Converting Between Mixed Numbers & Improper Fractions

Size: px

Start display at page:

Download "Converting Between Mixed Numbers & Improper Fractions"

Valerie Parsons
6 years ago
Views:

1 01 Converting Between Mixed Numbers & Improper Fractions A mixed number is a whole number and a fraction: An improper fraction is a fraction with a larger numerator than denominator: 9 2 You can convert between mixed numbers and improper fractions! + Mixed Number Improper Fraction * Step 1: Multiple the denominator and the whole number. Here: 2 * 4 = 8. Step 2: Add the numerator. Here: = 9 2 Step 3: Place that number as the new numerator, and keep the existing denominator. Here: * = 2 2 Improper Fraction Mixed Number

2 02 LCM & GCF (Fraction Simplification) LCM is the Least Common Multiple. Get two numbers, get their multiples, find the first common multiple among them, and you ve got it! 4 5 4, 8, 12, 16, 20, 24, 28, 32, 36, , 10, 15, 20, 25, 30, 35, 40, 45, There you go. 20 is the LCM for 4 and 5. (Note, 8 would be the LCM for 4 and 8.) GCF is the Greatest Common Factor. Get two numbers, get their factor pairs, find the largest common factor, and you ve got it! (Remember, a factor is the process of breaking down non-prime numbers into the prime numbers that can be multiplied together to get that number. 36 1*36 2*18 3*12 4*9 6*6 20 1*20 2*10 4*5 Bam! In all the factorpairs listed, 4 is the biggest on both lists. So 4 is the greatest common factor! Fraction Simplification is the process of reducing a fraction to an equivalent fraction by dividing the same number into both the numerator and denominator. A fraction is fully simplified when there is no longer any common factor among the numerator and denominator * 3 * 2 5 * 2 * 2 * 2 * (Note, you can also view the above process as just dividing 10 into both the numerator and denominator.)

3 03 Adding & Subtracting Fractions The key is that you have to have common denominators before adding or subtracting =? Adding Fractions List out the multiples of 4 and 5 (the denominators) looking for a common denominators. Hint! The product of the denominators (here, 4 * 5) will always be a common denominator. 4, 8, 12, 16, 20, 24, 28, 32, 36, , 10, 15, 20, 25, 30, 35, 40, 45, Convert fractions to new denominators and ADD * 5 * = and 1 5 = 4 20 * 5 * 4 15 so + 4 = Subtracting Fractions =? Same Thing! 15 4 = 11 Just Subtract at the End

4 04 Multiplying & Dividing Fractions The key is that you no longer need common denominators Multiplying Fractions 1 3 * 4 5 = 4 15 Dividing Fractions = Step 1 Step 2 Step 3 KEEP the first fraction as is CHANGE the division sign to multiplication FLIP the second fraction 1 * = 5 12

5 05 Adding & Subtracting Decimals The key is that you have to have line up the decimal points before adding or subtracting

6 06 Multiplying & Dividing Decimals The key is that you do not have to have line up the decimal points, but rather count the decimal places, then multiply or divide. 6.1 * * _ Multiply 4 by 61 and you get 244 Multiply 5 by 61 and you get 305 Don t forget the placeholder!! Add them up to get 3294 Because there are TWO total decimal places in the factors (6.1 and 5.4), you have to count two decimal places in the product. Thus, 3294 becomes If the products were 6.1 and 0.54, you would have THREE total decimal places in the factors, everything else would be the same, but the answer would be Because it is 4 1.6, the 4 goes inside the Do not make the mistake of thinking the answer is 2.8 because is 8. You have to bring down another ZERO for 80, and find a multiple of 16 into 80 (which here is 5). Because 1.6 has one decimal place, we have to move the decimal place after 4 one place to the right. Here, the answer is 2.5. If the divisor was 0.16, we would have to move the decimal place after 4 two places to the right, for 25 as an answer.

7 07 Converting Between Decimals & Fractions Going from decimals to fractions, the key is that you can think of: 0.8 as 8 10 (because the 8 is in the tenths place), and 0.65 as (because 65 is in the hundredths place) After that, you just have to simplify! Going from fractions to decimals, the key is that you can think of: 3 4 as 3 4, which is 0.75 as shown below:

