Elementary Mathematics

BasesIn each base, the numbers used are the whole numbers from zero up to, but not including the base. The base commonly used today and taught to elementary students in America is base ten. In each base, all rules of math still apply. Blocks: There are units, longs, flats, and cubes. The units always represent one. The longs are whatever number the base is. If the base is three, then the long is worth three. The flats are the base squared. If the base is three, the flat is nine. The cubes are the base squared. If the base is three, the cube is worth twenty-seven. Each of the blocks correspond with a place value. The units represent the ones place. The longs are the next place value up, and the pattern continues for flats and cubes in that order.In base three, the way to represent the number 43 base ten ( 43ten ) is with one cube, one flat, two longs, and one unit.One of the main benefits of using blocks is that they are physical. It is easy for the student to see what is happening and understand the concept better. However, it is difficult to represent more than four place values because there is no representation for a fourth dimension.Exponents: For each place value, there is an exponent that corresponds. For the ones place, the exponent is zero. For the next place value the exponent is one. This pattern is continual. In base three the way to represent the number 43ten is 1 x 33 + 1 x 32 + 2 x 31 + 1 x 30 When this problem is solved, it is 43ten . When written in base three, it is 1121three .The beneficial part of using exponents is that they make it easy to change from ten to any base, and back again. It doesn't matter the number, the digits are able to be completely represented by exponents. Exponents are a little more difficult to use because they are not physical or hands on in the slightest. To use exponents, the student must have a firm grasp on bases, or else they will get easily lost.

BinaryBinary is base two. As seen in the video, it is most commonly used in computer programing. Base two only has two digits, zero and one. The instructions below are given to people who know how to subtract in any other base than two. Addition: Adding in binary is just light adding in any other base, except that only the numbers zero and one are used. This means that whenever a two is created by adding, it must be carried to the next highest place value as a one. 1 11 11 10010 10010 10010 10010 +10111 +10111 +10111 +10111 1 01 1001 101001Subtraction: Subtracting in binary is similar to addition. The difference is that instead of carrying, borrowing is the difficult part. When borrowing from the next highest place value, it is important to remember that the one in the higher place value represents a two in the lower place value. 1 1 1 0 10 0 10 10 10 0 10 10 10 0 10 10 10 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 -1 0 1 1 1 -1 0 1 1 1 -1 0 1 1 1 -1 0 1 1 1 1 1 1 0 1 1 0 0 1 1 1 0 10 10 10 1 1 0 1 0 -1 0 1 1 1 0 0 0 1 1

Mayan Number SystemThe Mayan number system is in base twenty. The place values are vertical with the highest place value at the top. Stones are used to represent one, sticks are used to represent five, and the shell is used to represent zero. The only time the shell is used is to hold a place value.Base ten:17 + 5 = 22Mayan Number System: o o o ________________ + = ________ ________ o o

Order of OperationsThe order of when to solve which operation.P - Parentheses. Solve what is inside the innermost parenthesis first.E - Exponents. Next solve the exponents and roots.M & D - Multiply and Divide. Do whichever one comes first reading left to right, then continue doing so until all the multiplication and division are solvedA & S - Add and Subtract. These follow the same rules as multiplication and division, just after the multiply and divide steps are done.

AdditionAll of what is described below is in base ten. All the the principles apply in other bases as well however.Properties of Additions:Commutative property - Changing the order of the addends does not change the sum. Ex 1+2 = 2+1Associative property - Grouping the addends in any combination does not change the sum. Ex 1+(2+3) = (1+2)+3Identity property - The sum of any number and zero is the number. Ex 1+0= 1Closure property - When adding two whole numbers together, the sum cannot be decimal. Ex 1+2 = 3Inverse Property - For every number, there is an additive inverse. Ex 1-1= 0Addition Algorithms:In each algorithm, begin with setting up the addition problem vertically. The definition and examples given are for double digit number, but can be applied to all numbers.Standard - Add the ones first and then the tens. When needed, carry the ten to the next place value. 48 418 418+59 +5 9 +5 9 7 107Partial Sums - Separate the Addends into tens and ones. Then add the ones together. Add the tens together. Add the sum of the ones with the sum of the tens. 48 8 40 90+59 + 9 +50 +17 17 90 107Lattice - Create diagonal lines (from top right to bottom left) in the ones and tens answer spots. Then add the ones. If it equals a two digit number, place the tens digit in the top part of the diagonal and the ones in the bottom. Repeat with the tens. Then add the tens values in the answer together. 48 48 4 8 4 8 4 8 4 8+59 +59 + 5 9 + 5 9 + 5 9 + 5 9 / / / 1/7 0/9 1/7 1/0 /7 1 0 7 Column - Draw vertical line in between and around the tens and the ones place. Add the ones together. If carrying is needed, place the number bring carried in the tens column. Then add the tens. If carrying is needed, place the number being carried in the hundreds column (outside of the tens column to the left). | 1| | 1| | 1| |4 |8 |4 |8 |4 |8 |4 |8 +|5 |9 +|5 |9 +|5 |9 +|5 |9 | | | |7 1|0 |7 1 0 7

