Elementary Mathematics

Weeks 1-3

Weeks 4-6

Weeks 7-10

Solving integers with manipulatives or drawings

Adding Integers

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3 + (-5)= (+ + +) + (- - - - -) = -2Where 3 (+)s + 5(-)s = -2 because 1 - and 1+ makes a zero pair where they cancel each other out, leaving only 2 - = -2(-2)+ (-3)= (- -) + (- - -) = -5 because you add all negatives together

Subtracting Integers

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-2 - (3) = (- - ) - (+ + + - - -) in which you take away 3 positives(+) made by zero bank, then add together the negatives (-), giving you -5 as an answer.-3 - (-1) = (- - -) - (-), you take 1 negative (-) off and you get an answer of -2.

Multiplying Integers

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5(-2)= 5 groups of 2 negatives (- -) + (- -) + (- -) + (- -) + (- -) add the negatives= -103(1)= 3 groups of 1 positive (+) + (+) + (+) add the positives = 3-2(4)= take away 2 groups of 4 positive (+ + + +) (+ + + +) —>take away turning into negatives (- - - -) (- - - -) add all negatives up = -8-3(-5)= take away 3 groups of 5 negative (- - - - -) (- - - - -) (- - - - -) —>take away turning into positives (+ + + + +) (+ + + + +) (+ + + + +) add up all positives = 15

Factoring

Prime and Composite Numbers

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Prime numbers can only be multiplied by 1 and itself Ex: 3= 1 x 3 or 2 = 1 x 2 Composite numbers have other factors than itself and 1Ex: 4 = 1 x 4 and 2 x 2 or 0 = 0 x 7 and 0 x 4The number 1 is not composite or prime

Prime Factorization

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A way of listing the numbers that go into a certain number2 methods for prime factorization: Factor tree and Downward divisionFactor tree- 24= 3 x 8, then 8 is broken down further. 8= 2 x 4, then again to 4, 4 = 2 x 22 and 3 are factors, there are 3 number 2s and 1 number 3 which in prime factorization = 2^3 x 3Downward division- 36/3 = 12, then 12 is divided down until it is a prime number, 12/2 = 6 and 6/3 = 22 and 3 are te factors, there are 2 number 2s and 2 number 3s = 2^2 x 3^2

Lowest Common Multiple

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The lowest common multiple is the biggest number that 2 or more numbers can be divided by18 = 3 x 6 —> 6 = 2 x 3 and 54/3 = 18, 18/3= 6, and 6/3= 2The prime factorization of 18 is 2 x 3^2 and the prime factorization of 54 is 2 x 3^3 so the LCM= 2 X 3^3 because it has the bigger numbersAll factors of both numbers are written using numbers with greater exponent

Greatest Common Factor

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Greatest common factor is the smallest number that 2 or more numbers is able to be divided by12 = 3 x 4, 4 = 2 x 2 and 20 = 5 x 4, 4 = 2 x 2The prime factorization of 12 = 2^2 x 3 and the prime factorization of 20 = 2^2 x 5The Greatest common factor of 12 and 20 is 2^2 because that is the only common factorOnly smallest exponent of common factors

Fractions

About fractions

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Denominators must be the same when adding or subtracting fractions and numerators are the only thing added or subtractedIn multiplication the denominators of fractions do not have to be the same because both the numerator and denominator are multipliedNumerator = number of piecesDenominator = the size of the pieces

Adding Fractions

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Adding fractions- draw a rectangle and divide it into the number in the denominator of the fraction and fill in the amount based on the number in the numerator. Do the same for both fractions, ex: 3/4 and 1/53/4 = one rectangle divided into 4 horizontally and of those 4 there are 3 pieces shaded in1/5 = one rectangle divided into 5 vertically, and of those 5 there are 5 pieces shaded in For each box add the opposite fractions’ dividing lines, for the 3/4 fraction add the vertical lines from 1/5 to the box and in 1/5 add the horizontal lines from 3/4 which should result in 20 piecesIn rectangle put both the vertical and horizontal lines used to divide the first 2 rectangles and only shade the amount of boxes from the first two boxes = 19/20

Subtrcting fractions

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Using 2 boxes draw each fraction in their own boxesEx: 2/3 - 1/2 drawing a rectangle and vertically dividing into 3 only shading 2 pieces and doing the same for the other rectangle for 1/2Then add opposite lines to each fraction box which results in both boxes having 6 pieces each, then cross of the number shaded in 1/2 rectangle in the 2/3 rectangle = 1/6

Multiplying Fractions

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2/3(3/4) = 2/3 of a group or piece of 3/4Draw a rectangle divided vertically into 4 only shading in 3 pieces, then add vertical lines to split same rectangle into into 3 and then with another color shade in the first 2 horizontal rows of the rectangle, then count the number of pieces that are double-shaded, this gives you an answer of 6/12

