Learn Math with Caveman Chang

Hello! In this video, I will talk about arithmetic sequences and arithmetic series. First, I explain what an arithmetic sequence is. A sequence is a list of numbers with commas separating each term. There may or may not be a pattern between each term. A sequence does not need to begin with number 1. Just because the first term is 1, it does not mean that the second term cannot be 1 too. An arithmetic sequence is a sequence of numbers that have a common difference. This means the difference between the first and second, second and third, third and fourth term and so on are the same. This is what we call a common difference. If the sequence has a common difference, then it is considered an arithmetic sequence. Please note that there are other types of sequences, not just arithmetic. There is such thing called a geometric sequence and a sequence that does not follow any rules (not arithmetic or geometric). Common questions asked are what is the nth term, given the first few terms. Another question, they will give you two random non-consecutive term and ask for the common difference. Formulas mentioned in this video: a_n = a_1+ (n-1)d d = [(a_y) - (a_x)] / (y-x) a_1 + a_2 + ... + a_n = (n/2)(a_1 + a_n) Thanks for watching! Please drop a like and subscribe to my channel.Don’t forget to let me know if you have any questions in the comments below. For more math videos like this, be sure to subscribe to my channel https://rumble.com/user/CavemanChang Hope you enjoyed my video! Thanks for watching! Credits: Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsPt0xImHyuRGU7BntSqDQ

Hello, in today's video, I will talk about PERCENTAGES. Percentages are a wonderful topic but also a very scary one. Percentages are very strong in that they are simple but can distort the situation to make it look very exaggerated or very downplayed. In this video, you will see several scenarios where percentages causes misinformation, misinterpretation. Link to ratio video: https://rumble.com/vsj23m-ratio-and-proportions-practice-problems.html Chapters 0:00 Explaining Percentages 1:08 Percentage Scenario #1 (the lack of percentage makes it seem really good) 3:04 Percentage Scenario #2 (the inclusion of percentage makes it seem really bad) 5:37 What is Percentage? 8:38 Percentage Problem #1 10:36 Percentage Problem #2 14:32 Percentage Problem #3 Thanks for watching! Please drop a like and subscribe to my channel. Don’t forget to let me know if you have any questions in the comments below. For more math videos like this, be sure to subscribe to my channel https://rumble.com/user/CavemanChang Hope you enjoyed my video! Thanks for watching! Credits: Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsP...

Hello everyone, Happy new year! I wish you a safe, healthy and prosperous new year! Thank you so much for supporting my youtube channel in 2021. Cheers to 2022. For this video, I will do things a little differently. The focus of this video is to tell my story. What is the story behind Caveman Chang? Who is Caveman Chang? A key to how I solve math problems is using brute force. Brute force is best shown in solving a math problem 1+2+3+...+97+98+99+100 = ? I talk about my journey from a student to ultimately a math teacher. I graduated from UC Berkeley with a focus in mathematics and mathematics teaching. Thanks for watching! Please drop a like and subscribe to my channel. Don’t forget to let me know if you have any questions in the comments below. For more math videos like this, be sure to subscribe to my channel https://www.youtube.com/channel/UC97R... Hope you enjoyed my video! Thanks for watching! Credits: Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsPt0xImHyuRGU7BntSqDQ

Expressions are combinations of numbers, variables, and mathematical operations. Math operations are anything from addition, subtraction, multiplication, division, square root, etc. Expressions have no problems, questions, or anything to solve. The only thing you can do with expressions is to simplify it by combining like terms. Equations are a comparison between expressions. You have one expression equal another expression. In an equation, you can simplify the expressions by combining like terms to solve it. Equations usually have a problem for you to solve which includes finding the value for a variable. The point is to solve the equation so that it is true. Practice Problems: Level 1: 5x-7+3x+2+10-2x simplified is 6x+5. Level 2: Given 7(K+3)=56, K is 5. Level 3: (3x+10)+(4x+4) simplified is 7x+14 or 7(x+2). Given (3x+10)+(4x+4)=0, x is -2. Credits: Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsPt0xImHyuRGU7BntSqDQ

