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21:44:24 `q001. It will be very important in this course for your instructor to see and understand the process of visualization and reasoning you use when you solve problems. This exercise is designed to give you a first experience with these ideas, and your instructor a first look at your work.
Answer the following questions and explain in commonsense terms why your answer makes sense.......!!!!!!!!...................................
RESPONSE --> OK.
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21:46:45 For each question draw a picture to make sense out of the situation, and include a description of the picture.
Samples Sample question and response Question: If a bundle of shingles covers 30 square feet, how many bundles are required to cover a 600 square foot roof? Response: We might draw a picture of a rectangle representing the area, dividing the rectangle into a number of smaller rectangles each representing the area covered by a single bundle. This makes it clear that we are dividing the roof area into 1-bundle areas, and makes it clear why we are going to have to divide. Reasoning this problem out in words, we can say that a single bundle would cover 30 square feet. Two bundles would cover 60 square feet. Three bundles would cover 90 square feet. We could continue in this manner until we reach 600 square feet. However, this would be cumbersome. It is more efficient to use the ideas of multiplication and division. We imagine grouping the 600 square feet into 30 square foot patches. There will be 600 / 30 patches and each will require exactly one bundle. We therefore require 600 / 30 bundles = 20 bundles. {}Your responses might not be as clear as the above, though they might be even more clear. I won't be looking for perfection, though I wouldn't object to it, but for a first effort at visualizing a situation and communicating a reasoning process. This is not something you are used to doing and it might take a few attempts before you can achieve good results, but you will get better every time you try. {}You might be unsure of what to do on a specific question. In such a case specific questions and expressions of confusion are also acceptable responses. Such a response must include your attempts to come up with a picture and reason out an explanation. For example your response might be Sample expression of confusion: I've drawn a picture of a pile of bundles and a roof but I'm not sure how to connect the two. I tried multiplying the number of bundles by the square feet of the roof but I got 18,000, and I know it won't take 18,000 bundles to cover the roof. How do you put the area covered by a bundle together with the roof area to get the number of bundles required? A poor response would be something like 'I don't know how to do #17'. This response reveals nothing of your attempt to understand the question and the situation. Nor does it ask a specific question. Incidentally, you might be tempted to quote rules or formulas about rates and velocities in answering these questions. Don't. This exercise isn't about being able to memorize rules and quote them. It is about expanding your ability to visualize, reason and communicate.......!!!!!!!!...................................
RESPONSE --> OK.
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21:47:12 In your own words briefly summarize the instructions and the intent of this exercise.
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RESPONSE --> To summarize in my own words my thought process of solving a problem.
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21:51:27 `q001. If you earn 50 dollars in 5 hours, at what average rate are you earning money, in dollars per hour?
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RESPONSE --> I would draw a table with five columns labeled 1st hour, 2nd hour, 3rd hour, 4th hour, 5th hour. I would then visualize placing an equal amount of one dollar bills into each column until they were all distributed. The total number of one dollar bills in any one of the columns would be the dollars per hour.
This visualization would lead me to divide the total number of dollars by the total number of hours..................................................
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21:55:38 If you travel 300 miles in 6 hours, at what average rate are you traveling, in miles per hour?
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RESPONSE --> I would label six columns as hour 1, hour 2, hour 3, hour 4, hour 5, hour 6. I would visualize that I could not place an even hundred miles in each column (out of the 300 miles). I would then try to visualize placing groups of 50 miles in each column (while counting by 50 at each column; ex. 50, 100, 150, ...). By doing this, I would notice that I can place 50 miles in each column with none left over.
This visualization would lead me to divide the 300 miles by 6 hours to come up with 50 miles per hour..................................................
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22:00:10 `q002. If a ball rolling down a grooved track travels 40 centimeters in 5 seconds, at what average rate is the ball moving, in centimeters per second?
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RESPONSE --> I would draw a time line with marks for each second (1, 2, 3, 4, 5). I would then put 5 centimeter groups at each time line mark. This would equal 25 centimeters and I would know I needed to distribute 15 centimeters more. I would then place one centimeter at each mark until I had all 15 centimeters distributed (this would take three cycles of placing one at each of the 5 time line marks). I would then add all the centimeters at one time line mark (any one will work) and find the answer to be 8 centimeters per second.