8 08 Solving Ratios & Proportions (Unit Rates) Ratios and Proportions are simply two numbers in a relationship. The key is that with ratios and proportions, order matters! Ratios and Proportions can be written in three ways. For example, if 6 pints of yellow are combined with 2 pints of red to make an orange, the ratio of yellow paint to red paint would be: 6:2 or 6 to 2 or 6 2 Or, simplify it to... 3:1 or 3 to 1 or 3 1 (Note, there are no mixed numbers in ratio world) (Also, the ratio of orange paint to yellow paint would be 8:6, or simplified, 4:3) Ratios and Proportions can also be used to find equivalent ratios and proportions. For example, if an orange is formed with 5 pints of yellow with 4 pints of red, how much red would be needed for the same orange made with 15 gallons of yellow? * = 15 x 5 4 = 15 x x = 12 The ratio of yellow to red is 5 to 4 in the first orange, so the ratio will be the same in the second, regardless if speak of pints or gallons. Whatever you do to the top, you have to do to the bottom. You multiply 5 by 3 to get 15, so you have to multiply 4 by 3 to get your answer, which is 12. Unit rate is simply the equivalent ratio or proportion with 1 in the denominator. 5 4 = x = 1.25 or In other words, when you have 5 units of yellow paint to 4 units of red paint, the unit rate of yellow paint is 1.25 or

Cost (Dollars) 0 2 4 6 8 10 12 14 16 09 Graphing Ratios & Proportions Ratios and Proportions can be put into charts to help us solve problems.

9 Cost (Dollars) Graphing Ratios & Proportions Ratios and Proportions can be put into charts to help us solve problems. For example, consider the following chart showing the cost per pound of apples: Weight Cost 2 pounds $ pounds $ pounds $ pounds? 10 pounds? Following the pattern, you should be able to see that 8 pounds would cost $16.00, and 10 pounds would cost $ We can also get the unit rate (or unit cost) of each pound. $4 2 lbs. = $2 1 lb. We can also use the ratio or proportion to calculate the cost for any amount of apple (even if not on the chart) $4 2 lbs. = $ lb. We can also plot these points on a graph. We can also recognize that each Dollar (d) is each to 2 times each Pounds (p). Or, in other words: d = 2p Apples (Pounds)

10 10 Percents A percent is simply a ratio or proportion with 100 as the denominator: n % 100. In practice, if you have 12 girls on a 20 student math team, the percent is: = x% times 5 is 100, so we must multiply 12 by 5 as well. The answer is 60% Also, if you have 9 girls on a 15 student science team, the percent is: 9 15 = x% 100 Here, 15 is not easily multiplied into 100. You can still find the answer by: Calculating 9 divided by 15; and Multiplying the answer by * 100 = 60 The answer is 60%

11 11 Solving Algebraic Expressions The key is using inverse operations. In other words: if x is being told to multiple, she will divide (and vice versa); if x is being told to add, she will subtract (and vice versa). Also, remember that whatever you do to one side of the equation, you have to do the same thing to the other side. That is how you keep the equation in balance. Look below to see how it works in practice. x + 5 = 7 x = 7 5 x = 2 6x = 18 6x 6 = 18 6 x = 3 x + 7 = 10 x = 10 7 x = 3 x 3 = 12 x = x = 15 Important points to remember: equals 1 because anything divided by itself is 1; and 2. 1 times x equals x because any number times 1 is that same number x 5 = 3 x = x = 8 x 5 = 3 x 5 * 5 = 3 * 5 x = 15

12 12 Writing Algebraic Expressions in Sentences (Vocab) Here, we are translating between math speak and the real world. In other words, we are writing an algebraic expression (math speak) for word problem (the real world). Sample Question: Write an expression for Mindy has 3 times as many apples as Rachel where Mindy s total is given as M and Rachel s total is given as R. Answer: 3R = M Sample Question: Write an expression for Hannah has 6 less strawberries than Alice where Hannah s total is given as H and Alice s total is given as A. Answer: A 6 = H Here are some key words to help: Addition increased by... more than... combined together... total of... sum... plus... added to... Subtraction decreased by... minus... less... difference between/of... less than... fewer than... Multiplication times... multiplied by... product of... increased/decreased by a factor of... Division per, a... out of... ratio of, quotient of... In addition, here is some 6 th grade vocab you just gotta know: Sum The total amount resulting from the addition of two or more numbers. The sum of 4x and 3x is 7x. Term Number or variables (or combinations) separated by + or signs. There are 3 terms in 4x + 2y + 5. Coefficient A number multiplied with a variable. The coefficients in 4x + 2y + 5 are 4 and 2 (here, 5 is not a coefficient). Factor A number or variable that divides another term exactly. 1, 2, 3, 4, 6, 12 are the factors of 12. Product The result of multiplying two numbers or variables. 6 is the product of 2 and 3. 20x is the product of 4x and 5. Quotient The result of dividing two numbers or variables. 4 is the quotient of 12 divided by 3. 4x is quotient of 12x 3.

