6th grade math notes
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    6th grade math notes 6th grade math notes Document Transcript

    • One Step Equations6th Grade Math Notes<br />
      • *Rules*Perform the inverse operationCheck your answer (in order to check you must put your answer in place of the variable to see if the equation holds true)*One Step Equations*The goal of an equation is to isolate (get it by itself) the variable (the unknown value..usually is represented by an “x”). In order to do that you must do the inverse (opposite operation) which means instead of adding you subtract and instead of multiplying you divide.
      • *Example #2**Check*Instead of x put your answer (3). (7 x 3) + 1 does equal 22 so your answer is correct.*Example #1**Check*Instead of x put your answer (9). (3 x 9) + 7 does equal 34 so your answer is correct.Rule 2 (undo x/÷)Rule 1 (undo +/-)*Rules*Perform the inverse operation (undo addition or subtraction first)Then undo multiplication or divisionCheck your answer (in order to check you must put your answer in place of the variable to see if the equation holds true)*Two Step Equations*The goal of an equation is to isolate (get it by itself) the variable (the unknown value is always represented by a letter)In order to do that you must do the inverse (opposite operation) which means instead of adding you subtract and instead of multiplying you divide.Two Step Equations*Example #4**Check*Instead of x put your answer (24). 24 ÷ 8 does equal 3 so your answer is correct!*Example #3**Check*Instead of x put your answer (7). -7 x -8 does equal 56 so your answer is correct!*Example #2**Check*Instead of x put your answer (13). 13 - 8 does equal 5 so your answer is correct!*Example #1**Check*Instead of x put your answer (4). 4 + 3 does equal 7 so your answer is correct!
      Expressions<br />
      • *Expressions*There are four types of expressionsAddition: x + 5, or 5 + xSubtraction: x – 3, or 3 – xMultiplication: 6xDivision: x/7, or 7/x*Rules*Instead of the variable put the number that the problem tells you it equals.Solve the expression
      Directions: Solve the expression when x = 8<br />*Example #2*X - 2Instead of x put 88 - 2 = 6Answer = 6*Example #1*X + 5Instead of x put 88 + 5 = 13Answer = 13*Example #4*x/2 - 1Instead of x put 8(8/2) - 1 = 3Answer = 3*Example #3*5x + 2Instead of x put 8(5 x 8) + 2 = 42Answer = 42<br />Rounding<br />
      • *Example #1* (round to nearest thousand)5,6785, 6 78 (round up since 6 is 5 or higher)6,000 (5 becomes a 6 and everything else zeros)*Rules*Underline the place valueCircle the # on the rightRound up the circled # is 5 or greater. When you round up the underlined # goes up one # higherRound down if the circled # is 4 or less. When you round down the underlined # stays the sameEverything after the underlined # then becomes zeros.*Example #2* (round to nearest hundredth).562.5 6 2 (round down since 2 is 4or lower).560 (the 6 stays the same and everything else zeros)
      Place Value<br />-37719081915<br />Multiplication<br />
      • *Multiplying with DecimalsDo not line up decimals!!!MultiplyAdd all of the numbersCount the decimals*2 digit multiplication (34 x 23 or 345 x 56)*With 2 digit multiplication use Only Pizza Tastes Astonishing or OPTA.O = (Ones place) First multiply the # in the ones placeP = (Place Holder) Put one place holderT = (Tens Place) Next multiply the # in the tens placeA = (Add) Then add all of the numbers up*3 digit multiplication (342 x 823)*With 3 digit multiplication use Only Pirates Tickle Parrots Hairy Ankles or OPTPHA.O = (Ones place) First multiply the # in the ones placeP = (Place Holder) Put one place holderT = (Tens Place) Next multiply the # in the tens placeP = (Place Holder) Put two place holdersH = (Hundreds Place) Then multiply the # in the hundreds placeA = (Add) Finally add all of the numbers up
      *Example #1*<br />Division<br />
      • *Dividing w/ a decimal in the dividend and divisorMove the decimals over firstBring up the decimal in the dividend DivideYou cannot have a remainder..if there is a remainder you must add a zero, drop it, and continue dividing*Dividing w/ a decimal in the dividendBring the decimal up in the dividendDivideYou cannot have a remainder..if there is a remainder you must add a zero, drop it, and continue dividing
      *Example #1**Example #1*<br />Adding & Subtraction Decimals 3280865215928<br />
      • *Rules*Line up the decimals!!!Add or SubtractIf you have a whole # you must turn it into a decimal first (5 = 5.00, 7 = 7.00)
      Ratios Unit Rates<br />
      • *Ratios*Ratios explain how for every x units of one item there are x units of another.For example a ratio of 3 to 5 would mean for every 3 apples there are 5 oranges.A ratio of 2:6 would mean for every 2 girls there are 6 boys.Rations can be expressed three different ways: 1 to 3, 1:3, and 1/3.*Unit Rates*Unit rates compare unlike units to 1 unit.For example how much one item costs, how many miles a car travels in 1 hour, or how many laps a person runs in 1 day.
