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Reference PagesTable of contents (arranged alphabetically)Adding with decimals (9)Benchmark numbers (14)Circles (22)Combining like terms (17)Common reasons for setting up a proportion (4)Common reasons for using the operations (3)Converting between fractions, decimals, and percents (15)Distribution (17)Divisibility (5)Factoring (17)Fractions (14)Greatest common factor (7)Inequalities (20)Invisible things (14)Least common multiple (8)Long division (13)Measures of center, spread, and mode (23)Multiplication and division tables (5)Multiplying and dividing by powers of ten (4)Multiplying with decimals – matrix method (11)Multiplying with decimals – traditional method (12)Negatives (17)Percent expressions (16)Percent word problems (15)Place value (5)Probability (21)Properties (3)Proportional relationships (16)Representing division (4)Rounding (6)Scale drawings (18)Similar figures (20)Solving complicated equations (19)Solving simple equations (18)Subtracting with decimals (10)Surveys (24)The 5 actions of arithmetic (2)Unit rates (16)Writing equations and inequalities (25)The 5 Actions of Arithmetic (yes, five!) When the number of groups is a whole number, grouping can be thought of as repeated addition like in the examples above and the first example below… 5+3=8 3+5=8Five put together with three Three put together with five The commutative property holds. 14 1 2 3 4 8 2=4 groups 2 8=14 of a group How many groups How many groups of two fit into eight? of eight fit into two? Fit into The commutative property DOES NOT hold. “Fitting into” always gives you the same numerical answer as “splitting up” so people just use the ÷ symbol for everything. That’s a little confusing because whenever you see the ÷ symbol, it could mean splitting up but it could also mean fitting into. Notice that when we’re splitting up, the answer refers to the number of items in each group but when we’re fitting into, the answer refers to the number of groups. The commutative property DOES NOT hold. 14 1 2 3 4 8÷2=4 items per group 2÷8=14 of an item per group Eight split up Two split up into two groups into eight groups The commutative property still holds for fractions. Challenge question: Can you show that 12×34=34×12 ?Split it up into four parts and take three of them.Start with five. 34×5=154 Three fourths of a group of five.++++ When the number of groups is a fraction, grouping can be thought of as taking a piece of a number like in the example below… 5×34=154 Five groups of three fourths. 5×3=15 3×5=15 Five groups of three Three groups of five 3+3+3+3+3=15 5+5+5=15 The commutative property holds. 5-3=2 3-5=-2 Three taken away from five Five taken away from three The commutative property DOES NOT hold. Split up ÷ Group × Take away - Put together + Common reasons for using the operationsCommon reasons for subtractionTaking away (by definition)Finding how many more of something there is compared to something else.Finding how much greater or longer something is compared to something else.Finding the distance between two mon reasons for additionPutting together (by definition)Finding a totalCommon reasons for multiplicationGrouping as in repeated addition (by definition)Finding a fraction or percent of a number (by definition)Finding the area of a parallelogramFinding the volume of a rectangular prismCommon reasons for divisionSplitting up into equal parts (by definition)Seeing how many times one thing fits into another (by definition)Finding a unit ratePropertiesaCommutative property: a+b=b+a a×b=b×aSwitching the order for addition or multiplication does not change the result.Switching the order for subtraction gives you the opposite (commutative property does not hold).Switching the order for division gives you the reciprocal (commutative property does not hold).Examples: 12+4=16 12×4=48 12-4=8 12÷4=3 4+12=16 4×12=48 4-12=-8 4÷12=13Associative property: a+b+c=a+b+c a×b×c=a×b×cWhen adding or multiplying a string of numbers, we may group the numbers in any way without changing the result. The associative property does not hold for subtraction or division.Examples: 10+5+4 = 15+4 = 19 10×5×4 = 50×4 = 200 10+5+4 = 10+9 = 19 10×(5×4) = 10×20 = 200 10-5-4 = 5-4 = 1 10÷5÷4 = 2÷4 = 12 10-5-4 = 10-1 = 9 10÷(5÷4) = 10÷1.25 = 8 Distributive property: a×b+c=a×b+a×cThe products of sums is equal to the sums of products.Example: 3×6+2 = 3×8 = 24 3×6+3×2 = 18+6 = 24Common reasons for setting up a proportionConverting a fraction into a percent NumeratorDenominator=Percent (x)100Unit rateExample: When looking for miles per hour, miles go in the numerator and hours go in the denominator. MilesHours=Unit rate (x)1Similar figuresHere “small” and “big” refer to the small shape and big shape. Small Side ASmall Side B=Big Side ABig Side B Small Side ABig Side A=Small Side BBig BBasic proportionConverting units Small Unit ASmall Unit B=Big Unit ABig Unit B Small Unit ABig Unit A=Small Unit BBig Unit BTo solve proportions…Find the new valueFind the original value neworiginal=100±percent100Find the percent of change/error differenceoriginal/real=Percent (x)100Find the partFind the wholeFind the percent of the whole PartWhole=Percent100 25=x6 12=5x 12=5x 5 5 2.4 = x Cross multiply. Put an equal sign between the products. Divide both sides by whatever is next to the unknown.Surveys GoodTotal (sample)=Prediction (x)Total (population)Probability: Making predictions GoodTotal=Predition (x)AttemptsMultiplying and dividing by powers of 10Dividend: 2 Divisor: 8What is 2 divided by 8? What is 8 divided into 2?What is 2 split up into 8 equal parts? How many times can 8 fit into 2? Answer:4 82 8÷2 2 ) 8Standard Fraction LongDividend: 8 Divisor: 2What is 8 divided by 2? What is 2 divided into 8?What is 8 split up into 2 equal parts? How many times can 2 fit into 8?Answer: 14 28 2÷8 8 ) 2Standard Fraction LongThere are three ways of writing division. To convert between them, Simply FoLlow the rainbow. Make sure your rainbow has the Standard division symbol on the left, Fraction bar in the middle, and Long division symbol on the right. Make sure you also know which number is the dividend, which number is the divisor, and all the different ways of writing the division in words. Be careful! With certain phrases you say the dividend first and with others you say the divisor first.Representing divisionMultiplication and division tablesDivisibilityTo tell if a number is divisible by…5See if it ends in 0 or 5.2See if it ends in 0, 2, 4, 6, or 8.4Check the two-digit number at the end and see if it is divisible by 4. For example, to check 97,005,328, just check 28. 