POW #4: Dave Corner Sells Some Corn Dogs
The “Dave Corner Sells Some Corn Dogs” problem covered proportional and fractional concepts, advanced subtraction, and finding the values of different variables, as well as factoring. By completing the mini POW, which was a problem that cohered with the bigger picture, I was able to set up a ratio in order to transfer that information to the main POW. At first, I struggled to understand the pertinence between both activities – but as soon as I figured out the mini POW – I received a confidence boost and was able to start to solve the “Dave Corner Sells Some Corn Dogs” POW.
Part 1: Problem Statement
In this problem, I had to find how many corn dogs Dave could bring to the market while taking the parameters under consideration. Dave had 3000 corn dogs, and the market was 1000 miles away. However, Dave could not carry more than 1000 corn dogs at a time. In addition, Dave could not resist his product, and had to eat 1 corn dog for every mile that he walked. This was the case for walking both to and from the market. He had to have enough corn dogs to make it to the market without running out, but he also needed the right amount of corn dogs to eat on the way back.
Part 2: Process
To start on this task, I quickly found that simple back and forth trips would not work out. For example, if Dave were to take 1000 corn dogs to the end, he would have none left, because he would have eaten them all. If Dave were to take 1000 corn dogs to the halfway mark, he would have ran out on the way back to the start. This is where the mini-POW of “Dave Corner” Sells Some Corn Dogs” came in handy. Instead of having 3000 total corn dogs, he now had 45 corn dogs, and instead of the market being 1000 miles away, it was now just 15 miles away. He was also limited to carrying just 15 corn dogs at a time instead of 1000 of them.
For the mini-POW, I first needed to figure out the distances in which Dave would move to transport corn dogs from one place to another. After much trial and error, I discovered that simply guessing to find the appropriate mileage that Dave would walk would simply not cut it. To shed some light on the situation, I factored both of the numbers given in the problem, which were 15 and 45. This could help me to find commonalities between the two values. As depicted below, both numbers had factors of 5 and 3 once completely broken down.
Part 1: Problem Statement
In this problem, I had to find how many corn dogs Dave could bring to the market while taking the parameters under consideration. Dave had 3000 corn dogs, and the market was 1000 miles away. However, Dave could not carry more than 1000 corn dogs at a time. In addition, Dave could not resist his product, and had to eat 1 corn dog for every mile that he walked. This was the case for walking both to and from the market. He had to have enough corn dogs to make it to the market without running out, but he also needed the right amount of corn dogs to eat on the way back.
Part 2: Process
To start on this task, I quickly found that simple back and forth trips would not work out. For example, if Dave were to take 1000 corn dogs to the end, he would have none left, because he would have eaten them all. If Dave were to take 1000 corn dogs to the halfway mark, he would have ran out on the way back to the start. This is where the mini-POW of “Dave Corner” Sells Some Corn Dogs” came in handy. Instead of having 3000 total corn dogs, he now had 45 corn dogs, and instead of the market being 1000 miles away, it was now just 15 miles away. He was also limited to carrying just 15 corn dogs at a time instead of 1000 of them.
For the mini-POW, I first needed to figure out the distances in which Dave would move to transport corn dogs from one place to another. After much trial and error, I discovered that simply guessing to find the appropriate mileage that Dave would walk would simply not cut it. To shed some light on the situation, I factored both of the numbers given in the problem, which were 15 and 45. This could help me to find commonalities between the two values. As depicted below, both numbers had factors of 5 and 3 once completely broken down.
Once I had found the factors of both numbers given in the problem, I could proceed with the steps that I needed to take towards the correct answer. I started off in a cautious and careful manner, using the three mile marker as my “home base” of sorts. If Dave were to move five miles at the beginning, he would be limiting the number of corn dogs he would have further on. My first move for Dave was having him travel to the three mile marker while carrying fifteen corn dogs. Since Dave loses a corn dog for every mile he walks, he lost three on the way there, which left him with twelve corn dogs at the three mile marker. Dave also needs enough corn dogs to cover for his hunger on the way back, which means that he would lose another three on the way back. Because of the “round trip” scenario that he faced, I had Dave drop nine corn dogs at the three mile marker so that he would have enough to walk there and back. This was a really important concept to grasp for the purposes of understanding the the problem down the line.
