On July 12, 2006, the author had the honor to participate in the marking of Xiaoshengchu research in Suzhou Lida School, and corrected the fifth item of the fourth major question in the test paper. The original title is as follows: Qingfeng Beverage Company produced a granular orange beverage, which was cylindrical with a bottom radius of 4 cm and a height of 12 cm. The company wants to design a rectangular box that can hold 8 cans of this beverage, and try to save wrapping paper. What is the surface area of the packing box you designed?
After carefully reading, doing and marking the examination paper, the following are the author's thoughts on this topic:
I. Re-understanding of this issue
Mathematics comes from practice. Engels said: "The concepts of number and shape in mathematics come from the real world, not from anywhere else." Friedenthal, an internationally renowned mathematics educator, also believes that the concept, structure and thought of mathematics are the reflection of various concrete phenomena in the physical world, social existence and thinking world, and are also the tools to organize these phenomena. Therefore, applying mathematics in real life will make mathematics learning vivid. This is also the requirement of the latest version of mathematics curriculum standard for full-time compulsory education. The topic is based on the design of beverage packaging box, which is close to students' real life. And in the process of exploring this topic, students can feel that mathematics is not only boring addition, subtraction, multiplication and division, but also exists in real life and is valuable. Therefore, this question is in line with the spirit of the new curriculum reform, and it is a good question that can examine students' mathematical thinking ability and problem-solving ability.
Second, what comes from solving problems?
If we only consider this question from the perspective of designing a rectangular box that can hold 8 cans of drinks, the answer is not unique. Different placement methods will produce different surface areas. The solution is as follows:
Answer 1.
The length, width and height of a cuboid are 4, 4 and 96 respectively, so the surface area is
Answer 2.
The length, width and height of a cuboid are 32,4 and 12 respectively, so the surface area is
Answer 3.
The length, width and height of a cuboid are 8, 4 and 48 respectively, so the surface area is
Answer 4.
The length, width and height of a cuboid are 16, 4 and 2 respectively, so the surface area is
Answer 5.
The length, width and height of a cuboid are 16, 8 and 12 respectively, so the surface area is
Answer 6.
The length, width and height of a cuboid are 8, 8 and 24 respectively, so the surface area is
The above six solutions are the thinking methods commonly used by candidates. A candidate proposed the location of solution 7. Although they can't give accurate answers, it is not easy for them to have such profound mathematical thinking. The candidate's innovative solution won the appreciation of President Zhu Lin, who loves talents and talents.
Answer 7.
The cross-sectional view of the cuboid is as follows:
In the above figure, both △AFG and △EFD are equilateral triangles, and △AFE is an isosceles triangle with the vertex angle of 120 and the waist length of 4. According to the calculation, the width of the cuboid is about 1 1, and the volume of the cuboid is about.
In the process of marking papers, some colleagues came up with another answer, as follows:
Answer 8.
The length, width and height of a cuboid are 12, 4 and 32 respectively, so the surface area is
The above only lists some possible designs, but there is also a requirement for this problem, that is, "save packaging paper as much as possible", that is, minimize the surface area of the designed packaging box. The problem of finding the minimum value is generally solved by using the monotonicity of functions or the properties of some basic inequalities in middle schools. It looks a little out of line here, but it's not. You can divide the size by comparison, and all the possibilities should be compared together. The minimum is what you want. This comparative method is actually a basic scientific research method. Therefore, by comparing the above eight schemes, it can be concluded that if the wrapping paper is saved as much as possible, the surface area of the designed packaging box is about 8 16 cm2.
Some readers may feel that it is difficult for candidates to come up with eight different design methods in a limited time, and then calculate and compare them to get the answer. In fact, the above solutions only provide as many design ideas as possible. Do we have a more natural and smooth solution to this problem?
By calculating one or two design schemes that are the easiest to come up with (the first six are common packaging in our life, and are also easy for candidates to think of), we can find that the smaller the gap between the length, width and height of a cuboid, the smaller the surface area of the cuboid. This is an important guess of this problem. On this basis, we naturally think of the packaging method of Scheme 7. Maybe most students can't find out the width of this cuboid because they don't have enough math tools, so you can also standardize the picture before measuring it, or you can get the answer "The width of this cuboid is 1 1cm". This "primitive" method has embarked on the road of discovery by scientists again, which is what we expect.
The standard answer to this question is this. Get the answer 832, get a full score of 8 points; Get the answer 896, get 6 points; Get the answer 1 120, and get 5 points. This standard answer breaks through the previous practice of paying more attention to the test results (answers) and ignoring the thinking process in exam marking. Instead, it focuses on evaluating students' ability to find and solve problems. Suggestions on teaching evaluation under the new curriculum standards.
Combined with the actual answer situation, we changed the standard answer to: 8 16 and 832 can get full marks; Get the answer 896, get 6 points; The answer is 5 points between 1000- 1200; The answer is 4 points between 1200- 1568; Other answers are not scored.
This problem is not only close to students' life, but also a very divergent topic. There is no right answer, only a better answer. Students can carry out as many levels as they have. This type of topic makes us happy to see the endless creativity of students. In addition, it should be said that this problem provides a realistic mathematical situation, which requires students to experience a mathematical activity process of observation, experiment and guess independently. Therefore, we should not only examine students' basic knowledge, but also examine students' mathematical literacy.
Third, the feeling of reading the question
In the process of reading the question, we found that the answer to this question is as follows:
1, most candidates can come up with a packaging plan, but most of them are blind cats and dead mice, so they can answer whichever answer they think of. There is no mathematical process of comparison, analysis and discovery. This illustrates two problems. First, I didn't carefully examine the questions and didn't understand the meaning of "try to save wrapping paper"; Second, when analyzing and solving problems, there is a lack of the most basic or universal problem-solving strategies and methods. I think this also suggests that we should pay attention to cultivating these two abilities in the subsequent mathematics education.
2. This problem should have been calculated on the basis of drawing a good picture combined with graphics. But unfortunately, most candidates only left the calculation process on paper. This not only creates obstacles to the answering process, but also brings great inconvenience to the marking teacher. This reminds us that in the future teaching, we should strive to cultivate students' good habit of combining numbers with shapes.
Fourth, it can be more perfect.
I sat at my desk again and reflected on this question. I found that if I made two small changes, this question could be more perfect. Change 1 to clearly stipulate that the schematic diagram of the design should be drawn in brackets after the question, which is not only beneficial for candidates to find ideas, but also beneficial for marking teachers to mark papers. Change two, you can set three gradient questions like this. Question 1. Would you please design a rectangular box for your company and ask for its surface area? Question 2. If we want to save packaging paper as much as possible, how to design and find its surface area? Question 3. Is there the most economical design? Please draw a schematic diagram (without calculation). This is a simple way to start, and it is also helpful for students to correctly examine questions and divergent thinking.