Fractions are very helpful to work with but if they have a different denominator, it is hard to work with. Luckily, we can change the fraction so that they have common denominators. When changing a fraction, make sure to multiply the numerator by the same number that you multiply the denominator by. If the fractions have common denominators, we can add fractions, subtract fractions, and compare them. Practice Problems: Level 1: 3/4 - 4/9 = 11/36 Level 2: Both fractions are the same. 6/8 is equivalent to 48/64. Level 3: The improper fraction for 5 2/3 + 6 4/9 is 109/9. The mixed fraction is 12 1/9. Credits: Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsPt0xImHyuRGU7BntSqDQ

Fractions are a wonderful tool when working with segments or small parts of numbers. In a fraction, there are 3 parts. The number on the top is called the numerator. The line in the middle is called the fraction bar. The number on the bottom is called the denominator. The denominator tells us for an individual unit, how many equal segments do we split it into. The numerator tells us how many of those equal segments we care about. There are 3 types of fractions: proper, improper, and mixed. A proper fraction is a fraction where the numerator is less than the denominator. An improper fraction is a fraction where the numerator is greater than the denominator. A mixed fraction is a fraction where you see the whole units we work with and also the individual segments or fractions itself. Practice Problems: Level 1: If you take the individual unit between 0 and 1 on a number line and break it into 10 equal segments or parts, 7 of those parts represent 7/10. Level 2: 8/7 is an improper fraction. 9/14 is a proper fraction. 7 3/4 is a mixed fraction. Level 3: 26/5 is 5 1/5. Credits: Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsPt0xImHyuRGU7BntSqDQ

These are just some of the number sets you will encounter. Do not try to remember them all. Counting Numbers: All positive whole numbers Natural Numbers: 0 plus all the positive whole numbers Integers: All negative whole numbers, 0 and all positive whole numbers Rational Numbers: All numbers that can be written as a fraction Irrational Numbers: All numbers that cannot be written as a fraction Real Numbers: Including counting, natural, integers, rational, and irrational numbers. Practice Problems: Level 1: False Level 2: True Level 3: Yes Credits: Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsPt0xImHyuRGU7BntSqDQ

Factors and divisors are two words that mean the same thing. To be a factor or divisor, when you break down a number into multiplication, you have to be a possible number. Common factors refer to a group of two or more numbers. Break down each number into its factors. Factors that are shared in both numbers are called common factors. Greatest Common Factor (GCF) or Greatest Common Divisor (GCD) refers to the largest common factor that all the numbers share. To find a GCF or GCD, you break down the numbers into its prime factors. Select each individual prime factor with the smallest exponent and multiply them together. The end result is the GCF or GCD. If there are no common prime factors, the two or more numbers are relative prime which means the GCF or GCD is 1. Greatest Common Factor and Lowest Common Multiple are both tools that help you with divisibility, factoring and working with fractions. Practice Problems: Level 1: gcf[96,60] is 12. Level 2: The largest number that can be factored out of 48x+32y is 16 giving us 16(3x+2y). Level 3: 660,276 is divisible by 36. Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsPt0xImHyuRGU7BntSqDQ

For one number to be a multiple of another number, the one number multiplied by some whole number equals the other number. Finding the Least Common Multiple (LCM) of a group of number is to look at all the multiples and find the smallest number that each number shares. We take each number for which we are looking for the LCM and break it down into its prime factors. Look at all the unique prime factors and choose the ones that have the highest exponent. Multiply those unique prime factors together and the end result is the Least Common Multiple. This is a great tool that helps with adding and subtracting fractions. Link to Prime Factorization Video: https://rumble.com/vkpyg4-prime-factorization-composite-numbers-and-fundamental-theorem-of-arithmetic.html Practice Problems: Level 1: lcm[12,52] is 156. Level 2: lcm[8,20,48] is 240. Level 3: 7/54 + 5/12 + 3/8 = 199/216

Prime numbers are numbers greater than one such that the only positive whole numbers you can divide by so that it doesn’t have a remainder is one and itself. Composite numbers are numbers that you can actually divide by some other positive whole numbers. Number are either prime or composite. For composite numbers, you can break them down into multiplication of prime numbers. This process of breaking down composite numbers is called prime factorization. The prime numbers that they are broken down into are called prime factors. In prime factorization, it does not matter how you factor the number, the result will be the same. This is true because of the Fundamental Theorem of Arithmetic. Prime factorization helps us simplify fractions, work with exponents and square roots. Practice Problems: Level 1: 147 is composite because it is divisible by 3. Level 2: 1134/1512 simplified is 3/4. Level 3: x is 12.

