☰ Contents · Computing

Models and their types

Lessons 7–8 · 2 lessons · M. R. Fayziyeva, D. M. Sayfurov, N. S. Xaytullayeva. Informatics and Information Technologies, Grade 9. “Nashriyot uyi Tasvir”, Tashkent, 2020
8

Practical lesson: mathematical models

Textbook: p. 26
GoalWrites a problem as a mathematical model (formula), designs the algorithm for finding the unknown and checks the result by calculation.
New words
formalising: replacing the words of a problem with letters and a formula · formallashtirishunknown: the quantity we must find · noma’lumscale: the ratio of a length on the map to the real length · masshtabinformation volume: the amount of data, measured in bits and bytes · axborot hajmi
Explanation

The general way to build a mathematical model: read the problem, mark the given and unknown quantities with letters (formalising), write the link between them as a formula or equation, solve the equation for the unknown, and then check the result. Such a model gives an exact answer, but only for the factors it takes into account: for example, the formula E = m·v²/2 ignores air resistance. In information-volume problems the number of dots on the screen is multiplied by the number of bits needed for the colour; 4 different colours need 2 bits because 2² = 4, and 16 colours need 4 bits (2⁴ = 16). 8 bits = 1 byte. If the map scale is 1 : 500 000, 1 cm on the map is 500 000 cm = 5 km in reality.

Worked examples
“Think of a number: add 7, multiply by 2, subtract 4, divide by 2, subtract your number.” Model: ((x + 7) · 2 – 4) / 2 – x = (2x + 10) / 2 – x = x + 5 – x = 5. The answer does not depend on x – it is always 5.
A 640 × 480 screen uses 4 colours. Volume: 640 · 480 = 307 200 dots, 2 bits each: 614 400 bits = 76 800 bytes. Map: at scale 1 : 500 000, 4 cm is 4 · 5 = 20 km in reality.
Class activity

“Trick masters”: pairs invent their own number trick, write its model and show it to the class; the class checks the model with several numbers.

Practice
1
In the number trick take x = 9 and compute ((x + 7) · 2 – 4) / 2 – x.
2
A 320 × 240 screen uses 16 colours. How many bytes does the picture take?
3
The map scale is 1 : 500 000. A distance of 6 cm on the map is how many km in reality?
4
Why may a result found with E = m · v² / 2 differ a little from a real experiment?