This visualization would lead me to divide 40 centimeters by 5 seconds for the answer..................................................
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22:01:35 The preceding three questions illustrate the concept of a rate. In each case, to find the rate we divided the change in some quantity (the number of dollars or the distance, in these examples) by the time required for the change (the number of hours or seconds, in these examples). Explain in your own words what is meant by the idea of a rate.
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RESPONSE --> A rate is equal to the quantity divided by the time required to obtain that quantity.
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22:04:43 `q003. If you are earning money at the average rate of 15 dollars per hour, how much do you earn in 6 hours?
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RESPONSE --> I would use grid paper and draw the horizontal line across 15 boxes to represent dollars per hour. I would then draw a vertical line, starting from the endpoint of the horizontal line, across 6 boxes to represent hours. I would then draw the corresponding horizontal and vertical lines to form a box. I would then count the squares to find out how much money earned.
This would lead me to multiply 15 by 6 for an answer of 90 dollars earned..................................................
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22:07:10 If you are traveling at an average rate of 60 miles per hour, how far do you travel in 9 hours?
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RESPONSE --> I would draw a time line with marks for the 9 hours (1, 2, 3, 4, 5, 6, 7, 8, 9). I would then places a 60 above each time line mark to represent miles for that hour. I would place + signs in between each of the 60's to show that they need to be added together to get the total miles traveled.
This would lead me to multiply 60 by 9 for an answer of 540 miles traveled..................................................
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22:09:31 `q004. If a ball travels at and average rate of 13 centimeters per second, how far does it travel in 3 seconds?
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RESPONSE --> I would draw a time line with a hash-mark for each second (1, 2, 3). I would then place a 13 on each mark and place plus signs in between them to show they need to be added together to find the total distance.
This would lead me to multiply 13 by 3 for an answer of 39 centimeters..................................................
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22:12:36 In the preceding three exercises you turned the concept of a rate around. You were given the rate and the change in the clock time, and you calculated the change in the quantity. Explain in your own words how this increases your understanding of the concept of a rate.
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RESPONSE --> This increases my understanding of rate by showing me that I can find quantity by multiplying rate by time.
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22:16:26 `q005. How long does it take to earn 100 dollars at an average rate of 4 dollars per hour?
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RESPONSE --> For this, I would draw boxes with fours in them to represent 4 dollars per hour. I would continue drawing boxes and counting by fours (4, 8, 12, 16, ...) until I came to the number 100. I would then add the number of boxes and this would give me the total hours needed.
This would lead me to divide 100 by 4 for an answer of 25 hours..................................................
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22:18:22 How long does it take to travel 500 miles at an average rate of 25 miles per hour?
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RESPONSE --> For this, I would draw boxes with the number 25 in each one. I would continue to draw boxes and count by 25 (25, 50, 75, 100, ...) until I came to the number 500. Then I would count the number of boxes to find the time it would take.
This would lead me to divide 500 by 25 for an answer of 20 hours..................................................
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22:22:25 `q006. How long does it take a rolling ball to travel 80 centimeters at an average rate of 16 centimeters per second?
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RESPONSE --> I would use a piece of graph paper and draw a vertical line across 80 boxes. I would then proceed to count each mark from one of the endpoints until I came to 16. Here I would place a horizontal line showing there are sixteen boxes from the last mark. I would continue to do this until I came to the other end point. I would then count the sections of 16 boxes to find the time it takes.
This would lead me to divide 80 by 16 for an answer of 5 seconds..................................................
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22:23:17 In the preceding three exercises you again expanded your concept of the idea of a rate. Explain how these problems illustrate the concept of a rate.
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RESPONSE --> You can find the time needed by dividing the quantity by the rate.
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džڕwK} Student Name: assignment #001 001. Rates
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22:33:40 `q001. You should copy and paste these instructions to a word processor for reference. However you can always view them, as well as everything else that has appeared in this box, by clicking the 'Display Everything' button.
