*When multiplying regularly, line up your digits left to right it helps to have the number with the more digits on top.

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1 [6th Grade MIDTERM REVIEW NOTES] Multi-digit Numbers When multiplying multi-digit # s, place values are important and place holders need to be used. When multiplying regularly place holders are added to the partial products to make sure to line up the first part of the product under the digit you are multiplying by. *When multiplying regularly, line up your digits left to right it helps to have the number with the more digits on top. - Then multiply one digit at a time from the bottom right to left by all digits on top one by one right to left, carrying when necessary. -Next, put a place holder(s) to line up and continue. -After all digits on the bottom are multiplied by all the digits on top; add all partial products for final product. DIVISION DIVIDEND DIVISOR = QUOTIENT Divide, Multiply, Subtract, Check and Bring down 1. Divide- Divide the divisor into the partial dividend. 2. Multiply- multiply the partial quotient by the divisor. 3. Subtract subtract the partial dividend by the product. 4. Check- check that your difference is smaller than your divisor. 5. Bring down- bring down your next digit and continue until there is no remainder. Operations with Decimals Reminders: *All numbers can be decimals; if there are no decimals in the number it is at the end of the number. When moving a decimal we may need to add in zeros as place holders. Decimals can also be added at the end of a decimal if needed. To add or subtract decimals, line up your decimals, add place holders if needed, bring decimal straight down to your answer. Multiplying Decimals are done by lining up the digits place holders are not needed. Multiply normally and add up total amount of digits after each number. The decimal point moves from the end of your product left that total amount from factors. o Estimation can be used to check placement of decimals.

2 Dividing by decimals cannot be done you can have a decimal in your dividend but not in your divisor. o Make divisors whole numbers by moving decimal to the end and moving decimal same amount in dividend, then bring it straight up from dividend to your quotient and divide. o Divisors can also be made whole numbers by looking at it like a fraction and multiplying your divisor by a number to make it whole, but then you also must multiply your dividend by it and then divide normally. If there is no decimal in divisor bring decimal straight up from your dividend. *Division can also be made easier by treating the example as a fraction write it as a fraction, move the decimals and then try and reduce both numbers into simple/smaller numbers then divide. * Any decimal can be made into a fraction and any fraction into a decimal. Fractions turn into decimals by either: 1. Dividing the numerator by the denominator or 2. Getting an equivalent fraction with denominator of a power of ten (10, 100, 1000 ) your numerator tell you re your digits after the decimal point and your denominator tells your last decimal place your digit should end in. **MULTIPLICATION AND DIVISION HAVE VARIOUS STRATEGIES WE HAVE DISCUSSED, THE BEAUTY OF MATH IS TO DO A PROBLEM DIFFERENT WAYS AND ARRIVING AT THE RIGHT ANSWER USE WHICH WAY YOU ARE CONFIDENT IN BUT BE OPEN TO TRYING NEW WAYS TOO. Operations with Fractions ADDING/SUBTRACTING When adding or subtracting fractions make sure to get common denominators by changing one or more of the decimals into an equivalent fraction by multiplying or dividing the numerator and denominator by the same number Then add/subtract the numerators and keep the denominators Reduce/Simplify if possible o If you have a mixed number, you can either change to improper fractions or add/subtract your fractions then your whole numbers and carry/borrow when needed o Remember when you carry reduce your improper fraction to a mixed number and add the whole number from that to your sum from your whole numbers in your question and don t forget to pair with the fraction from your reduction o Remember when you borrow Borrow one whole from your whole numbers making that one less and add it to your fraction

3 Remember when you borrow one whole to borrow a fraction of your common denominator over your common denominator o Then subtract normally. MULTIPLYING When multiplying fractions, you do not need common denominators Multiply across the numerators, then multiply across the denominators and reduce o If possible, you are can cancel/reduce before you multiply as long as you reduce one top number/numerator and one denominator/denominator *When multiplying mixed numbers, you must change them to improper fractions before you multiply *To change a mixed number into an improper fraction you multiply the denominator and the whole number, then you add your numerator to get your new numerator, keep your original denominator. DIVIDING *You cannot multiply the whole numbers and the fractions separately. You cannot divide fractions, instead you multiply by the first fraction by the reciprocal/multiplicative inverse o The reciprocal is the fraction when multiplied by the original the product is 1 (the fraction flipped) When dividing fractions we use keep, change, flip to remind us to keep the first fraction, change division to multiplication and flip the second fraction. o After we keep, change, flip we follow the rules for multiplying fractions *REMEMBER if you have mixed numbers, you must first change to improper, then keep, change and flip following rules. ADDING/SUBTRACTING -COMMON DENOMINTAORS -add/subtract numerators, keep denominators -simplify MULTIPLYING -no common denominators - whole numbers over 1 - mixed numbers->improper - multiply numerators -multiply denominators -simplify DIVIDING -no common denominators - whole numbers over 1 - mixed numbers->improper *Keep first fraction *Change division to multiplication *Flip last fraction - multiply numerators -multiply denominators -simplify

