A puzzle from Sunzi Suanjing:
"今有雉、兔同笼，上有三十五头，下九十四足。问雉、兔各几何？"
Now we have chickens and rabbits in the same cage. There are 35 heads and 94 feet in total. How many chickens and rabbits do we have?
The solution from the book:
"上置三十五头，下置九十四足。半其足，得四十七。以少减多。"
Extensions:
What pattern can you find from the solutions?
What happens if we want to measure whole-number amounts from 1 to 41 grams? Do we need to have more weights?
I got into the state of “flow” when solving differential equations for 400 level math. I did not realize that the answer was almost 5 pages long before I finished.
To promote “flow” and happiness in math classes, I believe the top priority is to provide students with enough “weapons” such as mathematical thinking skills, calculation tricks and history of mathematics etc. My SA always teaches students how to solve the question in more than one way. When dealing with slopes, she not only showed the basic idea of “rise over run” to grade 9s but also “short-cut” for linear equations in standard form. She told students that this new method could be learned in Math 10 and complimented them that they could handle difficult problems for higher-level math. In this way, students build their confidence in math and reach the “flow” as they are equipped with skills and are able to solve challenging questions.
I think introducing puzzle-solving is a good activity for students to reach the “flow” of math. I was lucky to observe another math teacher’s class in the school. The topic was introducing a puzzle named nonograms to grade 8s. It was amazing that how well students follow the teacher and they even asked for more practice. Also, every time they finished a graph together, students would use their imagination and express what the graph looked like. I am more than willing to try this kind of activity in my own teaching later in the long practicum.
Given that the soup can has a diameter of 8cm and a height of 10cm and the height of the bike in the photo is 80cm. The diameter of the water tank is approximately three times the height of the bike, which is 240cm. Then use proportional reasoning to determine the height of the water tank: 240/8=x/10 and thus it is 300cm. As the water tank is a cylinder, the volume follows the formula: V=pi r^2 h. The radius for a circle is half of its diameter, so r=240/2=120cm. Therefore, the volume is pi*120^2*300 and the approximate result is 13571680 cm^3. The third part of the question requires research. According to a 15-year study in Scottsdale Report, 2935 gallons of water on average for firefighters to put out a house fire. To compare these two numbers, the same units for them are needed. Students could choose which unit to be used and personally, I decided to convert these two numbers into the unit of m^3 and keep two decimals. 13571680 cm^3 is around 13.57 m^3 and 2935 gallons is about 13.35 m^3. Thus, the answer is that yes it holds enough water!
I think there are three challenging points in this problem for students. The first one is how to make realistic assumptions. Then, they need to figure out which information is useful to them when determining the dimensions of the water tank. The last part is unifying the units. This kind of question is closely related to students' daily life so they could be more interested in and engaged in the problem-solving process. Also, the problem helps them to develop their mathematical thinking and gives them a sense that math is actually more useful than you think!
An extension to this puzzle is asking the following question: due to weathering, the water tank is out of the right cylinder shape, does it affect the volume? Students need to come up with assumptions of the possible shapes after weathering. For example, it could turn to be an oblique cylinder.
Solving without algebra:
The first step is to find a number that has factors 2, 3 and 4. The smallest number that meets the condition, i.e the least common multiple, is 12. Thus, the solution must be a multiple of 12. The possible solutions can be 12, 24, 36, 48, 60, 72 etc. The next step is to figure out which one meets the other condition: the total number of dishes is 65. By trials and errors, we can find 60 is the final answer as 60/2+60/3+60/4=65.
Math problems, puzzles and histories from diverse cultures foster students’ interests in math. Since these questions could have different approaches, students can practice their mathematical thinkings and have deeper understandings of the development of math. When students from other ethnic groups find that the math questions are from their cultures, they could be more excited and actively engaged in the class. During the interactions, other students could also benefit from the problem-solving process.
Word problems, puzzle stories and imageries play important roles in math learning and teaching. Students could develop essential skills such as understanding the backgrounds of problems, collecting useful information, visualizing problems and utilizing different approaches to solve problems. Moreover, they could feel connections to the ancient time and appreciate the efforts and wisdom of ancient people. By successfully solving these kinds of problems, they could gain more fulfillment and joy than figuring out the solutions to boring equations.
First, I had a sample of 10 lockers and 10 students and tried to observe some patterns. I drew the line that separated the final states and changing states. I also underlined numbers where the change of states happened.
Since there seemed to be no obvious patterns, I turned to collect information on the total number of open and closed lockers for each locker and each student. Unfortunately, I still could not see any special.
Then, I focused on the last row and surprisingly found that the only lockers that were closed are perfect square number lockers. However, this finding should be rationally supported before I made a conclusion.
Thus, I looked back to the sample of 10. I counted the numbers of changes in states for each locker and realized that they are the numbers of factors for each locker number.
Finally, I double-checked my solution with a sample of 20 that I made in a spreadsheet. It matched what I had got!
Reflection: Although I underlined numbers that change occurred, I did not get it in the first place since I forgot to include the initial states of lockers.
