Welcome Welcome to the to the

World of World of AlgebraAlgebra

AA

mathematical

mathematical

thinking race!

thinking race!

Two aliens, Xigmat and Two aliens, Xigmat and Yancy have just arrived on Yancy have just arrived on Earth. Xigmat and Yancy Earth. Xigmat and Yancy

disagreed about the answer disagreed about the answer to:to:

7 + 5 7 + 5 2 2What answers do you think What answers do you think

they got?they got?

Which answer is correct?Which answer is correct?

By the By the Order of OperationsOrder of Operations7 + 5 7 + 5 2 2

is equal to is equal to 7 + 2.57 + 2.5which is which is 9.59.5

Explain to X & Y, how the Explain to X & Y, how the mnemonic PEMDAS will help mnemonic PEMDAS will help

them! them!

““PPlease lease EExcuse xcuse MMy y DDear ear AAunt unt SSally”ally”

The The Order of OperationsOrder of OperationsP P Parentheses & Parentheses &

Grouping SymbolsGrouping SymbolsE E Exponents Exponents

MD MD Multiply & Divide Multiply & Divide from left to rightfrom left to right

AS AS Add & Subtract from Add & Subtract from left to right left to right

““PPlease lease EExcuse xcuse MMy y DDear ear AAunt unt SSally” ally”

Remember Parentheses are not the Remember Parentheses are not the only grouping symbol:only grouping symbol:

If a = 3 and b = 4, simplify:If a = 3 and b = 4, simplify:

““PPlease lease EExcuse xcuse MMy y DDear ear AAunt unt SSally”ally”

2 2a +b2 23 +4

9+16255

Square Root

symbol:What number times itself

results in this value?

Recall : Recall : Absolute ValueAbsolute Value is is the positive distance from zero.the positive distance from zero.

| -5.2 | and | 5.2 | are both equal to 5.2| -5.2 | and | 5.2 | are both equal to 5.2since they are each 5.2 units from the origin.since they are each 5.2 units from the origin.

0 2 4 6-4-6 -2-5.2

-5.2

5.25.2

Both distances equal 5.2Both distances equal 5.2

Origin

Remember Parentheses are not the only grouping symbol:Remember Parentheses are not the only grouping symbol: Recall : Recall : Absolute ValueAbsolute Value is the positive distance from is the positive distance from

zero.zero.

““PPlease lease EExcuse xcuse MMy y DDear ear AAunt unt SSally”ally”

| -4 | | -4 | 4 4Notice that you must simplify INSIDE Notice that you must simplify INSIDE the absolute value bars first!the absolute value bars first!

It would be It would be wrongwrong to do abs. value first: to do abs. value first: | -7 | + | 3 | | -7 | + | 3 | 7 + 3 7 + 3 10 10

Simplify: Simplify: | -7 + 3 || -7 + 3 |

Remember Parentheses are not the only grouping symbol:Remember Parentheses are not the only grouping symbol:

Xigmat used his calculator to simplify:Xigmat used his calculator to simplify:

His answer wasHis answer was 6.56.5What did he do wrong?What did he do wrong?

““PPlease lease EExcuse xcuse MMy y DDear ear AAunt unt SSally”ally”

He forgot the division line in a fraction is a He forgot the division line in a fraction is a grouping symbol!grouping symbol!

5 + 3/2 5 + 3/2 gives 6.5gives 6.5

5+32

You must enter parentheses in your You must enter parentheses in your calculator!calculator!

(5 + 3)/2 gives 4(5 + 3)/2 gives 4

Xigmat was stumped by this problem.Xigmat was stumped by this problem.Can you explain it?Can you explain it?

Solve to find x: Solve to find x: 22 xx = 64= 64

Since 2 Since 2 2 2 2 2 2 2 2 2 2 = 64 or 2 2 = 64 or 266 = = 6464

x must equal 6x must equal 6That is, 2 times itself six times will give That is, 2 times itself six times will give

64.64.MoreMoreExamples: Examples: 3322 = 9 10 = 9 10 55 = 100,000 3 = 100,000 34 4 = 81= 81

exponent

base

Yancy needs to learn how to translate Yancy needs to learn how to translate problems into algebra symbols.problems into algebra symbols. If you triple Xigmat’s age and subtract 2, If you triple Xigmat’s age and subtract 2,

the result is 601. the result is 601. How old is Xigmat? How old is Xigmat?

Let x represent Xigmat’s age.Let x represent Xigmat’s age. 3x – 2 = 6013x – 2 = 601

3x = 6033x = 603x = 201x = 201Xigmat is 201 years Xigmat is 201 years old.old.

6

4

2

-2

-4

-5 5 10

x

y

(6, 3)(3, 2)

(0, 1)

(-3, 0)(-6, -1)

X & Y also need to learn how to graph.X & Y also need to learn how to graph.Complete the table, plot the points Complete the table, plot the points and connect the dots to graph the and connect the dots to graph the

equation :equation : xx-6-6-3-300336699

-101234

xy= +13

Xigmat & Yancy have to visit 7 earth friends, Xigmat & Yancy have to visit 7 earth friends, (A(Alpha, lpha, BBeta, eta, CCee, ee, DDelta, elta, EEpsiplon, psiplon, FFred & red & GGamma),amma),

one each day for the next week.one each day for the next week.In how many ways, could they arrange their visits?In how many ways, could they arrange their visits?

