|
Homework Guidelines: Examples
1) Determine whether x = 0
is a solution of 5x – 3
= 3x + 5.
2) Write an algebraic equation for the verbal
expression: The distance traveled in t hours by a car traveling at 50 miles per hour.
[unacceptable solution]
| 3 50t |
Um, yes, but what's your
point? |
[acceptable solution]
time: t
rate r: 50
d = r · t,
so d = 50t
|
Much nicer! And now the
final answer
makes sense.
|
3) Suppose you are driving on a freeway to another
town that is 150 miles from your home. After 30 minutes, you pass a freeway exit that you know
is 25 miles from your home. Assuming that you continue at the same constant speed, how long will
it take for the entire trip?
[unacceptable solution]
150m 30m
25m ???? |
At a guess, I'd say that
this student doesn't
have a clue.Too bad he didn't at least write
down the appropriate
formula; it might have
given him a hint! |
[acceptable solution]
25
miles in 30 minutes (0.5 hours):
25 ÷ 0.5 = 50, so rate r = 50mph. |
Beautiful!
The work is done step-by-step,
the
reasoning is clear, and the
final answer is clearly marked.
|
d
= rt and d = 150, so
150 = 50t, and 3 = t. |
|
It will take three hours.
|
4) Subtract, as indicated.
[unacceptable solution]
| ? 12167 |
Why the question mark?
Where
does this number come from?
What on earth might the original
question have been?
|
[acceptable solution]
| Gross Profits |
$72,089 |
The
parentheses indicate that the
answer is a negative number, so that
this computation
reflects a loss.
Now everything is clear. |
| Operating Expenses |
–84,256 |
| ($12,167) |
5) Use equations to obtain the solution.
[unacceptable solution]
25% 78 1950 19.50 |
There is no equation, no
conversion of the percent to a number, and the wrong answer is scratched out instead
of erased. Also, there is little indication of what the actual question might have
been. |
[acceptable solution]
| 25% of 78 is how much? |
We
can clearly see what the original question was: "25% of 78 is how much?" Since
we can also see all the reasoning, we can see how to solve this problem.
And the final answer is marked.
|
| (0.25)(78) = x |
(0.25)(78) = 19.5, so
25% of 78 is 19.5
|
6) The value of a store’s inventory increased
from $50,000 to $62,000. By what percent did the value increase?
[unacceptable solution]
[acceptable solution]
| 62,000 – 50,000 = 12,000 increase |
The
work is done step-by-step, and the reasoning and answer are clear. Also, we can see that
this is computed correctly, as a twenty-four percent increase in value over the original
value, which will help when reviewing how to do this sort of problem for the Final. |
| 12,000 / 50,000 = 0.24, or 24% |
The inventory’s value
increased
by 24%.
|
7) Write an algebraic expression for the verbal
expression: The product of two natural numbers whose sum is 25.
[unacceptable solution]
| 25 x+y xy |
Huh? Nothing is explained,
the set-up is confusing, and as a result, the final answer is unclear and incomplete. |
[acceptable solution]
| x + y = 25. |
Now this makes sense!
The reasoning is clear,
and the answer is obviously marked. When you go back to study for the test, the clarity
of this worked exercise will really help!
|
| Then y
= 25 – x. |
| Then the product is:
(x)(25
– x)
|
8) Write a mathematical model for the problem,
and solve the problem:
One whole number is one-fifth of another whole
number. The difference between the two numbers is 76. Find the numbers.
[unacceptable solution]
| 1/5 76 x-y=76
?????? |
|
Why didn’t this student try
to get some help? He surely does not understand what is going on! If only he had started
by labelling, he might have made some progress. |
[acceptable solution]
one number: x
the other number: y |
|
Ahh, that’s much better!
By starting out with labelling, the problem became much clearer, and thus much easier.
Not only was this student
able to complete the problem correctly, but he is now much better prepared to study for
the test.
|
one number is one-fifth of
another number:
x = (1/5)y |
That means that x is the smaller
number; so y – x = 76. |
| Then y
– (1/5)y = 76 |
| (4/5)y = 76, y
= 95. |
| Then x
= (1/5)95 = 19. |
The numbers are 95 and 19.
|
|
|
|