28 is divisible by 4 so the original number 97,005,328 is as well. 3Add up all the digits and see if you get a number that is divisible by 3. For example, to check 825, do 8+2+5=15. 15 is divisible by 3 so the original number 825 is as well.10See if it ends in 0. 6Make sure it is divisible by 2 and 3. For example, 34 is divisible by 2 because it ends in 4. To check if it’s divisible by 3, do 3+4=7. 7 is not divisible by 3 so 34 is not divisible by 3 or 6. 9Add up all the digits and see if you get a number that is divisible by 9. For example, to check 559, do 5+5+9=19. 19 is not divisible by 9 so the original number 559 is not either.8Check the three-digit number at the end and see if it is divisible by 8. For example, to check 4,260,520, just check 520. 520 is divisible by 8 (you can check that with long division) so the original number 4,260,520 is as well. Place valuePlace values are based on how many spaces away from the decimal point a digit is. 4 0 3 , 9 7 4 . 0 5 5 6 0 100,000’s 10,000’s 1,000’s 100’s 10’s 1’s 110’s 1100’s 11,000’s 110,000’s 1100,000’shundred- ten- thousands hundreds tens ones tenths hundredths thousandths ten- hundred-thousands thousands thousandths thousandthsRoundingIf we don’t care about being exactly right, then we can round a number to get another number which is close enough.Example 2:Round 16.04281 to the nearest thousandth.Example 1:Round 587,403.92 to the nearest thousand.Find the digit in the place value you’re rounding to. Call it the “rounding digit”.Step 1: R 16.04281 R 587,403.92Find the digit one space to the right of the rounding digit. Call it the “decision digit”.Step 2: RD 16.04281 RD 587,403.92It the decision digit is 5, 6, 7, 8, or 9 then the rounding digit gets bumped up by 1. (Notice the 2 became a 3). RD 16.04381 16.043Step 3: If the decision digit is 0, 1, 2, 3, or 4 then the rounding digit stays the same (Notice the 7 is still a 7). RD 587,403.92 587,000Remove all digits after the rounding digit (even the decision digit). More specifically, digits after the rounding digit but before the decimal point become zeros. Digits after the rounding digit and after the decimal point get removed completely.Step 4: Round to what?Rounding to the nearest “whole number”, “percent”, or “dollar” means rounding to the ones place.Rounding to the nearest “cent” means rounding to the hundredths place because a penny is a hundredth of a dollar.Trouble with 9’sIf you have to bump up a 9, it becomes a 0 and the next digit on the left gets bumped up by one.Example 3:Round 38,497.20581 to the nearest tens place. RD The result is… 38,497.20581 38,500If you have to bump up a 9 and the digits directly to the left of it are also 9’s then they all become zeros and the digit to their left gets bumped up by one.Example 4:Round 999,959.99546 to the nearest hundredths place. RD The result is… 999,959.99546 999,960.00 Greatest common factor (GCF)The greatest common factor of two numbers is the biggest number that goes nicely into both of them. Note: A “factor” of a number is a number that goes into it.Simple method: This method is better when the numbers are small. Sometimes you can even do it in your head.Example: Find the greatest common factor of 24 and 36. Step 1: List the factors of each number. 24: 1 2 3 4 6 8 12 24 36: 1 2 3 4 6 9 12 18 36Step 2: Circle the factors they have in common. 24: 1 2 3 4 6 8 12 24 36: 1 2 3 4 6 9 12 18 36Step 3: Choose the greatest (biggest) common factor from Step 2. It’s 12!Prime factorization method: This method is better when the numbers are larger. The idea here is to build the biggest factor we can by using as many factors as possible. However, we can’t use all the factors because not all of them are factors of every number.Example: Find the greatest common factor of 126 and 162.Step 1: Find the prime factorization of each number. (Just keep breaking down the numbers into factors until you can’t break them down anymore.) 162 2 81 9 9 3 3 3 3 126 2 63 7 9 3 3Step 2: Find the factors that both numbers have in common. Pay attention to how many of them there are. Both numbers have at least ONE 2. Both numbers have at least TWO 3’s.Step 3: Multiply the factors from the last step together. 2×3×3=18.Shortcut method: If one of the numbers is a factor of the other one then that number is the greatest common factor.Example: Find the greatest common factor of 10 and 20.10 is a factor of 20 so the GCF is 10.Least common multiple (LCM)The least common multiple of two numbers is the smallest number that is a multiple of both of them. Note: A “multiple” of a number is a number that it goes into.Simple method: This method is better when the numbers are small. Sometimes you can even do it in your head.Example: Find the least common multiple of 2 and 5. Step 1: List the multiples of each number. 2: 2 4 6 8 10 12 14 16 18 20 22… 5: 5 10 15 20 25 30…Step 2: Circle the multiples they have in common. 2: 2 4 6 8 10 12 14 16 18 20 22… 5: 5 10 15 20 25…Step 3: Choose the least (smallest) common multiple from Step 2. It’s 10!Prime factorization method: The idea here is to use as few factors as possible to still end up with a multiple of all the numbers. If you do it correctly then each number’s prime factorization will be found within the prime factorization of the LCM.Example: Find the greatest common factor of 150 and 48.Step 1: Find the prime factorization of each number. (Just keep breaking down the numbers into factors until you can’t break them down anymore.) 48 2 24 2 12 3 4 2 2 150 3 50 2 25 5 5Step 2: For each factor that you see, write the largest number of times it occurs in any one number. The largest number of 2’s is FOUR (48 has this). The largest number of 5’s is TWO (150 has this). The largest number of 3’s is ONE (both have this).Step 3: Multiply the factors from the last step together. 2×2×2×2×5×5×3=1,200.*If there are no common factors in the prime factorizations then you can skip steps 2 and 3 and just multiply the original numbers together.Shortcut method: If one of the numbers is a multiple of the other one then that number is the least common multiple.Example: Find the least common multiple of 10 and 20.20 is a multiple of 10 so the LCM is 20. Adding with DecimalsWhen adding with decimals, keep in mind…1. The largest number does NOT have to go on top like it does when subtracting.2. The decimals MUST be lined up, including in the answer. If you don’t see a decimal, it’s invisible. It’s at the very end of the number.3. Blank spaces count as zeroes. You may put them in if you like, but it is not as important as it is with subtraction.4. Always add from right to left. 5. If you get a two-digit sum, add the first digit to the next column and write the second digit below. Once you get to the end, just write everything below. 