Elementary Mathematics

Classes consisted of going over the syllabus, course and class expectations, and learning our way around MindMap and MyLab.

There are several different bases to work with in mathematics. We started out by learning the standard base 10 both visually on paper and manipulatively with blocks.

How to create base 10: a Flat looks like a large square (100); a Long looks like a single line (10); a Units is a single cube (1).

You can work in any number base except 1. Some common bases to work with are:base 2 (binary)base 8 (octal)base 10 (denary)base 16 (hexadecimal)

"The Mayan culture used a base 20 number system. It was an additive positional system that used two symbols, a dot for 1, a horizontal bar for 5, and a cwory shell for a place holder..."

"A binary number system is one of the four types of number systems... binary numbers are represented only by two symbols or digits, i.e. 0 and 1..."

The order of operations is essential in solving equations that involve multiple math properties. It explains how to solve the equation in a specific order step by step. An easy way to remember the order is by using the abbreviation "PEMDAS": Paranethesis, exponents, multiply, divide (whichever comes first), add, subtract (whichever comes first). If you do not solve the quation in this order, your answer will be incorrect.Example:4 + 9 - (2^3) / 44 + 9 - 8 / 44 + 9 - 213 - 211

A few addition methods are: Addition using an open number lineAddition using partial sumAddition using regroupingExample:14 + 510 + 4 + 510 + 919

How to add in a different base: Rewrite the addition vertically, if not already.Start in the ones place (as usual), but find the number the sum represents in base b.If the sum is larger than base b, then carry over to the b1 place value.Repeat steps 2 and 3 for the b2, b3, … place values.

"The most common subtraction algorithm is the Right to Left Standard Subtraction Algoithm, which is where you start in the ones column and subtract, then move to the left and subtract at each column. The problem, of course, is when the top digit is less than the bottom digit and you have to regroup..."

How to subtract in different bases: Rewrite the subtraction vertically, if not already.Start in the ones place (as usual), but find the number the difference represents in base b.If the ones place of the minuend is smaller than the ones place of the subtrahend, then borrow from the place value to the left in that base b. Then subtract as usual.Repeat steps 2 and 3 for the b2, b3 place values.

You can practice subtraction algorithms using base 10 blocks.

"In math, divisibility refers to a number's quality of being evenly divided by another number, without a remainder left over. You can easily see the divisibility of 40 by 4, for example. The divisibility of a large piece of land means that it can be broken into smaller parcels to be sold separately, and the divisibility of a pan of brownies is clear when you slice into it and dole out equal portions to all of your friends."

"The Greatest Common Factor (also known as GCF) is the largest number that divides evenly into each number in a given set of numbers.The Least Common Multiple (also known as LCM) is the smallest positive multiple that is common to two or more numbers." GCF example: (prime factorization) 20 40 / \ / \ 10 2 20 2 / \ / \5 2 10 2GCF of 20 and 40: 20LCM example: (listing sets)8: 8, 16, 24, 32, 40, 48, 56, 64, 7224: 24, 48, 72LCM of 8 and 24: 72

"Prime factorization is a process of writing all numbers as a product of primes. So, for example, say if we have something like the number 20. We can break that down into two factors. We can say, “well, that’s 4 times 5.” And notice, 5 is a prime number." You can use a facor tree to do this process.Example: 81 / \ 9 9 / \ / \3 3 3 3

"Integer OperationsIntegers are all counting numbers, their negative counterparts, and zero. Integers include numbers such as 8, 342, -27, and -1. The four basic math operations - addition, subtraction, multiplication, and division - can be applied to all integers. Addition and subtraction are opposite functions and alter the value of integers by moving left or right on a number line. Multiplication is an abbreviated process of repeated addition. Division is the process of splitting an integer into equal parts. Multiplication and division are opposites. Integer Operations RulesEach integer operation has specific rules. When multiple operations exist in one expression, the operations must be performed in a specific order, called the order of operations. The order in which the operations are performed can be described by the acronym PEMDAS, which stands for parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right)."

If a number has a 5 or above in the ones place, you round up to the nearest number that ends in 0. If a number has a 4 or below in the ones place, you round down to the nearest number that ends in 0.

The Charged Field Model is a visual representation of how to solve equations using integers. This model represents integers by using plus signs for positive numbers and minus signs for negative numbers. The positive charges "neutralize" the negative charges. Example: 5 - 4 =+ + + + +- - - - =+

"Irrational number, any real number that cannot be expressed as the quotient of two integers—that is, p/q, where p and q are both integers."

Real Numbers include: Whole Numbers (like 0, 1, 2, 3, 4, etc) Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc ) Irrational Numbers (like π, √2, etc )Real Numbers can also be positive, negative or zero.So ... what is NOT a Real Number? Imaginary Numbers like √−1 (the square root of minus 1) are not Real Numbers Infinity is not a Real Number

Relationhips among mumbers include order of operations, "rule of ___", number properties, and more.