Please check out my first video for the Divisibility Rules for numbers 1 to 4. https://rumble.com/vjq9p9-caveman-chang-demonstrates-divisibility-rules-aka.-another-math-shortcut.html For this video, we went over the Divisibility Rules for numbers 5 to 12. Please know that there are many more divisibility rules (e.g. 13, 14, etc.) Check out the video for examples and practice problems! Rule for #5: Check the last digit of the original number to see if it ends in a 5 or 0. If it ends in a 5 or 0, the original number can be divisible by 5. Rule for #6: Check to see if the original number is divisible by 3 and 2. If the original number is divisible by 3 and 2, the original number can be divisible by 6. Rule for #7: Get the last digit of the original number and multiply it by 2. Take the remaining numbers and subtract the last digit that was multiplied by 2. If the result is divisible by 7, the original number can be divisible by 7. Rule for #8: If the original number is larger than 3 digits, take the last 3 digits and see if that number is divisible by 8. If the last 3 digits are divisible by 8, the original number can be divisible by 8. If the original number is 3 digits or less, you have to check by dividing the original number by 8. Rule for #9: Take the individual digits of the original number and add them together. If the sum of the digits is divisible by 9, the original number can be divisible by 9. Rule for #10: Check the last digit of the original number to see if it ends in a 0. If it ends in a 0, the original number is divisible by 10. Rule for #11: Get the alternating digits (1st, 3rd, 5th, etc.) and add them together. Add the rest of the alternating digits (2nd, 4th, 6th, etc.) together. Take the difference between the sum of the two sets of alternating digits and see if that number is divisible by 11. If the difference is divisible by 11, the original number is divisble by 11. It does matter in which order you subtract the sums of the alternating digits. Rule for #12: Check to see if the original number is divisible by 3 and 4. If the original number is divisible by 3 and 4, the original number can be divisible by 12. Practice Problems: Level 1: 636,482,790 is divisible by both 5 and 10. Level 2: 33,822 is divisible by 3, 6, and 9. Level 3: 9,506,112 is divisible by 77. Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/by-sa/3.0/ Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a3YPKYkhwlNRQKVBxj?si=_D7QUoF9TLOKN0LsqIxERA&dl_branch=1 Artist Youtube: https://www.youtube.com/channel/UCoZbM1a4PKQ6haa2Ap4TSdg Custom Titles: Simple Video Making Link: https://www.youtube.com/channel/UCOsPt0xImHyuRGU7BntSqDQ

Divisibility rules are a set of rules that let you very quickly determine whether there will be a remainder or not when you divide one number by another number. Rule for #1: Any number can be divisible by 1. Rule for #2: Check the last digit of the original number to see if it ends in an even number (0, 2, 6, 8). If it ends in an even number, the original number can be divisible by 2. Rule for #3: Take the individual digits of the original number and add them together. If the sum of the digits is divisible by 3, the original number can be divisible by 3. Rule for #4: Check the last 2 digits of the original number to see if it is divisible by 4. If the last 2 digits can be divided by 4, the original number can be divided by 4 with no remainder. If the original number can be divided by 2 twice, it can be divisible by 4. Practice Problems: Level 1: 561,334 is not divisible by 3. 9,862,332 is divisible by 3. Level 2: 43,630,134 is divisible by 2 and not divisible by 4. Level 3: 3,287,544 is divisible by 1,2,3, and 4. Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/by-sa/3.0/ Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a3YPKYkhwlNRQKVBxj?si=_D7QUoF9TLOKN0LsqIxERA&dl_branch=1 Artist Youtube: https://www.youtube.com/channel/UCoZbM1a4PKQ6haa2Ap4TSdg

Here are 3 properties of exponents. 1st Property: Any number to the exponent 0 equals 0. The only exception is if the base is 0 then 0^0 is undefined. (e.g. 1000^0 = 1) 2nd Property: Some number to the negative exponent is the same as 1 over that same base but to the positive exponent. The only exception is if the base is 0 because we can never divide by 0. (e.g. 7^-4 = 1/(7^4)) 3rd Property: If you want to distribute the exponent, make sure the function inside the parentheses is either all multiplication or all division. You cannot do this if there is addition or subtraction in the parentheses. Common mistakes: 1.) The zero exponent does not make that equal 0. Any number as the base to exponent 0 will equal to 1. Unless the base is 0, then the answer will be undefined. 2.) A negative exponent does not make the answer negative. It tells us how to orient the problem but does not tell if the solution is positive or negative. 3.) Distributing the exponent to the parentheses when it is addition or subtraction. Distributing exponents can only apply to muliplication or division. Practice Problems: Level 1: Answer is 1 Level 2: Answer is 4 Level 3: Answer is True. On the left side of the equation, the outside exponent 3 is multipled by -4, making it -12. On the right side of the equation, the inside exponent -3 is multiplied by 2, then multipled by 2 again, making it -12. If i replace the 7 by any number, it will still hold true. Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/by-sa/3.0/ Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a3YPKYkhwlNRQKVBxj?si=_D7QUoF9TLOKN0LsqIxERA&dl_branch=1 Artist Youtube: https://www.youtube.com/channel/UCoZbM1a4PKQ6haa2Ap4TSdg