Comparing fractions

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When comparing fractions you always want to choose the fraction that has the mostUse an anchor fraction to compare the fractions given to in comparison Ex: 2/3 and 3/8, use the anchor fraction of 1/2 where 2/3 is bigger than 1/2 and 3/8 is less than 1/2 so 2/3 > 3/8Another way to compare is to draw out each fraction and shade the amount in each fraction to see which fraction is missing less because that is the bigger of the fractions.The key to using that method is drawing all rectangles/boxes in as close in size as possible if not then it will not be to scale and possibly give an incorrect answer

Weeks 11-15

Order of operations

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-When solving problems with exponents, parenthesis, multiplication, subtraction, addition, division go in order according to GEMDAS1st-G=groups, at every plus or minus draw a line to group part of the problem together2nd-E= exponents, solve exponents next3rd-M/D= multiplication and division, whichever comes first going left to right4th-A/S= addition and subtraction, whichever comes first going from left to right

Division

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Division ex: 10/2 is asking how many are inside each of the 2 groups that equal 10(+++++) + (+++++)= 10In the 2 groups there is five in each which add up to 10Dividing fraction and whole number: ex- 3 divided by 1/3 you draw 3 groups with 3 in each group(+++)(+++)(+++) butyou only want one third of each so you end up with(1/3)(1/3)(1/3) which adds up to 3/3 or 1 wholeDividing fractions: KCF-Keep, change, flip24/9 divided by 10/6, you keep the first fraction the same, change the division sign into a multiplication sign, and flip the second fraction to its reciprocal, 6/10, the problem changes to 24/9 x 6/10. Then you multiply across and get 8/5 after reducing-Reduce fractions by writing out the numerator and denominators factors and crossing out same numbers, top can cross out bottom and vice versa but two similar numbers on top do not cross each other out, the same goes for the denominator-Dividing improper fractions: Turn improper fractions by multiplying denominator of fraction by whole number and then adding the numerator, put this number over the original denominator, do the same for both fractions the divide and reduce fractions as instructed above

Divisibility rules

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-2= number is divisible by 2 if it is an even number-3= number is divisible by 3 if the digits add up to a number that is divisible by 3 ex: 12, 1+2=3, 3/3=1-4=number is divisible by 4 as long as the last 2 numbers are divisible by 4ex:50328 last 2 numbers are 28, 28/4=7 so the whole number is divisible by 4-5=number is divisible by 5 if the number ends in 0 or 5-6=number is divisible by 6 if it i divisible by 2 and 3-8=number is divisible by 8 if the last 3 digits are divisible by 8-9=digits add up to multiple of 9 then the number is divisible by 9ex: 54, 5+4=9 so 54 is divisible by 9 and 54/9=6

Decimal algorithms

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-Adding or subtracting fractions= line up whole numbers o the decimal ends up n the right place -Multiplying fractions= multiply the whole numbers to get an estimate and you know where the decimal will goex: (3.2)(4.5), multiply whole numbers (3)(4)= 12 is the estimation, then multiply the numbers above without decimals so (32)(45)= 1440, then put decimal according to estimate, 14.40 is the closest that you could get to 12

Showing decimals

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-Showing adding decimals= draw two square and fill in according to decimals ex: 0.4 + 0.3, two rectangles drawn with 10 pieces because there is tenths in the decimals, in one only 4 pieces shaded in the other only 3 are shaded, count all pieces shaded, = .7-Showing subtracting decimals= draw 1 rectangle and shade number from first decimal and cross out number from second decimalex: 0.5 - 0.2, 1 triangle with 10 pieces and only 5 of them shaded, then go back and cross out 2, count the number of shaded peces that are not crossed off, = 0.3-Showing multiplying decimals= draw 1 rectangle and draw the number of pieces according to the place in the decimal, 0.3= 10 pieces, only 3 shaded. .33=100 pieces, only 33 shaded, then whatever the number being multiplied by, look at the decimal place and add those pieces to the first rectangleex: (0.7)(0.3) draw one rectangle with 10 pieces and shade in 7 pieces, then add 10 more pieces in the opposite direction and shade in 3 pieces in the direction drawn and count the total pieces that have been double shaded

Percentages

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-When finding any percentage always start out with 10% then multiply to get up to the desired percentageex: 40% of 8010% = 8X 4. X 440% = 32, 40% of 80= 32-Finding a percentage ending with 5Ex: do as above to get as close to percentage as possible then find 5% by dividing 10% in half and adding that to totalex: 45% of 8010% = 8X 4 X 440% = 32, then take 10% = 8 divide by 2, 5% =4 and add to total,40% = 32 5%. = 445% of 80 = 36-To find t how much you would pay if something is x % off subtract % off from 100 then solve as shown aboveex: How much would a $20 dollar shirt cost with 60% off?100%-60%= 40%10% of 20 = 2x4 X 440% of 20 = 8A $20 shirt that was 60% off would cost $8

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