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Algebraic Generalizations

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides sample problems designed to develop some concepts related to algebraic generalizations.

Exercise 1

  • 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?_____

Exercise 2

  • 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?

Exercise 3

Here are the first six powers of two.

  • 2 1 = 2 2 1 = 2 size 12{2 rSup { size 8{1} } =2} {}
  • {} 2 2 = 4 2 2 = 4 size 12{2 rSup { size 8{2} } =4} {}
  • 2 3 = 8 2 3 = 8 size 12{2 rSup { size 8{3} } =8} {}
  • 2 4 = 16 2 4 = 16 size 12{2 rSup { size 8{4} } ="16"} {}
  • {} 2 5 = 32 2 5 = 32 size 12{2 rSup { size 8{5} } ="32"} {}
  • 2 6 = 64 2 6 = 64 size 12{2 rSup { size 8{6} } ="64"} {}
  • a. If I asked you for 2 7 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.________________

Exercise 4

Look at the following pairs of statements.

  • 8×8=648×8=64
  • 7×9=637×9=63
  • 5×5=255×5=25
  • 4×6=244×6=24
  • 10×10=10010×10=100
  • 9×11=999×11=99
  • 3×3=93×3=9
  • 2×4=82×4=8
  • a. Based on these pairs, if I told you that 30×30=90030×30=900, could you tell me (immediately, without a calculator) what 29×3129×31 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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