59 . 9 988 .+ 6 . 74 .Example:Line up decimals.59.9 + 988 + 6.74 Invisible decimal 59 . 9 1+9+8+6=24 988 .+ 6 . 74 4 .64 59 . 9 9+7=16 988 .+ 6 . 74 . 64 59 . 9 4+0=4 988 .+ 6 . 74 . 4 59 . 9 1+9=10 988 .+ 6 . 74 1054 .64 59 . 9 2+5+8=15 988 .+ 6 . 74 54 . 64When subtracting, keep in mind…1. The larger number MUST go on top. The larger number is not necessarily the number with the most digits. For example…2. The decimals MUST be lined up, including in the answer. If you don’t see a decimal, it’s invisible. It’s at the very end of the number.3. Blank spaces count as zeroes so put them in.4. Always subtract from right to left.5. We cannot take a larger number away from a smaller number. If this happens, we must “borrow” from the next column by reducing the digit in that column by one and then placing a 1 in front of the number that was too small in the first place (by doing this we are really adding 10 to the number that was too small).6. We cannot borrow from 0. If this happens, we must instead borrow from the number in the next column, then put a 1 in front of the 0, turning it into a 10, then borrow from the 10 that we just created, turning it into a 9. Example 1: Subtracting with Decimals Example 2: Multiplying with Decimals – Matrix Method...Step 3: Multiply each digit of the horizontal number with each digit of the vertical number. Write the product inside the box. If the product only has one digit, put a 0 inside the upper left half of the box........Answer: 46.3125Answer: 2024.6Step 5: Draw a line down from the decimal in the horizontal number and to the left from the decimal in the vertical number. When the lines connect, continue along that diagonal. That’s where you will place the decimal in your final product. IMPORTANT: If you don’t see a decimal, it’s invisible. It comes at the end of the number and it’s a good idea to write it in. See Example 2. . Example 2: 382×5.3 Example 1: 6.25×7.4115374153740502062008243514421741221364505020620082435144212213645050206200824351442174 6 . 2 5 3 8 253062409104015531164202062409104015116420206240910401553 3 8 2Step 4: Add the digits along each diagonal. This is just like normal addition (from right to left) so, for example, if you got a sum of 17, you would have to carry the 1 into the next diagonal.Step 2: Draw diagonal lines going through the boxes. Each box should be split up into an upper left half and a lower right half.Step 1: Write one number horizontally and the other vertically (it doesn’t matter which number goes where) and draw boxes inside the two numbers. Each digit gets its own column or row of boxes. Each decimal should line up with a line. 6 . 2 5174 6 . 2 5 6 . 2 5 6 . 2 5 3 8 2 3 8 2 3 8 2Invisible decimalwritten in Multiplying with Decimals – Traditional MethodWhen multiplying with decimals, keep in mind…1. You do NOT have to line up the decimals. In fact, it’s easier if you don’t.2. Multiply from right to left. Multiply each digit in the bottom with each digit in the top. If you get a two-digit number as your product, put the first digit above the next number in the top and write the second digit on the bottom. If you put a digit in the top, add it to the next product. When you come to the end, just write everything in the bottom. 3. Whenever you move to the next digit in the bottom number, put an extra x (first 0, then 1, then 2, then 3, etc.) down below. 5. Once you’re done with the multiplying part, add everything up. Remember that x’s count as 0’s. After that, count how many spaces in from the right side the decimal was in the original numbers. Add the number of spaces together. That’s how many spaces from the right the decimal should be in the final answer. 6. It is easier to multiply by putting the number with more digits on top and the number with less digits below. You get the same answer no matter what, but you can see from the examples below, you’ll do a lot more work if you put the number with more digits on the bottom. Long DivisionA decimal in the dividend or quotient is alright, but we don’t want decimals in the divisor. If you see one, the first thing you should do is move it to the end. Next, however many spaces you moved it in the divisor, you have to move it that many spaces in the dividend as well.Step 1: How many times can 9 fit into 4? 0 times. Write that next to the 1.Now we’re done.Bring down another 0.Bring down a 0.Didn’t get a 0If there are no more digits to bring down but you didn’t get a 0 the last time you subtracted, place a decimal in the dividend and quotient and keep bringing down 0s until you’re done. Repeat the four basic steps of long division until there are no more digits to bring down AND you got a 0 the last time you subtracted. 1. into 2. × 3. - 4. Some students are afraid of 0s. Don’t be! If you get a 0 in Step 1, just put it in there like any other number. The decimal goes in the same place in the dividend and quotient. If we see that one or more digits in the quotient are going to keep repeating themselves forever, we write a line over the repeating part in our answer. Answer: 8.83 Answer: 0.75 Make sure you’re putting your digits in the correct place in the quotient. In the example below, 25 does not go into 1 or 13, but it goes into 137 five times. The 5 would go above the 7, not above the 1 or 3. FR7. Multiplying or dividing with mixed numbersTurn them into improper fractions first.FR6. BorrowingSimilar to borrowing with decimals but you have to add the denominator instead of putting a 1 next to the number. x2 x2 x3 x3 x3 x2 x3 x2FR5. Improper fraction to mixed numberDivide the numerator by the denominator. The remainder is the new numerator. Keep the original denominator. FR4. Mixed number to improper fractionMultiply denominator by whole number, add the numerator, keep the original denominator.FR3. DivideMultiply by the reciprocal (keep- change-flip).FR2. MultiplyMultiply numerators and multiply denominators. FR1. Add or subtractMake sure the denominators are the same first. 56 +14 = 1012+312=1312 56 +14 = 1012-312=712Fractions 165FR8. Multiplying a mixed number with a whole numberYou could turn the mixed number into an improper fraction but sometimes it’s easier to distribute. 