This module provides practice problems designed to develop some concepts related to algebraic generalizations.
In class, we found that if you multiply
by 2,
you get
. If you multiply
by 2, you get
. We expressed this as a general rule that
.
Now, we’re going to make that rule even more general. Suppose I want to multiply
times
. Well,
means
, and
means
. So we can write the whole thing out like this.
Using a similar drawing, demonstrate what
must be.
Now, write an algebraic generalization for this rule.________________
In class, we talked about the following four pairs of statements.
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
thing.
There are two numbers that are
oneaway from 10; these numbers are, of course, 9 and 11. As we saw,
is 99. It is
one less than 100.
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):
1 away from 10, the product is
1 less than 100
2 away from 10, the product is ____
less than 100
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
Do you see the pattern? What would you expect to be the next sentence in this sequence?
Write the algebraic generalization for this rule.
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!)