Elementary Mathematics

Spiraling content is revisiting content that has been presented alreadyThis allows students to be reminded of old content so they don't forget anythingGives better understanding of content

These algorithms build off of one anotherExpanded Form: Is a good transition into subtraction, not the most efficientEx: 425-204400+20+5- 200+0+4____________200+20+1 =221Equal Addends(Subtractends): Adding(subtracting) equal amounts to both numbers to keep the distance from the numbers the same, but create friendlier numbersEx: 53-2753 (+3) = 56- 27 (+3) - 30_____26Ex: 300-219 300 (-1) 299 =81 - 219 (-1) - 218

Timed Tests: Inefficient way of learning multiplication Progress Charts: Also inefficient 9's or 8's tricks: Poor method, it does not actually reinforce the concepts that need to be learned Ways to improve promote a growth mindset with charts timed tests can be efficient if used with growth mindsetFlashcards can be a great method to improve multiplication skillsDo not buy flashcards make your own Flashcards: Separate into learning by groupsGroup 1 (1's, 2's, 10's, 5's)Group 2 (3's, 9's, doubles)Group 3 (4's, 6's, 7's, 8's)

Expanded Form: Similar to when it was applied to subtractionEx: 35 x 2830 + 5 (Start with 20 x 30 then do 20 x 5 )x 20 + 8 (Then go to 8 x 5 and then 8 x 30)_________( Add all the digits together for the answer)600 + 100 = 700 = 980240 + 40 + 280Left to Right: Similar to expanded form, except there is no rewriting in expanded it just follows the same processEx: 47 x 5347 (Multiply 50 x 40 then 50 x 7)x 53 (Next multiply 3 x 40 followed by 3 x 7)_______2000 = 2491 350 120 21Lattice Method: Draw a box with diagonal cuts from each corner and write a number on the top and the second number on the right then multiply across working right to left.Ex:

Traditional/long division: Complicated, and can be easily messed up for instance, always believing the smallest number goes on the outside. If one mistake is made the whole thing can be messed up.Upwards division: This method is written like a fraction with space between the digits on the top to be able to write the remainder there. divide each digit by the number on the bottom and write the answer to the right of the problem and the remainder in the space you left between the digits. Repeated Subtraction: The set up is the same as the traditional method. This method uses subtraction to reach the answer.

Multiplying Fractions:When multiplying fractions break down the top and bottoms of both fractions by what they are made up of.Then cross out the similar numbers with a funky one if they are diagonal or vertical from one another.After this multiply across left to right taking into account the ones for both the top and bottomMultiplying with mixed numbersWhen multiplying fractions with whole numbers estimate the fractions to whole whole numbers and multiply those to get an estimate.Use the backwards c method by multiplying the whole number by the bottom of the fraction and then add it to the top numberOnce both numbers are converted to fractions apply the method used in multiplying regular fractions of finding common multiples and using the funky one.Then multiply as with regular fractions from left to right on the top and bottom. Compare the answer to EstimateDividing fractions: Apply the Keep change flip methodThe first fraction stays the same the sign is changed to multiplication and the second fraction is flipped so the top becomes the bottom and the bottom the topThen the funky ones method is applied to the fractions and multiplied as normallyDividing mixed numbers: The backwards C method is applied to the mixed numbers and then the steps are followed as with dividing the regular fractions with the keep change flip method.

Fractions are represented by a box divided into sections represented by each fractionAdding fractions: The fractions are used to create squares split up vertically by the fractions. The squares are shaded based on the fractionThen the squares are divided horizontally by the opposite fraction.The pieces are then counted and shaded in another square that is split up by both the first and second fraction.Subtraction: The steps are similar to addition except the subtracted fraction is circled from the final square createdMultiplication: One square is created that is divided by the first fraction vertically and shaded and then followed by the second section horizontally. The answer is the intersection of the two different shaded colors.

The process for showing adding/subtracting/multiplying is almost the same as the process for fractions.It also builds off of the base 10 block knowledge Completely shaded squares represent a whole or 1.Use different colors for each of the two numbers when shading Showing adding: A square is divided into 10 sections vertically for basic one number decimals in the tenths place such as 0.4 + 0.3 and shaded in accordingly. For two digit decimals such as 0.63 + 0.35 the square is divided into 10 sections vertically first to account for the tenths place. Then the square is divided into 10 more sections horizontally to account for the hundreths place digits. Showing Subtraction: Like in addition the square is divided into sections according to the number of digits in the problem. When subtracting the amount being taken away is circled from the section that is shaded in and removed. Showing multiplication: Showing multiplication is represented by dividing a square into 10 sections vertically and shading in the amount that the first digit represents. Then the square is divided into 10 sections horizontally and the amount represented by the second number is represented by shading in the horizontal sections. The answer is the intersection of the two different colors shaded is the answer.

