hisab al-jabr w’al-muqa-balah “science of the reunion and the opposition” or “the science of...
TRANSCRIPT
![Page 1: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/1.jpg)
Hisab al-jabr w’al-muqa-balah
“Science of the reunion and the opposition”
or
“The science of equations”
This is the title of a 300 year old book (in Arabic). Translated, it means:
Does that part of the title look familiar?
algebra
![Page 2: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/2.jpg)
For algebraic operations, we begin to mix together numbers and letters into our operations, which is a major challenge for students.
By now we know that a variable represents a quantity that can change….
![Page 3: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/3.jpg)
Think of a number from 1 to 5
Add 3
Multiply by 2
Subtract 4
Divide the number in halfSubtract the number you started
with…
A little math magic…..
![Page 4: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/4.jpg)
Multiplying and Dividing Powers
![Page 5: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/5.jpg)
Can you think of some examples of any short- cuts
It is our nature to search for more efficient ways to
do things
How about cleaning your room?Cutting the grass
Dishes? (eat over the sink)
![Page 6: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/6.jpg)
The Exponent Laws are an example of a mathematical short-cut.
We’ll learn the mechanics of the short cut first, then we’ll examine the applications.
Specifically, repeated operations can be compressed using the Exponent Laws
The following examples will illustrate
![Page 7: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/7.jpg)
Through the investigation of patterns, we are going to derive the first and second exponent laws….
But first, a few practice runs…
![Page 8: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/8.jpg)
Examine the following patterns to predict what the next
symbol will be….
O T T F F …..
![Page 9: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/9.jpg)
M T W T F S ….
JF M A M …
![Page 10: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/10.jpg)
Given any pattern, the simplest progression will be the implication.
![Page 11: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/11.jpg)
These examples are called Sequences
8 4 2 1 …
1 8 27 …
0 2 6 12 …1 3 5 7 9 …
2 4 8 1632 …
![Page 12: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/12.jpg)
We are multiplying 10 by itself 7 times, so this can be re-written in exponential form as:
10 X 10 X 10 X 10 X 10 X 10 X 10
107power
baseexponent
![Page 13: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/13.jpg)
23 X 25 = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2
= 28
Examine the exponents…Is there a short cut?
Since the bases can vary, we will use a variable to represent all cases
In general:
1. Xa X Xb = Xa + b
![Page 14: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/14.jpg)
For example:
(x3) (x8) = x11
(a4) (a3) = a7
(a3) =a9(a5) (a)
(y4) (x2)= x2y4
![Page 15: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/15.jpg)
25 22 = 2 X 2 X 2 X 2 X 2 2 X 2
= 23 Is there a short cut?
In general:
2. Xa Xb = Xa - b
![Page 16: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/16.jpg)
For example:
(x7)
(x2)= x5
(a4)
(a3)= a1 = a
![Page 17: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/17.jpg)
Evaluate for t = 3 and s = 2
t2 + s3
= (3)2 + (2)3
= 9 + 8
= 17
= 3 X 3 + 2 X 2 X 2
![Page 18: Hisab al-jabr w’al-muqa-balah “Science of the reunion and the opposition” or “The science of equations” This is the title of a 300 year old book (in Arabic)](https://reader035.vdocuments.us/reader035/viewer/2022062804/56649f305503460f94c4a46f/html5/thumbnails/18.jpg)
McGraw-Hill Ryerson
Pg 114 5,6,7
Pg 126 2,4