Elementary Mathematics (MTE280)

A is divisible by B if there's a number C that meets this requirement.c x b = aex: 10 is divisible by 5 2x5=10Divisibility RulesEnding: a # is divisible by 2 if it ends with..... 0,2,4,6,8 a # is divisible by 5 if it ends with..... 0,5 a # is divisible by 10 if it ends with..... 0Sum of digits:add all of the digits of a # to see if it is divisible by the #by 3: if the sum of the digits of a # is divisible by 3, then the whole # is divisible by 3by 6: a # is divisible by 6, if it's divisible by 2 and 3Last digits:by 4: a # is divisible by 4 if the last 2 digits form a # that's divisible by 4by 8: a # is divisible by 8 if the last 3 digits form a # that's divisible by 8 --> most likely will have to do long divisionBy 7: a # is divisible by 7 if you double the last digit and subtract the # from the original # (only the 100's place) to see if it's divisible by 7By 11:chop method ---> chop off the last two digits and add it to the remaining #---> if you double, you subtract---> if you chop off, you add

24: 1,2,3,4,6,8,12,2442: 1,2,3,6,7,14,21,4239: 1,3,13,39136: 1,2,4,8,17,34,68,13691: 1,7,13,9160: 1,2,3,4,5,6,12,15,30Prime numbers: #'s that have only 2 factors - 1 and itselfComposite numbers: #'s that have more than 2 factors

All prime numbers 1-60:2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59Best ways to factor: Use the factorization tree OR list method to find the Greatest Common Factor (GCF) and Least Common Multiple (LCM)

Models: Set (group of things)Fraction: Part-WholeQuotientRatio

Fractional parts are equivalent parts.When the numerator and denominator are the same, it equals one. EX: 6/6 = 1The more pieces you put, the smaller the pieces get. To simplify fractions, you need to:Find a common factorDivide by 1** The bigger the common factor, the less steps you will have to simplify a fractionEX:25/100Find a common factor: the greatest common factor is 25Divide by 1: DIVIDE 25/100 by 25/25ANSWER: 1/4

Adding fractions with common denominators:1/4 + 2/4 = 3/42/5 +2/5 = 4/53/8 + 4/8 = 7/8** add the numerators, don't change denominator.Subtracting fractions with common denominators:5/7 - 2/7 = 3/79/10 - 1/10 = 8/10** subtract the numerators, don't change denominator. If the denominators aren't the same?Find a common multiple for the denominators. Multiply the fraction by oneEX: 1/4 + 1/6Common multiple for denominators: 12Figure out what each denominator needs to be multiplied by in order to become 12Multiply 1/4 by 3/3 (1)Multiply 1/6 by 2/2ANSWER: 3/12 + 2/12 = 5/12

Multiplying fractionsMultiply ACROSSEX: 1/3 X 1/8 = 1/24Dividing fractionsKeepChangeFlipEX: 2/3 divided by 4/5Keep 2/3 the sameChange division sign to multiplication signFlip 4/5 to 5/4Solve2/3 X 5/4 = 10/12

In order to successfully teach children how to problem solve with fractions, you HAVE to draw pictures!Each fraction of a whole is equal in value.Example:If **** represents 2/7 of the whole, draw what the whole looks like.** = 1/7**** = 2/7****** = 3/7******** = 4/7 ********** = 5/7************ = 6/7************** = 7/7

3 7 5 . 3 2100's 10's ones 1/10 1/1003/10 is 3 dimes3/10 = 0.3 = 0.300.8 = 8/10 "eight tenths"0.123 = "123 thousandths" 0.003 = three thousandths72/100 = 0.7231/1000 = 0.0310.83 = 83/1005/6 = 5 divided by 6 (do long division) = 0.83 (3 repeating)

Pay attention to the wording of a problem - it will tell you what you need to solve for. 9/25 students voted for Marina, what percentage voted for Marina? 25n/25 = 100 x 9/25 n = 100 x 9 / 25 36% of students voted for Marina

Use square method in order to solve the problems.Use long divisionDetermine whether the answer is terminating or not.

Integers are positive and negative numbers.-------l--------l------l-------l-------l-------l-------l-------l-------l----- -4 -3 -2 -1 0 1 2 3 4Chip method: use two-colored tokens for probability, positive/negative numbers.red: negative (-)yellow: positive (+)(+) & (-) = "zero pair"Example:+5 + +1 = +6 (+)(+)(+)(+)(+) + (+) = (+)(+)(+)(+)(+)(+)

-5 + -1 = -6 (-)(-)(-)(-)(-) + (-) = (-)(-)(-)(-)(-)-5 + +1 = -4 (-)(-)(-)(-)(-) + (+) = (-)(-)(-)(-) **cross out (+) & one (-) because it is a zero pair

+5 - -1 = +4 (+)(+)(+)(+)(+) - (+) = (+)(+)(+)(+)-5 - +1 = -6 (-)(-)(-)(-)(-) - (+)(-) = (-)(-)(-)(-)(-)(-) **cross out (+)

+3 x -2 = -6 (-)(-)(-)(-)(-)(-) **3 groups of -2+6 / 2 = +3 (+)(+)(+)(+)(+)(+) ** 3 groups of 2 in 6

INTRODUCTIONThe syllabus includes just about all the information for this course that I could ever need. It includes contact information, policies, assignments, and information about the course itself.The syllabus can be found on canvas.

