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