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course MTH 151
10:25pm, 3/25/14
If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
018. Base-10 Place-value Number System
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Question: `q001. There are 7 questions in this set.
From lectures and textbook you will learn about some of the counting systems used by past cultures. Various systems enabled people to count objects and to do basic arithmetic, but the base-10 place value system almost universally used today has significant advantages over all these systems.
The key to the base-10 place value system is that each digit in a number tells us how many times a corresponding power of 10 is to be counted.
For example the number 347 tells us that we have seven 1's, 4 ten's and 3 one-hundred's, so 347 means 3 * 100 + 4 * 10 + 7 * 1.
Since 10^2 = 100, 10^1 = 10 and 10^0 = 1, this is also written as
3 * 10^2 + 4 * 10^1 + 7 * 10^0.
How would we write 836 in terms of powers of 10?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
8 * 100 + 3 * 10 + 6 * 1
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:
836 means 8 * 100 + 3 * 10 + 6 * 1, or 8 * 10^2 + 3 * 10^1 + 6 * 10^0.
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Self-critique (if necessary):
Could this be written out either way, or would it need to be written a certain way?
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Self-critique Rating: 3
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Question: `q002. How would we write 34,907 in terms of powers of 10?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
3 * 10000 + 4 * 1000 + 9 * 100 + 0 * 10 + 7 * 1
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:
34,907 means 3 * 10,000 + 4 * 1000 + 9 * 100 + 0 * 10 + 7 * 1, or 3 * 10^4 + 4 * 10^3 + 9 * 10^2 + 0 * 10 + 7 * 1.
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Self-critique (if necessary):
I had tried to remove the 0 after the 100, because we know that 100 + 0 = 100, but when I added it up, I realized how it completely changed the answer.
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Self-critique Rating: 3
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Question: `q003. How would we write .00326 in terms of powers of 10?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
.00001 * 6 + .0001 * 2 + .001 * 3 + .01 * 0 + .1 * 0
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:
First we note that
.1 = 1/10 = 1/10^1 = 10^-1,
.01 = 1/100 = 1/10^2 = 10^-2,
.001 = 1/1000 = 1/10^3 = 10^-3, etc..
Thus .00326 means
0 * .1 + 0 * .01 + 3 * .001 + 2 * .0001 + 6 * .00001 =
0 * 10^-1 + 0 * 10^-2 + 3 * 10^-3 + 2 * 10^-4 + 6 * 10^-5 .
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Self-critique (if necessary):
Is it okay if we write the problems like this, backwards? It helps me to start at the end, and then work my way back, and I got the same answer.
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Self-critique Rating: 3
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It's fine if you work it out that way, but in the last step you should then present the solution in the standard order.
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Question: `q004. How would we add 3 * 10^2 + 5 * 10^1 + 7 * 10^0 to 5 * 10^2 + 4 * 10^1 + 2 * 10^0?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
I would put each part in parenthesis, so that it would read (3 * 10^2 + 5 * 10^1 + 7 * 10^0) + (5 * 10^2 + 4 * 10^1 + 2 * 10^0), and then just follow the order of operations for each part. So (3 * 100 + 5 * 10 + 7 * 1) + (5 * 100 + 4 * 10 + 2 * 1) would be after the first step, then it would read (300 + 50 + 7) + (500 + 40 + 2), and 357 + 542 = 899.
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:
We would write the sum as
(3 * 10^2 + 5 * 10^1 + 7 * 10^0) + (5 * 10^2 + 4 * 10^1 + 2 * 10^0) ,
which we would then rearrange as
(3 * 10^2 + 5 * 10^2) + ( 5 * 10^1 + 4 * 10^1) + ( 7 * 10^0 + 2 * 10^0),
which gives us
8 * 10^2 + 9 * 10^1 + 9 * 10^0. This result would then be written as 899.
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Self-critique (if necessary):
ok
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&#Your response did not agree with the given solution in all details, and you should therefore have addressed the discrepancy with a full self-critique, detailing the discrepancy and demonstrating exactly what you do and do not understand about the parts of the given solution on which your solution didn't agree, and if necessary asking specific questions (to which I will respond).
&#
*@
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Self-critique Rating: 3
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Question: `q005. How would we add 4 * 10^2 + 7 * 10^1 + 8 * 10^0 to 5 * 10^2 + 6 * 10^1 + 4 * 10^0?