Adding and subtracting: First step is to roughly estimate the answer based off of the whole numbers. Then line up the numbers by whole numbers and add or subtract accordingly. Check that the decimal is placed correctly in the answer by comparing to the estimate. Ex: 48.14 - 24 ≈ 24 48.14- 24.00 Decimals may be added to show place_______ 24.14 Decimal is placed correctly since answer is close to estimateSolving Multiplication: The steps are similar to adding and subtraction and starts off with estimating the answer. The second step is to multiply the numbers without any decimals. Then the decimal is placed in position based on what best fits the answer of the estimateEx: 2.5(4.1) ≈ 8 25 the decimals are removed to multiply normally x 41______ 25 = 1025 the decimal is then applied 1000 where it best fits the estimate 10.25 Becomes the answerThis method does not work for numbers such as 0.02(0.6)In this instance multiplication is done by using the traditional method of counting decimals.Ex: 0.02(0.6) 0.02 x 0.6 decimals are counted for both of the numbers __________ 12 = 0.012 the decimal is placed by counting the numbers behind the decimals of both numbers and placing it in the answer

Order of operations is the order by which math is preformed to create the correct answerPEMDAS or Please Excuse My Dear Aunt Sally is the most well known version of the order of operations ParenthesisExponentsMultiplication however this method has been found to Division cause confusion and create incorrectAddition answersSubtractionMultiplication is on the same level as Division and Addition is on the same level as SubtractionA better alternative is using GEMDAS or GEDMAS which is the same thingG- stands for groupsE-ExponentsD/M- Divide and Multiply (from left to right if both are present)A/S -Add and subtract (also left to right when both are present)Groups are found where there are addition and subtraction signs present. Groups are found in between the signs

Standard form is the normal way that we write numbers, for instance 2,400,300 or even 221Basic Rules of Scientific notationAlways has a decimal always has an exponent always multiplied by 10 number multiplied by 10 has to be between 1 and 10 What is it used for? Usually used to represent really big numbers or really small numbers Steps to convert to scientificDetermine if the number is big or small (greater than 1 or less than 1) Big means a positive exponent a small means negative exponent.Place decimal point between the first two nonzero numbers Then put it as multiplied by 10 count the decimal places and use that as the exponent for 10Ex: 2,400,000 It is big so its going to be a positive exponent2.4 then the decimal is placed between the 2 and 4 2.4 x 106 since the decimal moved 6 places that is the exponent Ex: .00000065 This is a small number so it is a negative exponent6.5 The decimal was placed between the 6 and 5 6.5 x 10-7 since the decimal moved 7 places that is the exponentConverting scientific to Standard Form:First look at the exponent, a negative means it will be a small number and a positive means it will be a big numberThen the decimal will be moved the number of spaces as the exponent Check to make sure that the decimal is moved in direction that is based on wether or not the number will be small or big from the exponent sign Ex: 2.4 x 10-5 the negative means it will be small.000024 the decimal was moved over to the left 5 times to make it a small number, the zeros were added to fill the placesEx: 6.05 x 106 Positive exponent means it is a big number6,050,000 the decimal is moved to the right 6 spaces to make it a big number

Divisibility Rules2: any even number can be divided by 2 this means numbers ending in 0, 2, 4, 6, 8, 3 : A number is divisible by 3 if the sum of the digits is divisible by 34: A number is divisible by 4 if the last two digits are divisible by 4 5: Divisible by 5 if the last digit is a 5 or a 06: A number can be divisible by 6 if it is divisible by both 2 and 38: A number is divisible by 8 if the last three digits are divisible by 8 9: A number is divisible by 9 if the sum of the digits are divisible by 9 10: Can be divided by 10 if the number ends in a 0

Building integers with 2-color counters:The red side represents negatives and the yellow side represents positivesThe yellow sided counters go on the top of the red ones at all timesA zero pair is when there is a red tile and a yellow tile therefore adding 0 to the problemA zero bank is when there are the same number of red tiles as yellow causing the tiles to cancel and adding nothing to the problemEx: Show 3 using 5 tiles++++ - Addition with integersEx: Show 5 + 2+++++ ++ = 7 

Adding Integers AlgorithmIn the instance of 25 + (-12) hector's method is applied where the sign of the bigger group has two over the top of it and the smaller number has one of its sign above itOne of each of the signs are circled and the sign left not circled is going to be the sign of the answerIf the signs circled are opposite then the two numbers are subtracted, if they are the same then they are added