There must be an easier way than listing all the There must be an easier way than listing all the possible orders: possible orders:

ABCDEFG ABCDEFG or or DEABCGF DEABCGF or or GDCABEFGDCABEF...... Look for a mathematical shortcut:Look for a mathematical shortcut:Day 1: they have 7 friends to choose fromDay 1: they have 7 friends to choose fromDay 2: they have 6 friends left to choose Day 2: they have 6 friends left to choose fromfromDay 3: they have 5 choices leftDay 3: they have 5 choices left::Day 7: only 1 choice left by eliminationDay 7: only 1 choice left by elimination

Xigmat & Yancy have to visit 7 earth friends, Xigmat & Yancy have to visit 7 earth friends, Alpha, Beta, Cee, Delta, Epsiplon, Fred & Gamma, Alpha, Beta, Cee, Delta, Epsiplon, Fred & Gamma,

one each day for the next week.one each day for the next week.In how many ways, could they arrange their visits?In how many ways, could they arrange their visits?

Looking for a Mathematical Shortcut:Looking for a Mathematical Shortcut:Day 1: they have 7 friends to choose Day 1: they have 7 friends to choose fromfromDay 2: they have 6 choices leftDay 2: they have 6 choices left::Day 7: only 1 choice leftDay 7: only 1 choice leftSimply MULTIPLY Simply MULTIPLY to get the total number of different orders:to get the total number of different orders:

Day 1Day 1 Day 2Day 2 Day 3Day 3 Day 4Day 4 Day 5Day 5 Day 6Day 6 Day 7Day 7

7 7 6 6 5 5 4 4 3 3 2 2 1 = 5,040 1 = 5,040 waysways

In math symbols this is written as:In math symbols this is written as: 7 7!! = = 5,0405,040 The exclamation point is read “FACTORIAL”The exclamation point is read “FACTORIAL”

Want another example?Want another example?In how many different orders could you do In how many different orders could you do

homework for four classes: homework for four classes: AAlgebra, lgebra, BBiology, iology, CCivics and ivics and DDrawing? rawing?

You could list all the orders, if you are careful:You could list all the orders, if you are careful:ABCDABCD BACDBACD CABDCABD DABCDABCABDCABDC BACDBACD CADBCADB DACBDACB = 24 ways = 24 waysACBDACBD BCADBCAD CBADCBAD DBACDBACACDBACDB BCDABCDA CBDACBDA DBCADBCAADBCADBC BDACBDAC CDABCDAB DCABDCABADCBADCB BDCABDCA CDBACDBA DCBADCBA

But there has to be a mathematical shortcut But there has to be a mathematical shortcut using the COUNTING PRINCIPLE we saw using the COUNTING PRINCIPLE we saw previously!previously!

In how many different orders could you do In how many different orders could you do homework for four classes: homework for four classes:

AAlgebra, lgebra, BBiology, iology, CCivics and ivics and DDrawing? rawing?

What HW do you do first?What HW do you do first? 4 subjects to choose from4 subjects to choose fromWhat HW do you do second? What HW do you do second? 3 subjects left to choose from3 subjects left to choose fromWhat HW do you do third?What HW do you do third? 2 subjects left to choose from2 subjects left to choose fromWhat HW do you do fourth?What HW do you do fourth? Only 1 subjects left to complete!Only 1 subjects left to complete!

HW 1HW 1 HW 2HW 2 HW 3HW 3 HW 4HW 4

4 4 3 3 2 2 1 = 24 1 = 24 different ordersdifferent orders

This is written as 4This is written as 4!! = 24 Read: “4 factorial is = 24 Read: “4 factorial is 24.24.

Welcome to Algebra Quick Review: Welcome to Algebra Quick Review: | -5 | is read “Absolute Value of negative 5”| -5 | is read “Absolute Value of negative 5” | -5 | = 5 since it is five units away from the origin | -5 | = 5 since it is five units away from the origin (zero)(zero)

5! is read “5 Factorial” which is used in Counting 5! is read “5 Factorial” which is used in Counting PrinciplesPrinciples 5! = 5! = 5 5 4 4 3 3 2 2 1 = 120 1 = 120 8! = 8 8! = 8 7 7 6 6 5 5 4 4 3 3 2 2 1 = 40,320 1 = 40,3203355 is read “3 to the 5 is read “3 to the 5thth power.” power.” 3355 == 3 3 3 3 3 3 3 3 3=243 3=243 5522 = 25 which can also be read as “five = 25 which can also be read as “five squaredsquared.”.”4433 = 64 which can also be read as “four = 64 which can also be read as “four cubedcubed.”.”

PEMDAS PEMDAS (Please Excuse My Dear Aunt Sally)(Please Excuse My Dear Aunt Sally) is a mnemonic for is a mnemonic for the the

Order of OperationsOrder of Operations: : 1.1. Parentheses & Grouping SymbolsParentheses & Grouping Symbols 3. Multiply & Divide 3. Multiply & Divide 2.2. ExponentsExponents 4. Add & Subtract4. Add & Subtract