p. 99 only (for now) (3i)(4i) i(2i)(-4i) √-10 ∙ √-15 you must take out the i first!!!

Warm Up Factor and solve for x:

Can you factor this equation? If so, factor it and solve for x. If not, how else could you solve for x?

Homework Questions???

p. 99 only (for now)

Multiplying Imaginary Numbers

(3i)(4i)

i(2i)(-4i)

√-10 ∙ √-15

YOU MUST TAKE OUT THE i FIRST!!!

You Do

(4i)(-5i)(3i)(i)

Using imaginary numbers to solve equations

x2 – 16 = 0

3x2 + 48 = 0

2x2 + 12 = 0

6x2 + 72

You Do

5x2 = -125

Complex Numbers

a + bi

a is the real part

b is the imaginary part

Adding/subtracting complex numbers

2i + 3i

(3 + 5i) + (6 – 10i)

(5 – 2i) – (-4 – i)

You Do

(3 – 2i) – (6 + 3i)

Multiplying complex numbers

(2 – i)(3 + 4i)

(5 – 2i)(7 – i)

(2 – i)(2 + i)

You Do

(5 + i)(5 – i)

THINK/PAIR/SHARE

What are imaginary numbers? What are complex numbers? How are they similar and different? THINK silently for 30 seconds. PAIR discuss with your partner for 30

seconds. SHARE with the class

Homework

Finish the problems that were assigned on p. 113. We will go over the answers in a few minutes.

Arabic Mathematics & Completing the Square

Honors Algebra II

Arab Contributions to Mathematics

Arab Empire (632-end of 13th century) Main source of knowledge between Greeks and

European Renaissance Baghdad established as center of wisdom and

learning (9th century) Many contributions to

the study of algebra

(800-1450 )

The Father of Algebra: Al-Khwarizmi

“In the foremost rank of mathematicians of all time stands Al-Khwarizmi. He composed the oldest works on arithmetic and algebra. They were the principal source of mathematical knowledge for centuries to come in the East and the West. The work on arithmetic first introduced the Hindu numbers to Europe, as the very name algorism signifies; and the work on algebra ... gave the name to this important branch of mathematics in the European world...” -Mohammad Kahn

Life of Al-Khwarizmi

Full name: Abu Ja’far Muhammad ibn Musa Al-Khwarizmi

Lived 790-850 Of Persian descent May have been born in Baghdad

or modern-day Uzbekistan

Al-Khwarizmi’s Contributions to Algebra Wrote Algoritmi de numero Indorum (Al-

Khwarizmi on the Hindu Art of Reckoning) Only words (no symbols or numerals) This text was later studied in Europe for

centuries

Studied quadratic equations and their solutions

Invented “completing the square” and developed geometric representations of this process

The terms algebra and algorithm come from his name.

Mini-Quiz

Where was the center of wisdom and learning in the 9th century?

To what area of mathematics did Arabic mathematicians make the greatest contributions?

Why is Al-Khwarizmi also called the Father of Algebra?

Completing the Square

Goal is to manipulate an equation so that it can be factored nicely to solve for x

Two methods1) Al-Khwarizmi’s geometric representation

2) Algebraic representation

Example #1

From the warm-up:

Algebra Tiles

http://illuminations.nctm.org/ActivityDetail.aspx?ID=132

Example #1 (continued)

Another way:

Example #1 (continued)

Now we have

Example #2

THINK: Try to complete the square for the example below.

PAIR: Discuss with your partner. If you are stuck, try to work through it together.

SHARE: Discuss with the class. Who thinks they completed the square? How? Explain your thought process.

Now use your new equation to solve for x.

Practice

Work with your partner to complete the next 2 examples on your notes page. Be prepared to share with the class.

Completing the Square, Take Two

Given an equation in the form , we can use the following formula to complete the square:

Since we added here, we must subtract here to balance the equation.

Example #5

Complete the square using the second method:

Now solve for x.

Example #6

THINK-PAIR-SHARE:

Complete the square and solve for x:

Now work with your partner to complete the next two examples. Be prepared to share with the class.

Partner sticky note challenge!Partner on the left complete the square to solve for x:

Partner on the right complete the square to solve for x:

WHEN I SAY GO: Switch sticky notes with your partner and check their work. If you find a mistake, talk it out with them. When BOTH of you agree on BOTH problems, put your sticky notes on the board. Who can be the first pair to get the correct answer??

x2 – 8x – 1 = 0

x2 + 6x – 7 = 0

Resources

http://library.thinkquest.org/3526/facts/timeline.html

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html

http://www.math.tamu.edu/~dallen/masters/islamic/arab.pdf

www.google.com/maps http://www.algebra.com/algebra/about/history/Al

-Khwarizmi.wikipedia http://3.bp.blogspot.com/-LbVOcoKtCkI/TVl-4IV

3gLI/AAAAAAAAAAw/WUCQGD3JhLg/s1600/Algoritmi.jpg

http://www.storyofmathematics.com/islamic_alkhwarizmi.html

http://www.csames.illinois.edu/documents/outreach/Completing_the_Square_Lesson_Plan.pdf