Elementary Mathematics

Understanding IntegersHere you can find an explanation of Zero Pairs and Zero Banks.

Zero PairOne set of a positive and negative that add up to zero.Only 2 tiles that are opposite of each other make a zero PAIR.If there is more than one pair (ex: 4 tiles; 2 red, 2 blue) would make a zero BANK.Below is a picture of what a zero pair looks like:

Zero BankA zero bank is a representation of several zero pairs that make zero.Below you can see four zero pairs, which is also a zero bank.

Adding IntegersWhen you add integers make sure to have consistency with colors or shapes. When you add integers, it's important to remember that: Numbers with the SAME SIGN will be ADDEDNumbers with DIFFERENT SIGNS will be SUBTRACTED!Ex: -3+7= 4You would set up a visual representation of this like this: +++++++- - - You would cancel out the zero pairs as much as you can.You then count what you're left with. In this case, the three negatives cancel out with three positives leaving only four positives.

Subtracting IntegersWhen subtracting integers it is important to remember: When you take-away from a bigger number, the answer will be negativeIf you subtract two negatives, KFC KFC= Keep (the first sign: positive or negative), Flip (to addition or subtraction), Change (the sign: positive or negative)Ex: 3 - 5 = -2 + + +- - - - -In this case, three positives cancel out with three negatives. This leaves two negatives. Ex: 3 + 4 = 7 +++ ++++In this case, since their are no negatives, there are no zero pairs or a zero bank. You would just add normally. Ex:-2-(-6) --> KFC --> -2 + 6 = 4+ + + + + + - -In this case, two negatives and two positives would cancel out and leave four positives.

Multiplying IntegersWhen multiplying integers, it is important to remember: The triangle:2 negatives make 1 positive2 positives makes one positive 1 negative and 1 positive make a negativeEx: (3)(2) = 6 This is "three groups of two"++, ++, ++Together, you get six. Ex: (-3)(2) =-6This is "take away three groups of two"+ +, + +, + +, + + + + _ _ _ _ _ _ _ _ _ _Since the zero pairs cancel out, the answer would leave you with negative six.

Estimating AlgorithmsHere are a couple estimating algorithms that we learned in class.

Front EndFront end: Looking at the front of the number and converting it to the nearest tenth, hundredth, or thousandth.Ex: 47+39 =40+30=70Ex: 247+32 =240 + 30 = 270

RoundingRounding: what number is it closer to? Five, or greater round up to the next tenth Less than five, round downOdd numbers round upEven numbers round down Ex: 43+27= 40 + 30=70

Divisibility RulesHere are the divisibility rules for numbers 2, 3, 4, 5, 6, 8, 9, and 10.Please click the plus sign to expand.

Rule for 2:If the number is even it is divisible by 2 Ex: 2, 4, 6, 8, 10, etc

Rule for 3You add the number together and if the sum is divisible by 3 then it is divisible by 3Ex: 30: 3+0= 30/3=1021= 2+1= 21/3=7

Rule for 4If 4 goes into the last two digits evenly, then it is divisible by 4. Ex: 444 = 44/4 = 11308 = 8/4 = 2132 = 32/4 = 8

Rule for 5If it ends in 5 then it is divisible by 5.Ex:25 = 25/5 = 545 = 45/5 = 9If it ends in 0 it is divisible by 5.Ex: 10 = 10/5 = 220 = 20/5 = 4

Rule for 6If it is divisible by 2 and 3 then it is divisible by 6.Ex:18 = 18/2 = 9 and 18/3= 630 = 30/2 = 15 and 30/3=10

Rule for 8If 8 goes into the last 3 digits evenly then it is divisible by 8.Ex:640 = 640/8 = 80

Rule for 9If you add and the sum is divisible by 9, then it is divisible by 9. Ex: 36 = 3+6=9= 36/9= 481 = 8+1=9 = 81/9= 9

Rule for 10If it ends in zero then it is divisible by 10. Ex:10= 10/10 = 1100 = 100/10 = 10

Prime Numbers Prime: Numbers that are multiplied or divided by one and itself. It has exactly only 2 factors. Ex: 5 (1,5)7 (1, 7)

Composite NumbersHave more than 2 factors. Ex: 4: (1, 2, 4)35: (1, 5, 7, 35)

Prime FactorizationThere are three ways to factor numbers.Factor treeList MethodUpside down divisionPlease click the plus sign to expand.