Next, I had Dave take another fifteen corn dogs to the three mile marker like he had previously. Again, this gave him twelve corn dogs after three miles. On the contrary, he now had nine corn dogs at the three mile marker that he could take with him, as opposed to not having any. Thanks to this extra supply, Dave picked up three corn dogs from the pile of nine, which gave him a fresh set of fifteen corn dogs to move forward with. As one might remember from the diagram above with the factor trees, the two numbers not only had a factor of three, but there was also a factor of five for both values. Therefore, I used it to my advantage, and had Dave walk five miles from the three mile marker to the eight mile marker. With this move, Dave lost five corn dogs, which left him with ten corn dogs at the eight mile marker. However, he would need enough to get him back to where he came from (in this case the three mile marker). I decided to maximize the number of corn dogs that Dave would have later on, so he dropped five products at the eight mile marker and returned to the three mile marker. Since he had started this specific trip with fifteen corn dogs, he had just used the five remaining corn dogs to help him leave the eight mile marker for a five mile journey.
The next step for Dave was to recharge and refresh his supply of corn dogs. To achieve this, I had him take three corn dogs from the pile of dropped corn dogs at the three mile marker and return to the beginning, or zero. He now had the opportunity to take his last batch of fifteen corn dogs, which he did. Dave travelled to the three mile marker once more, lost three corn dogs along the way, and picked up three corn dogs from the dropped pile. This exhausted the dropped pile, and he no longer had any corn dogs at the three mile marker. With the fifteen corn dogs that he was carrying, Dave walked five miles to the eight mile marker, losing five along the way. This gave him ten corn dogs at eight miles. Just then, Dave noticed the pile of five corn dogs that he had dropped at the eight mile marker from earlier. He grabbed these corn dogs, which now gave him fifteen corn dogs. (He was carrying ten corn dogs and picked up five corn dogs.) From here, his adventure was all but over. All he had to do was reach the market with the remaining corn dogs. Travelling seven miles from the eight mile marker to the fifteen mile marker, Dave lost seven corn dogs, and arrived at the market with 8 corn dogs left.
Part 3: Solution
Now the process described above pertained to the information given in the mini-POW. But was it necessarily irrelevant to the main POW? Not at all. Just like the smaller example, the ratio of corn dogs that Dave could carry to the total number of corn dogs that Dave had in the main POW was 1:3, as illustrated below:
Next, I had Dave take another fifteen corn dogs to the three mile marker like he had previously. Again, this gave him twelve corn dogs after three miles. On the contrary, he now had nine corn dogs at the three mile marker that he could take with him, as opposed to not having any. Thanks to this extra supply, Dave picked up three corn dogs from the pile of nine, which gave him a fresh set of fifteen corn dogs to move forward with. As one might remember from the diagram above with the factor trees, the two numbers not only had a factor of three, but there was also a factor of five for both values. Therefore, I used it to my advantage, and had Dave walk five miles from the three mile marker to the eight mile marker. With this move, Dave lost five corn dogs, which left him with ten corn dogs at the eight mile marker. However, he would need enough to get him back to where he came from (in this case the three mile marker). I decided to maximize the number of corn dogs that Dave would have later on, so he dropped five products at the eight mile marker and returned to the three mile marker. Since he had started this specific trip with fifteen corn dogs, he had just used the five remaining corn dogs to help him leave the eight mile marker for a five mile journey.
The next step for Dave was to recharge and refresh his supply of corn dogs. To achieve this, I had him take three corn dogs from the pile of dropped corn dogs at the three mile marker and return to the beginning, or zero. He now had the opportunity to take his last batch of fifteen corn dogs, which he did. Dave travelled to the three mile marker once more, lost three corn dogs along the way, and picked up three corn dogs from the dropped pile. This exhausted the dropped pile, and he no longer had any corn dogs at the three mile marker. With the fifteen corn dogs that he was carrying, Dave walked five miles to the eight mile marker, losing five along the way. This gave him ten corn dogs at eight miles. Just then, Dave noticed the pile of five corn dogs that he had dropped at the eight mile marker from earlier. He grabbed these corn dogs, which now gave him fifteen corn dogs. (He was carrying ten corn dogs and picked up five corn dogs.) From here, his adventure was all but over. All he had to do was reach the market with the remaining corn dogs. Travelling seven miles from the eight mile marker to the fifteen mile marker, Dave lost seven corn dogs, and arrived at the market with 8 corn dogs left.
Part 3: Solution
Now the process described above pertained to the information given in the mini-POW. But was it necessarily irrelevant to the main POW? Not at all. Just like the smaller example, the ratio of corn dogs that Dave could carry to the total number of corn dogs that Dave had in the main POW was 1:3, as illustrated below:
Using this information, it would be correct to say that the other number and values involved in both problems could be connected, too. To prove this, I set up a proportion between the number of miles to the market and the final answer for the two POWs.
