6th Grade Midterm Review Notes

Rates, Ratios and Unit Rates

  • Ratios are a relationship between two different quantities or things.
  • Rates are relationships between two different quantities or things with different units.
  • Unit Rates are relationships between two different quantities or things with different units whose second unit is 1.
  • Unit rates can be found by dividing the first number by the second
  • Equivalent Ratios are two ratios that show the same relationship
  • They can be made by multiplying both parts of the ratio by the same number or by dividing both parts by the same number
  • To check if two ratios are equivalent, the ratios can be written as fractions and then reduced—they will reduce to the same fractions

Solving Proportions

  • To check if two fractions are proportional, you can either
  • Reduce both fractions to make sure they are the same
  • See if there is a number in which you can multiply or divide both the numerator by in order to get the second fraction
  • Cross multiply and see if you get the same number from each
  • To solve a proportion (when given 3 out of 4 parts of two equivalent fractions/ratios),
  • You can set up a proportion then cross multiply and solve for the variable. When you cross multiply with a variable, you want to multiply the two numbers on a diagonal and divide by the number that is diagonal the variable.

Prime Factorization

  • A prime number is a number that only has factors of 1 and itself.
  • A composite number is a number that has more factors other than 1 and itself.
  • PRIME FACTORIZATION-is when a number is broken down to all its prime factors. Prime factorization is written as a multiplication/product of all the prime factors in order from smallest to largest.
  • When a prime factor happens more than once it is written with exponents.

Greatest Common Factor

-A common factor between two or more numbers can be done by listing all the factors (numbers that go into it evenly).

  • To help find the largest factor and to make sure you do not forget any, you can start at 1 and write each set/pair of factors with one on each end of the line and work your way to the middle and there are no more factors.

-After you list all the factors look for the largest one that is in all of the sets of factors. A factor can be equal or smaller than the starting number.

-*Sometimes the greatest and only common factor between numbers is 1.

Least Common Multiple

-A common multiple between two or more numbers can be done by listing the multiples of each number. —the result of multiplying your number by 1, 2, 3 and so on. You should start with listing 5-10 multiples of each number. You can stop when you find the smallest number that is in all sets of numbers.

  • To help find the smallestmultiple write your multiples from smallest to largest.
  • You can always find a common multiple by multiplying the two or more numbers by each other but that will not always be the smallest.

Orders of Operation

Anytime there is more than one operation in an equation you must follow the order of operations:

  1. Parenthesis- solve anything and everything inside parenthesis first—if there is more than one operation inside the parenthesis, follow the order of operations inside the parenthesis.
  2. Exponents- solve any number that has an exponent attached to it
  3. Multiply or Divide in order from left to right
  4. Add or Subtract in order from left to right

*Make sure to solve one step/operation at a time and to bring down the rest of the equation in line each time you solve a piece.

Operations with Decimals

ADDING/SUBTRACTING

  • When adding or subtracting decimals make sure to line up the digits by the decimal
  • Remember if there is no decimal, the decimal is at the end of the whole number
  • After you line up your decimals, bring your decimal straight down to your answer and add or subtract normally

*To estimate with decimals –estimate/round the numbers to whole numbers and then add or subtract mentally, this will help you make sure you place your decimal correctly in your answer

MULTIPLYING

  • When multiplying decimals, line up your digits right to left (ignoring your decimal).
  • Multiply normally
  • Count the total number of digits after each decimal point
  • Then starting at the end of your product count that same number of places/digits to the left to place your decimal point correctly in your answer.

*To estimate with decimals –estimate/round the numbers to whole numbers and then add or subtract mentally, this will help you make sure you place your decimal correctly in your answer

DIVIDING

  • When dividing with decimals make sure you do not have a decimal in your divisor.
  • If there is no decimal in your divisor, bring your decimal straight up to your quotient and divide normally.
  • Don’t forget to add zeros at the end of your dividend if needed to divide without a remainder
  • If there is a decimal in your divisor, make your divisor a whole number by moving the decimal point all the way to the end of your dividend
  • Then move your decimal in your dividend the same number of places to the right as you just moved it in your divisor.
  • Add place holders if needed when moving your decimal in your dividend
  • Next, bring your decimal point straight up to your quotient and divide normally.
  • Don’t forget to add zeros at the end of your dividend if needed to divide without a remainder

*Remember if there is no decimal point in a number, it is at the end of the whole number. You must add it in/write it before either moving it or adding zeros to the end of a number.

*Sometimes dividing two whole numbers can result in a quotient that is a decimal.

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
  • 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
  • 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
  • Remember when you borrow—Borrow one whole from your whole numbers—making that one less and add it to your fraction
  • Remember when you borrow one whole to borrow a fraction of your common denominator over your common denominator
  • Then subtract normally.

*To estimate with fractions –estimate/round the numbers to either whole or half numbers

MULTIPLYING

  • When multiplying fractions, you do not need common denominators
  • Multiply across the numerators, then multiply across the denominators and reduce
  • 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.

*You cannot multiply the whole numbers and the fractions separately.

DIVIDING

  • You cannot divide fractions, instead you multiply by the first fraction by the reciprocal/multiplicative inverse
  • 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.
  • 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.

Percents

  • You can convert between percents, fractions and decimals
  • You change a percent to a decimal, divide by 100 or move the decimal two places to the left.
  • You change a percent to a fraction by putting it over 100 and reducing if possible.
  • You change a decimal to a percent by multiplying by 100 or moving the decimal two places to the right.
  • You change a fraction to a percent by getting an equivalent fraction with a denominator of 100 and the new numerator gives you your percent
  • You change your decimal to a fraction by putting your digits as your numerator over your last decimal place as the denominator and reduce if possible.
  • You change your fraction to a decimal by dividing your numerator by your denominator.
  • Percent questions can be found by changing the percent to a decimal or fraction and then either multiplying by the whole (of) or dividing by the part (is).

You can also always solve by setting up a proportion = and solving for the missing piece.

**Keep in mind these are only review notes/ cliff notes, you must study your notes, examples as well as your textbook and worksheets.