Exponents are another way to say repeated multiplication. The big number is called the base, the number that will be multiplied by itself. The little number on the top right of the base is the exponent, the number of times that the base will be multiplied. Here are 3 properties of exponents. 1st Property: If you have the same base and you multiply, just add the exponents. (e.g. 2^2 x 2^5 = 2^7) 2nd Property: If you have the same base and you divide, just subtract the exponents. (e.g. 4^7 ÷ 4^3 = 4^4) 3rd Property: If you have the same base and there is an exponent to the power of another exponent, just multiply the exponents. (e.g. (3^3)^2 = 3^6) Common Errors: 1.) Make sure the base is the same before attempting to add, subtract, or multiply the exponents. If the base is not the same, you cannot combine them in any way. (e.g. 2^2 x 3^5 cannot be combined in any way) 2.) When there are parentheses and exponents, you may not just distribute the exponent. (e.g. (a+b)^2 is not the same as a^2 + b^2) Practice Problems: Level 1: Answer is -200 Level 2: Answer is True. If you change the -1 and 1 to negative any number and the same positive number, this statement will still hold true. The first number is negative but if you multiply it an even number of times, it will become a positive number. In this example, -1 multiplied 6 times will become positive 1. The second side of the equation is 1^6 which will stay positive 1 no matter how many times you multiply it. As long the base on the left of the equation is the negative number of the positive number from the right of the equation, this statement will hold true. (e.g. (-10)^6 = 10^6 , (-25)^6 = 25^6, etc. Level 3: Answer in base 2 is 2^-1. Using the property, I subtract the exponents so 5 - 6 = -1. Final answer is 1/2 or 0.5 Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/by-sa/3.0/ Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a3YPKYkhwlNRQKVBxj?si=_D7QUoF9TLOKN0LsqIxERA&dl_branch=1 Artist Youtube: https://www.youtube.com/channel/UCoZbM1a4PKQ6haa2Ap4TSdg

From the rules of algebra, the additive inverse gives us negative numbers. Negative numbers are equal and opposite of positive numbers. Adding negative numbers and doing subtraction are the same thing. On a number line, all the positive numbers are on the right side of zero and all the negative numbers are on the left side of zero. Anytime we multiple any number by -1, we are flipping it across the number line across zero to the other side. We can flip the number across the number line as many times. If we flip the number once, it will go from negative to positive or positive to negative. If we flip the number an even amount of times, we will come back to the original side of zero (e.g. positive or negative) Practice Problems: Level 1: Answer is -200 Level 2: Answer is -96 Level 3: Answer is 105 Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/by-sa/3.0/ Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a3YPKYkhwlNRQKVBxj?si=_D7QUoF9TLOKN0LsqIxERA&dl_branch=1 Artist Youtube: https://www.youtube.com/channel/UCoZbM1a4PKQ6haa2Ap4TSdg