Factor TreeYou break down each number until as far as it can go.Ex:24/ \8 3/\4 2/\2 2

List MethodYou would list out all the factors for that individual number. Then, you would have to break down the numbers as far as they would go.Ex: 32=1, 322, 164, 8

Upside Down DivisionUpside down division looks like a division sign but upside down. REMEMBER!!! The number on the outside has to be a PRIME NUMBER!2 L 282 L 142 L 7= 2^3 * 7

Greatest Common FactorThe Greatest Common factor is between two numbers where you would compare to find the factor that is the greatest of the two numbers.Ex:24 and 3024/ \12 2/\6 2/\3 230/ \15 2/\3 5take the number that has the least exponents.2^3 * 32 * 3 * 52 * 3 = 6Also the list method:12 and 1612 - 1, 2, 3, 4, 6, 1216 - 1, 2, 4, 8, 16

Least Common MultipleWhen you find the multiple that is the least among two numbers. Ex: 3^2 * 5^4 * 72^2* 3 * 5^2 * 7^2 * 13Always take the numbers with the greatest exponent.= 2^2 * 3^2 * 5^4 * 7^2 * 13Also the list method: 12 and 1512: 12, 24, 36, 48, 6015: 15, 30, 45, 60

FactorsThere are multiple ways to find factors:Factor treeUpside down divisionListArray model**It's important to remember that factor trees and upside down division give prime factorization. Whereas, list and array models do not.

Array ModelAn array model is the visual representation of listing out the pairs that make the number.For example:Look at the number 12.List its factors1, 2, 3, 4, 6, 122. Draw a box to create the area of the factors.The problem with this method is that some students will miss some.However, they can use the divisibility rules to check if they did miss one.

IntegersHere are the rules that we came up with in class for integers. We learned rules for: AddingSubtractingMultiplyingDividing

Adding IntegersRules for when adding integers:When signs are DIFFERENT, SUBTRACT and keep sign of the BIGGER numberWhen signs are the SAME, ADD and keep the sign. Ex: 34+(-20)++-They cancel out which leaves us with a positive. Next step is to SUBTRACT! 34-20=14Therefore, the answer is POSITIVE 14. Ex:-42 + (-50)---Since the signs are all negative, they do not cancel out so you just add the numbers and keep the negative sign. Similar to when we add whole numbers.The answer is then Negative 92 = -92

Subtracting IntegersThe rule when subtracting integers:ALWAYS USE KCCKCC stands for Keep, Change, Change. Keep the sign of the first number, change the sign to the opposite of what the equation is asking you to do, and change the sign to a positive or negative of the last number.Ex: 6-(-4)Keep six positivechange the equation to additionchange the sign of the negative to positive Your new equation should look like this:6+4 which gives you 10.Ex:-41 - 10 Keep 41 negativechange the equation to additionchange the positive sign to negativeYour new equation should look like this:-41+-10REMEMBER: in the addition rules, when adding integers with the same sign we add and keep the sign the same. Therefore, the answer is -51.

Multiplication and Division of IntegersThe rule for both is:Same sign is positive Different sign is negativeEx:(2)(8) = POSITIVE 16(-2)(-8) = POSITIVE 16 (-2)(8) = NEGATIVE -16(2)(-8) = NEGATIVE -16The only time the rule does not apply is when you multiply by zero. This is because zero cannot be a positive or negative number; it has no value.

FractionsFractions are made up of a numerator and denominator. Numerators: tell us how many parts of the total that we have.Denominators: tell us how big the parts are in relation to sizelarger denominators mean smaller piecessmaller denominators mean bigger pieces Fractions MUST be divided into equal parts.

Comparing FractionsWhen comparing fractions we focus on the signs that show us Less thanGreater than Equal to When comparing we can use:Common Denominators Ex: 2/5 > 1/5Since 2 out of five pieces are greater than 1 out of five we use the greater than sign. (>)Cross MultiplicationEx: 4/7 < 7/104x10= 40 7x7=49 Therefore,40 < 49Anchor Fractions: think about it in sense of if it is under or over 1/2 (you can also use 1 as the anchor)Ex: 5/11 < 13/256.5 is half of 11 which means 5 is less than half.12.5 is half of 25 which means that 13 is over half.Therefore, the sign we use is the less than sign.

