Elementary Mathematics: By Nathan McCormick

Base 10 consists of numbers 0-9, once you hit 9 you get an additional place value in the tens spot, 10. Base 7 works the same way; Base 7 consists of 0-6 and once you hit 7 it is now represented by 10- this is saying that you one 7 and zero units… this is NOT the number 10!!!

There are several methods that can be used to complete addition in different base systems and they are as follows:Column Addition Opposite Change Standard algorithmPartial Sums Lattice Having multiple methods for a concept is important because when introducing new concepts its common for students to not understand the material so having multiple methods can ease the learning process.

Separation into two different problems using visual aids and manipulative’s. Using blocks is a good manipulative Multiplication using arrays- this means to understand that multiplication is repeated addition . 4+4+4= 12. This can and should be represented along side with visual aids.

Divisibility rules used to make solving division problems much easier and also provide a deeper understanding into the relationships between numbers. Rules:Divisor- 2- A number must be even, it must end in a 0,2,4,6,83- The sum of the digits is dividable by 3 4- the number formed by the last two digits must be divisible by 45- the last digit must be 5 or 06- the numbers must be divisible by 2 and 39- the sum of its digits must be divisible by 911- The sum of the digits in odd numbered places will be equal to the sum of the digits and even number places or will the fire by a multiple of 11. 1 1 1 the numbers go odd-even odd.

Finding the Greatest common Factor is all about finding a number that is the largest number within the original numbers that can be multiplied to make both the original numbers. Using Prime Factorization is the easiest way to do this. To use prime factorization start by making factor trees for each number; then identify all prime numbers and take what they have in common to make you GCFTo Find the Least common multiple use prime factorization as well for the quickest method. Start with your number trees and then from there identify all prime numbers- here you will take the largest sums of each factor tree from what they have in common- then you will either have your LCM or you will need to place the number into a equation and find it that way. See picture for example problems

Multiplication with fractions:Fraction bars can be used to represent addition with fractions: for addition the answer should be represented in the denominatorFraction bar: for subtraction the answer should be present in the numeratorUsing Pattern boxes- use shapes to represent fractions- say a hexagon is equal to 1 use the other shapes to represent fractions of that one whole.Another way of saying multiplication problems with fractions is to say the problem in reverse: 1/2 x 3/4 = 3/4 of 1/2By using the blocks you can make a constant shape representing the whole, then you can form the first fraction representing a potion of that whole- from here take you second fraction and take that portion from the second hope formed- now relate that new amount to the whole that is the constant.Dividing Fractions Diving is to find out how many of something can go into something else Division with fractions can be thought of in the same light as multiplication- 3/ 1/2= 6 not 1.5multiply the number by the denominator when dealing with a whole number and a fraction When dealing with 2 fractions flip and multiple the second fractionfractions are another way to write division this is why finding the reciprocal leads to flipping and multiplying Don’t use cross multiplication as a teaching method, it causes confusion.

Associative property - relating our grouping two values together. It does not matter witch values are associated with each other, you can group any two that you like.Identity property of multiplication - if you multiplied a value by 1 you will get the same numberAdding fractions with fraction bars- to find the common detonator place one line on top of one of the others.Do not accept improper fractions as the final answer- convert these types of answers to mixed numbers.We use the fraction bars to find and explain the common denominator.All fractions on a ruler are terminating decimalsWhen using a ruler for addition or subtraction use the largest fraction as your marker and then count the dash marks for the smaller fraction. Then add or subtraction the dash marks to the largest.Fraction Bars- fraction bars should have a reference as to what is the whole or 1.all numerators should be 1 numbers align when they are either possessing an even denominator or when they are an a multiple

To be able to understand and identify fractions with visual aids and manipulative's. Understanding how to identify the correct fraction portion when comparing it to a whole number can be aided or eased with the use of manipulative such as rods. Understand the difference between an improper fraction (5/2), proper (1/2) or a mixed number (1 1/2).Comparing Fractions both concrete and pictorial- When comparing fractions you can either ensure they have the same denominator or you can convert them to a percentage.number actors must be multiplied by the same number as the denominator you can use the lowest common multiple or you can just settle for a common denominatorThe above notes represents the abstract form Showing fractions pictorially can help change the denominator by giving a visual representation of what the fraction actually looks like

