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This module provides sample problems designed to develop some concepts related to algebraic generalizations.
  • A

    Pick a number:_____
  • B

    Add three:_____
  • C

    Subtract three from your answer in part (b):_____
  • D

    What happened?_________________________________
  • E

    Write an algebraic generalization to represent this rule._____
  • F

    Is there any number for which this rule will not work?_____
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  • A

    Pick a number:_____
  • B

    Subtract five:_____
  • C

    Double your answer in part (b):_____
  • D

    Add ten to your answer in part (c):_____
  • E

    Divide your answer in part (d) by your original number (a):_____
  • F

    Now, repeat that process for three different numbers. Record the number you started with (a) and the number you ended up with (e).
  • Started with:_____

  • Ended with:_____

  • Started with:_____

  • Ended with:_____

  • Started with:_____

  • Ended with:_____

  • G

    What happened?
  • H

    Write an algebraic generalization to represent this rule.
  • I

    Is there any number for which this rule will not work?
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Here are the first six powers of two.

  • 2 1 = 2 size 12{2 rSup { size 8{1} } =2} {}
  • {} 2 2 = 4 size 12{2 rSup { size 8{2} } =4} {}
  • 2 3 = 8 size 12{2 rSup { size 8{3} } =8} {}
  • 2 4 = 16 size 12{2 rSup { size 8{4} } ="16"} {}
  • {} 2 5 = 32 size 12{2 rSup { size 8{5} } ="32"} {}
  • 2 6 = 64 size 12{2 rSup { size 8{6} } ="64"} {}
  • A

    If I asked you for 2 7 size 12{2 rSup { size 8{7} } } {} (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term?________________
  • B

    Write an algebraic generalization to represent this rule.________________
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Look at the following pairs of statements.

  • 8 × 8 = 64
  • 7 × 9 = 63
  • 5 × 5 = 25
  • 4 × 6 = 24
  • 10 × 10 = 100
  • 9 × 11 = 99
  • 3 × 3 = 9
  • 2 × 4 = 8
  • A

    Based on these pairs, if I told you that 30 × 30 = 900 , could you tell me (immediately, without a calculator) what 29 × 31 is?________________
  • B

    Express this rule—the pattern in these numbers—in words.
  • C

    Whew! That was ugly, wasn’t it? Good thing we have math. Write the algebraic generalization for this rule.________________
  • D

    Try out this generalization with negative numbers , with zero , and with fractions . (Show your work below, trying all three of these cases separately.) Does it always work, or are there cases where it doesn’t?
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Source:  OpenStax, Advanced algebra ii: activities and homework. OpenStax CNX. Sep 15, 2009 Download for free at http://cnx.org/content/col10686/1.5
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