2 Enhancing Practicality and Strengthening Examination of Innovative Application Consciousness
Mathematics originates from life and practice. Mathematical knowledge is a powerful tool for solving practical problems, and mathematical ability is a mathematical literacy that every citizen must possess. The examination of innovative application ability in the college entrance examination requires examinees to combine materials from daily life, other disciplines, and learning practice to discover and raise problems; apply learned mathematical knowledge and thinking methods to think independently, explore and research, analyze and solve problems.
Paying attention to real social hot topics is an important characteristic of the 2014 College Entrance Examination mathematics papers. Through solving social hot topics, the papers examine the ability to apply mathematical tools and methods to solve practical problems. The examination papers closely integrate with social reality and examinees' actual lives, reflecting the tremendous role and application value of mathematics in solving practical problems, and embodying the characteristics of strengthening application and practicality in college entrance examination reform.
The 2014 national papers used ``air quality,'' a social hot topic, as the background: According to historical data, the probability that air quality is excellent on a given day is 0.75, the probability that it is excellent for two consecutive days is 0.6. If today's air quality is excellent, what is the probability that tomorrow's air quality will be excellent? The papers require examinees to apply statistical and probability methods to forecast tomorrow's weather, helping examinees understand the application value of statistics and probability.
The papers broadened the sources of test materials, extensively involving many aspects of social life. The 2014 mathematics papers involved backgrounds such as rural residents' per capita net income, participation in public welfare activities, and product quality indicators. These questions are grounded in reality, close to examinees' lives, allowing examinees to deeply feel that mathematics is alive and all around them, with mathematical thinking permeating their lives.
Example 2: Four students each choose one day from Saturday and Sunday to participate in public welfare activities. The probability that both Saturday and Sunday have students participating in public welfare activities is
(A) $\frac{1}{8}$
(B) $\frac{3}{8}$
(C) $\frac{5}{8}$
(D) $\frac{7}{8}$
The questions examine basic knowledge and methods of probability, testing examinees' ability to apply mathematical tools and methods to analyze practical problems and solve them. They guide examinees to pay attention to mathematical problems in life and strengthen their application consciousness of mathematics.
Example 3: The data for per capita net income $y$ (in units of thousand yuan) of rural residents' families in a certain region from 2007 to 2013 is shown in Table 1:
\begin{table}[h]
Table 1 | Year | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 |
| Year Code $t$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Per Capita Net Income $y$ | 2.9 | 3.3 | 3.6 | 4.4 | 4.8 | 5.2 | 5.9 |
\end{table}
(I) Find the linear regression equation of $y$ with respect to $t$; (II) Using the regression equation from (I), analyze the changes in per capita net income of rural residents' families in this region from 2007 to 2013, and predict the per capita net income of rural residents' families in this region for 2015.
The questions use rural residents' per capita net income, an important socioeconomic indicator, as the vehicle, providing data from a certain region from 2007 to 2013. Examinees are required to establish a linear regression equation, achieving the goal of examining students' ability to establish linear regression equations based on data, while also examining students' understanding of the practical significance of the slope of the regression line. The questions originate from reality, are close to life, combine theory with practice, and help examinees understand the application value of statistics and probability.