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Super Awesome Math Time!

Mr. Weidert. Super Awesome Math Time!. Grab paper and a pencil NO CALCULATORS!. Chat Rules Be respectful Do not share personal info Do not share outside links Do not troll/flood Be awesome. Math Notes!. By: Rachel. Want to share your awesome picture or poem?! Send it to Mr. W’s Kmail.

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Super Awesome Math Time!

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  1. Mr. Weidert Super Awesome Math Time! Grab paper and a pencil NO CALCULATORS! • Chat Rules • Be respectful • Do not share personal info • Do not share outside links • Do not troll/flood • Be awesome Math Notes! By: Rachel Want to share your awesome picture or poem?! Send it to Mr. W’s Kmail

  2. Why take notes? • Notes are a great way to organize what you have learned • Helps retain information • Easier to find things again • Helps to understand the material • Forces you to think differently

  3. When to take notes • Whenever you are learning new information • Working in the OLS • Listening to live lessons • Over time you will learn how to summarize notes. Fortunately with online schools, you can watch recordings! • When you come across new information

  4. What to put in my notebook • Important information • Definitions • Examples • Pictures that help explain the material • Do not do book work in the same book you use for your notes • Do not doodle in your notebook

  5. How to take notes There are many different ways to take notes, and one method is not necessarily better than another. Today we are going to start a math notebook.

  6. Your Name’s Math notes Definitions p. 6 - 20 Formulas p. 21 - 25 Indicator words p. 26 - 30 Fractions p. 31-34 Coordinate planes p. 35-36 Units of measurement p. 37 Factors and GCF p. 38 Order of operations PEMDAS p. 39-41 Solving for a variable in an equation p. 42-48 Page 1

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  11. Definitions Area – The amount of space inside a flat object. Area is measured in units² Absolute Value – The distance a number is from zero on a number line. It is always positive. Ex I-4I has an absolute value of 4 Ratio – The comparison of 2 or more numbers. A ratio 1:2 1 to 2 or 1/2 can be written three ways: Unit Rate – A ratio that compares an amount to a form of measurement. Ex miles per hour Integer – a number that can be written without a decimal or fraction ( 1, 23, -34, -43) (not .23, -4 3/5) Numerator– The top number of a fraction Denominator– The bottom number of a fraction Reciprocal – When you switch the numerator with the denominator in a fraction Ex: The reciprocal of ½ is 2/1 Inequalities - < Less than, > Greater than , = equal to , equal to ≤ Less than or equal to , ≥ Greater than or Fraction – A ratio that compares he number of parts we have to the number of parts we need to make one whole. Mixed number – A fraction and a whole number Invert – To flip upside down Multiple – A number that is found through multiplying. (Ex: the multiples Of 3 are 3, 6, 9, 12, 15, 18, etc) Page 6

  12. Definitions origin – (0,0) on a coordinate plane. This is where you start when finding an ordered pair. X-axis – On a coordinate plane, this is the number line that goes from left to right Y-axis – On a coordinate plane, this is the number line that goes from up and down Coordinate plane – a grid that is formed by two perpendicular number lines that intersect at (0,0). This is used to graph coordinates. Coordinates– for a typical coordinate plane, coordinates are points that are written in the form (x,y). Also called an ordered pair.” Quadrant – one part of an object that is broke into four parts. Ray – a line that starts at a given point and goes on forever in the other direction. Vertex – a point where two or more line segments or rays converge. Vertex Line segment – a line with two given end points. It is written AB where A and B are two points. Trapezoid – shape with at least one pair of parallel sides. For our math there will only be one pair of parallel sides. Parallelogram– shape with two pairs of parallel sides whose opposing angles are equal. Page 7

  13. Definitions Types of triangles Right Triangle – a triangle that has one right angle (an angle of 90 degrees) Obtuse Triangle – a triangle that has one angle over 90 degrees Equilateral Triangle – a triangle that has angles that are all 60 degrees. The segments are also the same length. Isosceles Triangle – a triangle that has two line segments that are the same length. Factors – numbers that can be multiplied together to form other numbers. For example, 1 and 7 are factors of 7 because 1x7=7 Prime number – A number whose factors are only 1 and the number itself (it can only be divided by 1 and the number itself (ex: 3, 7, 13, etc) Composite number – A number whose factors are only 1 and the number itself (it can only be divided by 1 and the number itself (ex: 3, 7, 13, etc) Greatest Common Factor – the largest factor that two or more numbers have in common. Page 8