1. For the next question or answer, you click on 'Next Question / Answer' button above the box at top left until a question has been posed. Once a question has been posed you are to answer before you click again on this button. 2. Before clicking for an answer, type your best answer to the current question into the box to the right, then clip on the 'Enter Answer' button. 3. After entering your answer you will click on 'Next Question / Answer' to view the answer to the question. Do not tamper with the information displayed in the left-hand box. 4. If your answer was incorrect, incomplete or would otherwise require revision, you will enter a self-critique. If you learned something from the answer, you need to restate it in your own words in order to reinforce your learning. If there is something you feel you should note for future reference, you should make a note in your own words. Go to the response box (the right-hand box) and type in a self-critique and/or notes, as appropriate. Do not copy and paste anything from the left-hand box, since that information will be saved in any case. 5. If you wish to save your response to your Notes file you may choose to click on the 'Save As Notes' button rather than the 'Enter Answer' button. Doing so will save your work for your future reference. Your work will be saved in a Notes file in the c:\vhmthphy folder. The title of the Notes file will also include the name you gave when you started the program. 6. After clicking either the 'Enter Response' or the 'Save as Notes' button, click on 'Next Question / Answer' and proceed in a similar manner. In the right-hand box briefly describe your understanding of these instructions, then click 'Enter Answer'.......!!!!!!!!...................................
RESPONSE --> I understand to enter a response and then enter a self critique if my initial response is incorrect.
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22:33:54 Your answer has been noted. Enter 'ok' in the Response Box and click on Enter Response, then click on Next Question/Answer for the first real question.
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RESPONSE --> OK.
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22:35:15 `q002. Note that there are 10 questions in this assignment. The questions are of increasing difficulty--the first questions are fairly easy but later questions are very tricky. The main purposes of these exercises are to refine your thinking about rates, and to see how you process challenging information. Continue as far as you can until you are completely lost. Students who are prepared for the highest-level math courses might not ever get lost.
If you make $50 in 5 hr, then at what rate are you earning money?......!!!!!!!!...................................
RESPONSE --> $50 / 5 hr = 10 dollars/hr
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22:35:53 The rate at which you are earning money is the number of dollars per hour you are earning. You are earning money at the rate of 50 dollars / (5 hours) = 10 dollars / hour. It is very likely that you immediately came up with the $10 / hour because almosteveryone is familiar with the concept of the pay rate, the number of dollars per hour. Note carefully that the pay rate is found by dividing the quantity earned by the time required to earn it. Time rates in general are found by dividing an accumulated quantity by the time required to accumulate it.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK.
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22:38:02 `q003.If you make $60,000 per year then how much do you make per month?
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RESPONSE --> $60,000 yr / (12 mn/yr) = 5000 dollars/mn
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22:38:20 Most people will very quickly see that we need to divide $60,000 by 12 months, giving us 60,000 dollars / (12 months) = 5000 dollars / month. Note that again we have found a time rate, dividing the accumulated quantity by the time required to accumulate it.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK.
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22:39:01 `q004. Suppose that the $60,000 is made in a year by a small business. Would be more appropriate to say that the business makes $5000 per month, or that the business makes an average of $5000 per month?
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RESPONSE --> The business makes an average of $5000 per month.
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22:39:20 Small businesses do not usually make the same amount of money every month. The amount made depends on the demand for the services or commodities provided by the business, and there are often seasonal fluctuations in addition to other market fluctuations. It is almost certain that a small business making $60,000 per year will make more than $5000 in some months and less than $5000 in others. Therefore it is much more appropriate to say that the business makes and average of $5000 per month.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK.
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22:40:49 `q005. If you travel 300 miles in 6 hours, at what average rate are you covering distance, and why do we say average rate instead of just plain rate?
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RESPONSE --> 300 miles / 6 hrs = 50 miles/hr
We say average rate because some hours we may travel more than 50 miles and some we may travel less than 50 miles..................................................
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22:41:42 The average rate is 50 miles per hour, or 50 miles / hour. This is obtained by dividing the accumulated quantity, the 300 miles, by the time required to accumulate it, obtaining ave rate = 300 miles / ( 6 hours) = 50 miles / hour. Note that the rate at which distance is covered is called speed. The car has an average speed of 50 miles/hour. We say 'average rate' in this case because it is almost certain that slight changes in pressure on the accelerator, traffic conditions and other factors ensure that the speed will sometimes be greater than 50 miles/hour and sometimes less than 50 miles/hour; the 50 miles/hour we obtain from the given information is clearly and overall average of the velocities.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK.
