The significance of positive proportion

☆ Knowledge points:

(1) ratio: two related quantities, one changes and the other changes. If the ratio (that is, quotient) of the two numbers corresponding to these two quantities is certain, these two quantities are called proportional quantities, and the relationship between them is called proportional relationship. ① Represented by letters: If two related quantities are represented by letters X and Y, and their ratio is represented by K,

(2) Positive proportion is related to the changing law of two related quantities: simultaneous expansion and simultaneous contraction, and the proportion remains unchanged. For example, if the speed of a car is constant, is the distance traveled directly proportional to the time spent?

The above manufacturers are certain, so dividend and divisor represent two related quantities, which are in direct proportion. Note: When judging whether two related quantities are directly proportional, we should pay attention to these two related quantities. Although they are also a quantity, they change with the change of another, but the proportion of the two numbers they correspond to is not necessarily, so they cannot be directly proportional. Such as a person's age and weight. The side length of a square is not proportional to the area. Inverse ratio: two related quantities, one of which changes and the other changes accordingly. If the product of two corresponding numbers is constant, these two quantities are called inverse proportional quantities, and the relationship between them is called inverse proportional relationship. Expressed in letters: two related quantities are "X" and "Y" respectively, and "K" is inversely related to: xy=k (certain) ② The changing law of the two related quantities in inverse relationship is that one quantity expands, the other quantity contracts, the other quantity expands, and the product remains unchanged. For example, if the distance on the map is fixed, is the actual distance inversely proportional to the scale? Because the actual distance × scale = distance on the map (certain), the actual distance is inversely proportional to the scale. 3. Similarity of positive and negative proportions: both quantities are related, one quantity changes and the other quantity changes. Difference: two quantities are in direct proportion, that is, one quantity expands, the other quantity expands, the other quantity shrinks, and the other quantity shrinks. The law of their expansion and contraction is that the ratio of the two numbers corresponding to these two quantities is different. The other quantity shrinks instead, and the other quantity expands instead. Their changing rule is that the corresponding product of two quantities is constant (definite).

☆ Basic exercises:

1. Fill in the blanks ① () There are two quantities, one quantity changes and the other quantity (). If the () of two quantities is fixed, these two quantities are called inverse proportional quantities, and their relationship is called ().

Judge the ratio of the following two quantities and explain the reasons.

Time is fixed, the number of meters woven per hour and the total number of meters woven.

② The parallelogram has a certain area and a high base.

③ The numerator is definite, the denominator and the fractional value.

The unit price of the newspaper is fixed, the total price and the number of subscriptions.

⑤ The perimeter and side length of a square.

⑥ The side length and area of a square.

⑦ The distance is fixed, the diameter of the wheel and the number of revolutions of the wheel.

8 Make a definite number, a number and a difference.

Pet-name ruby triangle height, bottom and area.

Participation in A and B is equivalent, and A and B are math hospitals:

(1) the total area is certain, and the area of each brick is directly proportional to the number of blocks required; ② The total number of students in the class is certain, and the attendance rate is directly proportional to the absenteeism rate; ③ The height of Xiao Gang's high jump is directly proportional to his body; ④ The circumference of a rectangle is constant, and its length and width are inversely proportional; The radius of a circle is proportional to its area.

proportion

Inverse relationship helps students understand through the relationship between the total number of application problems and the number of copies. In the relationship between total number and number of copies, including total number, number of copies and number of copies. When the total number is fixed, each copy and the number of copies are two related variables. If the number of copies is different, the number of copies will be different. Similarly, if the number of copies changes, each copy will also change. Their change, whether it is expansion or contraction, the product of the corresponding two quantities (that is, the sum) is certain. Specifically, when the total number of copies is fixed, each copy (or number of copies) is expanded or reduced by several times, while the number of copies (or number of copies) is reduced or expanded by the same multiple. Referred to as "one expansion and one contraction (or one contraction and one expansion)". The number of copies with this changing relationship is inversely proportional to the number of copies. Inverse proportional relation belongs to the inductive problem in typical application problems. Reflected in division, when the divisor is fixed, the divisor and quotient are inversely proportional. In a fraction, when the numerator of the fraction is constant, the denominator is inversely proportional to the fractional value. In proportion, the former term of the proportion is fixed, and the latter term is inversely proportional to the proportion. If the relationship between the total number and the number of copies is embodied as: in the shopping problem, the total price is fixed, and the unit price is inversely proportional to the quantity. In the travel problem, the distance is fixed and the speed is inversely proportional to the time. On the issue of work, the total amount of work is certain, and the work efficiency is inversely proportional to the working hours. If two quantities are inversely proportional, the ratio of any two numbers of one quantity is equal to the inverse ratio of two corresponding numbers of another quantity. For example, the total number of machined parts must be 600. If 10 pieces are processed every hour, it will take 60 hours to complete the task. If we process 20 pieces per hour, it will take 30 hours to complete the task. The ratio of hourly processing capacity is 1: 2, and the corresponding completion time ratio is 2: 1. 2∶ 1 is the inverse ratio of 1∶2.

The significance of inverse proportion teaching adopts analogy and reverse reasoning. That is, at the beginning of teaching, students first write the meaning of inverse proportion directly according to the meaning of positive proportion:

Two related quantities-→ two related quantities,

Quantitative change-→ quantitative change

The other quantity also changes-→ The other quantity also changes.

The ratio of the corresponding two numbers in these two quantities is determined-→ the product of the corresponding two numbers in these two quantities is determined.

Then, according to the meaning of inverse proportion written by students themselves, give an example to verify it.

After that, further understand the meaning of inverse proportion.

① Analyze the significance of inverse proportion.

Inverse proportional quantity includes three quantities, one quantitative and two variables. Study the relationship between the expansion (or contraction) of two variables. A change in one quantity causes an opposite change in another. These two quantities are inversely proportional, and their relationship is inversely proportional.

② Inverse ratio essence

Two related quantities, one of which changes and the other changes, and the product of the corresponding two numbers in these two quantities is certain. These two quantities are called inverse proportional quantities. Their relationship is called inverse relationship.

Compare positive and negative proportions:

Similarities: ① Both the direct proportion and the inverse proportion contain three quantities, including one quantification and two variables.

(2) Among the two variables with positive and negative proportions, if one variable changes, the other variable will also change. The change mode belongs to expansion (multiplied by a number) or contraction (divided by a number) several times.

Difference: Quantization of positive proportion is the ratio of two corresponding figures in two variables. Inverse proportional quantization is the product of two corresponding numbers in two variables.

Mutual transformation between positive proportion and inverse proportion: when the value of x in positive proportion (the value of independent variable) is transformed into its reciprocal, it is transformed from positive proportion to inverse proportion; When the value of inverse proportion x (the value of independent variable) is also converted into its reciprocal, it is converted from inverse proportion to positive proportion.