Elementary Math

Number Systems: a collection of properties and symbols agreed upon to represent numbers systematically All numerals are constructed from 10 digits and the place value is based on powers of 10 in base 10. Numerals: written symbols to represent cardinal numbers In the United States education system we use the Hindu-Arabic number system. All numerals are constructed from the 10 digits Place value is based on powers of 10Place value: assigns a value to a digit depending on its placement in a numeral1 long = 10^1 = 1 row of 10 units 1 flat = 10^2 = 1 row of 10 longs or 100 units 1 block = 10^3 = 1 row of 10 flats, or 100 longs, or 1000 unitsExploring Different Bases: Always check notations of base systems and converting to base ten or other basesThere are different kinds of bases that can be usedExample: Base 10: 0,1,2,3,4,5,6,7,8,9Base 2: 0,1Base 3: 0,1,2Base 4: 0,1,2,3

Continued to explore different bases and watched the video, Little 12 toes School House Rock to explore the use of different bases.SubtractionAlgorithms include the standard algorithm, and how to use themTake away model - views subtraction as a second set of objects being taken away from the original set 8 block - 3 blocks = 5 blocks Number-line Model- subtraction is represented by moving left on the number line in a given number of units 5-3=2Missing addends - a reasoning is used where students compute a difference by determining the value of an "unknown" addend 8-3= x Can be thought of as the number of blocks that must be added to 3 to get to 8 which would be 5

View: Operations with Whole Numbers - Studi MathematicsWhole Number Operations: AdditionSubtractionMultiplication Division For each operation, different methods can be used to get the same answerUsing different strategies for adding, subtracting, multiplying, and dividing different bases Example: 35+37 (base 9)= 103I sued lattice form to find the answer because it worked well to split up the place valuesBuild up strategy: start from something you know and build on that answer Base 10: use base ten blocks to find the area of the equationExample: Missing addends 8-3=_+3=8Problem Solving Skills 

Complete Subtraction AlgorithmsView: Subtraction standard algorithm  Multiple ways to show subtraction using base 10 blocks Example: 2 ✕ 102 + 4 ✕ 101 + 5 ✕ 100     ( 1 ✕ 101 + 8 ✕ 100)=2 ✕ 102  + 2 ✕ 101 + 7 ✕ 100227

This week we focused on divisibility and reviewing for the first exam for the class Integer n is divisive by m if n=km for some other integer kGreatest Common Divisor/Factor View: Greatest Common Factor | How to Find the Greatest Common Factor (GCF)Of two whole numbers a & b, both have one whole number that does not equal zero but can be divided equally by both a & bMethods: Color rodsPrime factorization Intersection of setsExample: The prime factorization of 8 Example: Intersection of sets between numbers a & b

Overview: Integer OperationsBasic operations on positive and negative numbers;contexts;representations;computational fluency;use of properties;mental computation;estimating results of computations;how to round;multiple algorithms;connections to whole number representations;CountingWhat is an Integer? View: Integer chips manipulation Integer Chips Virtual ManipulativeNegative and positive numbers Denoted by 1: 1= {...-4, -3, -2, -1, 0, 1, 2, 3}-4 is opposite of +4-3 is the opposite of +3Absolute value is the distance from zero Examples: |20|=20   |-9|=9    |-456|= 456How to show it as a picture or concrete example? The weather or a bank statementThe rule is multiples of 2 and 3 Exam 1

Multiplying Real Numbers Commutative property: Everything can be switched around regardless of the sign View: Commutative Property of Multiplication - MathHelp.comAssociative Property: way in which factors are grouped doesn’t change the product Multiplicative Identity Property: any number multiplied by one gives the same result as the Number itselfDistributive property: a(b+c)= ab+acMultiplying by zero always results in 0Dividing Real NumbersInverse property of multiplication Every real number besides zero has a reciprocal The product of any number and its reciprocal is one Dividing by a real number is the same as multiplying by its reciprocal Dividing by zero is impossible/no solution Sign rules Positive / Positive = positive Negative / negative = positive Positive / negative = negative Negative / positive = negative Adding and subtracting real numbers on a number line:

Flat = 1Long = 0.1Unit = 0.01In elementary school, we usually see this as described as “moving” the decimal point two places to the right in both the dividend and the divisor. We worked with a fake student’s test to correct their answers. View: Multiply a Decimal by a Decimal | Math with Mr. J

Grids can be a useful tool for exploring the structure and value of a decimal. Regardless of the base it will be assumed that each whole grid represents 1In base 10, we can use a square to represent the tenths and hundredths place of a base 10 decimal number the grid is usually drawn with either 10 columns or 10 rows Grids can be a useful tool for exploring the structure and value of a decimal number.  Regardless of theBase it will be assumed that each grid represents 1. This is a base 10 10x10 grid that can be used to solve addition and multiplication problems. W We assume that the entire grid is a representation of 1.  multiplying a decimal by a whole number using base 10 grid1 x 0.7 is 0.7 0.5 x 0.7 is 0.35We worked together to find and correct errors on a fake test. Using the techniques learned in class, I found the mistakes and recalled why they were wrong and then fixed them. 

