Elementary Mathematics

Units name the basei.e - 24 + 352 Longs & 4 Units + 3 Longs & 5 Units = 59

use FLU - Flat - Long - Unit i.e - 412 base seven 4 flats + 1 long + 2 units 196 + 10 + 2 = 208i.e - 346 base seven (smallest base it could be 7)3(72) + 4(71) + 6(70)3(49) +4(7) + 6(1) = 181

i.e- convert 172 to base five5/1725/34 r25/6 r41 r1 ------------------->1142 five1(53 ) + 1(52 ) + 4(5) +2(1) = base 10

Learn placements firstSteps:align the equationadd the numbers left to rightkeep in mind the correct placementi.e - 26 + 48 =26 twenty + forty = sixty-48 six + eight = fourteen60+1474i.e. 342+126=342+12640060+ 8468

steps:1) Take each number and separate it out.2) Add each row going down.3) Add left to right.i.e: 243 +125= 368200 + 40 + 3100 + 20 + 5300 + 60 + 8

1) Add the numbers in the units place starting at top.2) If sum ten or more put a scratch on the number, and write was is left over to the side.3) Count the scratches and add them to the next row, starting at the top.4) Continue until all rows are done being added.-Can add by tens or by the number given.

-Draw it out

Steps:1) Make the numbers "nice"2) If you change one change the other as well.3) subtract or add depending on the equation.Also known as sharing methodi.e. 24 - 18 =(24+2) =26(18+2) =2026-20=6

ie. 242--->+18 22 + 20 = 42

steps:1) Expand the numbers2) Add3) Multiply straight down and then across4) Add all results togetheri.e.25 x 36(20 + 5)+(30 + 6)60030120+150900

Draw it using the base ten blocks

Where there is a "+" make a line

EstimateMultiplySubtractThe biggest number doesn't always go on the insideIn n OutRemainder = to 1/2 a uniti.e.- 42 =345/ 6 ---> 45 = 7 3/4 6 x

estimatingsubtracting

Distributive: Always has multiplication and parenthesis- i.e. 6(5+3)Commutative: Addition and multiplication- i.e. 3+4 = 4+3Associative: Parenthesis- i.e. 3+(5+9) = (3+5) +9( changes who associates with each other)Identity: multiplication and addition- i.e. 5 x 1 - i.e. 5+1---------------------------------------------------------------------------  G    E   M.D. Lt -> Rt   A.S.  Lt -> Rt

Positive on top negative on bottomAlways start with what you want Real world Examples: - Thermometer - Bank Accounti.e. show -2 using 8{+ + +}{ - - -} - - Z.B. Z.P. Z.B. = Zero BankZ.P. = Zero Pairi.e. show 5 using 9+ + + + + + + - -

i.e. show 4+2+ + + + + +i.e. show 4+-2 (add two negatives)+ + + + - -i.e. 3 - 5 (add zero pairs)<---{+ + + + +} {+ + +} - - { - - -} --->i.e. -4 - (-1) (start with negative 4 and take away negative 1)- - - {-} --->

i.e. (3) (2) [three groups of two]+ + + + + +i.e. (3) (-2) [three groups of negative two]-- -- --i.e. (-3) (-2) + + + + + + {+ + +}{- -} {- -} {- - } {- - - }

Do not need the fake Zero place holderCan write numbers Left to Right

i.e. 47+39 = 47 + 39 40 + 30 = 70i.e. 247 + 32 = 247+ 32 200 + 30 = 230 240 + 30 = 270

number closest tooscientific roundingi.e. 43 + 27 43 + 27 40 + 30 = 70i.e. (55) (25) (55) (25) (60) (20) = 1,200

Rule for Two:If the last digit is evenRule for Three:If the sum of the digits is divisible by 3Rule for Four:If the last two digits are divisible by 4Rule for Five:If the last digit is a 5 or 0Rule for Six: If the number is divisible by 2 and 3Rule for Eight:If the last three digits form a number that is divisible by 8Rule for Nine:If the sum of all the digits are divisible by 9Rule for Ten:If the number ends in 0, divisible by 10

The ending branches show the prime factors of a numberi.e. Find the prime factorization of 24 24 / \ 4 6 /\ /\ 2 2 2 3 Prime Factorization = 2 x 2 x 2 x 3 or 23 x 3i.e. Find the prime factorization of 18 18 / \ 3 6 /\ 2 3Prime Factorization = 2 x 3 x 3 or 2 x 32

i.e. Find the prime factorization of 322 x 16 -> 4 x 4 = 2 x 2 x 2 x 2 or 25i.e. Find the prime factorization of 242 x 12 -> 2 x 6 -> 2 x 3 = 2 x 2 x 2 x 3 or 23 x 3

The number on the outside of the L has to be primei.e. Find the prime factorization of 28 2 I 28 2 I 14 7Prime factorization= 2 x 2 x 7 or 22 x 7i.e. Find the prime factorization of 56 2 I 56 7 I 28 2 I 4 2Prime factorization= 2 x 2 x 2 x or 23 x 7

