Elementary Mathematics

You can show multiplication through diagrams which can be a lot easier to visualize.To do this, start by creating a box that has the largest denominator of columns and the other denominator is the number of rows.Ex:3/4(2/3)The 4 in the first fraction's denominator will be the number of columns and the 3 in the second fraction's denominator will be the number of rows.Once you have the diagram drawn, you color in three columns and and 2 rows.The boxes in the diagram that have both colors colored in it, is the answer out of the total number of boxes. The answer to this problem would be 6/12.

This method is the easiest method in my opinion, but only with fractions that without a whole number.Example:3/4 x 4/5 = 12/20You just multiply across:3x4=124x5=20Therefore the answer is 12/20

When multiplying mixed number fractions, you have to do a few extra steps.Start by multiplying the whole number with the fraction denominator.Next you take that number and add it to the fraction numerator.Once you have the mixed number fraction converted to a new improper fraction, you have to set it up with the other fraction in the equation. (sometimes there may be 2 improper fractions that you have to convert)The next step is multiplying the fractions together like normal.Once you have the new fraction, you have to convert it back to a mixed fraction. You can do this by dividing the denominator by the numerator. Once you get the answer, you have to see if you can simplify it and if you can, then simplify it and you are done.Here is a video explaining it clearer:Here is also a video on how to simplify fractions:

This method is really hard to explain and show on this mindmap because I have very limited abilities as far as what I can insert and show for an example, so I am going to do my best to explain.First, you have to look at your fractions and find what two number (if applicable) multiply together to equal that number. You do this for all the numerators and denominators if it can be broken down (if the number was 9, the two numbers that equal that when multiplied would be 3 and 3, so you would write that over the 9 and cross out the 9).Next, once you have done that for all the numbers, you go back and cross out the ones that have the same factor from the diagonal. After that step, you circle the numbers that are left and you can multiply across which will give you the answer.

When dividing fractions you have to do something called the reciprocal which means to keep, change, flip. This means:keep the first fraction the way it ischange the sign from division to multiplicationflip the numerator and denominator of the second fraction After completing the keep, change, flip steps, you can multiply like normal.

The video I added to this section is the best way I can think of on how to explain showing adding decimals.

There are a couple methods to adding decimals:Estimate first then add (this helps students be able to tell if their answer is close to the estimated solution, or way off and it will help them easily determine if they have the correct answer or not).Line up decimals this could be difficult for students to understand if they have a whole number as part of their equation because they may not know where the decimal goes

There are a couple methods to subtracting decimals:Estimate first then subtract (this helps students be able to tell if their answer is close to the estimated solution, or way off and it will help them easily determine if they have the correct answer or not).Line up decimalsthis could be difficult for students to understand if they have a whole number as part of their equation because they may not know where the decimal goes

There are a couple methods to multiplying decimals:Estimate first (this helps students be able to tell if their answer is close to the estimated solution, or way off and it will help them easily determine if they have the correct answer or not).Then multiply without decimals (this allows students to multiply without worrying about the decimal and they can see once they have their answer where it goes based off of their estimated answer.Line up decimalsthis could be difficult for students to understand if they have a whole number as part of their equation because they may not know where the decimal goes

Why does it matter?Context----> we can't do it any way we want because the answer will turn out wrong, it was discovered to help us do it right!PEMDAS: Please Excuse My Dear Aunt SallyThis is a way of looking at how to do order of opperations, however sometimes the parenthesis part does not work. If the equation was 6/2(5), we do not do 2(5) first. Instead you would do 6/2 then multiply it by 5.Also, multiplication and division are interchangeable and so is addition and subtraction as long as you go from left to right.The better way to look at order of opperations is grouping, Exponents, Division/Multiplication or vise versa, and Addition/Subtraction, also vise versa.

-52= -25 = 0-52 = 0-25 = -25(-52)= -25 = (0-52) = (0-25) = -25-(52)= -25 = -(25) = -25(-5)2= 25 = (0-5)2 = 25Pretend that there is a 0 before the numbers with the exponent. If it does not change the sign that means that the "-" does not mean subtract, but it means a negative sign.

