 1/4 | 25/100 = 1/4Where to use LCMEx: 3 x 12 + 5 x 4 --> 36 + 20 = 56 4 12 12 4 --> 48 48 = 48Finding a common denominatorUsing the LCM method (deals with smaller numbers) Example: Refer to notes2) Prime Factorization Method: the prime factors are the DNA of the number. They are there alreadyExample: Factor Tree 24 36 / \ / \ 2 12 6 6 / \ / \ / \ 3 4 2 3 2 3 / \ 2 224 = 2 x 2 x 2 x 3 or 24 = 23 x 3 24 = 2 x 2 x 2 x 324 = 2 x 2 x 2 x 3 --> Prime Factorization of 24 36 = 2 x 2 x 3 x 3To find the GCF: 24 = 2 x 2 x 2 x 3 36 = 2 x 2 x 3 x 3 GCF = 2 x 2 x 3 = 12To find the LCM (GCF multiply by the numbers I didn't use) *In every case* 24 = 2 x 2 x 2 x 3 2 x 2 x 3 = 12 x 2 x 3 = 72 --> LCM = GFC x 2 x 3 36 = 2 x 2 x 3 x 3 LCM = 72

Fall break: Class was excused on 10/12

Meaning of fractionsDivision: 3/5 or 3 divided by 5Parts of a whole: 3 --> parts 5 --> whole*Fraction is a symbol or a representation of the relationship of parts and whole*Ratio: Not all ratios are fractionsExample: 20 students Fractions: 12 are girls What is the fraction of girl students in the class --> 12/20 8 are boys What is the fraction of boy students in the class --> 8/20*if you are referring to a ratio girls/boys --> this is not a fraction because you are comparing a part to a part, not a part to a whole. This is why a ratio is not a fraction**Use manipulative when talking about fractions to your students*3 types of manipulatives (refer to notes for visuals):Region (surface area)Length (number line, line segment) Sets (Groups of Things)*fractional parts are equivalent parts: 1 part of 4 are equal sizes*4 parts of 8 are all equal partswhen you multiply by 2/2 --> 2/2 = 1 Discovery:a/a = 1 whole the numerator and denominator are the samefractional parts are equivalent (same size)The more parts per whole, the smaller the piece

Integers:Positive and negative numbersnumber line --> while helpful, some students might find this confusing<----|----|----|----|---|----|----|----> -3 -2 -1 0 1 2 3"Chip Method" --> more recent method to help our students understand positive and negative numbers negative & positive together = zero pair- (red) = negative ( - )+ (orange) = positive ( + ) 1) Additiona) +5 + +1 = +6 b) +5 + -1 = +4 c) -5 + +1 = -4 d) -5 + -1 = -6 (+) (+) (+) (+) (-) (-) (-) (-) (+) (+) (+) (+) (-) (-) (-) (-) (+) (+) (+) (-) (-) (+) (-) (-)a) +3 + -2 = +1 b) -6 + -2 = -8 c) +3 + -4 = -1 d) -2 + -1 = -3 (+) (+) (-) (-) (-) (-) (-) (-) (-) (-) (-) (-) (+) (+) (-) (+) (-) (-) (+) (-) (-) (-) (-) 2) Subtraction a) +5 - +1 = +4 b) -5 - -1 = -4 c) -5 - +1 = -6 d) +5 - -1 = +6 (+) (+) (-) (-) (-) (-) (+) (+) (+) (+) (-) (-) (-) (-) (+) (+) (+) (-) (-) (-) (+) (+) (+) (-) *add zero pair and take *add a zero pair and take 1 orange token away away 1 red token *add a zero pair b/c you add a positive token that you can take away and it gives you the answera) +3 - -1 = +4 b) -6 - -2 = -4 c) +3 - -4 = +7 d) -2 - -3 = +1 (+) (+) (-) (-) (+) (-) (+) (-) (-) (+) (+) (-) (-) (+) (-) (+) (-) (-) (+) (-) (-) (-) (+) (-) (+) (+) (+) (-)

3) Multiplication3 x 2 = 3 groups of 2 a) +3 x +2 = +6 b) +3 x -2 = -6 c) -2 x +3 = -4 d) -2 x -3 = +6 (+) (+) (+) (-) (-) (-) communatitve prop. *When a negative sign is (+) (+) (+) (-) (-) (-) +3 x -2 = -6 first, treat this as "opposite (-) (-) (-) of" so the opposite of -6 is (-) (-) (-) +6. a) +2 x -4 = -8 b) +2 x +4 = +8 c) -4 x +2 = -8 d) -2 x -4 = +8 (-) (-) (-) (-) (+) (+) (+) (+) *opposite of* *opposite of* (-) (-) (-) (-) (+) (+) (+) (+) (+) (+) (+) (+) (-) (-) (-) (-) (+) (+) (+) (+) (-) (-) (-) (-) --------------------------------------------------------------------------------------------------------------------------4) Division - Inverse Operationa) +3 x +2 = +6 = +6 / +2 = +3 or +6 / +3 = +2b) +3 x -2 = -6 = -6 / -3 = +3 or -6 / +3 = -2