3×6417=181217What missing number makes two fractions equivalent?Strategy 1: Cross multiply and then divide both sides to isolate the unknown.Strategy 2: If you have numbers that go nicely into each other, you can symmetrically multiply or divide either horizontally or vertically.Examples for Strategy 2… How do you know if two fractions are equivalent?Strategy 1: Divide the numerator by the denominator for each fraction and see if you get the same number.Strategy 2: Put the fractions next to each other and see if you can multiply or divide horizontally or vertically by the same number.Examples for Strategy 2… Creating equivalent fractions / reducingIf you multiply the numerator and denominator of a fraction by the same number, the result is equivalent to the fraction you started with. This is because you really multiplied by 1! 27×55=1035 The same is true when you divide and then it is called “reducing”. 2842÷1414=23 What are equivalent fractions? Equivalent fractions are fractions that have the same value, even though they look different. This is just 1!This is just 1!Invisible thingsIT2. Invisible coefficientsIf you don’t see a coefficient, it’s 1. x 1xIT1. Invisible decimalsIf you don’t see a decimal, it’s at the end of the number. 47 47.IT4. Invisible operationsIt is often useful to think of a positive number as having a plus sign in front of it.7j-10y +7j-10y IT3. Invisible exponentsIf you don’t see an exponent, it’s 1. 5r 5r1Benchmark numbersBE4. Fifths 15=0.2 25=0.4 35=0.6 45=0.8 BE3. FourthsThink of quarters (25 cents each). 14=0.25 24=0.5 34=0.75 BE2. Thirds 13=0.3 23=0.6BE1. Half 12=0.5BE5. SixthsYou can remember some of these more easily by reducing them.16=0.16 26=13=0.3 36=12=0.5 46=23=0.6 56=0.83 BE6. EighthsYou can remember some of these more easily by reducing them.18=0.125 28=14=0.25 38=0.375 48=12=0.5 58=0.625 68=34=0.7578=0.875 BE7. NinthsThe ninths are easy. It’s just whatever the numerator is but repeating. 19=0.1 29=0.2 39=0.3 and so on…BE8. TenthsThe tenths are easy. It’s just whatever the numerator is. 110=0.1 210=0.2 310=0.3 and so on…Converting between fractions, decimals, and percentsCO6a. Fraction to percentDivide the numerator by the denominator, then move the decimal point two spaces to the right.CO6b. Fraction to percentfor “nice” numbersIf the denominator is a factor of 100, find an equivalent fraction with a denominator of 100. CO4. Decimal to peRcentMove the decimal point two spaces to the right.0 . 0 6 =6% 0.6 =60% 6. =600% CO3b. Decimal to mixed numberSimilar to “decimal to fraction” but the number in front of the decimal will also go in front of the fraction. 85.3=85310 85.73=857310085.273=852731,000 CO2. Percent to decimaLMove the decimal point two spaces to the left. 2.5%=0.025 2 5 . %=0.25 1 2 5 . %=1.25CO3a. Decimal to fractionPut it over the power of ten represented by the right-most place value. Do not keep the decimal point.85.3=85310 85.73=8,57310085.273=85,2731,000 CO5. Fraction to decimalDivide the numerator by the denominator. CO1. Percent to fractionPut it over 100. Keep the decimal point where it was.PW4a. Find the new value (basic strategy)Step 1: Multiply the original value by the percent of change (written as a decimal).Step 2: Add or subtract the result (depending on the problem) from the original value.Step 1: $50×0.2=$10Step 2: $50+$10=$60 $50? +20%PW4b. Find the new value (alternate strategy)Step 1: Add or subtract the percent of change (depending on the problem) from 100%.Step 2: Multiply the result (written as a decimal) by the original value.Step 1: 100%+20%=120%Step 2: $50×1.2=$60PW2. Find the wholeDivide the part by the percent (written as a decimal). 48 mL÷0.6=80 mL100%? mL48 mL60%0%0 mLPW1. Find the partMultiply the whole by the percent (written as a decimal). 80 mL×0.6=48 mL100%80 mL? mL60%0%0 mLPercent word problemsPW6a. Find the percent of changeStep 1: Subtract the two values.Step 2: Divide the difference by the original value (and convert into a percent). Step 1: $60-$50=$10Step 2: $10÷$50=0.2=20%PW6b. Find the percent errorThese are very similar to “find the percent of change” problems except you would be given an estimate and a real value.Step 1: Subtract the two values.Step 2: Divide the difference by the real value(and convert into a percent).$50 $60 +?%PW3. Find the percent of the wholeDivide the part by the whole and the convert into a percent. 48 mL÷80 mL=0.6=60% 0%0 mL?%48 mL80 mL100%PW5a. Find the original valueStep 1: Add or subtract the percent of change (depending on the problem) from 100%.Step 2: Divide the result (written as a decimal) by the new value.Step 1: 100%+20%=120%Step 2: $60÷1.2=$50PW5b. Using the P4a strategy backwards doesn’t work!A lot of students try to answer “find the original value” problems by using the P4a strategy in reverse. It won’t work.Step 1: $60×0.2=$12Step 2: $60-$12=$48 wrong answer ? $60 +20%Percent expressionsPE1. To write a two-term expression from a word problem, write the variable first, then a plus or minus sign (depending on the problem), then the percent of change (written as a decimal) times the variable.PE2. Going from a two-term expression to the one-term form is just a matter of combining like terms. Remember that x is the same as 1x.PE3. When writing an expression in words using the words “increase”, “decrease”, “more”, or “less”, use the percentage from the two-term form of the expression.PE4. When writing an expression in words using the words “of” or “as much as”, use the percentage from the one-term form of the expression. -40% +40%0% 60% 100% 140% 200%0x 0.6x x 1.4x 2x x-0.4x x+0.4x 140% of a number140% as much as a number60% of a number60% as much as a number Of / As Much As(follows one-term form)PR1. Definition of proportionalityTwo quantities are in a proportional relationship if the ratio between them is always the same.Proportional relationships / Unit ratesUR4. How to find a unit rate from a proportional tablePick any row and divide the numbers. Order matters! 