Fraction ModelsIn class we learned that students need representations of fractions. By representing fractions can use the three models:Area ModelSet ModelLinear Model

Area ModelArea Models are concerned about the size of the shapes. For example, each piece in the area model has to be the same size and the same shapes. This means that the shapes have to have the same AREA! Ex: This image shows that 2/3 of the circle is shaded in and in contrast, 1/3 of the circle is not shaded in.

Set ModelThe set model only focuses on similarities within shape and NOT size. Ex: In this image you can say that 4/9 of the shapes are circles; 2/9 of the shapes are triangles; and 3/9 out of 9 of the shapes are rectangles.

Linear ModelWhen using the linear model, it's important to make sure that each part of the line is split into equal parts. This is an image that shows that all of these segments are equal to one and they have an equal proportion.

Equivalent FractionsWhen finding equivalent fractions, follow these steps:look at your fraction and determine if it can be reduced. If it can be reducedreduce the fraction by:dividing both the numerator and denominator by the same number. If it cannot be reducedmultiple the fraction by the "weird one"Ex: 2/2, 3/3, 4/4, 5/5, etc are considered "weird ones" because they equal one.

Unlike DenominatorsDenominators are the part of the fraction that is a whole. Unlike denominators are when the the numbers are different than each other. In order to add or subtract you must find a common denominator. However, in class we learned an algorithm that works for each adding, subtracting and multiplying unlike denominators.

Adding Unlike DenominatorsWhen adding unlike denominators we need to find a common denominator.However, in class, we found a different algorithm to use when teaching this lesson.Here are the steps that need to be followed:Use the Area ModelFill in the area model according to the fraction - for BOTH fractions (you will have 2 boxes).Draw the lines of the other box so that it would "overlap" and make a new denominator.You would then count the number of shaded boxes in the second rectangle to add to the first original box which in turn gives you the answer.*IMPORTANT NOTE* When filling in one area model make the lines vertical; whereas, the second model would be filled in horizontally.Ex: if the fraction is 1/2 you would color one of the two pieces of the rectangle horizontally, whereas the other fraction would be filled in accordingly (1/4) but you would do this fraction vertically.

Subtracting DenominatorsWhen subtracting unlike denominators we need to find a common denominator.However, in class, we found a different algorithm to use when teaching this lesson.Here are the steps that need to be followed:Use the Area ModelFill in the area model according to the fraction - for BOTH fractions (2 boxes).Draw the lines of the other box so that it would "overlap" and make a new denominator.You would then count the number of shaded boxes in the second rectangle to determine how many boxes you would take away from the first box which in turn gives you the answer.*IMPORTANT NOTE* When filling in one area model make the lines vertical; whereas, the second model would be filled in horizontally.Ex: if the fraction is 1/2 you would color one of the two pieces of the rectangle horizontally, whereas the other fraction would be filled in accordingly (1/4) but you would do this fraction vertically.

Multiplying Unlike DenominatorsWhen Multiplying unlike denominators we DO NOT need to find a common denominator. We would just multiply the top part of the fraction first to get the numerator and then multiply the bottom part of the fraction second to get the denominator. IT IS OKAY THAT THE DENOMINATORS ARE DIFFERENT!However, in class, we found a different algorithm to use when teaching this lesson for students who are having difficulties understanding the concept and need an actual visualization.Here are the steps that need to be followed:Use the Area ModelFill in the area model according to the fraction - for BOTH fractions (This means you will have 2 boxes).Draw the lines of the other box so that it would "overlap" and make a new denominator.You would then count the number of shaded boxes from BOTH of the boxes (where they overlap when placed on top of each other) to determine the answer.*IMPORTANT NOTE* When filling in one area model make the lines vertical; whereas, the second model would be filled in horizontally.Ex: if the fraction is 1/2 you would color one of the two pieces of the rectangle horizontally, whereas the other fraction would be filled in accordingly (1/4) but you would do this fraction vertically.

FractionsFractions are made up of a numerator and denominator.Numerators: tell us how many parts of the total that we have.Denominators: tell us how big the parts are in relation to sizelarger denominators mean smaller piecessmaller denominators mean bigger piecesFractions MUST be divided into equal parts.