Greatest Common Factor the largest number that divides evenly into all of the numbersi.e. Find the GCF of 24 and 30 (factor tree) 24 30 / \ / \ 4 6 3 10 /\ /\ /\ 2 2 2 3 2 524: 23 x 3 30: 2 x 3 x 52 x 3 = 6-----------------------------------------------------------------------i.e. find the GCF - look @ common23 x 3 x 5 x 7 - look for small #'s22 x 33 x 5 x 11 2 x 2 x 2 x 3 x 5 x 7 2 x 2 x 3 x 3 x 5 x 11 GCF = 2 x 2 x 3 x 5 = 60------------------------------------------------------------------------i.e. find the GCF of 12 and 16 (listing method)12: 1, 2, 3, 3, 4, 6, 12 - look for biggest common factor16: 1, 2, 4, 8, 16GCF = 4

Least Common Multiplethe smallest number that the numbers are factors of.i.e. find the LCM of the given - list all the factors 32 x 54 x 7 - Bigger 22 x 3 x 52 x 72 x 13 - MultiplesLCM = 22 x 32 x 54 x 72 x 13-------------------------------------------------------------------------i.e. find the LCM of 12 and 15 12: 12, 24, 36, 48, 60 15: 15, 30, 45, 60LCM = 60-------------------------------------------------------------------------Advantages :no exponentscan see what they are looking at

Composite: more than 2 factorsDivisor: is the number that is the divider (i.e. 50/10)Dividend: the number that is being divided (i.e. 50/10)Prime Factorization: prime numbers being multiplied together Multiples: the number that two factors create, big numbers (i.e. 3 x 5 = 15 )Factors: The small numbers that multiply together to make the big numbers (i.e. 3 x 5 =15)Natural Numbers: The regular counting numbers, that are positive (i.e 1,2,3...)

Only for numbers greater than 10Steps:1. Look at the numbers pick the number that there is more of "biggest pile".2. The biggest pile gets two signs "depending on the sign of the biggest pile" 3. Circle one sign from both sets with one left out.4. If the signs in the circle are different,subtract. If the signs are the same, add.5. The sign leftover is the sign of the answer.i.e. -26+14 -- + * Different, Subtract 26 -14 -12i.e. -35+(-10) -- - *Same, Add 35 +10 -45

Steps:1. Use Keep, Change, Change (keep the first sign, change the second sign, change the third sign).2. Use the sign diagrams the same way as in addition.i.e. -10 - (-15)Keep, change, change: -10 + (+15) - ++ *Different, subtract 15 - 10 = +5i.e. 8 - (-12)Keep, change, change: 8 + (+12) + ++ *Same, Add 8+12=+20

look at signs.Same signs = positiveDifferent signs = negative

Draw out the fraction by splitting the shape into the sections total. (Denominator)Shade in the number that shows how many of sections of that size there are. (Numerator)

Less than <Greater than >Equal to =-The big numerator shows the bigger fraction ( 2/5 > 1/5 )-If denominators are different, but the numerators are the same: The smaller the number in the denominator, the bigger the piece. The bigger the number in the denominator, the smaller the piece. ( 3/8 < 3/7 )Anchor fractions: When the numerator is doubled (1/2 = anchor fractions), (1 whole = anchor fraction).(5/11 < 13/25)*when you double 5 it equals 10 which means 5/11 is less than 11. When you double 13 you get 26 which means 13/25 is more than half because 26 is greater than 25.Cross Multiply: Cross multiply the fractions.7 11-- < --8 1284 88

Area Model: same sizeSet Model: same number of objectsLinear Model: Same lengthI_I_I_I_I *1/4

Steps:Multiply or divide both the numerator and denominator by the same number.Draw diagrams that represent the original fraction.i.e.

Steps:Draw a model for both fractionsCombine the models to create oneIf denominators are different make on model horizontal and the other vertical... shade in the numerator with the corresponding model... draw vertical lines in the horizontal model and horizontal lines in the vertical model.Combine both models so it has both vertical and horizontal lines.Count the number of boxes shaded in on the two original models.That number become the number of boxes that should be shaded in on the combined model.i.e.

Steps:Draw Models for each fractionTake away the shades boxes on he second fraction's models after drawing the vertical and horizontal lines on to the model.i.e.

Steps:Draw one model and draw vertical and horizontal lines to represent both fractions in one combined model.The intersecting shaded areas equal the answer.i.e.

Steps:write out as a sentence (what does the problem say)Draw out the figure.Start off with what you have.OrganizeShow:i.e. 3(1/4) --> 3 groups of 1/4i.e. 1/2(8)

Steps:Write out as a sentence (what does the problem say)Draw out the figure.Start off with what you have.OrganizeSimplifyShow:

FractionsSteps:Look at both fractionsbreak down the denominatorMultiply each fraction by whats missingAdd the numerators togetheri.e.Mixed FractionsSteps:Add both whole numbers togetherLook at both fractionsbreak down the denominatorMultiply each fraction by whats missingAdd the numerators togetherput whole number and fraction togetheri.e.

FractionsSteps:Look at both fractionsbreak down the denominatorMultiply each fraction by whats missingSubtract the numerators togetheri.e.Mixed fractionsSteps:Subtract both whole numbers togetherLook at both fractionsbreak down the denominatorMultiply each fraction by whats missingSubtract the numerators togetherput whole number and fraction togetherIf needed do keep change change (If improper fraction take away from whole number and subtract by denominator)i.e.

Steps:Look at fractionsLook at numbers individuallyIf possible break numbers down. ( factors)create the weird onesmultiply whats left over i.e. Backward "C"Steps:Draw a "C" on the side of each fraction *Multiply denominator by whole number then add the numerator To get an improper fraction.Look at improper fractionsLook at each number individuallyIf possible break numbers down. ( factors)create the weird onesmultiply whats left overi.e.