"Decimals can be written in fraction form. To convert a decimal to a fraction, place the decimal number over its place value." Example:In 0.6, the six is in the tenths place, so we place 6 over 10 to create the equivalent fraction, 6/10. If needed, simplify the fraction. 6/10 simplifies to 3/5.

A terminating decimal means it repeats the last number on forever (.333333....). The way we mark a terminating decimal is by placing a single horizontal line over the number(s) being repeated. A non terminating decimal is a number that doesnt repeat forever (.56483934...).

To multiply with base 10 blocks, you fill in the units, longs, and flats, with the correct numbers in the matching places. For example, if you were multiplying .24 x .2, you would first add a zero to the .2 so that each number has the same amount of place values. Then you multiply 2 x 4, which is 8. After this you multiply 2 x 2, which is 4. You then place the decimal in the correct spot and you get the answer .48. Since you have a 4 in the tens place, you shade in 4 longs. Since there is an 8 in the ones place, you shade in 8 units.

Correcting errors is important to know for many reasons. One of those reasons is that when you are a teacher, you need to be able to identify and correct your students mistakes so tht they are able to learn where they went wrong and correct it in the future.

Just like with multiplying decimals using base 10 blocks, you shade in the correct blocks that correspond with the correct number places.

Desmos is a web application consisting of an advanced graphing calculator along with several math activities to do and problems to solve. We used Desmos many times throughout this last part of the course to help us review what we learned in class.

Percentage change can include a percentage increase, or a percentage decrease. A percentage increase means the numbers rose over a certain period of time, and a percentage decrease means the numbers fell over a certain period of time. The formula for calculating a percentage change is: New # - Old # / Old #.Example:In 2020, Sophie had 10 snowglobes in her collection. In 2022, she now has 18 snowglobes in her collection. What was the percentage change of snowglobes from 2020 to 2022?18 - 10 / 108 / 10.8There was an 80% increase in snowblobes over 2 years.

Interest is calculated as a percent of the principal in simplest form.Example:If you borrow $100 from a friend and agree to pay it back with 5% interest, then the amount of interest paid will be 5% of the $100, which is 100 x .05 = 5. $5 in interest will be paid back.

A rational number is any number that can be written as a fraction, where both the numerator and denominator are integers and the denominator is not equal to 0. A rational number can be expressed as p/q, where p and q are both integers and q ≠ 0.

Simplest form is when you break down a fraction into the smallest number it can be. For example, if I was asked what .5 was as a fraction, 5/10 would be the correct answer. But if I was asked what .5 was as a fraction in simplest form, I would divide both the numberator and denominator by their greatest common factor (5) to get 1/2.

Mixed numbers and improper fractions go hand in hand. Usually if you get an answer as an improper fraction, you will be asked to turn it into a mixed number. An improper fraction is a fraction where the numerator has a higher value than the denominator. A mixed number is a whole number combined with a fraction. Example of a Mixed Number:5 1/2Example of an Improper Fraction: 12/4

When adding and subtracting fractions, it is essential to make sure the denominator is the same. If you are adding or subtracting fractions with different denominators, you need to find the LCM between the numbers and then calculate. Example: 3/4 + 2/6 (LCM = 12)3/4 (3/3) + 2/6 (2/2)9/12 + 4/1213/12 (1 1/12)

Like and unlike denominators are a very important part of adding and subtracting fractions. If the denominators are unlike, you cannot operate the equation correctly. To calculate the correct answer, it is essential to find like denominators before operating.

Rounding can be done with whole numbers, fractions, decimals, and percentages. Any number that ends in a 4 or lower, will be rounded down to the nearest 10. Any number that ends in a 5 or higher, will be rounded up to the nearest 10. Example: 3.4 = 3.05.7 = 6.0

Multiplying and dividing fractions is a bit simpler than adding and subtracting them. Instead of making the denominators the same, you first multiply the numerators together and then the denominators together. Then you put the answer in simplest form. 2/4 x 4/62 x 4 = 8, 4 x 6 = 248/24 = 1/3

Converting improper fractions into mixed numbers is a very important thing to know how to do. When you have an improper fraction, the numerator is of a higher value than the denominator. So to convert this into a mixed number, you have to calculate how many times the denominator goes into the numerator and how much is left over at the end. Example: 14/6(6 goes into 14 (2) times with (2) left over)2 2/6 (simplify) = 2 1/3

There are plenty of ways to represent fractions, decimals, and percentages pictorially. One way is to use a diagram, with squares partly shaded and partly unshaded. Once you count the number of shaded squares out of all the unshaded squares, you have your fraction/decimal/percent.