Variables are nothing more than imaginary containers. The letters and symbols we use as variables are nothing more than labels for these imaginary containers. Variable have 2 main usages. The first one is that it represents one specific number. The second one is that it represents some group of numbers. There are 3 common mistakes when using variables. 1.) People try to multiply different variables. We cannot combine the different variables because we cannot guarantee the value of one variable is the same as that of the other variable. 2.) If we see the same variable in another problem, do not assume the same answer because we already solved for that variable. Unless otherwise stated, we should always solve for the variable all over again in each new problem. 3.) People may use letters and symbols as their variable that already represent something (e.g. pi, e, and i) Practice Problems: Level 1: Answer is x = 27 Level 2: Answer is any number that is greater than 20 will make the statement true. Level 3: 1.) x = 9 / 2.) x = 4 Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/by-sa/3.0/ Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a3YPKYkhwlNRQKVBxj?si=_D7QUoF9TLOKN0LsqIxERA&dl_branch=1 Artist Youtube: https://www.youtube.com/channel/UCoZbM1a4PKQ6haa2Ap4TSdgVariables are nothing more than imaginary containers. The letters and symbols we use as variables are nothing more than labels for these imaginary containers. Variable have 2 main usages. The first one is that it represents one specific number. The second one is that it represents some group of numbers. There are 3 common mistakes when using variables. 1.) People try to multiply different variables. We cannot combine the different variables because we cannot guarantee the value of one variable is the same as that of the other variable. 2.) If we see the same variable in another problem, do not assume the same answer because we already solved for that variable. Unless otherwise stated, we should always solve for the variable all over again in each new problem. 3.) People may use letters and symbols as their variable that already represent something (e.g. pi, e, and i) Practice Problems: Level 1: Answer is x = 27 Level 2: Answer is any number that is greater than 20 will make the statement true. Level 3: 1.) x = 9 / 2.) x = 4 Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/by-sa/3.0/ Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a3YPKYkhwlNRQKVBxj?si=_D7QUoF9TLOKN0LsqIxERA&dl_branch=1 Artist Youtube: https://www.youtube.com/channel/UCoZbM1a4PKQ6haa2Ap4TSdg

Associative Property allows us to group the numbers in whichever way we want as long as we are working with either all addition or multiplication. Commutatitve Property allow us to switch the order around in any way as long as we are working with either all addition or multiplication. On top of that, we have this trick where we can convert subtraction to addition and division to multiplication. We can break numbers up and combine them in whichever way we want to make our calculations easier. Practice Problems: Level 1: Answer = 400 Level 2: Answer = 1500 Level 3: Answer = 8 Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/by-sa/3.0/ Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a3YPKYkhwlNRQKVBxj?si=_D7QUoF9TLOKN0LsqIxERA&dl_branch=1 Artist Youtube: https://www.youtube.com/channel/UCoZbM1a4PKQ6haa2Ap4TSdg Image: red cloaks set Designed by: vectorpouch / Freepik License: Freepik for personal and commercial purpose with attribution. Attribution: https://www.freepik.com/vectors/waves Waves vector created by vectorpouch - www.freepik.com

These 7 rules are the foundation of Algebra. Everything else is derived and built from these 7 rules. Rule 1 (Additive Identity) says there exists the number 0. Rule 2 (Additive Inverse) says there exists negative numbers which gives us the concept of subtraction. Rule 3 (Multiplicative Identity) says there exists the number 1. Rule 4 (Multiplicative Inverse) says there exists fractions and gives us the concept of division. Rule 5 (Associative Property) allows for us to group the problems in any way and the solution will be the same. This only works when problems are adding or multiplying. Rule 6 (Commutative Property) allows for us to switch the order of the problems in any way and the solution will be the same. This also only works when problems are adding or multiplying. Rule 7 (Distributive Property) allows us to intermingle addition and multiplication. With this property, we can work with multiplication and addition in a single problem. Practice Problems: Level 1: Answer = 30 Level 2: Answer = 700 Level 3: Answer = 200 Song: Path Of The Fireflies By: AERØHEAD License: Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) Link to License: https://creativecommons.org/licenses/... Artist Soundcloud: https://soundcloud.com/aerohead Artist Spotify: https://open.spotify.com/artist/4DH6a... Artist Youtube: https://www.youtube.com/channel/UCoZb... Image: Devil horns and angel halo By: Macrovector - Freepik.com License: Freepik for personal and commercial purpose with attribution. Link to image: https://www.freepik.com/free-vector/d...

PEMDAS is a step-by-step guideline to help us solve problems and ensure there are no misunderstandings.This came from the idea that we are too lazy to put parentheses in their correct spot so we memorize an order of solving problems instead. PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition and Subtraction. Multiplication and Division are on the same level so when they are together, solve the problem from left to right. Addition and Subtraction are also on the same level so when they are together, solve the problem from left to right. When we are working with PEMDAS, we are going step-by-step only if possible. If the P E M D A S do not exist in the equation, skip over to the first operation that exists. Practice Problems: Level 1: Answer = 9 Level 2: Answer = 3 Level 3: Answer is (15 + 5 - 6 ÷ 3) x 0 = 0. Please note the answer may vary.