Generally speaking, the whole cake is divided into four parts, for example, a cake is divided into four parts, where 1 yes 1/4, two parts are 2/4, three parts are 3/4 and four parts are 4/4, which is a complete cake. It is particularly important to note here that the fractional unit obtained by equal division is 1/4, while 2/4 refers to two fractional units: 2/4 = 2× 1/4 = 1/4, and so on. At this time, it involves the comparison and operation of scores, which needs to be divided into two situations.

① The denominators of the two fractions are the same.

The denominator is the same, which means that the decimal units of the two fractions are the same. Just compare two scores directly: 3/4 > 1/4 (three 1/4 must be greater than11/4); For the addition and subtraction of two fractions, it is also easy to get: 3/4- 1/4 = 2/4 (3 decimal units minus 1 decimal unit equals 2 decimal units).

? ② The denominators of the two fractions are different.

Because the denominators of the two fractions are different, it shows that the decimal units of the two fractions are different, so it is necessary to further divide them on the basis of the original decimal units in order to compare the sizes of the two fractions and add and subtract them on the same decimal unit. For example, to compare the sizes of 1/2 and 1/5, you need to divide the cake into two parts and then divide the cake into five parts. For each cake divided into two parts, divide it into five equal parts. For each cake divided into five parts, refer to the figure below. The units obtained are all110 of the original integer, and a new decimal unit appears. For 1/2, the relationship between the original decimal unit and the new unit is 1/2 = 5. For 1/5, the relationship between the original decimal unit and the new unit is 1/5=2/ 10, so the decimal units are consistent and the sizes can be compared: because 5/ 10 > 2/ 10 >.

Since the decimal unit 1/5 is converted into the new decimal unit110, two copies of the original unit are equivalent to four points of the new unit: 2/5 = 2×1/5 = 2× 2//kloc-0 = 4/. It also demonstrates the nature of the fraction: the numerator and denominator of the fraction expand or shrink at the same time by the same multiple, and the size of the fraction remains unchanged. Other addition and subtraction operations of fractions without denominator can be solved by using the properties of general fractions.

? To put it simply, it is what we often say in class: the same denominator scores are directly added and subtracted, and the different denominator scores are divided first and then added and subtracted. (The principle of general division has been discussed above)

Second, the overall proportional relationship.

Fraction can also represent the integer ratio between the quantities of two things, or measure the quantity of another thing by integer multiples based on the quantity of one thing. We use an example to understand:

Here 1/3 is about proportion, and the key to solving the problem is to understand the meaning of 1/3.

Method 1: The number of geese is 65438+ 0/3 of the number of ducks (1 3 of the number of ducks is the number of geese), that is,1goose corresponds to 3 ducks, 2 geese correspond to 6 ducks, 3 geese correspond to 9 ducks and 4 geese correspond to 12 ducks.

Method 2: Backward deduction, the number of geese is 1/3 of that of ducks, and the number of ducks is 1/3 of that of geese, which means that the number of ducks is three times that of geese. The understanding of drawing is as follows.

? Method 3: Set the unknown number X, which is a general mathematical expression that we prefer: use X to represent the number of ducks, and get the proportional relationship of the number of geese and ducks 4:x= 1:3. With the help of this proportional relationship, we can get the desired result through two operation methods, one is the multiplication mentioned above (drawing is easier to understand), and the other is the division expected in the textbook: the number of ducks =4÷ 1/3=4×3= 12, which involves the topic 2/kloc-in the title article. (2) Why multiply the reciprocal?

(1) Why use division?

Many people are confused about this problem, mainly because of scores. Why use scores to score? According to the discussion of division, it is emphasized that division should also be used for the problem of "A is y times of B", and the operation form is: A ÷ B = Y. Because the divisor B and quotient Y are symmetrical in this operation form, the formula can be equivalent to: A ÷ Y = B. According to the latter formula, we can know that we should also use division for the problem of "Knowing A is y times of B and how much B is".

? The problem that the number of geese is 1/3 of the number of ducks is obviously solved by division: the number of ducks = the number of geese1/3, that is, 41/3.

? (2) Why multiply the reciprocal?

? The rule of 4÷ 1/3=4×3= 12 is very important and needs to be remembered, but in teaching, students should also be more or less aware of the topic and try to explain this rule. Personally, I think this rule can be well proved according to "division is the inverse operation of multiplication", as shown in the following figure.

This is a good argument: dividing by a fraction is equal to multiplying the reciprocal of this fraction.