Don Break the Law of Exponents by alicejenny

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									                                                                     TIMSS
                                                                                      Standard Indicators

                                                                     NAEP          8.7.1-8.7.12
 Don’t Break the Law (of Exponents)
  Purpose
   Students will solve problems by choosing strategies, explaining their
                                                                                  extending
   reasoning, making calculations, and checking results.                              THE
                                                                                           ACTIVITY
  Materials
                                                                                  Create a matching
    For the teacher: chalk, chalkboard                                            game for groups to
    For each student: paper, pencil, copies of Black Line Masters (BLMs)          play. On one set of
    Discovering Laws of Exponents and Understanding the Law, calculator           cards, write exponential
    (optional)                                                                    expressions. On the
  Activity                                                                        second set of cards,
                                                                                  write the simplified
    A. Introducing the Problem                                                    versions of those
      1. Tell students that today they will be discovering patterns that can      expressions. Ask groups
         be used to simplify expressions containing exponents.                    to use the laws of
      2. Explain that, in math, we frequently use laws found by other             exponents to match
         mathematicians to shorten the time it takes to solve problems.           equivalent expressions.
         Give examples such as the Pythagorean Theorem or the Quadratic
         Formula.
                                                             16 × c ÷ (b )
                                                                 2    5    −2 2

      3. Write the following expression on the chalkboard:       cb  2 3


      4. Call on volunteers to help you solve the problem at the
         chalkboard. Solve the problem by writing in expanded form.
           16 × 16 × c × c × c × c × c × b × b × b × b
         [              c×c×b×b×b                      ]
      5. Explain to students that they will discover laws of exponents that
         will help simplify expressions like this one.

    B. Solving the Problem
       1. Divide the class into groups of three students. Distribute one copy
          of the BLM Discovering Laws of Exponents to each student.
       2. Tell students to look at the first column of each row. Explain that
          students may use their scratch paper and/or their calculators to
          substitute values into the expressions for the terms a and b.
       3. Ask students to look for patterns and write algorithms based on
          those patterns.
       4. Tell students to test their algorithms for several different values.
          Once groups are convinced the algorithm is correct, write it in the
          second column on the BLM.
       5. Allow ample time for groups to complete their charts.
                                                                                                                 Standard 7




                                                                 (continued)
                                                                                   Standards Links
                                                                                     8.1.4, 8.1.5
Standard 7 / Activity 8
Indiana Mathematics Grade 8 Curriculum Framework, October 2002                                        page 199
                        Activity (continued)
                           6. Discuss the results with the class. Write algorithms on the
                              chalkboard. Make sure students correct any mistakes on their
                              chart before going on to section C. (Correct laws are found in
                              the answer key on the back side of the BLM.)

                         C. Understanding the Laws
                            1. Tell students that now they have discovered the Law of
                               Exponents, it is time to show they understand what the laws
                               mean.
                            2. Distribute one copy of the BLM Understanding the Law to each
                               student. Allow students to remain in groups of three to complete
                               the BLM.
                            3. Tell students to copy the correct algorithm for each law onto the
                               BLM. Ask the students to then verbalize the law. Refer students
                               to the example for Law One.
                            4. Allow students time to work on the BLM and then ask students
                               to share the results with the class.


                        Classroom Assessment

                         Basic Concepts and Processes
                         While groups work on solving the problem, ask students the following
                         questions to gauge their understanding of the Standard Indicator:

                              What patterns did you find for law [insert law number here]?

                              How did you go about finding the algorithm for law [insert
                              law number here]?

                              Which values did you use to test the law [insert law
                              number here]?
Standard 7




                                                                                          Standard 7/ Activity 8
             page 200                          Indiana Mathematics Grade 8 Curriculum Framework, October 2002
                                                                 Names:




                         Discovering Laws
                           of Exponents
     Work with your group to find a rule to simplify each exponential
     expression. Write the algorithm for each rule in the second column.



           bm × bn
           bm ÷ bn
           (bm)n
           (a × b)m
           (a ÷ b)m
           b−m
              a −n
           ( )b




Standard 7 / Activity 8                                                   Black Line Master 1
Indiana Mathematics Grade 8 Curriculum Framework, October 2002                      page 201
Discovering Laws
of Exponents
Teacher Directions
          Distribute one copy of the BLM Discovering Laws of Exponents to each student. Have students
          work in groups of three. Ask groups to substitute values for the variables and look for patterns in
          the solutions. Tell students to write an algorithm for the pattern and test the algorithm using
          several different values. Have students write the algorithm in the second column on the BLM.
          Discuss the results with the class. Make sure the students have the correct algorithms, as shown
          in the answer key, before continuing on to the next section of the activity.


Answer Key


                 bm × bn                                              bm+n
                 bm ÷ bn                                              bm-n
                 (bm)n                                                bmn
                 (a × b)m                                             am × bm
                 (a ÷ b)m                                             am ÷ bm
                                                                         1
                 b−m                                                     bm
                      a −n                                                b n
                ( )   b                                              ( )  a




Black Line Master 1                                                                                    Standard 7 / Activity 8
page 202                                                    Indiana Mathematics Grade 8 Curriculum Framework, October 2002
                                                                 Names:




            Understanding the Law
Copy the rule you found for each law. Underneath each rule,
write a verbal description of the law. Use Law One as an example.

    Law One: Product Law
    bm × bn = bm + n
    The product of two exponents of the same base is equal
    to the base raised to the sum of the exponents.

    Law Two: Quotient Law
    bm ÷ bn =



    Law Three: Power of a Power
    (bm)n =



    Law Four: Power of a Product
    (a × b)m =



    Law Five: Power of a Quotient
    (a ÷ b)m =



    Law Six: Negative Power
    b−m =



    Law Seven: Inverse Powers
      a −n
    (b) =




Standard 7 / Activity 8                                                   Black Line Master 2
Indiana Mathematics Grade 8 Curriculum Framework, October 2002                      page 203
Understanding the Law
Teacher Directions
          After groups complete the BLM Discovering the Laws of Exponents, review the laws with
          the class. Distribute one copy of the BLM Understanding the Law to each student. Allow them
          to work in their groups to copy the correct law from their earlier work and write a verbal
          expression for each law. Read the example given for Law One in class. Regroup and discuss
          the results with the class.



Answer Key
          Law One: Product Law
          bm × bn = bm+n
          The product of two exponents of the same base is equal to the base raised to the sum
          of the exponents.

          Law Two: Quotient Law
          bm ÷ bn = bm−n
          The quotient of two exponents of the same base is equal to the base raised to the difference
          of the exponents.

          Law Three: Power of a Power
          (bm)n = bmn
          The power of a power is equal to the product of the powers.

          Law Four: Power of a Product
          (a × b)m = am × bm
          The power of a product is equal to the product of the powers of the individual terms.

          Law Five: Power of a Quotient
          (a ÷ b)m = am ÷ bm
          The power of a quotient is equal to the quotient of the powers of the individual terms.

          Law Six: Negative Power
          b−m = 1mb
          A base number raised to a negative power is equal to the reciprocal of the base number raised
          to the positive power.

          Law Seven: Inverse Powers
                  −n       n
           a            b
          (b)     =
                       (a)
          The negative power of a fractional expression is equal to the reciprocal of the fraction raised
          to the positive power.



Black Line Master 2                                                                                    Standard 7 / Activity 8
page 204                                                    Indiana Mathematics Grade 8 Curriculum Framework, October 2002

								
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