Why do we need scientific notation?This helps us articulate and write out really really large numbers or really really small numbers. (found when we talk about atoms or distance to moon/stars/planets in science)Parts of a scientific notation equation: 1.36x104=A number between 1 and 10 (absolute value)The (x10) part of the equationThe exponentEasy way to determine between positive or negative exponent:Determine if the problem is large or small.large numbers = positive exponentsmall numbers = negative exponentExamples:204,000,000 = 2.04x1080.00000456 = 4.56x10-6To determine the decimal at the beginning of the equation, all you have to do is make sure that number is between 1 and 10 and that is the number. So in the first number we put the decimal after the 2 because that is the smallest number we could have between 1 and 10.

3,2560.0026Standard form = the normal way that we write numbersGoing from scientific notation to standard form:Make a decimal that is between 1 and 10.Now add the x10.Then, count how many place values there are either before or after the decimal (this depends on if it a large or small number).Lastly, put that in an exponent after the 10 with either a positive exponent for a large number, or a negative exponent for a small number.

Expanded Form425+314=? = 739400+20+5300+10+4-----------------700+30+9 = 739Equal Addends: Add equal amount to each numbers (can be any number as long as it is equal).42-35=? = 7 42 + 5 = 47 - 35 + 5 = 40------------------- 7

Timed Test: Ineffective because of the stress they put on our young studentsProgress Chart: Chart that shows the students what score they got--->Students will be able to see their scores and if they are falling below, they will start to dread the timed test which is what makes them ineffective.--->Instead we should use a progress chart to show students how they are improving that way it will motivate and encourage them instead of shame them.Flash Cards: Increases automaticity and is highly encouraged(should make your own flashcards)Tricks (9s): Not great to teach because it is only a one number strategy

Expanded Form:Left to Right: Lattice:

Long Division:This is a harder way to divide that most schools teach their students.Long division requires students to remember a whole series of steps that can get confusing and they are more susceptible to making mistakes.

Repeated Subtraction: This method is great for students who are learning division because it allows them to subtract whatever number instead of having to know their multiplication facts by memory. It also decreases the amount of steps they have to accomplish and replacing it with only subtraction.

Upwards Division: 532-------- = 177 and 1/33First you are going to see how many times 3 goes into 5. In this problem, 3 goes into 5 one time, so you put a 1 as the first part of the solution.Next, you subtract 3 from the 5 in the numerator. This gives you the number 2. Now you have to take the 2 and put it in front of the 3 (5232) to make it 23.Next, you see how many times 3 goes into 23. Three goes into 23 seven times, so you put the 7 next to the number 1 in the solution.Now you do the same step as before and do 23-21 which equals 2 and you have to put that 2 in front of the 2 (5322).Just like before, you now see how many times 3 can go into 22 and it is 7. Add the 7 to the solution and subtract 22-21. This leaves you with 1 over 3 or 1/3.Add the 1/3 to the solution and now you have the completed answer which is 177 and 1/3.

Divisibility Rules:2- Even number (end in 0, 2, 4, 6, 8, ...) Example: 4,862 3- The sum of the digits is divisible by 3Example: 564 = 15 when added up and 15/3 = 54- If the last 2 digits are divided by 4 Example: 6,7365- Last digit is a 5 or a 0Example: 3,530 or 5,6856- If both divisibility rules 2 and 3 work, then 6 works tooExample: 342 = 9 when added up and 9/3 = 38- If the last three digits divide by 8Example: 13,6489- The sum of the digits is divisible by 9Example: 6,327 = 18 when added up and 18/9 = 210- Last digit is 0Example: 2,980

To get these problems correct, you have to add zeros which means adding in the addition and the subtraction symbols to cancel each other out. What I do when completing these problems is start by counting out the first number given because that is what we are being asked to show.After that, I add in my zeros by counting each symbol to make sure I am using the correct amount. Show 6 using 8 symbols --> For this problem I typed out 6 + + + + + + + plus signs and then added in another - plus and minus sign. This used 8 symbols, but shows 6 positives because the extra + and - cancel out.Show -3 using 7 symbols - - - - -+ +Show 5 using 9 symbols+ + + + + + +- -