To convert a decimal to a percentage divide the number by 100-to convert the decimal to an actual percentage just move the decimal 2 spaces to the right. Rounding decimals is even more simple- when asked to round a decimal keep in mind what place value you are rounding to, for a percentage you will always be rounding to the hundredth (the second place value to the left of the decimal) then will make a binary choice as to whether or not a number is at or above 5 (example: .485); if it is at or above 5 then you will round up making the new number .49. on the other hand if the number reads .484 then you will round down making the new number .48

The relationship between fractions decimals and percentages- A fraction can be written as a decimal and a decimal can be converted to a percentage because they percentage is just a ratio the same as what a fraction represents. Decimals fractions in percentages are all just variations of the same concept.Using 10 x 10 grids can be a good tool for learning the relationship between decimals fractions in percentages.Percentages are based off 100 so using a 10 x 10 grid allows you to see the exact decimal and therefore percentage of a numberFractions can be converted to decimals by dividing the numerator by the denominator a decimal can then be converted into a fraction by multiplying the numerator by 100 and then dividing by the denominator.Repeating decimals can occur in this will be marked with a line above the place values that are repeatingWhen working with exponents remember that a positive exponent move the decimal place to the right however many times the exponent is represented by and if the exponent is negative then it will be moved to the left creating a smaller number.

Teaching Decimals:12.61843twelve AND sixty-one-thousand…..Identifying place Value-. Tens, Hundreds, thousands…AND means the decimal point in written formWhen you go to teach decimals start by explaining place value with whole numbersdescribe decimals as the numbers that are in between other numbersnext go over the place values and use a 10 x 10 grid to explain how these numbers work and then relayed them to something that they previously know such as fractionsshow students how to convert one to another by using another concepts that they are familiar with such as multiplication and division

Blocks simplify multiplication with decimals by giving a physical manipulative for a physical representation. blocks can be used to group together both numbers in the equation and then be reused by adding them together to get the answer for the multiplication equation.AND - represents the decimal when put in written form.expanded form with decimals will look like this-21.3420+1+.30+.04

Add and subtract on a number line; use the first number as your starting point, move left if middle sign is/are negative, move right if middle sign is/are positive, answer is at the end point. positive x positive = positive positive x negative = negative negative x negative = positive charge field model use dots that represent positive and negative valuegroup any dots that may cancel out and circle said dotsthen any remaining dots should be the answer to the equationCircle everything togetherCommutative property- This means that you can switch numbers around and still get the same result Associative property Dash when the expression has three terms they can be grouped in any way to Xpress the same result Multiplicative identity property - Any number multiplied by one will remain the same Distributive property – when a number is distributed to others within parentheses distributive property – when a number is distributed to others within parentheses multiplying by zero results in zeroDividing real numbers: Inverse property of multiplication –Every real number besides zero has a reciprocalThe product of any number and its reciprocal is oneDividing by number is the same as multiplying by its reciprocaldividing by zero is impossible this will result in no solution

Real: any number on the positive and negative number lineNatural: so Counting numbers (one, two, three, four,…..)Rational: Any number that can be written as a fraction of two integers a/bIrrational: A number that cannot be written as a fraction - Pie, e, fImaginary:Integers: A whole number both positive and negativerational numbers can be made by dividing two integers- you can identify whether or not a number is irrational or rational by converting the number into its decimal form if the number in decimal form is terminating or non-terminating with a repeating pattern then the number is rational if it is anything other than less than a number is irrational

Integers- these are whole numbers that can booth positive and negative. ( …. -4, -3, -2, -1, 0, 1, 2, 3, 4,…..)-4 is opposite of 4Absolute value is simply the distance from zero20 = 20-5 = 50 = 0when teaching negative numbers try to have the student develop the concept that a -3 is just three spaces away from zeroHave students use a number line and have them plays zero at the center of that number line 1st go over the positive numberthen show them that numbers exist to the right of the zero and these are the negative numbersLearning negative numbers is important because they often come in real life situation such as discounts on a receipt or building things