Elementary Mathemetics

8 base nine + 13 base nine = 21 base nine12 base six + 20 base six = 32 base six232 base four + 101 base four = 333 base four

25 base seven - 15 base seven =10 base seven34 base six - 15 base six = 15 Base sixConvert one Long into Units to be subtracted21 base four - 13 base four = 2 base fourSubtract one long (21-13), then convert the other into four units to subtract the three, after which there will be one unit left, in addition to the one in 21

UnitsThe number of individual unitsLongsA collection of units. Number determined by base, or how many units fit into that particular long. If 6 units made up the long it would be a base six long.FlatsA collection of longs. Number also determined by the base. If 6 longs could fit into the flat, then it would be a base 6 and hold 36 units.Any digit in a number may not be equal to or exceed the number of the base.

Multiply the number of Longs by the Base, then add the Units34 base six = 223 longs plus 4 units, 3*6= 18 + 4= 22423 base seven4 (7^2) + 2 (7^1) + 3 (7^0)4 (49) + 2 (7) +3196 + 14 + 3213any number to the power of zero is 1132 base four1 flat, 3 longs, 2 units1(4^2) + 3(4^1) + 2(4^0) 1(16) + 3(4) +216+12+230

Base ten # Base five Base six Base Nine8 13 12 812 22 20 1319 34 31 2141 131 105 45Converting 8 into base five you have 1 long and 3 units which is 13 base fiveConverting 19 into base six you have 3 longs and 1 unit which is 31 base sixConverting 41 into base nine take 41 divided by 9 which is 4 with 5 remaining, so 45 base nineConverting 19 into base four . . . ~ . . . ~ . . . ~ . . . ~ . . .. = 1 and ~ = a long, so that's 4 longs a 3 units which equal 43 base fourLLLLLLLLLeaving base 10 7|_152 =305 base seven

Make sense of problem and persevere                             Reason Abstractly/QuantitativelyConstruct Viable ArgumentsUse Appropriate ToolsLook for & make sense of StructureLook for Repeated ReasoningModelAttend to Precision

Understand the ProblemDevelop a PlanCarry out the PlanLook Back

Look for a PatternDo a Similar/Easier ProblemIdentify What Makes the Problem Hard, and Get Rid of the Hard part.Draw a Diagram (units, longs, flats)Guess and CheckWrite an Equation

Change one number to be a factor of tenAdd same number to both sides.Subtract normally.Ex: 64-22 = 64(+8)-22(+8)72-30 = 42

Each number is represented by a + or - sign, depending on their sign, positive or negative.Positive and negative signs = zero pair; collection is a zero bank.Ex: + + + - - -Can make as many zero pairs as desired/needed.SUBTRACT or TAKE AWAY is different from NEGATIVE.For multiplication, separate numbers into groups and units.Each group contains a number of positives or negatives.For negative numbers, TAKE AWAY groups of units.

2+(-3) = (++) + (---)3 + and 3 - create a zero bank + +- - - One - left overAnswer = -1(-5)+(-3) = (-----) + (---)Add negatives togetherNo zero bankAnswer = -8

(-2) -5 = (--) - (+++++ - - - - - )Take away the five positives created by zero bank(+++++) -->Left with seven negatives (-------)Answer = -7(-2)-(-1) = (--) - (-)Take away one negative (-) -->No need to create a zero bank Answer = -1

3(2) = Create 3 groups of 2 positives(++) + (++) + (++)Answer = 6-3(-2) = (++(--) -->) + (++(--) -->) + (++(--) -->)Take away three groups of two negativesLeft with only positivesAnswer = -6

Prime NumbersNumbers that only have exactly 2 factorsWhich are 1 and that number Ex: 2, 3, 5, 7, 11, 13Composite NumbersEvery other number; numbers with more than 2 factors4, 6, 9, 10, 12A number can't be both

The smallest number that goes into two or more numbers.Use smallest exponents for each number.Ex: 30 = 5x6, 3x10, 1x30, 2x15 / 12 = 2x6, 4x3, 1x126 is the Greatest Common Factor

The greatest number that two or more numbers are divisible by.Use greatest exponents for each number.Ex: 12 = 24, 36, 48, 60, 72 / 30 = 60, 90, 120, 15060 is the Least Common Multiple

The numerator shows how many pieces you have and how big the pieces areThe Denominator shows how many total and how big they are

Same Numerator5/11 < 5/7 a denominator of 7 is bigger pieces so 5 of those pieces is more than 5 of little piecesSame Denominator5/7 > 2/7Anchor Fractions4/9 < 6/114/9 is less than half, 6/11 is more than half ex: 1/2 , 1/4

If there's common denominator just add the numerators togetherEx: 3/7 + 1/7 = 4/7When adding, draw two boxes and divide them up per the fraction, shade in the amount of the numerators, draw both lines on each box, count up how many boxes are shaded from each, then draw a third box to add up how many individual spaces are shaded in both original models.Ex: 1/4 + 2/3 = 11/12

When subtracting, only two boxes are used, draw those two boxes and divide them up per the fraction, shade in the amount of the numerators, then draw the second line on each box, 'x' out the shading from the second box (second fraction, you're 'taking it' away), count the amount of boxes crossed out and crossed the same amount out from the first box, count up how many boxes are still shaded in the first box and that's your answer over how many total boxesEx: 1/2 - 1/5 + 3/10

When multiplying, only one box is used. Draw the box, draw the first set of lines per the first fraction, shade in the numerator, then add the second set of lines, shade in for the second fraction, count up how many boxes have both shading, and that's the answer over total amount of boxes.Ex: (1/2)(3/4)= 3/8When multiplying a fraction by a whole number, the first number is how many groups, and then the second number is how many is inside each group.Ex: 3 (1/3) 3 groups with 1/3 in each =1

Flip second fraction and multiply across

Count to base then scratch

243 200+40+3+133 + 100+30+3 = 300+70+6 = 376

579+864________1300 130 13________1443add hundreds spot, then tens, then one

Also called compatible numbers36+18+42+24+17+11+4336+18+42+24+17+11+4340+20+40+20+20+11+40

35 33+28 = +30 ____ ____ 63 47 46+19 = +20____ ______ 66

(157)(64) 100 + 50 + 7x 60 + 4______________60003000 420 400 200 28_______10,048

Order stays the same 5 + (7 + 1) = (5 + 7) +1

Order doesn't matter5 + 7 + 1 = 1 + 7 +5

7(x*2)6(5*4)

Order of operation redone - GEM/DA/SGGroupingEExponentsM/DMultiple/DivideA/SAdd/Subtract

The "show" for adding, subtracting, and multiplying decimals is very similar for the "show" for fractionsyou make one box for each you split it into 10 adding you fill in the amountif it goes into the hundredths place you have to make it split into tenths horizontally, toofor subtracting you cross out what you're taking away (just like fractions)and multiplying you count everything that's shaded with two colors

40% of 6010% of 60=66 x 4= 2445% of 60 10% = 6x 4% =x4____________40% = 24+ 5%= + 3____________ 45% = 27

This is the basic area model Now can we do it with unlike termsSuch as (2x+4)(4x+7)Top left box would be: 8x^2Top right box: 14xBottom left box: 16xBottom right box: 28= 8x^2+30x+ 28Diagonals always are like terms