Yu Yanshan of Liutang Primary School

Some time ago, I saw an article in the fourth issue of "Primary School Mathematics Teachers", "Teachers should explain the wrong questions and listen to the wrong reasons", which involved such a question:

For a cup of sugar water, the ratio of sugar to water is 1: 20. After drinking half, the ratio of sugar to water is ().

A, 1: 20b, 1: 10 C, uncertain.

This paper emphasizes that teachers should pay more attention to students' mistakes when explaining wrong questions. The teacher is still listening to the cause of the mistake and has got an explanation that can make the students understand the problem better. But I am more concerned about the problem itself. The problem of "sugar and syrup" and "salt and brine" is also encountered in the content of "Percentage" taught by Beijing Normal University Edition this semester, and it appears very frequently. Unfortunately, students make mistakes again and again, and few children can really understand this topic. Some "sharp-eyed" children will remember the method when they appear many times. Although the result is correct, their understanding is still not in place. The topic is as follows:

After 20g of sugar is completely dissolved in100g of water, sugar accounts for ()% of sugar water.

Misunderstanding: 20 ÷ 100 = 20%

Positive solution: 20 ÷ (100+20) ≈16.7%

For adults, this problem is easy to understand, but for fifth-grade primary school students, it is a "difficult problem". This kind of problem has appeared many times, whether in homework or in exams, but the effect is not ideal. What happens is that students often make mistakes gracefully, teachers speak gracefully, and students still speak gracefully.

I reflected on this situation and listened carefully to the students' ideas, summed up the following reasons and tried to explain it more simply.

1, students are not sure about the whole 1. What percentage of sugar is in syrup? The sugar water here is the whole "1". And sugar water = sugar+water = 20+ 100 = 120g. In addition, the data in the topic is quite special. When sugar is dissolved in 100g of water, it is easy to induce students to calculate 20 ÷ 100 = 20% immediately, without considering its rationality deeply.

2. About this whole "1" sugar water = sugar+water = 20+ 100 = 120g, the students' understanding is also vague. Because it involves chemical concepts such as solution, solvent and solute (now it is science). But I think there is no need to explain these three concepts here. If you explain it, students may not understand it, but it will be more vague. Therefore, there is no need to have these three concepts that are more professional and can't help students understand. So how to understand it? Although the concept has not been established, I have some life experience. Sugar is completely dissolved in water and becomes syrup. First of all, understand the complete solvent, which means uniform and average (to avoid the phenomenon that students encounter sweeter sugar water in their lives). In other words, the first sip of a cup of sugar water tastes the same as the last sip, because it is completely dissolved and has an average sugar content. Then, the whole "1" is understood. Because sugar dissolves in water and becomes syrup, syrup contains sugar and water. The mathematical formula shows that syrup = sugar+water. As a whole, "1" is composed of sugar and water, so the whole "1"= 20+100 = 65400.

I think the above two explanations can better help students understand the problems of sugar and syrup, salt and brine. Back to the proportion of sugar and water, because the proportion is the content of grade six in Beijing Normal University, students have already learned the relevant knowledge of percentage when learning this content, so we can understand the proportion problem with the help of the knowledge of percentage. The ratio of sugar to water is 1: 20. Assume that 1 g sugar is completely dissolved in 20 g water, and sugar accounts for 4.8% of syrup. Because it is completely dissolved, the percentage of sugar in syrup remains unchanged after drinking half, in other words, the ratio of sugar to water remains unchanged. Understanding new knowledge with what you have learned can get twice the result with half the effort, but the premise is to master old knowledge comprehensively.

These are my views on sugar and syrup. Please criticize and correct me.