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​

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Algebra: Restoring the Value of an Unknown Number

 الكتاب المختصر في حساب الجبر والمقابلة‎‎ 

al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala
The Compendious Book on Calculation by Restoring and Balancing 
by Muhammad ibn Musa al-Khwarizmi (circa year 830 of the Common Era)
          Some middle-school students will complain about algebra, saying, "When will I ever use this in real life?" But algebra was constructed 1,187 years ago as a tool to help solve real-world problems. When Muhammad ibn Musa al-Khwarizmi wrote his famous text, "The Compendious Book on Calculation by Restoring (al-jabr) and Balancing, (al-muqabala)" his examples came from problems like trading between countries with different money and goods, or constructing streets, buildings, and canals in Persia. This was important because Persia was a center of trade as the middle of the route for goods between Western Europe and Eastern Asia. 
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Variables

Big Idea: In many jobs and tasks in the world, success depends on solving similar-but-not-quite-the-same problems in similar ways. After years--or sometimes centuries--of work, scientists or engineers or bankers or business leaders have created mathematical models that explain how some part of our world, culture, society, or economy works. With this knowledge, they can be confident in the parts of the model that stay the same and concentrate their attention on the parts that vary, or change.

Variables are symbols or letters that people in a certain profession (science, engineering, or business, for example) have agreed will represent the parts of the mathematical model that change. The rest of the equation, the pattern that stays the same, will help find answers for any missing variables.


Patterns

Big Idea: When there is a pattern to numbers in a series, observers can often use the pattern to figure out relationships between variables.
​
 

Equations

Big Idea: The most common way to express patterns between variables is in the form of an equation, a mathematical statement that uses arithmetic operations (addition, subtraction, multiplication, division, and exponents) to show that two amounts are equal. 

Constants

Big Idea: While many students and some adults focus on the use of letters or symbols in algebra, many algebra equations rely on the pattern of relationships between variables to identify constants in an equation, numbers that do not change from problem-to-problem.
​

Formulas

Big Idea: When an equation has been shown to be effective in solving a mathematical problem  over time, it is accepted by people who work on that type of problem as a formula that will work for future problems of that type.

Restoring (al-jabr)

Big Idea: making an equation simpler by . . .
  • getting rid of parentheses by performing the required operations (usually multiplication or division), 
  • reducing the terms (individual values or variables) through combining (addition or subtraction),
  • gathering or isolating all the variables (unknown numbers, usually represented by letters) on one side (usually the left) of an equation, and the constants on the other side (usually the right) helps make an equation clearer.

Balancing (al-muqabala)

Big Idea: a mathematician can add a number to an expression, or multiply an expression by a factor, as long as the the same operation is applied to both sides of the equals sign.
  • Multiply both expressions (sides) by the reciprocal of the coefficient (the number multiplying the variable). This is the same function as dividing by the coefficient. This is often a good strategy when fractions are part of one of the expressions.
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A statue of Al-Khwarizmi in front of Baghdad's House of Wisdom Museum.
Baghdad, Iraq
          Muhammad ibn Musa Al-Khwarizmi wrote other valuable books on geography, astronomy and numerals as well. His Latinized name Algoritmi (from al-Khwarizmi) in later translations give us the words--

Algorism--the placement of digits in place-value positions and using powers of 10 (10, 100, 1,000 . . . ) to describe them.

Algorithm--in computer science and higher mathematics, a 
step-by-step set of operations to be performed the same way in response to certain starting conditions to get a result.


Scholarly Pursuits Always Reveal Knowledge
​​SPARK Mission:
To provide students with experiences and foundational skills necessary to pursue knowledge, develop character, and strengthen personal motivation to engage and persist in more rigorous academic challenges.

