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					Dr. Cheryl Malm is an Associate Professor in the Department of Mathematics and Statistics and
    is Co-PI for the PRISM Project.
    Northwest Missouri State University, 800 University Drive, Maryville, Missouri, 64468
    Phone: 660 562 1206 (work)              660 582 2851 (home)
    FAX:      660 562 1188

Note: PRISM is an acronym for Promoting Reasoning and Inquiry in Science and Mathematics
      and is funded by NSF grant #ESI0098792 and NWMSU.

                                         Learning Cycle

Chapter or Unit Topic: Systems of Equations
Grade Level: 8th - 9th
Learning Cycle Topic: Dependent Systems
Big Idea or Broad Concept: A dependent system of equation does not have a unique answer.
   Rather, the values of the dependent variables will depend upon the value selected for the
   independent variable. Which variable is designated as independent is determined by the
   context of the problem.

Performance Outcomes Related to Big Idea:
  The student will be able to state a definition for dependent systems in their own words.
  The student will be able to solve a dependent system of equation for the designated variable

NCTM Standards Addressed:
  Represent and analyze mathematical situation and structures using algebraic symbols
  Use mathematics models to represent and understand quantitative relationships
Problem Solving:
  Build new mathematical knowledge through problem solving
  Apply and adapt a variety of appropriate strategies to solve problems
  Analyze and evaluate the mathematical thinking an strategies of others
  Recognize and use connections among mathematical idea
  Recognize and apply mathematics in contexts outside of mathematics
  Use representations to model and interpret physical, social, and mathematical phenomena

Materials: calculators
Vocabulary to be Introduced: dependent systems, dependent variable, independent variable

Engagement: Create interest, foster curiosity:
Review systems of equations with the students by reiterating the solution to a system is the point
of intersection. Have the class consider the equations 2x + y = 8 and 3x – y = 2.
Engagement: (continued)
Have the students graph these equations on the same graph. Ask students to discuss the steps
they followed to graph the equations and what they discovered about these two graphs. Have
students explain why (2,4) is a solution to this system.

Exploration Activity: (Manipulatives, demonstrations, brainstorming)

Then, give the students the following scenario:

A tax collector comes into a poor village to collect the taxes. The village is so poor, the tax
collector feels sorry for the people. So, he makes them the following offer. He says, "I will give
you $100 of this tax money if you will follow my instructions exactly and use the money to buy
animals to bring back to the Village. In the town, they are selling horses for $10.00 each, cows
for $1.00 each, and sheep for $0.50 each. You must go into town and spend the $100 buying
horses, cattle, and sheep. You must spend exactly $100 and buy exactly 100 animals. You must
also buy some of each animal. If you do this, I will not ask for the money back." The villagers
agree to this deal and accept the $100 and the tax collector leaves. Now the villagers must
decide who will go into the town and buy these animals. They agree the Sven should go, as he is
the leader of the village. Sven is reluctant to do this, however, as he doesn't know how many of
each animal he should buy. He tells the villagers that they must help him decide. You are the
rest of the villagers. It is your task to tell Sven how many of each animal he should buy, to
insure that he spends exactly $100 and brings back exactly 100 animals. Remember, you must
buy some of each animal.
        Have students work in small groups to solve the villager's problem with any appropriate
strategy they may devise. Tell students you will chose a random member of their group to
explain the group’s findings to the class. Each group should get a chance to present their ideas,
explaining their plan and showing any work needed to illustrate their solution.
Explanation: Discuss activity/ explain / introduce vocabulary / reading
               (Check for Understanding and modeling)
        Begin by having the groups present their ideas. Discuss the effectiveness of each method
presented, relative merits, etc., to evaluate the problem solving strategies.
        Discuss with students why there is more than one answer to this problem. Do they think
this actually happens in the "real world"? If there is more than one answer like this, how do you
decide what is the "best" answer?
Lead students to the definition of a dependent system of equations and discuss the terms
independent and dependent variables. Have students explain which variable they would chose as
independent in the villager problem and why they would make this choice.
Elaboration Activity: Develop idea (Application of Knowledge)

Review solving systems of equations algebraically, if necessary, from the engagement.
        Have student write equations for the information in the problem. One equation will deal
with the number of animals (h+c+s=100) and the other equation will deal with the amount of
money (10h+1c+.5s=100). Review what students know about solving a system of equations with
substitution or elimination. Ask why those methods don't seem to work here (3 variables, only 2
equations).                                                          Date:
Work with them to solve the # of animals equation for each animal:

                        h  c  s  100          10h  c  .5s  100
                        h  100  c  s          h  10  .1c  .05s
                        c  100  h  s          c  100  10h  .5s
                        s  100  h  c          s  200  20h  2c

Using these literal equations, solve for c in terms of h:

                                   c  100  h  s
                                   c  100  h  200  20h  2c 
                                   c  100  h  200  20h  2c
                                    c   100  19h
                                   c  100  19h

Then solve for s:
                                   s  100  h  c
                                   s  100  h  100  19h 
                                   s  100  h  100  19h
                                   s  18h

 Discuss that the number of cattle and sheep depends on the number of horses: h is the
independent variable; s and c are the dependent variable. Ask if c or s could have been the
independent variable. How would you decide?
Evaluation: match performance outcomes
         Have students work, symbolically, on a dependent system of equation, explaining which
is the independent variable and which is the dependent - and why they solved it that way.

                                     SOLDIER ARITHMETIC


      A) There is a set of things…..these things need not be numbers
      B) There are one or more operations
      C) There are some properties concerning the operations and the sets of things such as
         commutative, associative, closure, identities, inverses, and distribution

The “rules” for soldier arithmetic are as follows:

         R = right face                AF = about face
         L = left face                 A = attention (don’t move)

The operation * indicates that you do the “rules” in the given order, one after another.

Complete the table for the * operation

                          *        R          L          AF          A





10.      Is this operation cummutative?     ___________________________________________

11.      Is this operation associative? _______________________________________________

12.      Is there an identity element for this operation? _______________________________

13.      Is there an inverse for this operation? _____________________________________

14.      Is this system closed under this operation? _____________________________________
                           Transformation with an Equilateral Triangle

The “rules” of the system are the transformations that come from line and rotational symmetry of
the equilateral triangle. The rotations are assumed to be in a clockwise direction.

R1 is a rotation of 120      T is a reflection about the altitude from “top” vertex to the opposite

R2 is a rotation of 240      L is a reflection about the altitude from the bottom left corner

I is a rotation of 360       R is a reflection about the altitude from the bottom right corner

Complete the table for the * operation, defined as one transformation followed by another.

                             *          I         T          L          R         R1         R2







Which properties does this operation have? Commutative? Associative? Identity? Inverse?

Is this system closed?

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