4 SIMPLIFYING/REDUCING -When simplifying a fraction(less than 1) divide the numerator and the denominator by the same number - if there is no other common number that goes into both numbers except for 1 then the fraction is in simplest form - When simplifying an improper fraction you divide the numerator by the denominator and you will get either a whole number (if no remainder) or a mixed number. **REMEMBER, IF EITHER OF THE NUMERATORS AND EITHER OF THE DENOMINATORS OF TWO FRACTIONS BEING MULTIPLIED HAVE A FACTOR IN COMMON THEN YOUR ANSWER WILL BE ABLE TO BE REDUCED Greatest Common Factor Keep in mind a factor is a number that is smaller to or equal to your starting number it goes into/divides into your starting number evenly. Common factor it goes into/divides into both starting numbers evenly. To find the common factors, LIST ALL THE FACTORS OF EACH NUMBER least to greatest o Do this by checking which numbers go into it starting with 1 times your number, then move on to check 2 times what number, continue to do so until you reach one of the factors you have already listed *When you get to certain factors such as 9 and you already have a factor higher and that is a multiple of that factor such as 27 then you know that 9 has be a factor since one of its multiples already is o REMEMBER DIVISIBILITY RULES: Even numbers are divisible by 2 Numbers whose digits add to a number divisible by 3 then the whole number is divisible by 3 Numbers that end in 0 or 5 are divisible by 5 Numbers whose digits add to a number divisible by 9 then the whole number is divisible by 9 and therefore divisible by 3 as well Numbers that end in 0 are divisible by 10 and therefore also divisible by 2 and 5 as well. Once you have all the common factors listed starting with the greatest of each look for a factor that is found in both the largest to be in both lists is your GREATEST COMMON FACTOR

5 Least Common Multiple Keep in mind a multiple is a number that is larger to or equal to your starting number it is your starting number multiplied by another number You can find multiples by taking your starting number and multiplying it by 1, then starting number multiplying by 2, then it multiplied by 3 and so on o You do this a few times to get a list of multiples then do the same for your second starting number After you do this for both of your starting number, start looking at your list which should be in order from least to greatest and look for the smallest number that is in both list, this is your least common multiple. Ratios and Rates A ratio is a comparison of two quantities by division and can be written in three ways as a fraction, using the word to or using a colon. A rate is a ratio that compares two quantities that have different units of measure. A unit rate is a rate that makes a comparison to 1 unit. o To find the unit rate of something you divide the first number/unit by the second number/unit in the order of which it is looking for Example: miles per hour miles divided by hours Unit rates always have the second unit as a one so you can also make an equivalent fraction of the original rate with a denominator of 1. o Unit price is the price per item dollars divided by item Equivalent ratios are ratios that name the same comparison. *To make equivalent ratios or equivalent rates you can write your rate/ratio as a fraction and make an equivalent fraction by: Multiplying or dividing both quantities (the numerator and the denominator) by the same number. *If you are trying to determine a missing quantity in a set of two ratios you can set up a proportion (two equivalent fractions) and either: Determine what you multiply or divide one numerator or denominator by to get the new numerator or denominator and then do the same to the other part to determine the missing quantity. Or you can cross multiply to find the missing quantity. Multiply the two numbers that are diagonal to each other and divide by the third number. Percents A percent is a ratio, or rate that compares a number to 100. A percent can also be rewritten as a fraction with a denominator of 100 and as a decimal.

6 o To make a percent a fraction, drop the percent sign and put the digits as your numerator and 100 as your denominator. Then reduce if possible. o To make a percent a decimal, drop the percent sign and move the decimal, which is at the end of whole numbers, two places to the left. The last whole number digit of your percent will be in the hundredths place. When solving percent questions you can either: - Set up an equation and translate the equation to solve for the missing number o Remember that of means multiply and is means equals and that your percent needs to be changed to a fraction or decimal before multiplying of dividing. - Another way to solve percent problems is by setting up a proportion where =, fill in the information given, and cross multiply to solve for the missing number. *Remember to read questions carefully and determine if you are finding the part or the whole of the ratio/quantities the percent is always the part and 100 is always the whole when working with the percent portion of the equivalent ratio/fraction. *Remember when working with more than one percent, to break apart the question to see the different steps they are asking. * Be careful when they give you a percent and are asking for the remaining percent instead when this happens you can either subtract your given percent from 100 or you can find the part they give you and subtract your part from the whole of the total number.