SubtractionAll of what is described below is in base ten. All the the principles apply in other bases as well however.Subtraction Algorithms:Blocks - Using base ten blocks, represent the number with the least amount of blocks possible. Then take away the least amount of blocks possible to represent the amount being subtracted. 352-171Using blocks to represent 352, use three flats, five longs, and two units. 352-171 1Then take away one unit. There should be one unit remaining.2 15 352 -171 81In order to subtract seven, one must borrow from the next highest place value. When using blocks, take the flat and create ten longs out it. Then take away seven out of the now fifteen longs. There should be eight longs remaining.2 15 352 -171 181Then last step is to take away one the the flats. This should leave one flat left. The solution to the example is 181.Expanded - Split the number being subtracted into smaller, easier numbers to subtract.

Multiplication methods

Greatest Common Factor (GCF)Also known as greatest common divisor. Of two whole numbers a and b not both 0 is the greatest whole number that divides both a and b. If the greatest common factor of two numbers is 1, then the two numbers are relatively prime.Prime Factorization - Find the prime factorization of a and b. Do this by finding the set of prime numbers that multiply to a and b. Begin by dividing a and b by a prime number (not including 1) that goes evenly into it. Then divide the quotient of that division by another prime number. Continue this until there is only a prime number left. Then organize all of the prime numbers used when finding the prime factorization of a and b. Find the prime numbers and how many of each multiply into both a and b. The multiplication of these shared prime numbers is the GCF. The GCF can include multiple of the same prime number. Ex. a = 25 b = 40 GCF = 5  Colored Rods - Take the length of rod for both a and b. Using smaller length rods. line up multiple of them until the smaller length rods reach the lengths of a and b. The longest length rod that lines up perfectly with a and b is the GCF. Ex. a = 6 b = 8 GCF = 2Intersection of Sets - List all of the divisors of a and b. Find the set of all common divisors. The greatest element is the GCF. Ex. a = 20 b = 32 GCF = 4Least Common Multiple (LCM)All of what is described below is in base ten. All the the principles apply in other bases as well however.Prime Factorization - Find the prime factorization of a and b. Do this by finding the set of prime numbers that multiply to a and b. Begin by dividing a and b by a prime number (not including 1) that goes evenly into it. Then divide the quotient of that division by another prime number. Continue this until there is only a prime number left. Then organize all of the prime numbers used when finding the prime factorization of a and b. Find the prime numbers and how many of each multiply into both a and b. The LCM includes all of the prime numbers for each, but the prime numbers do not need to be used twice if in both a and b. Ex. a = 25 b = 40 LCM = 200Colored Rods - Line up rods of the length a and b until they line up to the same length. Multiply the number of a rods by a and do the same with b. The products of both should equal the same number. This number is the LCM. Ex. a = 4 b = 3 LCM = 12Intersection of Sets - List all members of the set of positive multiples of a and b, then find the set of all common multiples. Finally, pick the smallest element in that set. Ex. a = 6 b = 8 LCM = 24Number Line - Using a number line, make lines the length of a and b beginning at 0. Once the multiples of a and b match up, that is the LCM. Ex. a = 4 b = 3 LCM = 12Division by Primes - Divide a and b by the smallest prime number that goes into both of them. Divide the quotient by a prime number that goes into both of them. If the quotients are divisible by numbers that are the other quotient isn't divisible by, just divide the one quotient by it while keeping the other quotient the same. Continue until the quotients are prime numbers. Then multiply all of the prime numbers used to find the LCM. Ex. a = 20 b = 45 LCM = 180