Comparing Old to New Percentage Change: subtract old value from the new value Example: You have 5 books buy now have 7, the change is 7-5=2Percentage Change: show that change as a percent of an old value... so divide the old value and make it a percentage So the percentage change from 5 to 7 is 2/5 = 0.4 or 40% There are two ways to calculate percentage change:Method 1Calculate the change Divide that change by the old value Convert that to a percentage Note: when the new value is greater than the old value, it is a percentage increase, otherwise it is a decrease. Method 2Divide the new value by the old value convert that to a percentageSubtract from 100%ExamplesA pair of sicks went from $5 to $6, what is the % change?Step 1: $5 to $6 is $1 increase Step 2: divide by the old value: $1/$5 = 0.2 Step 3: convert 0.2 to percent: 0.2x100 = 20% raise in price A discount is an amount by which an original price is reduced. Discount is a % of the original price. A markup is an amount by which an original price is increased. This is an example of using change of percentage to find the salary of someone.

Review for Exam 2Write six hundred seventy five and thirty five thousandths a. as a numeral - 675.035b. in expanded form - 600 + 70 + 5 + 0.03+0.005Terminating decimal = 0.50Non-terminating decimal = 0.555555555

Rational Number (fraction) OperationsFractions represented by words, diagrams, and symbols;Modeling fractions as parts of a whole or as a count of a subset;Placing fractions on a number line; equivalent fractions; common denominatorsSimplest form; mixed numbers and improper fractionsFraction TalkAdding and Subtracting Multiply and divideimproper and properRational Numberspropertiesexponents representing fractionspictorial and concrete models of fractions Fractions can be represented on a number line Picture RepresentationThe rods represent different parts of a whole which can be used to solve addition and subtraction of fractions Improper Fractions Proper Fractions

Adding and Subtracting FractionsLike and unlike denominators; contexts;representations;computational fluency;use of properties;mental computation;estimating results of computations;how to round; countingProperty Reminders Commutative PropertyThe commutative property states that the numbers on which we perform the operation can be moved or swapped from their position without making any difference to the answer.Associative PropertyThe associative property gets its name from the word “Associate” and it refers to the grouping of numbers. This property states that when three or more numbers are added (or multiplied), the sum(or product) is the same regardless of the grouping of the addends (or multiplicands). Distributive PropertyDistributive property helps us to simplify the multiplication of a number by a sum or difference. As the name suggests, it distributes the expression.Identity PropertyAdditive Identity An additive identity is a number, which when added to any number, gives the sum as the number itself. This means, the additive identity is “0” as adding 0 to any number, gives the sum as the number itself.Multiplication Identity A multiplicative identity is a number, which when multiplied by any number, gives the product as the number itself. This means, the multiplicative identity is “1” as multiplying any number by 1, gives the product as the number itself.Fraction bars helps when defining the part of a whole for fractions Example

Multiplication and Divisionthe meaning of multiplication for fractions;the procedures for multiplying fractions;mixed number answers to whole number division problems;using division to convert improper fractions to mixed numbers;interpreting division for fractions, the “invert and multiply” procedure for division;contexts;representations; computational fluency;use of properties;mental computation;estimating results of computations; how to round; counting.Multiplying FractionsThree simple steps are required to multiply two fractions:Step 1: Multiply the numerators from each fraction by each other (the numbers on top). The result is the numerator of the answer.Step 2: Multiply the denominators of each fraction by each other (the numbers on the bottom). The result is the denominator of the answer.Step 3: Simplify or reduce the answer.For other types of fractions Improper fractions - With improper fractions (where the numerator is greater than the denominator) you may need to change the answer into a mixed number. For example, if the answer you get is 17/4, your teacher may want you to change this to the mixed number 4 ¼.Mixed numbers - Mixed numbers are numbers that have a whole number and a fraction, like 2 ½. When multiplying mixed numbers you need to change the mixed number into an improper fraction before you multiply. For example, if the number is 2 1/3, you will need to change this to 7/3 before you multiply.Dividing FractionsDividing fractions is very similar to multiplying fractions, you even use multiplication. The one change is that you have to take the reciprocal of the divisor. Then you proceed with the problem just as if you were multiplying.Step 1: Take the reciprocal of the divisor.Step 2: Multiply the numerators.Step 3: Multiply the denominators.Step 4: Simplify the answer.Taking the reciprocal: To get the reciprocal, invert the fraction. This is the same as taking 1 divided by the fraction. For example, if the fraction is 2/3 then the reciprocal is 3/2.