Technically, I had my final answer right here for the main POW: 533 1/3 corn dogs. But I needed a way to prove it. Therefore, I replicated the steps that I took for the mini-POW in terms of the information given in the main POW using the ratio I had discovered, and checked to see if I had to correct answer. I knew that some of the numbers would be fractional, since the values were being translated to a larger scale:
Finally, to further prove the relationship between the mini-POW and the “Dave Corner Sells Some Corn Dogs” POW, refer to the two diagrams below that display the trips taken on a number line:
Both images show the similarities in trips that Dave had to take in order to reach the market, and are scaled according to the ratio I developed earlier. The arrows are placed on the number line based on the order of the steps I took in overlapping fashion to signify repetition.
Part 4: Extensions
Reflection
The “Dave Corner Sells Some Corn Dogs” problem was crucial in my ability to understand concepts surrounding factoring, proportions, and fractions. It was important for me to use visualizations along with written explanations for the steps that went into solving the problem, so that I could see the work being done in a variety of ways. If I had not had a way to conceptualize my ideas, I definitely would not have retained as much knowledge regarding the material related to the POWs. In addition, I learned how to set up and apply proper ratios to two elaborate problems, and how breaking a problem down could help to build my skills and mastery of another one. I laid out in-depth instructions on how to find the bigger picture after reviewing the mini-POW, and it worked with the help of a few visual aids and some explanations.
I honestly believe that I deserve a 10 out of 10 for my work on this problem because I delved deep into the content behind the question and the solution. I went into detail about how I found the correct answer for the mini-POW, and then used that information to explain how that correlated with the main POW. I incorporated drawings, factor trees, complex instructions, and a comprehensive examination of the problems provided, and I do not think I could have completed a more exhaustive analysis of the work I did. I expanded on basic ideas with more complicated ones, and I truly understood the math concepts presented in both POWs. Thanks to patience, effort and hard mathematical thinking, I found the correct answer(s) after a series of lengthy steps and proofs.
The most prominent Habit of a Mathematician shown in my work was Seek Why & Prove. Sure, I also utilized the trait of starting small, but that was a part of the problem that I was instructed to include. For my work, I came up with a way to merge the information given in two separate ways in an efficient and effective manner, but I needed to make sure that I was correct. From proportions to diagrams to mathematical calculations, I tried to use various methods in order to explain my process and solution(s), and I feel that I was able to do that. It is easy to state an answer without any proof or reasoning behind it, but it is difficult to take that extra step and thoroughly analyze the work behind that answer. I am very happy with what I was able to achieve for both the mini-POW and the regular POW, and how I was able to tie both problems together in a clean, precise way.
Part 4: Extensions
- Simple: If Dave had a total of 60 corn dogs and the market was 20 miles away, how many corn dogs could he get to the market?
- Moderate: If Dave had a total of 4200 corn dogs and the market was 1400 miles away, how many corn dogs could he get to the market?
- Difficult: If Dave could only eat or carry whole corn dogs, (without fractions), what is the greatest number of corn dogs he could get to the market?
Reflection
The “Dave Corner Sells Some Corn Dogs” problem was crucial in my ability to understand concepts surrounding factoring, proportions, and fractions. It was important for me to use visualizations along with written explanations for the steps that went into solving the problem, so that I could see the work being done in a variety of ways. If I had not had a way to conceptualize my ideas, I definitely would not have retained as much knowledge regarding the material related to the POWs. In addition, I learned how to set up and apply proper ratios to two elaborate problems, and how breaking a problem down could help to build my skills and mastery of another one. I laid out in-depth instructions on how to find the bigger picture after reviewing the mini-POW, and it worked with the help of a few visual aids and some explanations.
I honestly believe that I deserve a 10 out of 10 for my work on this problem because I delved deep into the content behind the question and the solution. I went into detail about how I found the correct answer for the mini-POW, and then used that information to explain how that correlated with the main POW. I incorporated drawings, factor trees, complex instructions, and a comprehensive examination of the problems provided, and I do not think I could have completed a more exhaustive analysis of the work I did. I expanded on basic ideas with more complicated ones, and I truly understood the math concepts presented in both POWs. Thanks to patience, effort and hard mathematical thinking, I found the correct answer(s) after a series of lengthy steps and proofs.
The most prominent Habit of a Mathematician shown in my work was Seek Why & Prove. Sure, I also utilized the trait of starting small, but that was a part of the problem that I was instructed to include. For my work, I came up with a way to merge the information given in two separate ways in an efficient and effective manner, but I needed to make sure that I was correct. From proportions to diagrams to mathematical calculations, I tried to use various methods in order to explain my process and solution(s), and I feel that I was able to do that. It is easy to state an answer without any proof or reasoning behind it, but it is difficult to take that extra step and thoroughly analyze the work behind that answer. I am very happy with what I was able to achieve for both the mini-POW and the regular POW, and how I was able to tie both problems together in a clean, precise way.