Building Subtraction: When building subtraction problems it is the same as in the instance of building addition problemsEx: 8-3++++++++ = 5_ _ _However in the instance of a problem like 4 - 7 the use of zero pairs or a zero bank is applied. Zero pairs can be added to create a zero bank that will allow for extra units to be present to take away fromEx:++++ becomes +++++++++ then the 7 units are taken away_ _ _ _ _++_ _ _ _ _ the remaining two positives are a zero pair with their negatives and taken away from the problem leaving only 3 negatives_ _ _Building Multiplication: Building multiplication problems are represented by creating groups. The first number represents the number of groups and the second represents what is in those groupsEx: 5(-3)_ _ _ _ _ _ _ _ _ _ _ _ _ _ _Ex: -4(3) When the first number is a negative it is best to think of it as 0 - 4(3). Then create a zero bank big enough for the problem++++++++++ + + + _ _ _ _ _ _ _ _ _ _ _ _ _Then the 4 groups of three are removed from the positive side + Then count the negatives_ _ _ _ _ _ _ _ _ _ _ _ _ excluding the zero pairthe answer is -12

Showing Subtraction: diagrams are drawn to show the number of digits starting off with and then the amount being subtracted is circled from the group and taken awayEx: 5-3+++++ three of these positives are circled and removed to reveal the answer of 2 In the event of a problem where the signs are opposite a zero bank is created Ex:-2 -5_ _ will become +++++++ _ _ _ _ _ _ _ _ _five of the positives are circled and taken away creating++_ _ _ _ _ _ _ _ _ Then by counting the negatives excluding the zero pair the answer of -7 is reached Showing Multiplication: The first number represents the number of groups to draw and the second is how many are in each group

Solving Subtraction: In the problem 15-5 it is the same as the problem 15 + (-5) it is just adding the oppositeIn subtraction problems the method of Keep Change Change is applied. Where the first number remains the same, the change the second sign, and change the third sign K C CEx: -43 - 20 -43 + -20 Hector's mini diagram can be applied here_( _ _ ) The signs are the same and means that the numbers are added the left over negative means the answer will be negative-63Multiplication/ Division Rules:Same signs equal a positiveDifferent signs equals a negativeEx: -24 = 6 -4 same signs here creates a positive answerEx: 6(-4) = -24Opposite signs here creates a negative answer

The top number of a fraction tells us the number of thingsThe bottom number tells the size of the piecesThe bigger the bottom number the smaller the piecesThe bigger the top number the more pieces you have

Solving Fractions: With fractions that have the same size pieces or same denominator the top number is added or subtracted like a normal equationEx: 4/11 + 9/11=5/11When adding fractions that have whole numbers start with the whole numbersEx: 8 4/10 + 2 2/10 then do the fractions 4/10 + 2/1010 6/10When subtracting a fraction from a whole number turn the whole number into another whole number that has the same value as before and has the same size pieces as what is subtracted from itEx: 8 - 4/11 8 becomes 7 11/11then subtract the 4/11 from that7 11/11 - 4/11 = 7 7/11When two fractions have different size pieces, what makes up the bottom numbers can be found to see what is missing from each fraction. That number is multiplied by both the top and bottom so that it is multiplied by one and stays the same value.Ex: 6/18 - 1/618 consists of 6x3 which means that 1/6 is missing the value of 31/6 (3/3) 1/6 is multiplied by 3/3 to keep the same value6/18 - 3/183/18Ex: 8/12 - 2/6(2/2) 8/12 - 2/6 (4/4) 12 consists of 3x4 and 6 consists of 3x2the left side is missing 2 and the right side is missing 4 so they are multiplied to their respective sides by both the top and bottom16/24 - 8/24 = 8/24once multiplied the size of pieces should be the same and be easier to subtract

Comparing fractionsUsing anchor fractions like 1/2 can help to compare fractions. If one fraction is greater than 1/2 and the other is not then the first fraction is bigger Ex: 7/11 > 4/9If the fraction has the same number of pieces the fraction with the smaller size of pieces is the smaller fraction Ex: 2/7 > 2/9If both fractions are missing one piece then the fraction with the smaller number of pieces is greater because it is missing less than the fraction with bigger pieces.Ex: 14/15 > 9/10When there are the same size pieces in both fractions the one with more pieces is biggerEx: 5/12 > 4/12When there are whole numbers and fractions the one with the bigger whole number is the bigger fraction Ex: 4 3/12 < 6 1/12