A Whole Number Multiplied by a FractionIt's important to remember that when multiplying, the first number is the number of groups. The following number is the number of items within one group.When multiplying a whole number by a fraction, there will be whole groups multiplied by a fraction. Show this by: Drawing the amount of circles necessary for the amount of groupsWithin the circles, draw an area model that represents the fraction presented.Add all of the shaded parts together and put it out of the denominator. You may have to simplify to simplest terms.

Fractions Multiplied by a Whole NumberIt's important to remember that when multiplying, the first number is the number of groups. The following number is the number of items within one group.When multiplying a fraction by a whole number, the fraction will be the group and the whole number will be the number within the group.Show this by:Drawing a rectangleDivide the rectangle according to the fractionDivide the amount of items (the whole number) evenly within each group. Count the amount of items within the space.

Fraction Multiplied by a FractionIt's important to remember that when multiplying, the first number is the number of groups. The following number is the number of items within one group.When multiplying a fraction by a whole number, the first fraction will be the group and the following fraction will be the number within the group.Show this by:Draw a rectangle Divide it according to the SECOND fractionTake however much of the SHADED region according to the FIRST fractionPut how many spaces you took over the total amount of boxes (shaded and unshaded)

Algorithm for Adding FractionsWhen adding fractions with similar denominators follow the following steps: Set up the problem horizontally Be sure that the numerators and denominators line up.Add the NUMERATORS straight across. DO NOT ADD the denominators! Keep the denominator the same! Get your answer. You may have to simplify in simplest form. When adding fractions with unlike denominators follow the following steps: Set up the problem horizontally.Be sure that the numerators and denominators line up. Find the greatest common factor for the denominators you would then find the other number that is used to make that number. Next, you would use that number on the opposite side and then you would multiply by the "weird one" and do the same for the other side. You will then find the common denominator and add these NUMERATORS together but LEAVE THE DENOMINATOR THE SAME! When adding proper fractions follow these steps: Begin by adding the whole numbers.You would then be focusing on the fractions.follow the steps according to above: if the fractions have like or unlike denominators.

Algorithm for Subtracting FractionsFor fractions with like denominators:Set up the problem horizontallyBe sure that the numerators and denominators line up.Subtract the NUMERATORS straight across.DO NOT SUBTRACT the denominators!Keep the denominator the same!Get your answer.You may have to simplify in simplest form.For fractions with unlike denominators:Set up the problem horizontally.Be sure that the numerators and denominators line up.Find the greatest common factor for the denominatorsyou would then find the other number that is used when multiplied to make that number.Next, you would use that number on the opposite side and then you would multiply by the "weird one" and do the same for the other side.You will then find the common denominator and SUBTRACT these NUMERATORS together but LEAVE THE DENOMINATOR THE SAME!When subtracting fractions with whole numbers:Begin by subtracting the whole numbersYou will be left with the fraction partYou must find the common denominator to do this:Find the greatest common factor for the denominatorsYou would then find the other number that is used when multiplied to make that number.Next, you would use that number on the opposite side and then you would multiply by the "weird one" and do the same for the other side.You can subtract.KEEP IN MIND YOU MAY NEED TO USE KEEP CHANGE CHANGE IF THE NUMBER THAT COMES SECOND IS BIGGER THAN THE FIRST NUMBER.Make sure that you keep the denominator the same, but the numerator must change.

Algorithm for Multiplying FractionsFor this method, we use something similar to cross simplifying.We use this method only when multiplying fractions. find the factors for each numberif there are similar factors from the numbers across from each other, then you can cross out the similar numbers. you are then left with the simplified numbers multiply straight across. get your answer.When multiplying fractions with a whole number in front:Convert the fraction and whole number into an improper fraction. find the factors for each numberif there are similar factors from the numbers across from each other, then you can cross out the similar numbers. you are then left with the simplified numbers multiply straight across. get your answer.

Adding BasesWhen adding base numbers, you would do it similar to the way one normally adds.With the rule that the unit number has to be less than the base number (other wise it would be changed to an additional number for the long).Ex: add base 32 base seven + 21 base seven.In this case, the answer would be 53 base seven because 3 is less than seven and therefore can be accounted for as units.