  14. Definitions Prime factorization - the expression of a positive integer as a product of prime numbers EX: 20 = 2 × 2 × 5 = 22 × 51 Commutative Property – an operation is commutative if the order of the terms can be changed without affecting the result. Addition and multiplication are both commutative. Examples Addition: 3 + 4 = 4 + 3 Multiplication: 2(8) = (8)2 Subtraction is not commutative: 9-5 does not equal 5-9 Division is not commutative: 8/4 does not equal 4/8 Associative Property – An operation is associative if the numbers can be grouped in any way without changing. It doesn’t matter how they are combined, the answer will always be the same. Addition and multiplication are both associative. Examples Addition: (3+2) + 4 = 3+(2+4) Multiplication: (3 x 4) x 5 = 3 x (4 x 5) Subtraction is not associative: (7 – 3) – 1 does not equal 7 – (3 – 1) Division is not associative: (16/4)/2 does not equal 16(4/2) Volume – the quantity of three-dimensional space enclosed by some closed boundary. (how much of space an object takes up) Page 9

  15. Definitions In this example, the 5 is being “distributed” to 2 and the 5. That is how We got from the left side To the right side. The second and third steps are showing that they are equivalent. Page 10

  16. Area – Area is the amount of space a two dimensional object takes up. It is always measured in terms of units² Diameter – A line segment that passes through the center of the circle and whose endpoints lie on the circle. Circumference – The distance around a circle Radius – The distance from the center of a circle to any point on the circle’s edge • Net - A two-dimensional pattern of a three-dimensional figure that can be folded to form the figure. • Surface Area - The measure of the number of square units needed to cover the outside of a three dimensional figure. Page 10

  17. Mean – the average of a data set. Add all of the numbers in the data set together and divide by the total number of numbers in the set. Median – the number in the middle of the data set when the numbers are put in order from least to greatest. If there is an even number in the data set, average the two numbers in the middle. Mode – the number that occurs most often in a data set. Data set – a group of numbers that are somehow related. Range – the difference between the largest and smallest number in a data set Mid-Range – the average of the largest and smallest numbers in a data set first quartile(lower quartile)– the median of the smaller half of the data. Everything smaller than this number represents 25% of the dataThe second quartile(median) – this is the median of the entire set of data. The data between the first quartile, and the second quartile is 25% of the data. Also, the data from the second quartile to the third quartile represents 25% of the data.The third quartile(upper quartile) – this is the median of the upper half of the data. Everything larger than this number represents 25% of this dataThe interquartile range is calculated by subtracting the 1st quartile from the 3rd quartile. It is the range of the middle 50% of a distribution. Mean absolute deviation is the mean of the differences of the values in the data set from the mean. It is best used to measure spread when the numbers are farther apart. Page 11

  18. Vocab examples 25% 25% 25% 25% median

  19. Formulas Area of a parallelogram – a=bh 2m 3m Page 21 a = 2(3) a = 6m²

  20. Formulas b₁ Area of A trapezoid h b₂ A=.5(b₁+b₂)h Diameter= C/π , R(2) Radius= .5(D) , D/2 Circumference = 2Rπ , Dπ Page 22

  21. Volume For cuboids and rectangular prims, use the formula V=lwh where lw & h are all lengths For prisms like the ones to the left, use the formula V =bh where b is an area, and h is a length. You must find the area of the base first! Page 23

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  24. Indicator words The following words indicate that a number is positive. The following words indicate that a number is negative. credit, added, gained, deposited, debt, subtracted, lost, lose, withdrew, below, under, beneath above, over, received, Page 26

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  29. Fractions Converting fractions to decimals Every fraction can be converted to a division problem by dividing the numerator by the denominator 1 = 2 2 1 1 2 After setting up the long division problem, divide Normally to produce a decimal. 0.5 .0 -0 10 -10 0