      • *example* what is the unit rate if 8 apples cost $16To find your answer divide (16/8). The unit rate ends up being $2/apple.
      Multiplying Fractions<br />Dividing Fractions<br />
      • *Rules*Change all mixed #’s and whole #’s into improper fractionsThe denominators do not have to be the sameSimply multiply across (numerator x numerator…denominator x denominator)Make sure your answer is a proper fraction and in simplest form..*Rules*If there are mixed #’s or whole #’s you must first change them into improper fractionsChange the division sign into a multiplication signUse the reciprocal or flip the second fraction upside downMultiplyMake sure your answer is a proper fraction and in simplest form
      -390525250190<br />
      • Adding & Subtracting Fractions*Rules*If the denominators are not the same then you must first make them the sameAdd or subtract (borrow if you have do)Make sure your answer is a proper fraction and in simplest form.
      51435007658100<br />5143500765810051435007658100<br />
      • *Prime Numbers*Prime #’s are those that have only 2 factors like 5, 7, 3, 11.*Composite Numbers*Composite #’s are those that have more than 2 factors like 4, 6, 8, 10.Prime and Composite Numbers8699512001551435007658100
      • Subtracting Fractions w/ borrowing
      In this case you can’t subtract 3/12 from 5/12 so you have to borrow which makes the whole # 10 a 9. Again look at the denominator which is a 12 so you are borrowing 12/12. Add 12/12 to 3/12 and you get 15/12. Now you can subtract and make sure your answer is in simplest formThere is not fraction on top to subtract 1/3You must borrow from the 6 which makes the whole # a 5. When you borrow you are borrowing a whole #. Since the denominator in 1/3 is a 3 you are borrowing 3/3.Now you can subtract & make sure you answer is in simplest form <br />Greatest Common Factor (GCF)<br />1) List all of the factors for each #2) Find the greatest factor that the #’s both haveExample: 12 = 12, 1, 2, 6, 3, 4 16 = 1, 16, 2, 8, 4 GCF = 4 <br />Prime Factorization<br />726757530480<br />-41767871092<br />Factors, Multiples, Prime #’s, & Composite #’s<br />Factors: the 2 #’s you multiply to get the product. In 2 x 3 = 6, 2 & 3 are the factors.Factors of 8 = 8, 1, 4, 2Multiples: Multiples of 8 = 8, 16, 24, 32, 40, 48 etc..Multiples of 3 = 3, 6, 9, 12, 15, 18, 21 etc..Prime #’s: Have only 2 factorsComposite #’s: Have more than 2 factors<br />Least Common Multiple (LCM)<br />1) List all of the multiples for each #2) Find the smallest multiple that the #’s both haveExample: 3 = 3, 6, 9, 12, 15, 18, 21, 24 5 = 5, 10, 15, 20, 25, 30, 35 LCM = 15<br />Improper fractions into mixed #’s318135084455<br />1) In the box out of the box (numerator in, denominator out)2) Whole #: Quotient Numerator: Remainder Denominator: Divisor3) Make sure your answer is in simplest form<br />Mixed #’s into Improper Fractions<br />1) Multiply the denominator and the whole # together2) Add the answer from step one to the numerator3) The denominator from the mixed # will be the denominator in your improper fraction318533190445<br />Ordered Pairs (x,y)Proper Fractions, Improper Fractions, Mixed #’s1) A proper fraction is one where the numerator is smaller than the denominator: 1/3, 3/5, or 6/82) An improper fraction is one where the numerator is larger than the denominator: 5/3, 8/2, or 13/43) A mixed # will have a whole # and a fraction: 3 2/4, 5 1/6, or 7 4/9.