Base CountingCounting in base twelve is different from counting in base 5.Ex: In base 5 you would count 1, 2, 3, 4 units before it would be come one long.In the base 12 case, 10 is commonly known as one long and no units and 11 is known as one long and one unit. So when we write base 12 two digit numbers we would right 1, 2, 3, ... 8, 9, T (for Ten), E (for eleven), 10 (for one long and no units), 11 (for one long and one unit), etcFurthermore, for base 3 the counting would be much similar to base 12.You would count: 1, 2, 10 (for one long and no units), 11 (for one long one unit), 12 (for one long 2 units), 20 (for two longs and no units) and so on.Fun Fact: The highest three digit number in base 3 would be 222. This is true because you cannot write down the number three since it would be counted as a long, or a flat.

Addition AlgorithmsHere are a couple algorithms for addition that we learned in class.

Adding Expanded FormThe expanded method of addition is best for students to use when they do not understand place value because it keeps the place value while adding. Although, expanded version is longer than the traditional way of adding, it is easier for students to visualize and see the place value of the numbers they are adding.Ex:243+ 125 =(200 + 40 + 3) + (100 + 20+ 5)= 300 + 60 + 8= 368

Adding Lattice MethodThe lattice method of adding is easier for students to see how they would begin to "carry the one". Although some students get confused when adding with the diagonals, it is easy to get students to understand the concept of adding the same units.Below, you will find a picture of how the boxes in the lattice method are supposed to look.

Adding Scratch MethodThe scratch method is best when adding more than two numbers. This method is similar to the traditional method of addition. Thus, before using the scratch method, you must be sure that students understand place value. Students must remember: Every time you reach the number 10 you put a scratch through the number and write down the remainderWhen you have left overs at the final number, you write it down in the place value it is supposed to be inThen count however many scratches you made and continue counting on the next place value.

Adding Left-to-RightThe traditional way of adding two numbers together usually consists of setting up numbers vertically by lining the ones. It is common that we would say "carry the one," when in reality it is carrying over a ten, hundred, thousand, and so on. Students are confused with what "carry the one" means.Left-to-right adding can be the solution to the confusion! In left-to-right adding, students begin solving an addition problem by setting up the numbers being added vertically so that the ones line up. It is critical to that student understands place value.Start by making sure the ones add up.Add the highest number together (tens, hundreds, thousands, etc).Proceed by adding the next highest number together.Repeat this step as needed.Lastly, add the ones together.You would stack up the numbers so that you can get a total at the end.* As you can see left-to-right addition is a longer method than the traditional method of adding numbers together, but students understand it better.

Subtraction AlgorithmsHere are the couple subtraction algorithms that I learned in class

Using Base 10 BlocksUsing Base 10 Blocks is the visual representation of subtraction. In order for students to complete this method, they must understand the terms "take away," "minus" or "fewer than". Students will need to understand that they might need to "break-up" some flats, or units. to do this, they must know:10 units = 1 long10 longs = 1 flat Students should group what they are "taking-away" from the number that they originally started with.Ex:

Division AlgorithmsHere are some division algorithms that we learned in class.**NOTE:Students get confused on which number goes inside the house when a division is written like this:26 / 2To avoid confusion, we teach them theIn ~ N ~ Out Method:The first number is In, and (n) the second number goes outside.

Upwards DivisionThe Upward Division Method is meant for students to understand that the remainder always goes over the denominator. Also, they understand what the denominator is instead of getting confused. This method requires the understanding of estimation, multiplication, and subtraction of students.Ex:45/645/6 = 7(7)(6) = 4245-42/6 = 7= 7 3/6Steps:Estimate: guess the number of times six goes into 45.Multiply: multiply 6 by 7 and get the product 42.Subtract: subtract 45-42 = 3 (which is the remainder)Move remainder to the other side and put it over the denominator.

Repeated SubtractionThe Repeated Subtraction Division method gives the students who do not know their multiplication a chance at getting the answer right. This is because it is basically guessing. It is best to start with this method, and then it is best to teach them long division, since this method seems to have students guess.Ex:156 / 2020 / 156 5 - 10056 1 - 2036 1 -20165 + 1 + 1 = 7 and 16 / 20

Multiplication AlgorithmsHere are some multiplication Algorithms that we learned in class.

