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Homework: Algebraic Generalizations

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

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

Exercise 1

In class, we found that if you multiply 2 6 2 6 by 2, you get 2 7 2 7 . If you multiply 2 10 2 10 by 2, you get 2 11 2 11 . We expressed this as a general rule that ( 2 x ) ( 2 ) = 2 x + 1 ( 2 x )(2)= 2 x + 1 .

Now, we’re going to make that rule even more general. Suppose I want to multiply 2 5 2 5 size 12{2 rSup { size 8{5} } } {} times 2 3 2 3 size 12{2 rSup { size 8{3} } } {} . Well, 2 5 2 5 size 12{2 rSup { size 8{5} } } {} means 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 2 ∗ 2 ∗ 2 ∗ 2 ∗ 2 size 12{2 * 2 * 2 * 2 * 2} {} , and 2 3 2 3 size 12{2 rSup { size 8{3} } } {} means {} 2 ∗ 2 ∗ 2 2 ∗ 2 ∗ 2 size 12{2 * 2 * 2} {} . So we can write the whole thing out like this.

Figure 1
Figure 1 (algebraic.png)

( 2 5 ) ( 2 3 ) = 2 8 ( 2 5 ) ( 2 3 ) = 2 8 size 12{ \( 2 rSup { size 8{5} } \) \( 2 rSup { size 8{3} } \) =2 rSup { size 8{8} } } {}

  • a. Using a similar drawing, demonstrate what ( 10 3 ) ( 10 4 ) ( 10 3 ) ( 10 4 ) size 12{ \( "10" rSup { size 8{3} } \) \( "10" rSup { size 8{4} } \) } {} must be.
  • b. Now, write an algebraic generalization for this rule.________________

Exercise 2

The following statements are true.

  • 3 × 4 = 4 × 3 3 × 4 = 4 × 3 size 12{3 times 4=4 times 3} {}
  • 7 × − 3 = − 3 × 7 7 × − 3 = − 3 × 7 size 12{7 times - 3= - 3 times 7} {}
  • 1 / 2 × 8 = 8 × 1 / 2 1 / 2 × 8 = 8 × 1 / 2 size 12{1/2 times 8=8 times 1/2} {}

Write an algebraic generalization for this rule.________________

Exercise 3

In class, we talked about the following four 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. You made an algebraic generalization about these statements: write that generalization again below. Now, we are going to generalize it further. Let’s focus on the 10×1010×10 thing. 10×10=10010×10=100 There are two numbers that are one away from 10; these numbers are, of course, 9 and 11. As we saw, 9×119×11 is 99. It is one less than 100.
  • b. Now, suppose we look at the two numbers that are two away from 10? Or three away? Or four away? We get a sequence like this (fill in all the missing numbers):
    Table 1
    10×10=10010×10=100  
    9×11=999×11=99 1 away from 10, the product is 1 less than 100
    8×12=____8×12=____ 2 away from 10, the product is ____ less than 100
    7×13=____7×13=____ 3 away from 10, the product is ____ less than 100
    __×__=___ __ away from 10, the product is ____ less than 100
    __×__=___ __ away from 10, the product is ____ less than 100
  • c. Do you see the pattern? What would you expect to be the next sentence in this sequence?
  • d. Write the algebraic generalization for this rule.
  • e. Does that generalization work when the “___away from 10” is 0? Is a fraction? Is a negative number? Test all three cases. (Show your work!)

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