10 dogs ÷6 cats=123 dogs per cat 6 cats ÷10 dogs=0.6 cats per dogUR2. Constant of proportionalityFor proportional relationships, “constant of proportionality” means the same thing as unit rate.UR1. Definition of a unit rateA unit rate is a ratio between two quantities with one of the quantities having a value of 1. For example, 25 miles per hour means I drive 25 miles for every 1 hour that goes by.PR5. How to tell if an equation is proportionalTo represent a proportional relationship, an equation should not have more than one term on a side. Also, if one of the variables is in the denominator of a fraction, the equation is not proportional. NotProportional Proportional y=-4x y=-4x+6 y=x10 y=10xPR4. How to tell if a graph is proportionalThe line must be straight AND it must go through the origin (0,0).PR3. How to tell if a table is proportionalReduce the numbers in each row. If you always get the same number then the relationship is proportional.You can also divide the numbers in each row. If you always get the same number, the relationship is proportional. PR2. Consequence of proportionalityIf two quantities are in a proportional relationship, then if one gets multiplied by a number the other will get multiplied by the same number.UR3. To calculate a unit rate Divide!But be careful of the order. To calculate miles per hour, divide miles by hours. To calculate hours per mile, divide hours by miles.UR6. How to find a unit rate from an equationThe number that is next to the variable is the unit rate Equation Unit rate y=7x 7 y=-7x -7 y=35x 35 y=-35x -35UR5b. How to find a unit rate from a proportional graph: (1,r) strategyFind 1 on the x-axis and draw a line straight up until you hit the line. The y-coordinate of that point is the unit rate.UR5a. How to find a unit rate from a proportional graph: Dividing coordinates strategyPick any point on the line (except the origin) and divide the y-coordinate by the x-coordinate.NegativesNE3. Multiplication / Division rulesSame signs (both positive or both negative): The answer will be positive.Different signs (one positive and one negative): The answer will be negative.NE2. Adding / Subtracting a positive rules.*Use these rules after rewriting.**These rules are most useful when the first number is negative.Same symbols: Add and keep the symbol. Different symbols: Subtract and take the symbol of the “larger” number (the number with the greater absolute value). * Note When doing subtraction, always put the larger number on top, even if it was negative in the expression.NE1. Rewriting rulesNE4. Finding the distance between two points on a number lineUse subtraction and absolute value.Distance between dog and shirt:5--2=5+2=7=7 or -2-5=-7=7 Distance between house and shirt:-5--2=-5+2=-3=3 or -2--5=-2+5=3=3 NE5. Signs of fractions Combining like terms / Distribution / FactoringCT2. What does “combining” mean?Combining like terms means adding or subtracting them. Only like terms may be combined. If a term cannot be combined, leave it the way it is. Example 1: Example 2: CT1. What are like terms? Like terms are terms that all have the exact same variable(s) raised to the same exponent(s) or terms that all do not have a variable.DI3. Factoring Factoring is like distribution in reverse. We are given an expression without parentheses and we have to rewrite it with parentheses, typically using the greatest common factor.DI2. Distributing just a plus sign or minus signWhen a plus sign is next to the parentheses, nothing happens to the terms inside. When a minus sign is next to the parentheses, all of the terms inside change sign. Example 1: Example 2: DI1. Distribution Multiply the number next to the parentheses with every term inside the parentheses. Pay very close attention to signs. Treat plus signs as positives and minus signs as negatives.Example 1: Example 2: Scale drawingsSD2. Unit scale To find the unit scale, divide both sides of the scale so that one of the numbers becomes 1. 2 cm=45 ft ÷2 ÷2 1 cm=22.5 ftSD1. Scale factor The real life house will be how many times wider and taller than the blueprint? To make this comparison, we first need to make sure the units are the same. The key says every 2 cm on the blueprint corresponds to 45 ft in real life but 1 foot is about 30 centimeters. Therefore, 45 ft is about 45×30=1,350 cm. Now we can divide 1,350 cm by 2 cm to get 675, which is the scale factor. The width and height of the house will be about 675 times larger in real life than they appear on the blueprint.SD4. Converting units: Proportion strategySet up a proportion with the scale written as a fraction on one side. Make sure the units in the numerators match as well as the units in the denominators.How wide is the building in real life? 2 cm45 ft=6 cmx ft 2 cm45 ft=6 cmx ft 2x=270 x=135SD3. Converting units: Multiplication strategyMultiply the number given with the numbers in the scale written as a fraction. Make sure you write the fraction so that the units cancel.How wide is the building in real life? 6 cm × 45 ft2 cm=6×22.5 ft=135ftThe questions that follow refer to the above picture.2 cmSD5. When finding area, you need to convert the length and the widthExample: What is the area of the face of the house (not including the roof) in real life?Solution: 6 cm×22.5 ftcm=135 ft 2 cm×22.5 ftcm=45 ft 135 ft×45 ft=6,075 ft2Common mistake:6 cm×2 cm=12 cm2 12 cm2×22.5 ftcm=270 ft cm Solving simple equationsSE1. Our goal is to get the unknown by itself.SE2. We get the unknown by itself by using inverse operations. Multiplication and division are inverses of each other and addition and subtraction are inverses of each other.SE3. The equal sign is what separates the two sides of an equation.SE4. Whenever an inverse operation is used on one side of an equation, the exact same thing must be done on the other side.SE5. To check to see if we got the right answer, we can take the value we got and plug it back into the original equation. If both sides are the same, our answer is correct. CE1. Rules SE1-SE5 also apply to complicated equations.CE2. If a term is being added/subtracted while inside parentheses or a fraction, you cannot immediately get rid of it by using subtraction/addition. Example 1: Incorrect CorrectCorrect 30=10(4x+2) 30=10(4x+2) 30=10(4x+2) -2 -2 10 10 28=10(4x) 30=40x+20 -20 -20 3=4x+2 10=40x -2 -2 1=4x Example 2: IncorrectCorrect Correct 7=30x-105 7=30x-105 7=30x-105 +10 +10 7=6x-2 35=30x-10 17 = 30x5 +2 +2 +10 +10 9=6x 45=30xCE3: If you choose to multiply or divide an equation, every term on both sides of the equation must get multiplied or divided. However, don’t forget to read CE4 below. Example: Incorrect Correct Incorrect Correct 12-3x=8 12-3x=8 9+x5=20 9+x5 =20 3 3 3 3 ×5 ×5 45+x=100 12-x=223 4-x=223 9+x=100CE4: When dividing an equation, terms in parentheses stay the way they are. At first, you might think this goes against rule CE3 which says that every term gets divided. Actually we are not breaking rule CE3. What’s happening is that the terms are first getting multiplied by the number outside the parentheses and then getting divided; the operations cancel each other. The same is true when we multiply a fraction. Example: Incorrect Correct Incorrect Correct 50=2(8-6x) 50=2(8-6x) 6=8x+364 6=8x+364 2 2 2 2 25=4-3x 25=8-6x 24=32x+144 24=8x+36 because… 50=2(8-6x) because… 6=8x+364 50=16-12x 6=2x+9 and then… 2 2 and then… 6= 2x+9 25=8-6x 24=8x+36Solving complicated equationsInequalitiesIN3: Reading inequalities You can read inequalities from left to right OR from right to left. 7<10Left to right: Seven is less than ten.Right to left: Ten is greater than seven. 