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22:42:42 `q006. If you use 60 gallons of gasoline on a 1200 mile trip, then at what average rate are you using gasoline, with respect to miles traveled?
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RESPONSE --> 1200 miles / 60 gal = 20 miles/gal
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22:45:10 The rate of change of one quantity with respect to another is the change in the first quantity, divided by the change in the second. As in previous examples, we found the rate at which money was made with respect to time by dividing the amount of money made by the time required to make it.
By analogy, the rate at which we use fuel with respect to miles traveled is the change in the amount of fuel divided by the number of miles traveled. In this case we use 60 gallons of fuel in 1200 miles, so the average rate it 60 gal / (1200 miles) = .05 gallons / mile. Note that this question didn't ask for miles per gallon. Miles per gallon is an appropriate and common calculation, but it measures the rate at which miles are covered with respect to the amount of fuel used. Be sure you see the difference. Note that in this problem we again have here an example of a rate, but unlike previous instances this rate is not calculated with respect to time. This rate is calculated with respect to the amount of fuel used. We divide the accumulated quantity, in this case miles, by the amount of fuel required to cover those miles. Note that again we call the result of this problem an average rate because there are always at least subtle differences in driving conditions that require the use of more fuel on some miles than on others. It's very important to understand the phrase 'with respect to'. Whether the calculation makes sense or not, it is defined by the order of the terms. In this case gallons / mile tells you how many gallons you are burning, on the average, per mile. This concept is not as familiar as miles / gallon, but except for familiarity it's technically no more difficult. You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> I solved for miles/gallon instead of gallons/mile. I do understand the difference and see where I misread the question.
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22:46:32 `q007. The word 'average' generally connotes something like adding two quantities and dividing by 2, or adding several quantities and dividing by the number of quantities we added. Why is it that we are calculating average rates but we aren't adding anything?
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RESPONSE --> Because we do not have the quantity for each measure of time, but instead we have the total amount and the time taken to reach it.
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22:48:39 The word 'average' in the context of the dollars / month, miles / gallon types of questions we have been answering was used because we expect that in different months different amounts were earned, or that over different parts of the trip the gas mileage might have varied, but that if we knew all the individual quantities (e.g., the dollars earned each month, the number of gallons used with each mile) and averaged them in the usual manner, we would get the .05 gallons / mile, or the $5000 / month. In a sense we have already added up all the dollars earned in each month, or the miles traveled on each gallon, and we have obtained the total $60,000 or 1200 miles. Thus when we divide by the number of months or the number of gallons, we are in fact calculating an average rate.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK.
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22:57:24 `q008. In a study of how lifting strength is influenced by various ways of training, a study group was divided into 2 subgroups of equally matched individuals. The first group did 10 pushups per day for a year and the second group did 50 pushups per day for year. At the end of the year to lifting strength of the first group averaged 147 pounds, while that of the second group averaged 162 pounds. At what average rate did lifting strength increase per daily pushup?
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RESPONSE --> Group 1 10 pushups/day * 365 days/yr = 3,650 pushups/yr
Group 2 50 pushups/day * 365 days/yr = 18,250 pushups/yr (Group 1 pushups/yr) + (Group 2 pushups/yr) 3,650 + 18,250 = 21,900 pushups/yr (Group 1 average strenth) + (Group 2 average strenth) 147 lbs + 162 lbs = 309 lbs 309 lbs / 21,900 pushups = 0.01 lb/pushup.................................................
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23:00:09 The second group had 15 pounds more lifting strength as a result of doing 40 more daily pushups than the first. The desired rate is therefore 15 pounds / 40 pushups = .375 pounds / pushup.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> I did not understand the question when I first solved this problem (I thought I did). I do understand that the lbs/pushup was found by dividing the difference in lifting strength by extra pushup performed.
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23:02:26 `q009. In another part of the study, participants all did 30 pushups per day, but one group did pushups with a 10-pound weight on their shoulders while the other used a 30-pound weight. At the end of the study, the first group had an average lifting strength of 171 pounds, while the second had an average lifting strength of 188 pounds. At what average rate did lifting strength increase with respect to the added shoulder weight?