DivisibilityDivisibility Rules:Divisor 2 - A number must be even. Ex. 2, 4, 6, 8, 10 are all divisible by 2 because they are evenDivisor 3 - The sum of the digits is divisible by 3. Ex. 1+2=3 so 12 is divisible by three.Divisor 4 - The number formed by the last two digits must be divisible by 4. Ex. 3312 is divisible by 4 because 12 is divisible by 4.Divisor 5 - The last digit must be 5 or 0. Ex. 40 ends in 0, so it is divisible by 5.Divisor 6 - The number must be divisible by 2 and 3. Ex. 36 is divisible by 2 and 3, so is divisible by 6.Divisor 9 - The sum of its digits is divisible by 9. Ex. 4+5=9 , so 45 is divisible by 9.Divisor 10 - The last digit must be zero. Ex. 40 ends in 0, so it is divisible by 10.Divisor 11 - The sum of the digits in odd-numbered places will be equal to the sum of the digits in even numbered places or will differ by a multiple of 11. Ex. 2+1=3 3+0=3 , so 2310 is divisible by 11.Dividing Real Numbers:Inverse Property of Division - Every real number besides zero has a reciprocal. Ex. The reciprocal of 8 is 1/8.The product of any number and its reciprocal is one. Ex. 8x(1/8)=1.Dividing by a number is the same as multiplying by its reciprocal. Ex. 8x(1/8)=1 and 8/8=1.Dividing by zero is impossible/ no solution.Sign Rules - Positive divided by a positive equals a positive.Negative divided by a negative equals a positive.Positive divided by a negative equals a negative.Negative divided by a positive equals a negative.

Types of NumbersDivisibility Rules:Integer - Any whole number, positive or negative.Real - Any number on the positive or negative number line.Natural - All positive numbers divisible by 1; not including 0.Rational - Any number that can be written as a fraction of two integers.Irrational - A number that cannot be written as a fraction of two integers.Imaginary Numbers - Any number with no concrete existence or value.

Positive and Negative IntegersInteger Rules - When adding positive integers, the value is going to increase. Ex. 9+5=14When subtracting positive integers, the value is going to decrease. Ex 9-5=4When adding negative integers, the value is going to decrease. Ex. 9+ (-5)=4When subtracting negative integers, the value is going to decrease. Ex. 9- (-5)= 14Chip Model Method - Use two different colors of chips. One color represents, positive and the other negative. Use the chips to represent adding and subtracting positive and negative integers. Ex. Number line method - Put a dot on the first integer in the problem. Then use an line and arrow to either add or subtract the other integer. Move right for addition and Left for subtraction. Remember that when subtracting a negative number, change the direction of the line and arrow. The answer is the end point Ex.