Decimals*refer to notes for visual representations*100 boxes = 1 whole unit 1 = 10 boxes of the whole unit (100 boxes)10 1 = 1 small box of the whole unit (100 boxes)100 tens ^ Hundreds<--3 7 5-->ones$375 = 3 - 100 dollar bills 7 - 10 dollar bills 5 - one dollar billsIn regards to money, what does the decimal represent?Example: $375.25 --> .25 = dimes and pennies. .25 = 2: tenths, 5: hundredths. Pennies --> 1/100 of $1 dollar & 1?10 of a dimeDimes --> 1/10 of $1 dollarA penny is 1/10 of a dime : 1/10 --> Equation will read as: 1 x 1 = 1 A dime is 1/10 of $1 dollar: 1/10 --> 10 10 100 -->tenth of a tenthDecimals allow us to show quantity in more specific detailsmore precisemore accurate 3 = 0.7634 --> more accurate 4 --------------------------------------------------------------------------------------------------------------------------------------1 small box = 1/100 Break up the small box into 10 pieces and you get 1/1000 1 x 1 = 1 100 10 1000 --> tenth of a hundredth = thousandthsExample:3/10 = 3 hundredths 3/10 = .03Zero (0) placement is important for the placement after the decimal. This is important because we can understand the value after the decimal. Example: 12/100 = .12 --> 1 tenth and 2 hundredthsWhich is bigger? .9 or .12?.9 > .12 --> why?.90 is bigger than .12 students will think .12 is a bigger number because 12 is a bigger number than 9. *This is why manipulatives are important to understand place value of tenths and hundredths*Example:What numbers are between 0.3 and 0.4?0.31, 0.32, 0.33, 0.34, 0.35, 0.36, 0.37, 0.38, 0.39 2. What numbers are between 0.32 and 0.3?0.321, 0.322, 0.323, 0.324, 0.325, 0.326, 0.327, 0.328, 0.329 ^ this reads as 3 hundred twenty-one thousandthsExample:when reading numbers AND is where the decimal is <--.40

Adding/Subtracting Decimals*We must tell our students to line up the decimals*Example: 3.21 (Wrong Way) 3.21 (Correct Way)+ 1.2 + 1.20

Multiplying DecimalsExample: 2.9 --> 3x 2.1 --> x 2 6 --> Answer will be roughly 6 2.9 x 2.1 29 + 580 609 --> move decimal over 2 spots = 6.09Example: 2.3 --> 3/10 x 3.1 -->1/10 3/10 x 1/10 = 3/100 --> why we move the decimal over 2 spotsExample: 2.31 x 2.2 462 + 4620 5082 --> 5.082-------------------------------------------------------------------------------------------------------------------------------------Dividing DecimalsExample:369.36 divided by 3 friends/students*Kids are told to move the decimals straight up*You bring it up because you are shoring the whole number as well as the tenths and hundredths 123.12 --> 123.12 --> what each student will receive3 |369.36 3 06 6 09 9 03 3 06 6 0 Example:When dividing by a number with a decimal, move the decimal over to make a whole number. You will also move the decimal over in the number in the division bracket as well. (x10) makes it 10 times bigger <-- 3.1 | 369.36 --> (x10) need to make dividend bigger Whole number now <-- 31 | 3693.6 *The answer will be the same regardless if the decimal is moved*

Percentages% --> per cent = per 100 out of 100Example:30% of 300 means you pay 70% of 100 30% off of 300 --> 300 / | \ 100 100 100 = 70 + 70 +70 = $210Example:Fractions that can be converted to a denominator of 100 2 (x 2) = 4 (x 10) = 40 = 0.40 = 40% 5 (x 2) = 10 (x 10) = 100 3 (x 5) = 15 = 0.15 = 15% 20 (x 5) = 100 *With fractions without a denominator that can be easily be converted to 100, carry out the division*Example: 0.125 1 = 1 divided by 8 --> 8|1.000 8 - 8 20 --> 20 hundredths (2 tenths left over) - 16 40 --> 4 hundredths (40 thousandths) - 40 01/8 = 0.125 --> thousandths1/8 = terminated decimal (there is no remainder)0.125 --> round up to .13 --> 13% we round up because this is a 5th grade math problem, teach them to round up to make a percentages easier. Example: 0.8333 5 = 5 divided by 6 --> 6 | 5.0000 6 - 48 20 - 18 20 - 18 205/6 = 0.8333 = 0.83 (put a bar over the digit that repeats, not the whole answer 0.83 = 83% 5/6 = repeating decimal 0.353535 = 0.35 = 35% or 36% -------------------------------------------------------------------------------------------------------------------------------------pie or 3.14 --> pie is not a decimal because no numbers repeatAs of today, there are more than 3 trillion digits, and there are no repeating numbers or patterns--------------------------------------------------------------------------------------------------------------------------------------Dr. Spanias Rules for word problems:is : =of : x (multiple sign)what : n (variable)change % to a decimal 8 is what % of 22? = 8 ---> n x 22 --> 8 = .368% of 22 is what number? ---> .08 x 22 = n --> n = 1.76 8% of what number is 22? ---> .08 x n = 22 --> 275

On Tuesday our work papers were returned and together as a class, we went over all the questions on the exam.

Refer to Notes for fraction practice problems*In the notes, there are better visual representations*

Fractions have the same denominator, you can add and subtract easily --> common denominatorStudents will begin to have difficulty when the denominators are differentExample: (Refer to notes for better visual representations) 1 + 1 --> 3 + 2 = 5 4 6 --> 12 12 = 12 b. Equivalent Fractions: 4 and 8 6 12

Multiplication of Fractions: 4 x 3 = 4 groups of 3*Drawing pictures will be your best friend when dealing with fractions*When multiplying fractions, the product is smaller and smaller numbersMultiplying fractions = numerator x numerator denominator x denominator*Numerator by denominator* Examples: Refer to notes for better visual representationsDivision of Fractions: 6 / 3 (3 goes into 6 how many times?)Examples: *Refer to notes for better visual representations*

Refer to Notes for fraction practice problems*In the notes, there are better visual representations*