7 + 4+ + + + + + + + + + + = 11-3 + 6- - - + + + + + + = 35 + (-2)+ + + + + = 3- - -2 + (-8)- - - - - - - - - - = -106 + (-4)+ + + + + +- - - -

If the signs are the same, we addIf the signs are different, we subtractThe sign that is left out is the sign that we put in front of our answer, so if it is a - then the answer is negative, but if the sign left is + then it is positive.+ + - = subtract25 + (-12) = 13+ - - = subtract38 + (-54) = -16 - - - = add-40 + (-64) = -104

To start these problems off, I start by putting the amount of symbols that the first number shows. If the number is a negative then I use the subtraction sign and if the number is positive, I use the addition sign.Then I add zeros as necessary to make sure that I am able to take away what it is asking me to take away. Sometimes the problem will have you take away from a number that is bigger, so in those cases (like the first problem below) you won't have to add any zeros.5 - 2 = 3 (5 positive take away 3 positive)+ + + + + -3 - (-2) = -1 (3 negative take away 2 negative)- - - -3 - 6 = -9 (3 negative take away 6 positive)- - - - - - - - -+ + + + + + -2 + 7 = 5 (2 negative plus 7 positive)- - + + + + + + +

For these problems, you also have to add zeros if necessary in order to get the correct answer. You can add as many zeros as you want as long as it is enough. For the first and last problem below, I did not have to add any zeros, I just had to put the correct number in the correct number of groups.3(-3) = -9 (3 groups of 3 negative)- - - - - -- - --5(2) = -10 (Take away 5 groups of 2 positive) + + + + + + + + + +- - - - - - - - - --2(-4) = 8 (Take away 2 groups of 4 negative)+ + + + + + + + - - - - - - - - 2(3) = 6 (2 groups of 3 positive)+ + + + + +

Adding the oppositeKeep change-change - short way to add the oppositeExamples: K C C34 - (-52) ---> 34 + (+52) = 34 + 52 = 86K C C -45 - 12 --->. -45 + (-12) = -57For this problem, we could use Hector's diagram as well and see that 45 would have 2 - signs over it and the 12 would have only 1. Upon determining that, we could see that we are going to add them together and the solution would be a negative number.K C C25 - 60 ---> 25 + (-60) = -35This problem is a lot like the last one, all you have to do is see that one sign is the same and one is different so we are going to subtract them and because the sign on the bigger one leave a negative, the answer will then be negative.

-3(-6) - A negative TIMES a negative = a positive2(-4) - A positive TIMES a negative = a negative7(5) - A positive TIMES a positive = a positive Examples: 75---- = 15 Same sign (both numbers are positive), so it is a positive 5-3(5) = -15 Different sign (one is positive and one is negative), so it is negative-17(-24) = 408 Same sign, so it is a positive

When is 6 bigger than 10?NEGATIVE there is no context... 6 whats?6 trains and 10 bikes (This is context and it helps students be able to identify when 6 is bigger than 10)Is 25 big or small?Again, this needs context.25 marbles or 25 basketballs (The context helps the students to visualize what they are going to answer)Given 5 We know that we have 5 of somethingHow big are these 5 things though?Numerator vs. DenominatorNumerator ---> number of things we have---------------Denominator ---> the total possibleORNumerator ---> number of things we have----------------Denominator ---> the size of the piecesPractice:?--- ----> The size of the pieces are different10 (big or twice)?--- 20 (small or half) The size of the pieces is inversely related to the size of the denominator4 (more)--- ----> How many of each size piece we have?7 (less)--- ?

This topic is super hard to type out, so I linked a video that has 7 tricks on how to solve different fractions.

Determining which fraction is bigger using <, >, or = and our reason behind it...7/13 > 11/23 - Reason: Anchor fractions 1/22/9 < 2/7 - Reason: same number of pieces/size of each piece14/15 > 9/10 - Reason: missing one small piece4/12 < 5/12 - Reason: more of the same size piece13/16 < 25/28 - Reason: missing three small pieces4 9/10 < 5 1/3 - Reason: comparing whole numbers7/16 > 3/8 - Reason: multiply to make the same size pieces