9≥2Left to right: Nine is greater than or equal to two.Right to left: Two is less than or equal to nine.IN1: What are inequality symbols?Inequality symbols are symbols that tell you what’s bigger and what’s smaller. Whatever is next to the “mouth” is bigger and whatever is next to the “point” is smaller.IN2: The line underneathIf you see a line underneath the inequality symbol it should be read “or equal to” telling you that the two sides could be equal. IN4: Solving inequalities Solving inequalities is exactly like solving equations with one extra rule. Whenever we multiply or divide both sides of an inequality by a negative number, we have to flip the inequality symbol. Briefly, this is because multiplying or dividing by a negative changes the signs so that the side that was bigger is now smaller and the side that was smaller is now bigger.IN5: Graphing inequalitiesa. If the variable is next to the mouth, that means it’s bigger; shade to the right. If the variable is next to the point, that means it’s smaller; shade to the left. b. If there is a line underneath the inequality symbol, that means both sides could be equal to each other; fill in the circle.Examples: Similar figuresIf two proportional shapes are given, you can find a missing side length by setting up a proportion. To figure out how to set up the proportion, it’s a good idea to draw arrows and put the number at the beginning of the arrow in the numerator and the number at the end of the arrow in the denominator. Arrows can be drawn within shapes or between them. Be careful! One of the shapes might be drawn upside-down, flipped around, etc.SF2. Between shapesExample: Each arrow connects the same side (hypotenuse or base) and goes from big to small.First arrow: Small hypotenuse to big hypotenuseSecond arrow: Small base to big base SF1. Within shapesExample: Each arrow stays within the same shape (small or big) and goes from hypotenuse to base. First arrow: Small hypotenuse to small baseSecond arrow: Big hypotenuse to big base PB3. Counting principle To find how many combinations there are, multiply the number of ways each thing can happen.Example: I have 3 hats, 4 shirts, and 2 pairs of pants. How many different outfits can I make?Solution: 3×4×2=24PB2. What do probability values mean?P = 0: The event is impossible.0<P<0.5 : The event probably won’t happen but it might.P = 0.5: The event is exactly as likely to happen as not to happen.0.5<P<1: The event will probably happen but it might not.P=1: The event will definitely happen.Note that numbers could also be written as fractions or decimals:0=0% 0.5=1/2=50% 1=100%PB1. What is probability? How do you find a simple probability? Probability can be defined as the ratio of events that are considered “good” to the total number of possible events. It can also be defined as the chance of an event being successful. Based on the first definition, we can find the probability of an event by using the equation P(event) = GoodTotal .ProbabilityPB4. How do you find a simple probability with more than one good outcome If there is more than one outcome which is considered “good” then the probability of the event being successful is the sum of the probabilities of each good outcome.Example 1: Jessica will win a game if she spins the spinner and lands on 7 or 20. What is the probability of her winning?Solution: P7 or 20=24+14=34Example 2: Victor will win a game if he reaches into the bag and pulls out a black, gray, or triangle marble. What is his probability of winning?Solution: Pblack or gray or triangle=28+28+18=58The problems the follow refer to the spinner below which has four sections of equal sizeand the bag of marbles which has eight marbles of equal size.77c20PB5a. How do you find a compound probability (more than one thing has to go right)? If you need more than one independent event to be successful then the probability of being successful overall is the product of the probability of success for each event. Example: Jonathan will win a game if he spins the spinner and lands on c and pulls a white marble out of the bag. What is the probability of him winning?Solution: Pc and white=14×38=332PB5b. How do you find the probability of something happening multiple times in a row? This is really just a compound probability for which each event happens to be the same. You would need to multiply the probability of the event with itself however many times in a row it is supposed to happen.Example: Rajab will win a game if she spins the spinner and it lands on c five times in a row. What is the probability of her winning?Solution: Pc five times in a row =P(c and c and c and c and c) =14×14×14×14×14=11,024Warning! Pay close attention to “or” and “and”. “Or” implies a simple event with more than one good outcome (you should add) while “and” implies a compound event (you should multiply). Notice that the probabilities in P4 all have “or” in them while the probabilities in P5a and P5b all have “and” in them. You can even have a probability with both “or” and “and”…Example: Sven will win a game if he spins the spinner and it lands on 7 or c and he pulls a black or white marble out of the bag. What is the probability of him winning?Solution: P(7 or c and black or white) =24+14×28+38=34×58=1532PB7. Making predictionsStep 1: Find the probability (using P1, P4, or P5)Step 2: Multiply the probability with the number of attempts.OR set up a proportion: Probability=xAttempts Example: Michelle will win a penny every time the spinner lands on 20. If she spins the spinner 16 times, what is the best prediction you can make for how many pennies she will win? Solution: Step 1: P(20)=14 OR Step 2: 14×16=4 P(20) = 14 14=x16 x=4PB6. Be careful when things are not split up equally AFor the spinner on the left…What is P(A)? It’s not 14 !What is P(B)? It’s not 14 !DBPB8. RepresentationsExample: Here’s how you could represent all possible outcomes from flipping a coin (heads or tails) and rolling a six-sided die (1, 2, 3, 4, 5, 6). CTo find the probabilities, we have to imagine what the spinner would look like if it was split up into equal sections. Now we can see that P(A)=36=12 and PB=16 .ABDCAACI2: Pi (π) Pi is the ratio of the circumference of a circle to its diameter. That is, if you divide the circumference of any circle by its diameter you always get the same number… Cd=π=3.141592653589793238462643383….. It’s impossible for you to write this number all out because it never ends and never repeats. That’s why we have to represent it with a symbol.CI1: Parts of a