course Mth 163
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12:26:30 `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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12:29:12 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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RESPONSE --> Ok
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12:31:22 In your own words briefly summarize the instructions and the intent of this exercise.
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RESPONSE --> This exercise is meant to train me to visualize and reason out a problem, so thjat i might be able to communicate my thoughts in better detail with you. I am supposed to draw a picture based on the problem and then describe what i have drawn. I am to explain my use of any mathematics used, ie addition, subtraction, multiplying and division.
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12:35:56 `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 divide 50 dollars per hour by 5 hours total to give me the average per hour. The answer is ten dollars per hour. The picture i vizualised was a rectangle divided into 5 blocks. than I divided the totsl 50 into 1/5 of the total time.
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12:39:40 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 divide 300 total miles by the time 6 hours to reach the average of 20 miles per hour. I visualized this by thinking about a pie divided into 6 sections. I put the total miles, 300 and multiplied by 1/6, which is 1 hour. This gave me 20 miles per hour.
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12:42:27 `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 divided 40 total centimeters by 5 seconds to get 8 cent. per sec. I visualized this by drawing a grooved line, or V and dividing it into 5 sections. then I multiplied one section, or 1/5 by 40 total centimeters to get 8 cent per. sec.
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12:43:45 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 --> rate is when you divide the total quantity by the time it takes to change.
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12:46:19 `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 visualized a clock and I put $15 in it. I thought if i make $15 in 1 hour i should multiply by 6 to see what I would make in 6 of these clocks. the answer is $90.
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12:48:32 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 --> Same concept. I thought about a clock and I put 60 miles in it. If I go 60 miles in 1 clock, then I should mulitply by 9 to see how far I will go in 9 clocks. The answer is 60 miles * 9 hours = 540 miles.
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12:50:36 `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 thought of a line. I visualized one second being 13 centimeters, then i multiplied by three to see how long my line would be when more 13 centimeter sections were added. The answer is 39 centimeters.
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12:52:43 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 --> I took the quantity of something at a certain the rate at which it occurred and multiplied the # of times that quantity ocurred based the length of rate or time.
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12:57:16 `q005. How long does it take to earn 100 dollars at an average rate of 4 dollars per hour?
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RESPONSE --> Divide 100 by 4 to equal 25 hours. I visualized 100 dollar bills and 1 clock. I put $4 in that clock. I did this until all of my money was gone. I had 25 clocks.
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12:59:56 How long does it take to travel 500 miles at an average rate of 25 miles per hour?
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RESPONSE --> Again divide 500 miles by 25 miles per hour to get 20 hours. I thought about a clock. I put 25 miles in that clock. I drew clocks until I had enough to equal 500. Then I added the clocks, there were 20.
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13:01:36 `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 --> 80/16=5 seconds. I drew a line that represented 16 centimeters. I continued to draw lines until I reached 80. Then I added the lines. There were 5 lines.
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13:02:56 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 --> I am still trying to take a given quantity and a given time and figuring out a section of it and how that section adds up to the average rate.
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