  30. Fractions Changing an improper fraction into a mixed number Set your fraction up as a long division problem Complete the problem (find a remainder, not a decimal) 3. Write the number of wholes you removed as the whole Number 4. Write the remainder as your new numerator. Write the Originaldenominator as your denominator 5 23 5 4 r3 23 20 3 - Changing a mixed number to an improper fraction Multiply your denominator by your whole number Add this number to your numerator and write this number 3 5 2 6 5 4 in as the numerator to your improper fraction 3. Write the denominator from the original problem in as The denominator to your improper fraction 32 6 6(5)=30 30+2=32 Page 32

  31. Fractions Dividing fractions When dividing fractions, we are going to change the problem Into a multiplication problem. 1. Change all mixed numbers to improper fractions 2. Re-write the first fraction as is 3. Change the division sign to a multiplication sign 4. Re-write the second fraction as its reciprocal 5. Solve the multiplication problem. The answer to this problem is the answer to your division problem. 8 9 1 5 = ÷ 5 1 8 9 40 9 = Lowest common denominators When finding the LCD, you need to find the multiples of the Denominators, and locate the smallest multiple they have in Common. 8 9 1 5 Multiples 9 : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 Take the first fraction. Find what we multiplied the denominator by to get to the LCD, and multiply the numerator and denominator by this number. 5 5 8 9 8 9 40 45 9 (x) = 45 x = 5 = Page 33

  32. Fractions Lowest common denominators - cont 1 5 40 45 8 9 = = Multiples 9 : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 Take the second fraction. Find what we multiplied the denominator by to get to the LCD, and multiply the numerator and denominator by this number. 9 9 1 5 1 5 9 45 5 (x) = 45 x = 9 = 1 5 40 45 8 9 = = 9 45 Page 34

  33. Coordinate planes Q1 Q2 (-,+) (+,+) - always runs left to right Origin (0,0) Q4 Q3 (+,-) (-,-) - always runs up and down X axis Coordinates are always listed as an “ordered pair” and presented in the form (x,y). To find an ordered pair, first locate the number in the “x” part of the ordered pair on the x-axis. Next go up or down to “y” number of the ordered pair using the y-axis Reflecting When reflecting a coordinate. Look at the coordinate. If you are reflecting over the x-axis, change the y coordinate in the ordered pair from positive to negative or negative to positive. Ex: The point (2,-3) reflected over the x-axis is (2,3) If you are reflecting over the y-axis, change the x coordinate in the ordered pair from positive to negative or negative to positive. Ex: The point (3,-4) reflected over the y-axis is (-3,-4) Y axis Page 35

  34. Coordinate planes Reflecting When reflecting a coordinate. Look at the coordinate. If you are reflecting across the origin, change the x and y coordinate in the Ordered pair to their opposite Ex: The point (2,-3) reflected over the origin is (-2,3) Distance on a coordinate plane When locating the difference between two points on a coordinate plane, look at the two coordinates. Either the x or the y coordinate will not change. Find the sum of the absolute value of the other two numbers. (-4, 6) (-4, -7) Ex: The x coordinate (-4) does not change. So add the Absolute value between the two y coordinates. The absolute value of 6 is 6. The absolute value of -7 is 7. 6+7 is 13. The distance between the points is 13 units. Page 36

  35. Units of Measurement When converting from smaller units to larger units, divide. When converting from larger units to smaller units, multiply. Page 37

  36. Factors, GCF Finding Factors Always start with 1 and the number Try 2 and see if it goes evenly into the number • If it does, then 2 and the quotient are factors of the number • Try 3, 4, 5, etc (increasing by 1) until you get to a number you have already written down • Once you have reached a number you have written down, you have written all of the factors for that number. Finding Greatest Common Factors List out all of the factors of the numbers you are trying to find the GCF for Look at the list, and find the LARGEST factor that is in both numbers lists 24: 1, 2, 3, 4, 6, 8, 12, 24 8: 1, 2, 4, 8 GCF of 24 and 8 = 4 Page 38

  37. Oh that Sally! Always getting into trouble… PEMDAS – Please Excuse My Dear Aunt Sally Parentheses Exponents Multiplication Division Addition Subtraction How to remember the order What the letters stand for Page 39