Fraction and Decimal PatternsBasics of Decimals:Expanded Notation - The number in the tenths place is multiplied by 10-1 then add the number in the hundredths place multiplied by 10-2. The pattern continues. Ex. 52.36419 5 x101+ 2 x100+ 3 x10-1+ 6 x10-2+ 4 x10-3+ 1x10-4+ 9 x10-5Speaking Decimals - Say the decimal as a whole number, but at the end say the smallest place value of the decimal. When changing from a whole number to the decimal part of the number use and. Ex. 52.36419 fifty-two and thirty-six thousand four hundred nineteen hundredth thousandths.Decimal Block Value - When using blocks to represent decimals, the flat represents 1, a long represents .1, and the unit represents .01 . Some of the downsides to using blocks is that it can be hard to represent numbers that have thousandths or hundreds. The blocks are good to use as a physical representation of a number though. They can also be used to show different base systems.Multiplying Decimals Using Blocks: Multiplying a Decimal by a Whole Number - Represent the number with the decimal in blocks. Then repeat that representation the number of the whole number. Count the number of flats, longs, and units. Ex. 3 x 1.4 = 4.2Multiplying a Decimal by a Decimal - Create a rectangle with one side representing the first number and the other side representing the second number. Then fill in the spaces until a full rectangle is created. This is done by using the flats, longs, and units to fill in the gaps.Ex. 2.1x 3.3 = 6.93Multiplying a Decimal by a Number Less than 1 - The main point to remember is that when multiplying a positive number by a number less than 1 that the product will be smaller than the original number. When using the block representation, instead of adding more blocks, blocks will be taken away. Ex. 1.2 x .4 = .48Divisibility Rules:1/5 - The decimal is double the value of the numerator. Ex. 2/5 = .41/7 - The decimal is a sequence of the numbers 142857. The order stays the same for each fraction, but the number that begins the sequence changes. The number the begins the sequence increases as the numerator increase. Ex. 2/7 = .285714 (repeating). 1/9 - The decimal is the numerator repeating. Ex. 2/9 = .2 (repeating).1/11 - The value of the decimal is the numerator multiplied by 9 repeating. Ex. 2/11 = .18 (repeating).1/13 - There are two sequences of numbers used for the decimal. the first one is 076923 and the second one is 153846. The decimal is repeating one of these sequences. The order of the numbers do not change, but the number that begins the sequence does change. The sequences are in order of least to greatest as the numerator increases from least to greatest. Ex. 4/13 = .307692 (repeating) and 5/13 = .384616

DecimalsThe principles stated here are set in base ten, however it can be changed to any base.Decimals and Grids:Tenths - The basic base 10 grid is drawn with 10 columns or 10 rows and 10 columns. The place value of one column is .1 Ex. Hundredths - The square must be split using 10 columns and 10 rows. Each square is .01 and the column is .1Multiplication - Shade in the area that is described by the multiplication. The product of a natural number and a decimal is a larger decimal. This rule has the exception of 1 and 0. The product of a rational number that is a decimal and another rational number that is a decimal is a smaller decimal. The exception to that rule is when two negative decimals are multiplied.Keys to Understanding DecimalsThere are no oneths. When learning place values for whole numbers, it begins with ones then tens then hundreds. However, with decimals it begins with tenths then hundredths. There are no oneths.When contrasting decimals, the one with the greater tenths place will be ther greater decimal. Many people see the decimal .58 as smaller than .6 because it seems like 58 is bigger than 6. However .6 is actually bigger because it has the greater tenths place. The tenths place holds more value than the hundredths place.When finding the number of place values for decimals, count all of the place values upto the last natural number. the decimal .708 has three place values. However, the decimal .780 only has two because the 0 at the end of the decimal holds no value.

Percent ChangeA percent change is an increase or decrease given as a percent of the original value.Changing from a percent to a decimal:When changing from a percent to a fraction, it is important to remember that any fraction less than 100% is going to be a decimal less than 1 and any percent greater than 100% is going to be a decimal greater than 1. When making a percent a fraction, move the decimal point two places to the left. Ex. 93% changes to .93 5% changes to .05 324% changes to 3.24 55.7% changes to .557The next part is changing from a decimal to a fraction. For this, move the decimal point two places to the right. Ex. .93 -> 93% .05 -> 5% 3.24 -> 324% .557 -> 55.7%Calculating the Percent Change:If the the new amount minus the original amount is negative then the percent change is a decrease. If the new amount minus the original amount is positive then the percent change is an increase. It is important to also add absolute value bars around the numerator so that the percent will be positive. It is impossible to have a negative percent.Word problems:I had seventy apples last year. This year I have 100 apples. What is the percent change?100-70 = 30 = .3 or 30% 100 100Was it an increase or a decrease? It was an increase because the numerator is positive.A more complicated example using variables:

