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Welcome to the World of Algebra

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  1. Welcome to the World of Algebra A mathematical thinking race!

  2. Two aliens, Xigmat and Yancy have just arrived on Earth. Xigmat and Yancy disagreed about the answer to: 7 + 5  2 What answers do you think they got? Which answer is correct?

  3. By the Order of Operations 7 + 5  2 is equal to 7 + 2.5 which is 9.5 Explain to X & Y, how the mnemonic PEMDAS will help them! “Please Excuse My Dear Aunt Sally”

  4. The Order of Operations P  Parentheses & Grouping Symbols E  Exponents MD  Multiply & Divide from left to right AS  Add & Subtract from left to right “Please Excuse My Dear Aunt Sally”

  5. Remember Parentheses are not the only grouping symbol: If a = 3 and b = 4, simplify: Square Root symbol: What number times itself results in this value? “Please Excuse My Dear Aunt Sally”

  6. Recall : Absolute Value is the positive distance from zero. | -5.2 | and | 5.2 | are both equal to 5.2 since they are each 5.2 units from the origin. -6 -4 -2 0 2 4 6 Origin -5.2 5.2 Both distances equal 5.2

  7. Remember Parentheses are not the only grouping symbol: Recall : Absolute Value is the positive distance from zero. Simplify: | -7 + 3 | | -4 |  4 Notice that you must simplify INSIDE the absolute value bars first! It would be wrong to do abs. value first: | -7 | + | 3 |  7 + 3  10 “Please Excuse My Dear Aunt Sally”

  8. Remember Parentheses are not the only grouping symbol: Xigmat used his calculator to simplify: His answer was6.5 What did he do wrong? He forgot the division line in a fraction is a grouping symbol! 5 + 3/2 gives 6.5 You must enter parentheses in your calculator! (5 + 3)/2 gives 4 “Please Excuse My Dear Aunt Sally”

  9. Xigmat was stumped by this problem. Can you explain it? Solve to find x: 2x= 64 exponent base Since 2  2  2  2  2  2 = 64 or 26 = 64 x must equal 6 That is, 2 times itself six times will give 64. More Examples: 32 = 9 105 = 100,000 34 = 81

  10. Yancy needs to learn how to translate problems into algebra symbols. If you triple Xigmat’s age and subtract 2, the result is 601. How old is Xigmat? Let x represent Xigmat’s age. 3x – 2 = 601 3x = 603 x = 201 Xigmat is 201 years old.

  11. X & Y also need to learn how to graph. Complete the table, plot the points and connect the dots to graph the equation : -1 0 1 2 3 4

  12. Xigmat & Yancy have to visit 7 earth friends, (Alpha, Beta, Cee, Delta, Epsiplon, Fred & Gamma), one each day for the next week. In how many ways, could they arrange their visits? There must be an easier way than listing all the possible orders: ABCDEFG or DEABCGF or GDCABEF... Look for a mathematical shortcut: Day 1: they have 7 friends to choose from Day 2: they have 6 friends left to choose from Day 3: they have 5 choices left : Day 7: only 1 choice left by elimination

  13. Xigmat & Yancy have to visit 7 earth friends, Alpha, Beta, Cee, Delta, Epsiplon, Fred & Gamma, one each day for the next week. In how many ways, could they arrange their visits? Looking for a Mathematical Shortcut: Day 1: they have 7 friends to choose from Day 2: they have 6 choices left : Day 7: only 1 choice left Simply MULTIPLY to get the total number of different orders: Day 1Day 2Day 3Day 4Day 5Day 6Day 7 7  6  5  4  3  2  1 = 5,040 ways In math symbols this is written as: 7! = 5,040 The exclamation point is read “FACTORIAL”

  14. Want another example? In how many different orders could you do homework for four classes: Algebra, Biology, Civics and Drawing? You could list all the orders, if you are careful: ABCD BACD CABD DABC ABDC BACD CADB DACB = 24 ways ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA But there has to be a mathematical shortcut using the COUNTING PRINCIPLE we saw previously!

  15. In how many different orders could you do homework for four classes: Algebra, Biology, Civics and Drawing? What HW do you do first? 4 subjects to choose from What HW do you do second? 3 subjects left to choose from What HW do you do third? 2 subjects left to choose from What HW do you do fourth? Only 1 subjects left to complete! HW 1HW 2HW 3HW 4 4  3  2  1 = 24 different orders This is written as 4! = 24 Read: “4 factorial is 24.

  16. Welcome to Algebra Quick Review: | -5 | is read “Absolute Value of negative 5” | -5 | = 5 since it is five units away from the origin (zero) 5! is read “5 Factorial” which is used in Counting Principles 5! = 5 4  3  2  1 = 120 8! = 8 7  6 5 4  3  2  1 = 40,320 35 is read “3 to the 5th power.” 35=3 3  3  3  3=243 52 = 25 which can also be read as “five squared.” 43 = 64 which can also be read as “four cubed.” • PEMDAS (Please Excuse My Dear Aunt Sally) is a mnemonic for the • Order of Operations: • Parentheses & Grouping Symbols 3. Multiply & Divide • Exponents 4. Add & Subtract