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Separate each part into parenthesis: (4 * 10^2 + 7 * 10^1 + 8 * 10^0) + (5 * 10^2 + 6 * 10^1 + 4 * 10^0). Then follow the order of operations: (4 * 100 + 7 * 10 + 8 * 1) + (5 * 100 + 6 * 10 + 4 * 1). Next: (400 + 70 + 8) + (500 + 60 + 4), and 478 + 564 = 1042.
confidence rating #$&*: 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution:
We would write the sum as
(4 * 10^2 + 7 * 10^1 + 8 * 10^0) + (5 * 10^2 + 6 * 10^1 + 4 * 10^0) ,
which we would then rearrange as
(4 * 10^2 + 5 * 10^2) + ( 7 * 10^1 + 6 * 10^1) + ( 8 * 10^0 + 4 * 10^0),
which gives us
9 * 10^2 + 13 * 10^1 + 12 * 10^0.
Since 12 * 10^0 = (2 + 10 ) * 10^0 = 2 * 10^0 + 10^1, we have
9 * 10^2 + 13 * 10^1 + 1 * 10^1 + 2 * 10^0 =
9 * 10^2 + 14 * 10^1 + 2 * 10^0.
Since 14 * 10^1 = 10 * 10^1 + 4 * 10^1 = 10^2 + 4 * 10^1, we have
9 * 10^2 + 1 * 10^2 + 4 * 10^1 + 2 * 10^0 =
10^10^2 + 4 * 10^1 + 2 * 10^0.
Since 10*10^2 = 10^3, we rewrite this as 1 * 10^3 + 0 * 10^2 + 4 * 10^1 + 2 * 10^0.
This number would be expressed as 1042.
STUDENT SOLUTION
(4 x 10^2 + 5 x 10^2) + (7 x 10^1 + 6 + 10^1) + (8 x 10^0 + 4 x 10^0)
adds up to
9 x 10^2 + 13 x 10^1 + 12 x 10^0 = 1042
INSTRUCTOR RESPONSE
You got
9 x 10^2 + 13 x 10^1 + 12 x 10^0 = 1042
But this isn't in its final powers-of-10 notation.
13 * 10^1 isn't a legal expression. Since 13 is greater than 9, you would use the fact that 13 * 10^1 = 10^2 + 3 * 10^1 to write this in correct notation.
Your expression would then become
9 x 10^2 + 10^2 + 3 x 10^1 + 12 x 10^0
Also 12 * 10^0 = 10^1 + 2 * 10^0, so your expression is equivalent to
9 x 10^2 + 1 * 10^2 + 3 x 10^1 + 10^1 + 2 x 10^0
When we add the like powers of 10 we find that 9 * 10^2 + 10^2 = 10 * 10^2, which is 10^3.
Since 3 * 10^1 + 10^1 = 4 * 10^1.
your final expression should be
10^3 + 4 * 10^1 + 2 * 10^0.
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Self-critique (if necessary):
Could you not just arrange the two parts into parenthesis, and then follow the order of operations? I got the right answer when I did this.
@&
You got the right answer, but you did not use powers of 10 to get it. Your solution should be substantially the same as the given solution.
Saying 478 + 564 = 1042, without reasoning it out completely in terms of powers of 10, is not a valid solution.
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Self-critique Rating: 3
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Question: `q006. Write each of the following in expanded notation using the highest possible powers of 10, and explain your reasoning for each, using only expanded notation in your explanations:
• 14 * 10^3
• 4 * 10^4 + 14 * 10^3
• 8 * 10^3 + 17 * 10^2 + 21 * 10^1 + 15 * 10^0
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
For all of these problems, I’m having trouble being able to expand them any more than they already are. I’d be tempted to just go ahead and write out the problems, or just use the order of operations to solve them. How can they be expanded any further?
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For example, 17 * 10^2 does not use the highest possible power of 10. 17 = 10 + 7, so this product could be written (10 + 7) * 10^2, which is equal to 10 * 10^2 + 7 * 10^2, and thus to 10^3 + 7 * 10^2.
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confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Question: `q007. Show how we would add 8 * 10^2 + 7 * 10^0 to 4 * 10^2 + 5 * 10^0, performing all steps in expanded powers of 10.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
I would separate the parts into parenthesis, and then go from there. I’m not sure how to go about expanding the problem further.
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Self-critique Rating:
@&
See my inserted notes.
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