course Phy 201
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10:46:12 `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.
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RESPONSE --> Ok
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10:53:23 06-02-2006 10:53:23 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.
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10:58:12 In your own words briefly summarize the instructions and the intent of this exercise.
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RESPONSE --> The instructions are to draw out a visualization of the problem posed and write out a logically reasoned thought process decribing what was visualized. The intent is to have us be able to think through the problems and ""see"" through our visualizations why the calculations are done a certain way, instead of simply memorizing formulas. Also, this type of exercise aides in our communication skills which is especially essential for clearing up misunderstandings.
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11:09:35 `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 drew 1 square block of time to represent 1 hour, and then drew 5 square blocks of time to represent 5 hours. The 1 hr. block is at a rate of x, but the total of the 5 1-hour blocks is equal to a rate of $50. This clearly shows a division. The five 1-hour blocks must divide the $50 up into equal amounts. $50 / 5 is equal to $10. Therefore, each 1-hour block is at a rate of $10. $10 per hour. This can be checked by adding the $10 five times to make sure the total amount is $50.
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11:22:17 If you travel 300 miles in 6 hours, at what average rate are you traveling, in miles per hour?
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RESPONSE --> Similarly to the last question, I drew one block to represent 1 hour at x miles. I then drew the 1 hour blocks 6 times to show 6 hours. The 6 1-hour blocks sum up to 300 miles. Therefore, the 300 miles must be broken up into 6 equal amounts to determine the number of miles in 1 hour. 300 divided by 6 shows that 50 mile 1-hour blocks add up to 300 miles. The average miles driven per hour is 50 miles. Checking the answer by adding 50 six times (or 50 times 6) gives the total sum of 300 miles.
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11:41:34 `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 drew one block to represent 1 second at an unknown cm traveled. I then drew 5 1-second blocks which can be added together to form a total of 40 cm traveled. Dividing the 40 cm into 5 equal amounts will give the average rate traveled in 1 second. 40 cm / 5 s gives an average of 8 cm traveled per 1 second. To check, 8 cm times 5 1-second blocks (8cm added 5 times) does equal to 40 cm.
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11:55:02 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 --> The idea of rate is a quatity of something (anything) that can be broken up by units of time. The units of time can very by seconds, minutes, hours, days, weeks, years, etc., as long as the unit represents time in some way. We can find the change in rate by breaking the quantity up into equal amounts where one unit of time added together to sum up to the changed unit of time. This shows that if you divide the known quantity up by the unit of time it took to get that quantity, you'll get the average quantity obtained at one unit of time.
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12:00:27 `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 drew one square block to represent 1 hour. 1 hour also represents $15. To determined how much is earned in 6 hours, I drew 6 1-hour blocks. Since the blocks also represent $15 each, we can add each block's dollar amounts up to find the total (so add $15 six times) or multiply $15 by 6 to get $90. Therefore, $90 is earned in 6 hours.
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12:05:25 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 drew one block to represent 1 hour. The same block also represents 60 miles since during one hour we travel 60 miles. To find the number of miles traveled in 9 hours, I drew 9 1-hour blocks which is also 9 60-mile blocks. We can add the 60 miles up 9 times to find the miles or we can multiply 60 miles times 9. 60 miles * 9 equals 540 miles. Therefore, 540 miles is traveled in 9 hours.
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12:10:20 `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 --> One block can be drawn to represent 1 second. The same block also represents 13 centimeters since that is how far the ball travels in one second. To find rate for 3 seconds, 3 blocks can be drawn to represent 1 second each and 13 cm each. We can add the 3 13cm blocks together or simply multiply 3 * 13 cm to figure out how far the ball traveled in 3 seconds. Multiplying 3 * 13 we get a rate of 39 cm traveled per 3 seconds.
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12:16:59 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 --> The idea of rate has not change. It is still some quantity that is divided up into units (or blocks) of time. This concept can be used to determine the average unit per one block of time through division of the quatity by the known change in time, or the concept can be used to determine a change in quanity by using the known quantity of one unit of time and multiplying it by the change in time.
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12:24:34 `q005. How long does it take to earn 100 dollars at an average rate of 4 dollars per hour?
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RESPONSE --> To visualize this, we know that one block may represent 1 hour and also represent $4. We need to know how many blocks does it take to get to $100. We can write out $4 blocks adding them as we go until we get to $100 and count up how many blocks we need to get there, or we can simply divide the $100 by $4 (since $4 times that amount will give us the $100). We get 25 blocks at $4 1-hour each. Therefore, it would take 25 hours to earn $100 at the average rate of $4 per hour.
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12:31:13 How long does it take to travel 500 miles at an average rate of 25 miles per hour?
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RESPONSE --> We can visualize this in the same way as the previous question or we can take one big block to represent 500 miles and break that block down into 25 mile blocks. Then we can count up how many 25 mile blocks the 500 mile block can be broken down to, which clearly shows division so we known we can instead simply divide the 500 miles up by 25 miles. We then see that 500 miles can be divided up into 20 25-mile blocks. Since each 25-mile block also represents one hour, it would take 20 hours (1 hour 20 times) to travel 500 miles.
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12:41:15 `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 --> Again, we can take one block to represent 1 second and also 16 cm and keep adding blocks while adding the new cm amounts as we go until we get to 80 cm, or we can draw one big block representing 80 cm total and break that block down into smaller 16 cm blocks and count how many 16 cm blocks it took to create that big 80 cm block. Or, we can simply divide 80 by 16. Doing any of these will show that it takes 5 16cm blocks to make up 80cm. Since one block is also 1 second, we multiply 5 blocks * 1 second per block to get 5 seconds. Therefore, it takes 5 seconds for a ball rolling at 16cm/s to travel 80cm.
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12:46:22 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 --> These problems expanded the concept of rate because instead of having 2 known time amounts we are asked to find the time, as opposed to the varying quanity. Using a standard of the known amount of some quantity per one unit of time and a known change in the quanity of the variable, we can determine the new time amount by dividing the changed variable quanity by its quanity at one unit of time.
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