First of all, from the perspective of grammatical meaning and the coherence of mathematics teaching (mainly involving junior high school mathematics)
1. Enlarge and enlarge are the same meaning, both of which mean how much more than the original.
Example:
① Enlarge 2 by 5 times to 2+2×5.
(2) Enlarge by 2 times 5 times, which is also 2+2×5.
2. The expansion to is different, indicating how much has been reached (or now).
Example 1: Extend 2 to 5 times to 2×5.
Example 2: from 0.256 to 25.6, which means "expanding to 100 times of the original number, or 99 times".
Example 3: The change from 25.6 to 0.256 is "reduced to the original number of1100, or reduced by 99 times".
2. Refute the view that "expand, expand and expand to" has the same meaning. Does "expanding to" mean "expanding" or "expanding"
One is the fallacy of "expanding A by n times to na", that is, understanding "expanding" as "expanding to".
As we all know, in mathematics, a false proposition can be demonstrated by giving counterexamples. If "A magnifies N times to na", 2 magnifies 1 times to 2× 1=2, there is no amplification; Enlarge 2 by 0. 1 times to 2×0. 1=0.2, but reduce it. This violates the dictionary's analysis of the word "expand", so it is wrong to "expand A by n times into na".
Second, determine the rationality of "expanding A by n times to (n+ 1)a", that is, understand "expanding" as "expanding".
1. "Enlarge A by n times to (n+ 1)a" is in line with the original meaning of the word "enlargement" in the dictionary. For example, multiply 2 times 1 times to 2+2×1= 2× (1+1) = 4; Enlarge 2 by 0. 1 times to 2+2× 0.1= 2× (1+0.1) = 2.1,and it must be enlarged after amplification, otherwise it will violate the common sense of three-year-old children.
2. The regulation of "expanding A by n times to (n+ 1)a" does not contradict most of the expressions in the textbook.
Example 1: when summing up the law of quotient invariance, the primary school mathematics textbook says: "In division, the divisor and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged."
When divisor A and divisor B are expanded by n times at the same time, (n+ 1)a÷(n+ 1)b=a÷b, and the quotient remains unchanged.
Example 2: In summing up the changing law of the product, the primary school mathematics textbook says, "If one factor remains unchanged, the other factor is expanded (or reduced) several times, and the product is also expanded (or reduced) by the same multiple".
Let ab=c, when a is enlarged by n times, b remains unchanged.
(n+ 1)ab=(n+ 1)c, the product expansion is (n+ 1) times, (n+1) c–c = NC, and the product expansion is n times "same multiple".
Example 3: If the vehicle speed is constant and the distance is expanded by five times, how many times will the time taken be expanded?
If the speed is t=s÷v and the original distance is s, then the original time is t = s ÷ v, and the currently used time is =6s÷v=6t, which is expanded to 6 times and 6t–t = 5t, that is, the used time is expanded to 5 times.
Third, why are there still many people and many teaching AIDS who think that "multiplying several times is multiplying several times"?
Because of the influence of the old textbooks, it is the product of the old textbooks, and now it has been "abolished". In the reply of People's Education Publishing House, there is such a sentence: "If the number from A to na or from na to A changes (n is greater than 1), it is expressed by expanding N times or reducing N times; Expanding n times is multiplying n, and reducing n times is dividing n. "
These "knowledge points" that have been used for many years have aroused the doubts of many experts and scholars. I look forward to your analysis and exchange.