Website Updated: 27 September 2019

Webpage maintained by the Scholarly Pursuits Always Reveal Knowledge (SPARK) classes of Richland School District Two, Columbia, South Carolina.
Please direct queries to Mr. Kevin Durden.
  • SPARK Overview
    • SPARK Goals >
      • 6 Cs of Education--Golinkoff & Hirsh-Pasek
      • Collaboration
      • Communication
      • Content
      • Critical Thinking
      • Creative Innovation
      • Confidence
      • SPARK Curriculum Framework
      • SPARK Assessment
      • Mindsets
      • The Renzulli Model
    • About Mr. Durden
  • 4th Grade
    • SPARKed!
    • Jumpstart! Word Play >
      • Analogies
      • Idioms
      • Phonetics
      • Spelling
      • Vocabulary and Word Work
    • Word Power! Latin Roots >
      • Latin Study Tools
      • Latin Stems Video Library
      • A Game of Stems
    • Literature Ladder >
      • Share and Request Writing
      • Expository-Informative Writing
      • Narrative-Storytelling Writing
      • Persuasive-Argument Writing
      • Poetry Month!
    • Critical Thinking Practices >
      • A History of Numbers
      • Algebra: Restoring by Balancing >
        • Algebra Patterns and Formulas
        • Algebra Variables and Expressions
      • Benjamin Franklin's Magic Squares
      • Logic Puzzles >
        • Logic Puzzles Coaching
      • Perimeter Magic Polygons
      • PI Day
      • A Game of Asteroids
      • Critical Mathematical Thinking
    • Digital, Collaborative Investigations >
      • Digital Projects Spring 2017
      • Digital Projects Fall 2016
      • SPARK Research Checklist
      • SPARK Digital Research
      • SPARK: Getting More Out of Google
      • SPARK Presentation Checklist
      • SPARK Digital Presentation Tools
      • 4th Grade Digital Exemplars
      • 5th Grade Digital Exemplars
      • SPARK Student Website Design
      • NEW GOOGLE SITES! Tutorial
    • Mindsets Moments
  • Breakout EDU
    • Breakout EDU Coaching
    • Breakout EDU Codes and Ciphers
    • Breakout EDU Timers
  • e-Reading
  • Extra Time
    • SC MESAS Contest
    • Healthy Soils Poster Contest
    • Poetry Contest: Young Minds Dreaming
    • Actors and Actresses >
      • People Smart: Multiple Intelligences
      • R2 Film Festival
      • Recitations and Readers' Theater
    • Artists >
      • Art & Images
      • Amazing Artists: Multiple Intelligences
    • Authors at Play >
      • Verbal-Linguistic Intelligence
      • The Art of Cursive Handwriting
      • Kids in Print (Jan 10-11)
      • Letters About Literature (Jan 3-4)
      • Narrative Writing Challenges
      • Paragraph of the Week
      • Poetry
    • Go >
      • Go--Math is Fun
      • Go Puzzles
      • Go Strategies and Tactics
      • Go in Art and Film
      • American Go Foundation
    • Chess >
      • Chess Advice for New Players
      • ChessKid.com
      • Chess.com
      • Chess: The Scholar's Mate
      • Chess in Art and Film
      • Chess Puzzles and Mathematics >
        • Chess: A Difference of Opinions
      • ​A Chess Novice Challenged Magnus Carlsen
      • The Most Instructive Games of Chess Ever Played >
        • Rook on 7th Rank: Capablanca-Tartakower, ​New York 1924
        • The King Is a Strong Piece: Tal – Lissitzin, Leningrad 1956
        • Knight in Center: Boleslavsky – Lissitzin, Moscow 1956
        • Aggressive Rook in the Ending Tarrasch – Thorold, Manchester 1890
        • Passed Pawn: Akiba Rubinstein vs Oldrich Duras Vienna 1908
        • Mighty Knights: Mattison – Nimzovich, Carlsbad 1929
        • Finesse Ending: Rafael Domenech vs Salomon Flohr B44 Sicilian Taimanov Rosas 1935
        • Pawn Phalanx: Tigran V Petrosian vs Kozali Montevideo 1954
        • Passed Pawns: Fischer-Berliner 1960
        • Rook and Pawn Ending: Smyslov – Reshevsky, Moscow 1948
        • King in the Center: Tartakover-Frentz, Paris 1933
        • Shifting Attacks: Reshevsky-Najdorf, Dallas, 1957
    • Computer Coding, Gaming, and Making >
      • botLogic
      • Code.org
      • Game Design and Makers
      • Logic and Brain Games: Multiple Intelligences
      • SCRATCH
      • Snapp
      • Stencyl
    • GeoBee and GeoGuessr
    • Kinesthetics: Sports and Activity
    • Music >
      • Music Library
      • Music Smart: Multiple Intelligences
    • Renzulli Learning Guidelines
    • Renzulli Learning Online
    • Self-Paced Math Review >
      • Khan Academy (Self=Paced Math Skills and More)
      • The Beauty of Mathematics
      • Coordinate Grids, Geometric Forms
      • Data and Graphing
      • Decimals and Percents
      • Division
      • Exponents and Scientific Notation
      • Fraction Operations
      • Geometric Forms
      • Greg Tang Math
      • Measurement and Scale
      • Multiplication
      • Pi!
      • Nature Smart: Multiple Intelligences
      • Right Triangles and Pythagoras
    • Mensa Kids
    • Reflect and Recharge
    • World Languages >
      • Duolingo