<br />3881366168152<br />Probability1) Probability is always a fraction2) Numerator: what you want to get3) Denominator: Total possible outcomesExample: Probability of landing on a 3Answer: 1/8Example: Probability of landing on an even #Answer: 4/8 = 1/21) To find the probability of 2 or more events u must add the probabilities of each event together & then simplifyExample: Probability of landing on a 3 then a 4. P(3,4)Answer: 1/8 + 1/8 = 2/8 = 1/4Example: Probability of landing on an even # then 3. P(even#,3)Answer: 4/8 + 1/8 = 5/8Adding Probabilities<br />59436006400800<br />59436006400800<br />Median, Range, OutlierMode & MeanMode: The # that occurs the most often (there can be no mode or several modes)Example: 5,6,2,6,3,4,9mode = 6 Example: 4,5,2,1,4,6,5mode = 4, 5Mean: 1) add all of the 3’s2) divide by how many #’s there are3) no remainders you must add a decimal and bring it up4) round your answer to the nearest tenth<br />Median:1) Put the #’s in order from least to greatest2) Cross the #’s out two at a time from the outside3) If there r 2 #’s left, add them together and divide by 2 (no remainders and round to nearest tenth)Range: Subtract the largest # from the tiniest #Outlier: The # in a set of #’s that does not belong, it either way larger or smaller than the rest of the #’s. ex: 2,4,5,6,22…outlier would be 22.<br />Subtracting IntegersAdding Integers<br />1) Change the subtraction sign into a plus sign2) Change the sign of the 2nd integer3) use addition rules to addExample: (-10) - (+3) = (-10) + (-3) = -13Example: (+7) – (+2) = (+7) + (-2) = +51) If the integers have the same sign (either both positive or negative) then you just addExample: (+5) + (+4) = +9 Example: (-4) + (-3) = -72) If the signs of the integers are different (one is positive and one negative) then you must subtract. The larger # goes on top and use the sign of the larger #.Example: (-10) + (+3) = -7Example: (+15) + (-6) = +9<br />Area of a rectangle or square<br />Area = Length x Width (A=lw)(area is always squared)Example: Area = Length x WidthArea = 14 x 7Area = 98cm²Multiply & Dividing Integers<br />1) If the signs of the integers are the same then your answer is always positive (+)Example: (+5) x (+4) = +20 Example: (-12) ÷ (-3) = +42) If the signs of the integers are different then your answer is always negative (-)Example: (-10) x (+3) = -30Example: (+54) ÷ (-6) = -9<br />Area of a parallelogramArea of a triangle<br />Area = base x height (A= bh)(area is always squared)Example: Area = base x heightArea = 15 x 5Area = 75cm²Area = 1/2 x base x height (A= 1/2bh)(area is always squared)Example: Area = 1/2 x base x heightArea = 1/2 x 12 x 15Area = 180/2Area = 90m²<br />Area of a CircleArea of a CircleCircles<br />Area = TTr² (TT = 3.14)(area is always squared)Example: Area = TTr²Area = 3.14 x 4²(4 x 4)Area = 3.14 x 16Area = 50.24m²Area = TTr² (TT = 3.14)(area is always squared)Example: Area = TTr²Area = 3.14 x 6²(6 x 6)Area = 3.14 x 36Area = 113.04m²<br />Volume of a rectangular prismCircumference of a CircleCircumference of a Circle<br />Volume = (area of the base) x height (volume is always cubed)Example: Volume = (area of the base) x heightVolume = (length x width) x heightVolume = (8 x 3) x 12Volume = 288in³C = TT x Diameter (TTd)(TT = 3.14)(area is always squared)Example: C = TTdC = 3.14 x 8C = 25.12mC = TT x Diameter (TTd)(TT = 3.14)(area is always squared)Example: C = TTdC = 3.14 x 12C = 37.68m<br />Volume of a triangular prism<br />Volume = (area of the base) x height (connects the bases) (volume is always cubed)Example: <br />Volume = (area of the base) x heightVolume = (1/2 x base x height) x heightVolume = (1/2 x 6 x 3) x 12Volume = 9 x 12Volume = 108cm³<br />Volume of a cylinderVolume = (area of the base) x height (volume is always cubed)Example: Volume = (area of the base) x heightVolume = (TTr²) x heightVolume = (3.14 x 4²) x 12Volume = 50.24 x 12Volume = 602.88in³Volume = (area of the base) x height (volume is always cubed)Example: Volume = (area of the base) x heightVolume = (TTr²) x heightVolume = (3.14 x 4²) x 12Volume = 50.24 x 12Volume = 602.88in³Volume of a cylinder<br />1) Change the percent into a decimal (add a decimal to the end of the number and then move the decimal to the left twice)2) Larger # goes on top, multiply, count decimalsExample: 25% of 35 = .25 x 35 = 8.75Example: 34% of 125 = 125 x .34 = 42.5Order of