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RESPONSE --> 188 lbs - 171 lbs = 17 lbs (difference if strength) 30 lbs - 10 lbs = 20 lbs (difference in shoulder weight)
17 lbs / 20 lbs = 0.85.................................................
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23:02:55 The difference in lifting strength was 17 pounds, as a result of a 20 pound difference in added weight. The average rate at which strength increases with respect added weight would therefore be 17 lifting pounds / (20 added pounds) = .85 lifting pounds / added pound. The strength advantage was .85 lifting pounds per pound of added weight, on the average.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK.
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23:05:56 `q010. During a race, a runner passes the 100-meter mark 12 seconds after the start and the 200-meter mark 22 seconds after the start. At what average rate was the runner covering distance between those two positions?
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RESPONSE --> 100 m / 12 sec = 8.3 m/sec for the first mark 200 m / (22 sec - 12 sec) = 20 m/sec for the first to second mark.
200 m / 22 sec = 9.1 m/sec from start to 200 mark..................................................
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23:07:28 The runner traveled 100 meters between the two positions, and required 10 seconds to do so. The average rate at which the runner was covering distance was therefore 100 meters / (10 seconds) = 10 meters / second. Again this is an average rate; at different positions in his stride the runner would clearly be traveling at slightly different speeds.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> 100 m / 12 sec = 8.3 m/sec for the first mark 200 m / (22 sec - 12 sec) = 20 m/sec for the first to second mark.
I see my error (and it was a careless error). I should have used 100 meters in the second calculation instead of 200 meters..................................................
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23:10:53 `q011. During a race, a runner passes the 100-meter mark moving at 10 meters / second, and the 200-meter mark moving at 9 meters / second. What is your best estimate of how long it takes the runner to cover the 100 meter distance?
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RESPONSE --> First mark 100 m / 10 m/sec = 10 sec
Second mark 200 m / 9 m/sec = 22.2 sec 22.2 sec - 10 sec = 12.2 seconds from the 100 mark to the 200 mark..................................................
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23:13:42 At 10 meters/sec, the runner would require 10 seconds to travel 100 meters. However the runner seems to be slowing, and will therefore require more than 10 seconds to travel the 100 meters. We don't know what the runner's average speed is, we only know that it goes from 10 m/s to 9 m/s. The simplest estimate we could make would be that the average speed is the average of 10 m/s and 9 m/s, or (10 m/s + 9 m/s ) / 2 = 9.5 m/s. Taking this approximation as the average rate, the time required to travel 100 meters will be (100 meters) / (9.5 m/s) = 10.5 sec, approx.. Note that simply averaging the 10 m/s and the 9 m/s might not be the best way to approximate the average rate--for example we if we knew enough about the situation we might expect that this runner would maintain the 10 m/s for most of the remaining 100 meters, and simply tire during the last few seconds. However we were not given this information, and we don't add extraneous assumptions without good cause. So the approximation we used here is pretty close to the best we can do with the given information.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> I understood the information to be that when he reached the 200 mark he was averaging 9 m/sec from start to that point. But seeing that you meant he was traveling 9 m/sec from the 100 Mark to the 200 Mark I understand your answer.
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23:14:16 `q012. We just averaged two quantities, adding them in dividing by 2, to find an average rate. We didn't do that before. Why we do it now?
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RESPONSE --> Because we are given the individual quantities.
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23:14:36 In previous examples the quantities weren't rates. We were given the amount of change of some accumulating quantity, and the change in time or in some other quantity on which the first was dependent (e.g., dollars and months, miles and gallons). Here we are given 2 rates, 10 m/s and 9 m/s, in a situation where we need an average rate in order to answer a question. Within this context, averaging the 2 rates was an appropriate tactic.
You need to make note of anything in the given solution that you didn't understand when you solved the problem. If new ideas have been introduced in the solution, you need to note them. If you notice an error in your own thinking then you need to note that. In your own words, explain anything you didn't already understand and save your response as Notes.......!!!!!!!!...................................
RESPONSE --> OK.
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