  38. Order of Operations • Find the value of expressions within grouping symbols, such as parentheses and fraction bars. (use PEMDAS within parentheses to solve) • Find the value of all powers. • Multiply and divide from left to right. • Add and subtract from left to right. Page 40

  39. VEEEERRRYYY Important… Without PEMDAS With PEMDAS 12 + 8 ÷ 2 · 2 - 2 DO NOT DO IT THIS WAY!!! THIS IS WHY ORDER (not left to right) MATTERS!!! 12 + 8 ÷ 2 · 2 - 2 20÷ 2 · 2 - 2 12 + 4· 2 - 2 10 · 2 - 2 12 + 2- 2 20- 2 14 - 2 18 12 Page 41

  40. Solving for a variable • When you are solving for a variable you need to “isolate the variable.” This means that you need to get the variable on one side of the equal side by itself. • Before you can isolate the variable, you need to know opposite operations: The opposite of division is multiplication The opposite of multiplication is division The opposite of addition is subtraction The opposite of subtraction is addition Page 42

  41. Solving for a variable in an addition problem 12 + x = 23 12 + x = 23 -12 -12 x = 11 • The variable is “x” and is not alone. We need to get rid of the “12” • We need to get rid of the “12” by subtracting it (we subtract because it is positive). We also subtract 12 from 23 because what we do to one side of an equals sign we need to do to the other. • In this case we find that x=11 Page 43

  42. Solving for a variable in a subtraction problem When you are solving for “x” in a subtraction problem, it is important to notice which comes first, the variable or the number. This instance shows how to solve a subtraction problem when the variable is listed first. x - 9 = 43 x - 9 = 43 +9 +9 x = 52 • The variable is “x” and is not alone. We need to get rid of the “-9.” Note that it is a -9, not a 9 because the sign in front of a number is part of the term. • We need to get rid of the “-9” by adding a positive 9. We also add 9 to 43 because what we do to one side of an equals sign we need to do to the other. • In this case we find that x=52 Page 44

  43. Solving for a variable in a subtraction problem When you are solving for “x” in a subtraction problem, it is important to notice which comes first, the variable or the number. This instance shows how to solve a subtraction problem when the variable is listed second. • 7 - x = 23 • 7 - x = 23 -7 -7 • - x = 16 • x = -16 • The variable is “x” and is not alone. We need to get rid of the “7.” Note that in this instance “x” is negative, not a positive because the sign in front of “x” is part of the term. • We need to get rid of the “7” by subtracting 7. We also subtract 7 from 23 because what we do to one side of an equals sign we do to the other • In this case we find that –x = 16 however we do not want to know what “–x” is, we want to know what x” is. To change from a negative “x” to a positive “x”, we need to multiply both sides of the equals sign by “-1.” This tells us that x = -16 Page 45

  44. Solving for a variable in a multiplication problem 8x=104 8x=104 ÷8 ÷8 x=13 • The variable is “x” and is not alone. We need to get rid of the “8.” The order of terms does not affect how we solve this type of problem. • We need to get rid of the “8” by dividing by 8 because it is the opposite operation of multiplication. We also divide 104 by 8 because what we do to one side of an equals sign we do to the other • In this case we find that x=13 Page 46

  45. Solving for a variable in a division problem • To solve for a variable in a division problem, you need to set up both sides of the equals sign as fractions. • The first number of a division problem (dividend) is made the numerator, and the second number (divisor) Ex: 72 ÷x can be made into 72/x • A whole number can be made into a fraction by placing it over 1 Ex: 12 can be changed to 12/1 Once you have the problem set up, find the number in whatever fraction has the variable in it. In the example below, we would start with the number 72. Then locate the number caddy corner to the fraction and multiply them together (you will always multiply first, and it will always be the two numbers caddy corner to one another). 72 x 72 x 12 1 12 1 = = 72(1)=72 Page 47

  46. Solving for a variable in a division problem • Next find the number above or below and divide. In this case we will divide our product (72 which we found by doing 72 times 1) by 12. • The quotient to this problem tells us what “x” is. In this case, “x” is six. Place this number in the spot for “x” The order we do things make a “fish shape.” This concept is hard to understand without being taught it. Ask Mr. W if you need help. 72/12=6 72 x 72 6 12 1 12 1 = = Page 48

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