Defining FractionsFractions are set up to represent rational numbers. The numerator is the number above the fraction bar and the denominator is the number below the fraction bar.Improper, proper, and mixed numbers:Improper - When the numerator is greater than the denominator. Ex. 9 5Proper - When the numerator is less than the denominator. Ex. 3 5Mixed Number - When there is a larger number in front of the fraction representing the amount of ones in the number. Ex. 2 3 5Exponents representing fractions:10-1 = -1 10-2 = -1 101 = 1 102 = 1 4-1 = -1 10 100 10 100 4Comparing fractions using rods:place the larger fraction on top and place the smaller fraction beneath. Then add more of the smaller fraction until it equals the length of the larger fraction. Then count how many of the smaller rods were used. That number is the denominator and one is the numerator when comparing fractions. An interesting thing to note is that the fraction is a simplified version of the smaller rod in the numerator and the larger one in the denominator. Ex.Another way to use the rods is to have the length of the smaller rod and the fraction that represents its relationship to the larger fraction. Use this to create an equation that equals the variable. Ex. The pink rod is 2/5 of the white rod. The first thing needed to solve this problem is to find the value of the pink rod. measure the amount of units that fit in it. The pink rod equals four. The next step is to multiply the fraction 2/5 by a number that will get four on top. In this case by multiplying 2/5 by 2/2 you get 4/10. The top of the fraction equals the pink rod and the bottom equals the white rod. In this case, the white rod equals 10. Number Line:A number line is another good way to represent and compare fractions. Create a number line from 0 to 1 to represent proper fractions. Then create the marks for the fractions.  Another way to compare fractions is to find greater and less than. When trying to do this, it is easier if the fractions have the same denominator. Here is an example of using a number line.How to pictorially make fractions have the same denominator. Here is an example of two fractions with different denominators. Make sure to have one fraction with the lines vertical and the other horizontal.Here is how to pictorially create the same denominator. Put the number of horizontal and vertical lines on top of each other. The amount of rectangles in the whole is the new denominator.With the picture as shown, it is easy to count the number of rectangles in the whole and find which one has the most and least, therefore comparing the fractions.

Adding and Subtracting FractionsThe properties of addition (commutative, associative, and the the identity property of addition) still apply when adding fractions. The concrete examples are shown drawn on paper to represent the physical objects that would be used.Addition and Subtraction:The most important thing to remember is that the denominators have to be the same in order to add or subtract fractions. When the denominators are the same, then just add or subtract the numerators and do not change the denominator. In the definition of fractions I demonstrated how to help fractions to have the same denominator pictorially, so I will not be repeating it here. Adding/ Subtracting fractions pictorially:First step. Create like denominators. Then add/ subtract the numerators.Adding/ subtracting fractions abstractly:Create like denominators by finding the LCM of the two denominators. Then multiply both fractions to get the same denominator. Make sure to multiply both the numerator and the denominator by the same number. Then add/ subtract the two numerators together. Ex. Adding/ subtracting fractions concretely:Using the hexagon, trapezoid, rhombus, and triangle pieces, it can be easy to add/ subtract fractions. Here are the values of each of the pieces.When adding/ subtracting fractions just use the pieces that represent the fractions. Then add or subtract pieces accordingly. Addition:Subtraction:

Multiplying and Dividing FractionsThe properties of multiplication (commutative, associative, and the the identity property of addition) still apply when multiplying fractions. The concrete examples are shown drawn on paper to represent the physical objects that would be used.Multiplying fractions pictorially:When multiplying fractions, first create the first fraction out of the area model with the lines vertically. Then split that area model into the denominator of the second fraction using horizontal lines. Then shade the second fraction into the area model. The amount that is shaded in by both fractions is the numerator. The amount of rectangles total is the denominator.Dividing fractions pictorially:Create an area model for both fractions. One horizontally and the other vertically and then combine the lines for both area models. find the amount of shaded rectangles for each fraction. Then find how many times you can subtract the divisor's number from the dividend. The amount is the whole number answer. If there are any of the dividend's rectangles left over, create a fraction using the left over as the numerator and the divisor's are as the denominator.Multiplying fractions abstractly:Multiply the numerators. Multiply the denominators. Keep them in the respective places.Dividing fractions abstractly:Multiply the first fraction by the reciprocal of the second. Below is the proof for this method.Multiplying fractions Concretely:When multiplying fractions, first create the first fraction out of the pieces. Then split that number into the denominator of the second fraction. Then using the newly created parts, take pieces of it equaling the amount of the numerator of the second fraction. Dividing fractions Concretely:When dividing fractions, you can ask yourself: How many ____ fit in _____? Create the Dividend out of the pieces. Then place the divisor over the Dividend. The amount of the divisor that fits in the dividend is the answer.