operations1)Brackets + Parenthesis first2) exponents3) multiply or divide from left to right4) add or subtract from left to rightExample:(48 ÷ 8) x 5 + 3²6 x 5 + 3²6 x 5 + 930 + 9Answer: 39Percent (%) of a # (of = multiply)<br />Tax & Total Cost<br />Example:Subtotal: $25Tax : 7%Tax Amount: $1.75Total Cost: $26.751) Change the tax from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 7% = .072) then multiply the tax by the subtotal (amount you are spending). This gives you the tax amount.25 x .07 = $1.753) To find the total cost you must then add the tax amount to your subtotal25.00 + 1.75 = $26.75<br />Discount & Sale Price<br />Example:Subtotal: $52Discount : 25%Discount Amount: $13.00Sale Price: $39.001) Change the discount from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 25% = .252) then multiply the discount by the subtotal (amount you are spending). This gives you the discount amount.52 x .25 = $13.003) To find the total cost you must then subtract the discount amount from your subtotal52.00 - 13.00 = $39.00<br />Tip & total Cost<br />1) Change the tip from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 15% = .152) then multiply the tip by the subtotal (amount you are spending). This gives you the tip amount.85 x .15 = $12.753) To find the total cost you must then add the tip amount to your subtotal85.00 + 12.75 = $97.75Example:Subtotal: $85Tip : 15%Tip Amount: $12.75Total Cost: $97.75<br />Proportions<br />If Alex ran 12 miles in 3 hours then how many did he run in 1 hour?*in order to solve this problem you must use a proportion.*When using a proportion the labels in the numerators must be the same as well as the label in the numerators.*Once your proportion is set up you must then cross multiply (always start with the variable) and solve accordingly.With Shapes*When doing proportions with shapes…each side of the proportion corresponds with the exact same side on each shape.*Same shape has to be on top and bottom of the proportion then cross multiply and then solveX miles12 miles1 hour3 hours3x = 12X = 12/3X = 4 miles<br />=<br /> 4m x 8m 6m8x = 24X = 24/8X = 3 meters<br />=<br />x<br />Missing Angles: TrianglesMissing Angles: Triangles<br />The 3 angles of a triangle add up to 180̊°Example: 1) add the 2 angles together 36 + 57 = 932) Then subtract your answer from 180 180 – 93 = 873) angle a = 87̊4) If your answer is correct then all 3 angles should add up to 180<br />The 3 angles of a triangle add up to 180̊°Example: 1) Subtract 180 - 40 180 - 40 = 1402) 140 is how many degrees both missing angles add up to. To find each individual angle’s degrees you must divide by 2 140 ÷ 2 = 703) So each missing angle (x) is 70°4) If your answer is correct then all 3 angles should add up to 180<br />x40°<br />x<br />Missing Angles: Quadrilaterals<br />The 4 angles of a quadrilateral add up to 360°Example: 1) add the 3 angles together 50 + 50 + 130 = 2302) Then subtract your answer from 360 360 – 230 = 1303) missing angle = 1304) If your answer is correct then all 4 angles should add up to 360<br />Missing Angles: Quadrilaterals<br />The 4 angles of a quadrilateral add up to 360°Example: 1) add the 2 angles together 48 + 64 = 1122) Then subtract your answer from 360 360 – 112 = 2483) 248 is the total of both missing angles together4) To find each angle you must divide by 2248 ÷ 2 = 1245) See each missing angle is 124°6) If your answer is correct then all 4 angles should add up to 360<br />x48°x64°<br />Perimeter<br />1) add all of the sides2) line up the decimals if there are anyExample:Perimeter = add all sidesPerimeter = 1 + 5 + 4 + 2 + 7Perimeter = 19<br />Decimals into fractions (always simplify)Fractions into Decimals<br />Example: 1.251) the # before the decimal is always the whole #2) look at the last #..the 5 is in the hundredths place so the fraction would be:1 25/100 = 1 1/4Example: .81) the 8 is in the tenths place so the fraction would be:8/10 = 4/5Example: .051) the 5 is in the hundredths place so the fraction is: 5/100 = 1/201) In the box out of the box (numerator in, denominator out)2) Add a decimal to the end of the # (dividend) and bring it up3) There are NO remainders, if you have a remainder then you must add a zero, drop it, and continue dividing*To change a fraction into a % you must change it to a decimal then to a %<br />1) If there is no decimal in the percent then you must add a decimal at the end of the percent 45% = 45.