circle Radius Diameter Circumference Area1-dimensional 1-dimensional 1-dimensional 2-dimensionalCirclesCI3: Going from radius to diameter and vice versaLook at the pictures of radius and diameter in P1. The diameter is twice as long as the radius. Therefore…If you know the radius, multiply by 2 to get the diameter.If you know the diameter, divide by 2 to get the radius.CI5: Using the circle equations Starting with the information you’re given, follow the paths, and use the equations or operations along those paths, to find anything else.Example 1: If you know d and you want C, use C=πd.Example 2: If you know d and you want A, first divide by 2 to find r, then plug r into A=πr2.Example 3: If you know A and you want C, first use A=πr2 to find r, then plug r into C=2πr .CI4: The circle equations C=πd C=2πr A=πr2where r is radius, d is diameter, C is circumference, and A is area.Type equation here.CI6. Getting rid of the exponent in A=πr2 An exponent of 2 means we are squaring a number. That means multiplying the number with itself. The inverse of this is taking the square root ( ). For example, 169 is 13 because 132 is 169. When you take the square root, ask yourself, “What number multiplied with itself will give me this number?”Example: The area of a circle is 50π. What is the circle’s radius?Solution: A=πr2 169=r2 169π=πr2 169=r2 169π=πr2 13=r π πCI8a: Cancel π's if there is a π in the numerator and the denominator. Don’t cancel it if there’s only one.Example: The circumference of a circle is 18 inches long. What is the radius?Incorrect Solution: This is WRONG!CI8b: Use parentheses when dividing by 2π in a calculator.Example: To answer the question above, put 18÷(2π) into a calculator. The answer is ≈2.86. If you put 18÷2π in the calculator then the calculator will do 18÷2 first and then multiply by π and will give you an answer of ≈28.27 which is wrong for this problem.CI7: After you multiply by π, the π is not there anymore.Example: The radius of a circle is 4 inches long. What is its circumference?Solution: In terms of pi Without piThis student cancelled the pi on the left even though there was no pi in the numerator to cancel it with. Measures of Center and Spread (and Mode) Measures of Spread Range and mean absolute deviation are called measures of spread because they tell you how spread out the data is.CS3. Range: Subtract the smallest number from the largest number. 97-73=24 RangeThe largest number is 24 units away from the smallest number.CS4. Mean absolute deviation: Find the mean of the data set. After that, find how far away each individual point of data is from the mean by taking the absolute value of the differences between them and the mean. After that, find the mean of those differences.Step 1: The mean of the data set is 85.8 .Step 2: 84-85.8=1.8 97-85.8=11.2 73-85.8=12.8 80-85.8=5.8 95-85.8=9.2Step 3: 1.8+11.2+12.8+5.8+9.2=40.8 40.8÷5=8.16 Mean absolute deviationOn average, each point of data is 8.16 units away from the mean. When we are shown data plotted on a number line, it makes it easier to compare measures of center and spread just by using our eyes! On the dot plots below, the center of Class B’s test scores is clearly higher than that of Class A. However, the scores for Class A are more spread out. Therefore, the mean and median would be greater for Class B but the range and mean absolute deviation would be greater for Class A.Visually Comparing Measures of Center and SpreadThe median is halfway between 84 and 88.Each dot on the number line represents a number in the data set. 70 75 80 85 90 95 100 In all of the examples below, except for the second median example and mode, the following data set was used… Data set: 84 97 73 80 95 Measures of Center Mean and median are called measures of center because they will give us a number that lies somewhere in the middle of the data set.CS1. Mean (average): Add up all the numbers and then divide by how many numbers there are. 84+97+73+80+95=429 429÷5=85.8 MeanCS2a. Median: First put the numbers in order from least to greatest! After that, simply find the middle number. 73 80 84 95 97 MedianCS2b. Finding the median when there are two middle numbers: If there are an even number of numbers, then there will be two middle numbers. If this happens, find the mean of the two middle numbers (add them up and divide by two). For example, if the number 88 were inserted into the data set above, here’s how we would calculate the median… 73 80 84 88 95 97 84+88=172 172÷2=86 ModeCS5. Mode: Find the number that occurs the most. Example: In the data set below, themode is 4 because that is the numberthat occurs the most. 9 4 5 9 4 2 4 On a dot plot, the mode would have the greatest number of dots above it.CS6.Mr. Frydman wants to know if the students in class 711 prefer dogs or cats. The population (the group of interest) is class 711. Mr. Frydman wants to make sure his sample (the students he asks) is chosen randomly (Mr. Frydman should not be able to predict who will get chosen) and is representative of the whole class (conclusions drawn from the sample are similar to conclusions that would have been drawn had everyone been asked). This will ensure that the inferences (conclusions) that Mr. Frydman draws are valid (legitimate). What are some examples of good and bad sampling methods?SurveysSU1. Random sampling leads to valid inferencesGood: Mr. Frydman puts all of the students’ names into a hat. Each name is written on equal sized strips of paper. He then closes his eyes and begins pulling names out of the hat.Bad: Mr. Frydman puts all of the students’ names into a hat. The girls’ names are on larger strips of paper than the boys’ names. Mr. Frydman then closes his eyes and begins pulling names out of the hat. This is a bad sampling method because the girls in the class have a higher likelihood of getting picked than the boys.Good: Mr. Frydman assigns every student a number. He then uses a computer program to randomly generate numbers and chooses those students.Bad: Mr. Frydman assigns all the students on the checkers team a number. He then uses a computer program to randomly generate numbers and chooses those students. This is a bad sampling method because the students who are not on the checkers team have no chance of being picked.Good: Mr. Frydman asks all the students in the class to form a line. He then chooses every other person (second in line, fourth, sixth, eighth, and so on).Bad: Mr. Frydman arranges the students in a line so that they alternate girl boy girl boy. He then chooses every other person. This is a bad sampling method because the girls have no chance of being