Hewitt Video Rough Thought

First thing I noticed is in creating an engaging classroom environment, Hewitt did some incorporation of visual methods and activities stands as the initial key strategy. By leveraging visual aids, educators can captivate students' attention, fostering a dynamic and interactive learning atmosphere.

Moving beyond conventional teaching approaches, the second crucial part I have noticed is Hewitt involves transferring the judgment responsibility to students, empowering them to actively participate in the learning process. This shift not only encourages critical thinking but also cultivates a sense of ownership over their educational journey. 

Third thing I noticed Hewitt did was to enhance student involvement and autonomy, he implemented some sort of student-led approach, Hewitt's questions are supportive, guiding, but students engagement is the key factor to lead the class on going. By granting students the opportunity to take charge of the class and encouraging them to freely respond to questions, a more collaborative and participatory learning environment is fostered, nurturing the development of essential skills and a deeper engagement with the subject matter.

From the Hewitt's 2020 video, he asked how to find fractions between 5/7, 3/4, it is easy to compare 5/7 and 6/8 instead of 3/4. Then I noticed that 5 and 7, 6 and 8 have common difference 2. And it is easy to see there is a trend if we start from 0/2, 1/3, 2/4, 3/5, etc ..., so I can easily tell any number between 5 and 6 as numerator, then add 2 as its denominator, then make the fraction as whole number by multiply multiples of 10. For instance, I can easily find 5.5/7.5 → 11/15 as a solution. By making common factor of 5/7 and 3/4, which tells me that as long as I can find number between 20/28 and 21/28, I can make 40/56, 42/56, by so 41/56 must be in between the two fractions, and I can keep on and on, find infinitely many different fractions that satisfies this. Both methods are about comparing, with determining upper and lower bounds by different approach. 

Now my question turns to, since which number will there always be a solution?

Considering the numerator, by common the numerator, 5/7 and 3/4 are equivalent to check 15/21 and 15/20, so starting n=16, we can find a series such that n/[(n+5)(n+6)]=n*[1/(n+5)-1/(n+6)] < 15*(1/20-1/21), and I can simply see the descending trend with increasing the value of n, by derivatives or by telescoping comparing technique, which means since 16, there will always be a fraction with such numerator between 5/7 and 3/4. 

By checking 16, I find that 16/22=8/11, which 8 is a numerator less than 11 and the fraction between 5/7 and 3/4.

Above methods I used are required either understanding of trending or comparing by common denominator or numerator. By checking the trend: I need to know how to find the pattern of numbers; and comparing, I need to understand the basic idea and definition of fraction. All these tell me that: awareness of knowledge require deep understanding. So in the future teaching, I would consider how to teach the necessary of knowledge and support with relational use of the knowledge to help students use awareness in order to support them with better understanding.