%2) Move the decimal twice to the left 45.% = .45Example: 4.5%4.5% = .045Example: 125%125.% = 1.25Decimals into PercentsPercents into Decimals1) Move the decimal twice to the right .38 = 38%Example: .8.8 = 80%Example: 99.00 = 900%<br />Complementary AnglesesSupplementary AnglesVertical Angles<br />1) Two angles that add up to 90°<br />1) Angles that are opposite to each other (are always congruent)1) Two angles that add up to 180°<br />Types of Triangles<br />QuadrilateralsPolygons<br />Simplifying Fractions<br />1) List all of the factors of the numerator2) Start w/ the largest factor and see which one you can divide the denominator by3) Once u find the factor you must divide both the numerator and denominator by it 4) Repeat until u can no longer simplify anymore<br />Comparing Fractions<br />1) U can only compare fractions when the denominators are the same2) If the denominators are different then make them the same 3) Compare the fractions <br />Least GreatestNumber LinesDistance = Rate x Time 1) Plug in known values into the formula then solve the equation.Ex: Erik ran 24 miles at a rate of 4 miles per hour. How long did it take him to run 24 miles?D = R x T24 = 4T24/4 = T6 hours = TLeast to Greatest<br />
      • 1) If there r fractions u must change them all to decimals first.2) Line up your decimals and compare your #’s (be very careful if there are negative integers)3) Remember that the # furthest left on the # line is always the smaller #.4) Write down the original #’s for your answer.Example (least – Greatest) 1/2, .67, 2, 3/41/2, .67, 3/4 = .50, .67, 2.00, .75Put in order= .50, .67, .75, 2.00Answer = 1/2, .67, 3/4, 2
      Permutations<br />
      • Example 2: how many 4 letter codes can u make from the letters a,b,c,d,e (repeat letters can be used)?Since there r repeat letters…once a letter is chosen it can be used again (so a code of aaaa would be allowed)For the 1st letter in the code u can choose from 5 lettersFor the 2nd u can choose from 5 lettersThe 3rd 5 letters and the 4th 5 lettersSo your permutation would look like this: 5 x 5 x 5 x 5 = 625 possible codes If a problem is a permutation then the order of the items matters.For example if you were to make a 4 letter code from the letters a, b, c, d, e……then the code abcde is totally different than the code abced.Therefore in this case order matters and the problem would be a permutation.Example 1: how many 4 letter codes can u make from the letters a,b,c,d,e (no repeat letters can be used)?Since there r no repeat letters…once a letter is chosen it cannot be used again (so a code of aaaa would not be allowed)For the 1st letter in the code u can choose from 5 lettersFor the 2nd u can choose from 4 lettersThe 3rd 3 letters and the 4th 2 lettersSo your permutation would look like this: 5 x 4 x 3 x 2 = 80 possible codes