13 13 Simplifying Algebraic Expressions The key is combining like terms and carrying the sign to the left of the term. 10x + 5y 2x + 4 = 8x + 5y + 4 and 10x 5y 2x 4 = 8x 5y 4 Also note that x = 1x because any number times 1 is that same number 10x + 5y x + y = 9x + 6y

14 14 The Distributive Property The key is to multiply and the outside term to both terms in the parenthesis. 5 ( 2x 3 ) = 10x 15 Note that you carry forward the sign in the parenthesis 4 ( 3m + 5 ) = 12m + 20 You can also go in reverse, and factor an algebraic expression. 18s + 12 = 6 ( 3s + 2 ) 12xy 3x = 3x ( 4y 1)

15 15 Exponents Exponents are shorthand for repeated multiplication. The key is to take the base (big number) and multiply it by itself as many times as the exponent tells you to. 2 4 = 2 * 2 * 2 * 2 = = 4 * 4 * 4 = = 3 * 3 = 9 Below are the weird cases. Any number raised to the 1 power is that same number. For example: 8 1 = 8. Also, any number raised to the 0 power is 1. For example: 5 0 = 1.

16 16 Order of Operations The key is that you don t just do math left to right. You do math in a specific order. P - Parenthesis E - Exponents MD - Multiplication & Division AS - Addition & Subtraction (One way to remember the order is Please Excuse My Dear Aunt Sally ) The basic is that * 5 is 22 not 30 (we multiply 4 times 5 first, then add the 2) Note! There is a mistake everyone makes with the last two of the list, Addition and Subtraction. Try to use PEMDAS for 5 * Some are tempted to do 5 * 4 first (20), then add (19), then subtract (20 19) and get 1. That is wrong! You are really adding + 7 to 12 for 5. The correct answer is 20 then minus 12 (8) then plus 7. The answer is 15. YOU GO LEFT TO RIGHT WITH ADDITION AND SUBTRACTION

17 17 2 Dimensional Figures Area & Perimeter Perimeter = Distance around a shape Area = Total space occupied by a shape For Triangles Perimeter = Side + Side + Side 5 in. 5 in. 4 in. Perimeter 16 in. Area = ( Height * Base ) 2 6 in. Area 12 in. 2 For Rectangles (including squares) Perimeter = ( 2 * One side ) + ( 2 * Adjacent side ) Area = Height * Base 3 in. 7 in. Perimeter 20 in. Area 21 in. 2 How to Handle Weird Shapes? Quadrilaterals (four straight sides) 8 in. Polygons (any number of straight sides) Break them down into Rectangles and Triangles! 5 in. 4 in. 11 in. 8 in. 5 in. 4 in. 3 in. Perimeter 28 in. Area 38 in. 2

18 18 3 Dimensional Figures Surface Area, Volume & Nets Volume = Space inside of a shape Surface Area = Total space occupied by all faces of a shape This is a right rectangual prism by the way: Volume = Height * Base * Width 5 in. 7 in. 2 in. Volume = 5 in. * 7 in. * 2 in. Volume = 70 in. 3 Surface Area = Area of Front (Back) + Area of Right (Left) + Area of Top (Bottom) Surface Area = (5 in. * 7 in.) ( *2) + (2 in. * 5 in.) (*2) + (2 in. * 7 in.) (*2) Surface Area = 35 in. 2 * in. 2 * in. 2 * 2 Surface Area = 70 in in in. 2 Surface Area = 118 in. 2 A net is a two-dimensional flat version of a three-dimensional figure Find the surface area of the following net: 4 in. Each of the 4 triangles has a base of 4 in. and a height of 3 in. So the total is: 4 in. 3 in. Triangle Surface Area: 4 * ( ( 4 in. * 3 in. ) 2 ) Triangle Surface Area: 24 in. 2 Square Surface Area: 4 in. * 4 in. = 16 in. 2 Total Surface Area 40 in. 2

19 19 Numberlines (Absolute Values) The key is that when you are calculating the difference (or distance) between two points on either side of ZERO, remember to add together the distance of each point from ZERO. The distance from -3 to 4 is = 7 The key is to remember that you always go lower to the LEFT and go higher to the RIGHT is between -4 and -5, not between -4 and -3 The Absolute Value of -3 is 3 or ǀ -3 ǀ = 3 The Absolute Value of 4 is 4 or ǀ 4 ǀ = 4 The Opposite of -3 is 3 The Opposite of 4 is -4

20 20 The Coordinate Plane (3, 5) (3, 5) is in the First Quadrant (-6, -5) is in the Third Quadrant (-6, -5) (-6, -5) reflected over the x-axis would be (-6, 5) (3, 5) reflected over the y-axis would be (-3, 5)

Integer Operations. Summer Packet 7 th into 8 th grade 1. Name = = = = = 6.

Integer Operations. Summer Packet 7 th into 8 th grade 1. Name = = = = = 6. Summer Packet 7 th into 8 th grade 1 Integer Operations Name Adding Integers If the signs are the same, add the numbers and keep the sign. 7 + 9 = 16-2 + -6 = -8 If the signs are different, find the difference