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Concepts in Advanced Algebra
Concepts in Advanced Algebra
Grade level: 10-12 Subject: Mathematics Duration: Three class periods
 


lesson plan support
Objectives
Students will
  • Discover Chinese achievements in mathematics.
  • Convert Chinese numbers and decimal numbers.
  • Solve excess and deficiency problems.
  • Solve 3 x 3 systems of equations.
Materials
  • Discovering Math: Concepts in Advanced Algebra video
  • Computer with Internet access
  • Print resources about the history of Chinese mathematics
Procedures
  1. Have students research The Nine Chapters on the Mathematical Art and create an illustrated table of contents for each chapter using print and Web resources. The following Web sites are a good starting point:
  2. Have students research Chinese contributions to mathematics using print and Web resources. Useful information is in the following Web sites:
  3. When students have completed their research, ask them to summarize their findings in a one-page report.
  4. Ask students to choose a partner to share their report and answer any questions. Then have each student summarize the partner's report for the class, making sure to include at least three interesting facts.
  5. Show students examples of converting Chinese numbers to decimal numbers and converting decimal numbers to Chinese numbers. Be sure to include numbers that contain the digit 0. Allow enough time for practice.
  6. Provide students with an example of an excess and deficiency problem:
    Suppose a group of people is purchasing merchandise. If each person contributes six coins, there is an excess of two coins. If each person contributes five coins, there is a deficit of one coin. How many people are in the group? What is the price of the merchandise?
    Solution: Solve the problem by substitution. The price of the merchandise is 16 coins and the number of people in the group is three.
  7. Use the Chinese method to solve the above problem: If each person contributes a1, then there is b1 either in excess or deficit. If each person contributes a2, then there is b2 either in excess or deficit. The price of the merchandise is a1b2 + b1a2. The number of people is b1 + b2. Each person should pay a1b2 + b1a2 divided by b1 + b2.
  8. Give students an example of problem involving a 3 ´ 3 system of equations:
    There are three different types of corn: corn x, corn y, and corn z.

    3 bundles of corn x plus 2 bundles of corn y plus 1 bundle of corn z equal 22 pounds of corn.

    3x + 2y + z = 22

    4 bundles of corn x plus 6 bundles of corn y plus 2 bundles of corn z equal 48 pounds of corn.
    4x + 6y + 2z = 48

    1 bundle of corn x plus 2 bundles of corn y plus 3 bundles of corn z equal 34 pounds of corn.

    x + 2y + 3z = 34

    1 4 3     1 12 3     1 0 3     3 0 3 C1 = column 1
    2 6 2     2 18 2     2 10 2     6 10 2 C2 = column 2
    3 2 1     3 6 1     3 2 1     9 2 1 C3 = column 3
    34 48 22     34 144 22     34 56 22     102 56 22  
      3C2     C2 - 4C3     3C1  


    0 0 3     0 0 3     0 0 3
    4 10 2     40 10 2     0 10 2
    8 2 1     80 2 1     72 2 1
    80 56 22     800 56 22     576 56 22
    C1 - C3     10C1     C1 - 4C2

    From column 1, 72z = 576. z = 8

    From column 2, 0 + 10y + 2(8) = 56. y = 4

    From column 3, 3x +2(4) + 8 = 22. x = 2

  9. Give students an example of a problem involving a 3 ´ 3 system of equations that uses negative numbers to reach the solution.

    Solve the system of equations:

    3x + y = 1

    y + z = 1

    2x + 2z = 1

    Set up and solve the problem using the Chinese method.
    2 0 3     6 0 3     0 0 3     0 0 3     0 0 3
    0 1 1     0 1 1     -2 1 1     -2 2 1     0 2 1
    2 1 0     6 1 0     6 1 0     6 2 0     8 2 0
    1 1 1     3 1 1     1 1 1     1 2 1     3 2 1
      3C1     C1 - 2C3     2C2     C1 + C2

    From column 1, 8z=3. z=3/8

    From column 2, 2y + 2(3/8)=2. y=5/8

    From column 3, 3x + 5/8=1. x=1/8

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Evaluation
Use the following three-point rubric to evaluate students' work during this lesson.
  • Three points: Students were highly engaged in class discussions; produced complete reports, including all of the requested information; clearly demonstrated the ability to convert between Chinese and decimal numbers, the ability to solve excess and deficiency problems, and the ability to solve 3 ´ 3 systems of equations.
  • Two points: Students participated in class discussions; produced an adequate report, including most of the requested information; satisfactorily demonstrated the ability to convert between Chinese and decimal numbers, the ability to solve excess and deficiency problems, and the ability to solve 3 ´ 3 systems of equations.
  • One point: Students participated minimally in class discussions; created an incomplete report with little or none of the requested information; were not able to convert between Chinese and decimal numbers, solve excess and deficiency problems, or solve 3 ´ 3 systems of equations.

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Vocabulary
Chinese counting board
Definition:A checkerboard with rows and columns where numbers were represented using small rods made from bamboo or ivory: A number was formed in a row with the units placed in the rightmost column, the tens in the next column to the left, the hundreds in the next column to the left, and so on.
Context:The most significant attribute of the counting board was that it presented a place-value number system.

column of a rectangular array
Definition:One of the vertical lines of elements in a rectangular array
Context:Using a counting board, the Chinese would solve a 3 x 3 system of equations by setting up the coefficients of each equation in side-by-side columns, creating a rectangular array.

rectangular array
Definition:An arrangement of mathematical elements into rows and columns
Context:Chapter 8 of The Nine Chapters on the Mathematical Art is "Rectangular Arrays," offering a method for solving systems of simultaneous linear equations.

row of a rectangular array
Definition:One of the horizontal lines of elements in a rectangular array
Context:To solve a 3 x 3 system of equations, the Chinese set up the coefficients of each equation into a rectangular array of three rows and three columns.

3 x 3 system of linear equations
Definition:A collection of three linear equations with three unknowns
Context:Chapter 8 of The Nine Chapters on the Mathematical Art solves 3 x 3 systems of equations.

2 x 2 system of linear equations
Definition:A collection of two linear equations with two unknowns
Context:Excess and deficiency problems can be represented by 2 x 2 systems of equations.

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Academic Standards

National Council of Teachers of Mathematics (NCTM)
The National Council of Teachers of Mathematics provides guidelines for teaching mathematics in grades K-12 to promote mathematical literacy. To view the standards, visit this Web site:http://standards.nctm.org/document/chapter3/index.htm
This lesson plan addresses the following national standards:

  • Understand patterns, relations, and functions
  • Represent and analyze mathematical situations and structures using algebraic symbols
  • Use mathematical models to represent and understand quantitative relationships

Mid-continent Research for Education and Learning (McREL)
McREL's Content Knowledge: A Compendium of Standards and Benchmarks for K-12 Education addresses 14 content areas. To view the standards and benchmarks, visithttp://www.mcrel.org/compendium/browse.asp.
This lesson plan addresses the following national standards:

  • Mathematics: Understands counting procedures and reasoning (e.g., use of the counting principles to find the number of ways of arranging objects in a set, the use of permutations and combinations to solve counting problems); Understands basic applications of and operations on matrices; Understands and applies basic and advanced properties of functions and algebra
  • Science: Physical Science: Understands the structure and properties of matter; Understands the sources and properties of energy
  • World History: Understands Chinese achievements in mathematics
  • Historical Understanding: Understands evidence of social and cultural development of Chinese civilization

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