Expanded FormWhen using expanded form, you must be sure that you multiply each number by every number. This form is similar to using the FOIL method, but you do not have to use the FOIL method. Ex: 25 x 36 =(20 + 5) (30 +6)= (20)(30) +(5)(6) + (20)(6) + (30)(5)= 600 + 30 + 120 + 150 = 720 + 180= 900

Building the AreaWhen building the area, students must understand that when multiplication is adding one number the other number's times. For example, if you have 3 x 4 you could multiply: 3 four times: 3+ 3 + 3 +3 = 12or 4 three times: 4 + 4 + 4 = 12This is because both multiplying and addition are commutative. Which means that you can switch the numbers around and still get the same answer. For building the area, one draws a representation using units, longs and flats. They usually fill a rectangle when its only two numbers being multiplied Here is an example:

PropertiesHere are some properties that we learned in class.

Distributive PropertyThe distributive property always has multiplication or parenthesis within the problem.It involves multiplying the outside digit by all of the digits inside the parenthesis.Ex:6(5+3) =30 + 18 =48

Commutative (MU) PropertyMU = Mix upThe commutative property can be used for both the addition and multiplication. It involves the mixing up of the numbers, while the sign remains in the same spot. Although you mix the numbers, you will still get the same answer.Ex: 4 + 3 = 3 + 4 = 74 * 3 = 3 * 4 = 12

Associative (SO) PropertySO = Same OrderThe associative property can be used for both multiplication and addition. You can remember this property because associate means partner, and you pair the numbers up with different 'partners' through parenthesis and get the same answer. Ex: 3 + (5+9) = (3 + 5) + 9

Identity PropertyThe Identity property is used for addition and multiplication. Addition rule: Anything added to zero is itself. Ex: 0 + 1 = 10 + 2 = 20 + 3 = 30 + 4 = 40 + 5 = 5Multiplication Rule: Anything times one is itself, with the exception of zero (because anything times zero is zero).Ex:1 * 1 = 11 * 2 = 21 * 3 = 31 * 4 = 41 * 5 = 5

Bizz-BuzzBizz-Buzz was a counting activity that involved the participation of every classmate. The game requires the understanding of following rules in a certain order. For example, we were to say "buzz" on multiples of 7, or any number that had a 7 in it. Whereas, we were to say "bizz" when we landed on a multiple of 11, whereas it would then reverse the order.In my future classroom I can accredit Mr. Miltenberger and play bizz-buzz with my class. This activity would be fun for students to play and it would help me determine if they understand their multiples and numbers.

Base 10 BlocksEvery base is determined by the number of units make a long.Unit: a single block.Long: a couple or several units to make a straight line (Ex: in base ten there are 10 units to form one long).Flat: longs put together to form a square; based on how many units there are. (Ex: in base ten there are 10 longs to form one flat, which means there are 100 units in a base ten flat).If you see a number (ex: 27) you can assume that it is base ten. However if you see a number and transcript (27 base eight), it is telling you that they want to see 27 in base eight.If you were to complete 27 base ten, it would look like two longs made up of ten units with seven single units.Whereas, to complete the other, you would have two longs that are made up of eight units and a set of seven single units.

Converting BasesI found the method shown in the video easier than the method shown in the book.Depending on how long your number is, begin by writing the number one on the upper right box.Proceed by going to the next box and multiplying it by the base number (if its base 5, put 5).Continue to the next box (if there is one) and multiply by 5 again. Put that answer in the next open box to the rightContinue step 3 as needed.Write down the number you are given, starting left to right. Make sure each number has their own space and there are no empty spaces.Multiply by columns.Add all the products.Get the base number!

Converting to Base 10When converting to a base you would use the exponents.You would use the number that is trying to be converted to multiply each of them by the single number (based on the place value) and then add the products together.Ex:1142 base five = _____ base ten1 ( 5^3) + 1 (5^2) + 4 (5) + 2 (1) =125 + 25 + 20 + 2 =172

Converting From Base 10You would create an L division that would basically be backwards from normal division.On the outside of the division symbol you would write the base number that you are transforming to.On the inside of the division symbol you would write the number you are transferring from.You would divide regularly using remainders.Keep going until the base number cannot go into the number that is being divided by.You would go upwards using the last number that you could divide by and the rest of the remainders.You would then get your answer.