picked. SU5. Valid surveysQuestion: Mr. Frydman wanted to know if the students in class 711 preferred dogs or cats. He randomly selected 20 students in the class and asked them, “Do you prefer dogs or cats?” Would the inferences that Mr. Frydman draws from his survey be valid?Answer: Yes, the inferences would be valid. Mr. Frydman chose the students randomly, he asked a large number of students, he did not try to persuade them, and the students were not biased in any way. This is a good survey.SU4. Poor surveys: Asking biased peopleQuestion: Mr. Frydman wanted to know if the people in New York City generally preferred dogs or cats. He went to a dog park and asked 50 people, “Do you prefer dogs or cats?” Would the inferences that Mr. Frydman draws from his survey be valid?Answer: No, the inferences would not be valid. People in a dog park are more likely than the general population to prefer dogs over cats. Their opinion would not be representative of people in New York City generally.SU3. Poor surveys: Survey biasQuestion: Mr. Frydman wanted to know if the students in class 711 preferred dogs or cats. He randomly selected 20 students in the class and asked them, “Do you prefer noisy, smelly dogs or clean, cuddly cats?” Would the inferences that Mr. Frydman draws from his survey be valid?Answer: No, the inferences would not be valid. There is bias within the survey itself because Mr. Frydman put his own opinion into the question. Calling dogs noisy and smelly and cats clean and cuddly would persuade more people to choose cats over dogs, even if that was not their own opinion before Mr. Frydman asked the question.SU2. Poor surveys: Not asking enough peopleQuestion: Mr. Frydman wanted to know if the students in class 711 preferred dogs or cats. He asked Jamie, “Do you prefer dogs or cats?” Would the inferences that Mr. Frydman draws from his survey be valid?Answer: No, the inferences would not be valid. To get a good idea of how students in class 711 feel, Mr. Frydman should ask a large number of them. It doesn’t have to be everyone, but certainly more than just one or even a few individuals.WE3a. y=q(x+r)Equations in this form are also linear. If we wanted to, we could write them like WE2 but writing them as above is easier for some problems.Example: There are 6 buckets. Each bucket originally had d dandelions in it. 2 more dandelions were then added to each bucket. What is the total number, t, of dandelions in all buckets now?Solution: Originally, there were 6 buckets and each bucket contained d dandelions…Original total: t=6dAfter two dandelions were added to each bucket, each of the 6 buckets contained d + 2 dandelions…New total in all six buckets: t=6(d+2)WE3b. Parentheses are important When we write t=6(d+2) we are saying there are 6 buckets and each bucket has d+2 dandelions in it. If we wrote t=6d+2 then we would be saying that each bucket only has d dandelions in it and there are an extra 2 dandelions somewhere… This is incorrect for the problem. However, it is possible to write the correct equation without parentheses by distributing: t=6(d+2) t=6d+12In other words, there are an extra 12 dandelions total because we put an extra 2 in each bucket. By distributing, we can make a WE3a equation look more like a WE2 equation.WE1. y=mxEquations in this form show a proportional relationship between the variables x and y. The constant m is the unit rate (constant of proportionality).Example: Fiona studies 2 hours per day. If d represents the number of days that go by and t represents the total number of hours studied, write an equation that relates these two quantities.Solution: Figure it out by thinking about what the total would be after a certain number of days.After 0 days: t=0=2(0)After 1 day: t=2=2(1)After 2 days: t=2+2=2(2)After 3 days: t=2+2+2=2(3)After 97 days: t=2(97)After d days: t=2d Writing equations and inequalitiesdd+2d+2d+2d+2d+2d+2WE2. y=b+mxEquations in this form show a linear relationship between the variables x and y but they are not proportional (on a graph we would see a straight line but it would not go through the origin unless b is 0).m is the unit rate (but there is no constant of proportionality because the relationship between x and y is not proportional).b is the initial value (starting point). Example: Fiona has already studied for 6 hours. She will continue studying 2 hours per day. Write and equation relating t, the total number of hours studied, and d, the number of days studied after the original 6 hours.Solution: Figure it out the same way as in WE1 but remember that Fiona started having already studied for 6 hours.After 0 more days: t=6=6+2(0)After 1 day: t=6+2=6+2(1)After 2 days: t=6+2+2=6+2(2)After 3 days: t=6+2+2+2=6+2(3)After 584 days: t=6+2(584)After d days: t=6+2ddddddWE4. Writing inequalities When deciding which inequality symbol to use, try translating a phrase into another phrase with the words “less” or “more”. Pay close attention to when an inequality should have a line underneath.Examples:The number of students should be more than 10.No translation necessary: s>10The number of students should be no more than 10.Translation: …10 or less: s≤10The number of students should be less than 10.No translation necessary: s<10 The number of students should be no less than 10.Translation: …10 or more: s≥10The number of students should exceed 10.Translation: …more than 10: s>10The number of students should not exceed 10.Translation: …10 or less: s≤10The number of students is at least 10.Translation: …is 10 or more: s≥10The number of students should be at most 10.Translation: …10 or less: s≤10The minimum number of students is 10.Translation: The number is 10 or more: s≥10The maximum number of students is 10.Translation: The number is 10 or less: s≤10WE5. Always check by plugging in For the example in WE1, what if a student is not sure if she should write d=2t or t=2d ? She can decide by thinking of an example and then plugging numbers into both equations and seeing which one makes sense. For instance, after 4 days Fiona will have studied a total of 8 hours. Let’s plug in 4 for d and 8 for t and see which equation makes sense… d=2t t=2d 4=2(8) 8=2(4) 4=16 8=8 The first equation gives a nonsense result but the second equation checks out!Warning: There could be negatives and minus signs! None of the examples on this page had minus signs or negatives but that was just to keep things simple. You will certainly have to include minus signs and negatives in some of your own equations and inequalities depending on the situation. ................
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