      • Example 2: how many ways can u give 3 cans to 8 kids?The order of the cans does not matter so we have a combinationSince there r 8 kids….u can choose 8 different kids for the 1st can, 7 kids for the 2nd can, & 6 kids for the 3rdSo far your combination looks like this 8 x 7 x 6..Since we r using 3 cans we must divide by 3! (3 factorial)So we now have 8 x 7 x 6 divided by 3! = 336/6 = 56 ways to give 3 cans.If a problem is a combintation then the order of the items does not matter.For example if you were to make teams of 2 from matt, sara, tim, rob, and ally……then the team of tim & rob is the same as rob & tim Therefore in this case order does not matter and the problem would be a combination.To solve combinations you must use factorials…..4 factorial (4!) = 4 x 3 x 2 x 1 and 5 factorial (5!) = 5 x 4 x 3 x 2 x 1In combinations u must divide by the factorial of the # in each team or group to get rid of duplicate groups like team rob/tim & team tim/robExample 1: how many teams of 2 can u make from sara, tim, rob, and ally?Since there r 4 kids….u can choose 4 different kids for the 1st spot on the team and then 3 kids for the 2nd spot on the team….So far your combination looks like this 4 x 3..Now we r choosing teams of 2 so we must divide by 2! (2 factorial)So we now have 4 x 3 divided by 2! = 12/2 = 6 teams of 2.Combinations
      • Probability w/replacementProbability with replacement means that once an item is picked it is then put back into the sample space so that it can be picked againExample 1: What the probability of picking a black marble and then a stripped marble (p) black,strpiped w/ replacementThe probability of getting a black marble = 1/6The probability of getting a stripped marble = 3/6Add the probabilities for each event1/6 + 3/6 = 4/6 = 2/3
      Probability w/o replacement<br />
      • Probability without replacement means that once an item is picked it is not put back into the sample space so that it cannot be picked againExample 1: What the probability of picking a black marble and then a stripped marble (p) black,strpiped w/o replacementThe probability of getting a black marble = 1/6The probability of getting a stripped marble = 3/5 (remember there is one less marble now)Add the probabilities for each event1/6 + 3/5 = 5/30 + 18/30 = 23/30
      Figuring out percents (out of)<br />1) Change to a fraction2) Change the fraction into a decimal3) Change the decimal into a percentExample 1: Find the % for 1 out of 41) change to a fraction = 1/42) change to a decimal = .253) change to a percent = 25%<br />Probability (Finding all Possible Combinations): List, Tree Diagram, Counting Principal<br />ShirtsSizesRedTree DiagramSmallBlueMediumWhiteLargeX-Large<br />Counting Principal1) Multiply the total # of choices from each category2) 3(shirts) x 4 (sizes) = 12 possible combinationsList1) In a list you do just that make a list of all of the possible combinations..this is also usually done at the end of a tree diagram.Ex.Red, small blue, small white, smallRed,medium blue,medium white,mediumRed,large blue, large white,largeRed,xlarge blue,xlarge white,xlarge<br />Graphing Equations<br />Ex: Graph the equation y = 2x1) Make a table…in place of x pick 3 #’s usually pick 3’s that r easy to calculate…in this case we will pick 1, 2, -2XY12-22) Fill out the table by solving the equation with each value u have for x (y = 2 x 1…so y = 2)XY1222-2-43) Plot each set of points on a graph…connect your dots and then label your equation on the line.<br />y = 2x<br />AdditionSubtractionMultiplicationDivisionSumDifferenceOfEachDepositWithdrawAboveBelowForwardBackwardIncreaseDecreaseGainLossAltogetherWords and their operation<br />You can divide any number by…1) 2 if the number is even2) 3 if the individual numbers in the number add up to a multiple of 33) 5 if the number ends in a 5 or 04) 10 if the number ends in a 05) if you can divide by 2 then always try a 4 and an 86) if you can divide by 3 three then always try a 6 and a 9Hidden Integers<br />1) Change the rate from a percent into a decimal (add a decimal to the end of the # and then move the decimal to the left twice) 4% = .042) then multiply the principal by the rate. 52 x .05 = 2.083) Now multiply that answer by the time.2.08 x 3 = $6.24Calculating Interest…..Interest = Principal x rate x time<br />Example:Principal (amount u have in bank): $52Rate (interest rate): 4%Time (amount of time interest is accruing): 3 yearsInterest (amount gained over x amount of time): ?<br />