POLYA'S RULESStep 1: Understand the problem.Step 2: Devise a plan (translate).Step 3: Carry out the plan (solve).Step 4: Look back (check and interpret).We solved a few problems in class using these steps. Multiple students would go up to the board and write out how they solved problems to show different ways of interpreting the problems.

POLYA'S RULES Step 1: Understand the problem.Step 2: Devise a plan (translate).Step 3: Carry out the plan (solve).Step 4: Look back (check and interpret).This week we solved more problems while applying this method. The biggest thing I took away from this week is that MULTIPLICATION tables are the key to solving a lot of problems!Our professor had us start off as doing problems the "hard" way, then at the end of class revealed the "easy" way. It was very helpful to start our problem solving without having the easier method so that we could understand how things are broken down and how different students could interpret problems differently.

BASE-10 SYSTEMIn the US, we use the Base-10 system (also referred to as a decimal or positional system)Numbers always relate to each other on a 1-10 relationshipThe spot where a number is placed indicates the number's valueWhen you move to the left, the number gets bigger by 10 timesWhen you move to the right, the number gets smaller by 10 timesFor example:375the 5 is considered the "ones-units"the 7 is considered the "tens-units"the 3 is considered the "hundreds-units"The same applies for different units such as decimals, money, miles, etc.

Expanded Notationthe simple definition is to break down or expand numbers.For example in Base-10:375 = 300 + 70 + 5375 = (3x100)+(7x10)+(5x1)375 = (3x10^2)+(7x10^1)+(5x10^0)

BASE-5 SYSTEMAustralians use the Base-5 System because there are 5 fingers on a handThe place values differ from Base-10Digits used: 0,1, 2, 3, 4, 10(base 5), 11(base 5), 12(base 5), 13(base 5), 14(base 5), 20(base 5)How to convert Base-5 to Base-10:232 (base 5) = (2x25)+(3x5)+(2x1)232 (base 5) = (2x5^2)+(3x5^1)+(2x5^0)232 (base 5) = 50 + 15 + 2232 (base 5) = 67The video shows a much faster, straight forward way to convert without showing as much work. But, it shows what the thinking process should be while converting

Addition (aka putting things together)Identity property: for any number a, a+0=a (when you add zero, a number's identity does not change)Commutative property (aka "order" property): for any two numbers a & b, a+b=b+a (the order you add the numbers does not matter)Associative property (aka "grouping" property): for number a, b, c: (a+b)+c=a+(b+c) - the way you group any numbers does not matter.Addition AlgorithmsAmerican Standard - should be the last stepPartial Sums - emphasizes place value, right to leftPartial Sums (b) - adds them by actual place value, right to leftLeft-to-Right - emphasis on place value, kids read left to rightExpanded Notation - emphasis on place value, separate by values, the BEST way to start teaching additionLattice - just for fun. Not in traditional textbooks

SubtractionTake away: 4-1=3Comparison (?): 4-1=3 - I read 4 books and she read 3 books. How many more books did I read than her? (3)Missing addend (?): 4+( )=7 ----} 7-(4)=( ) - Ivan has 4 cookies, mom gave her more and now she had 7. How many cookies did mom give Ivan?Subtraction Algorithms

Multiplication (aka groups of things)Identity property: a, a x 1 = aCommutative property: a, b: a x b = b x aAssociative property: a, b, c: (a x b) x c = a x (b x c)Zero property: a, a x 0 = 0Distributive property: a x (b x c) = (a x b) + (a x c)Arrays---------} #2________________l 0 0 0 0 0 0 0 l l l 0 0 0 0 0 0 0 l ll 0 0 0 0 0 0 0 l l #1 3 x 7 = 0l 0 0 0 0 0 0 0 l ll 0 0 0 0 0 0 0. l ll 0 0 0 0 0 0 0 l V ________________

7 minuend-3 subtrahend---- 4 differenceSubtraction AlgorithmsAmerican StandardEuropean/MexicanReverse IndianLeft-to-RightInteger SubtractionExpanded Notation*** Refer to notebook on how to do these.

Multiplication AlgorithmsStandardPlace ValueLattice*** Refer to notebook on how to do these.

8 = dividend _________ 2 = divisor 2 l 8 the answer = qoutient 8 / 2 = 4 R0Division AlgorithmsStandardPlace ValueAlternate Algorithm*** Refer to notebook on how to do these.