|
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.
|
| |
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