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Unit 18: Computational Thinking
Pre-enrolment
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Unit introduction Computers influence much of the world around us
used to solve a range of problems, generating the visual effects in films landing the Curiosity Rover (unmanned robot) on Mars. use mathematics and logic.
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Numbers – an invention No problem, I have food, clothing
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More than one… Still not a problem, just give them names
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Lots more than one Now need a system
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Egyptians
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Decimal system Introduces a zero Invented by independently by:
Babylonians, Mayans Indians Uses place value The value of a symbol depends on where it appears in the number: eg 33 333 3333 etc
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Decimal system 10 symbols 0 – 9 Click over to new place value after 9
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Other systems Binary Hexadecimal Base 2 Uses 1 & 0 Base 16
Uses additional symbols (A,B,C,D,E,F)
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Binary Basis of electrical data transmission Decimal Binary
0 = no current flowing 1 = current flowing Decimal Binary
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Task 7, 5, 102 11, 1011,
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three types of computational thinking skills
Logical thinking Algorithmic thinking Optimal thinking
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Logical thinking Work out how logical thinking is applied to the following games/puzzles: Sudoku Minesweeper
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Sudoku What are the ‘rules’?
Using the rules, what is this number? (There is only one possibility).
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Minesweeper The numbers indicate how many of the 8 surrounding squares have ‘mines’. There are 10 mines here, how many can you place from this start?
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Finished game 4 of the mines could be placed From the original screen.
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Logical thinking Don’t need to know Just use the available information
number bonds Multiplication tables Complex maths Just use the available information Don’t guess!
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Algorithmic thinking A bit like logical thinking
uses a predefined set of instructions to solve problems E.g. strategy games Chess
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Algorithmic thinking Don’t need to know Just follow the instructions
number bonds Multiplication tables Complex maths Just follow the instructions
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Optimal thinking refining one of many possible solutions to a problem to make it more efficient E.g. Rubik’s Cube Towers of Hanoi
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Towers of Hanoi http://www.mathsisfun.com/games/towerofhanoi.html
Give it a go – what is your best score? What if there were 4 pegs? Best score is 7 The minimum number of moves required to solve a Tower of Hanoi puzzle is 2n - 1, where n is the number of disks Frame–Stewart algorithm[edit] The Frame–Stewart algorithm, giving a presumably optimal solution for four (or even more) pegs, is described below: Let be the number of disks. Let be the number of pegs. Define to be the minimum number of moves required to transfer n disks using r pegs The algorithm can be described recursively: For some , , transfer the top disks to a single peg other than the start or destination pegs, taking moves. Without disturbing the peg that now contains the top disks, transfer the remaining disks to the destination peg, using only the remaining pegs, taking moves. Finally, transfer the top disks to the destination peg, taking moves. The entire process takes moves. Therefore, the count should be picked for which this quantity is minimum. This algorithm (with the above choice for ) is presumed to be optimal, and no counterexamples are known.
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Boolean Algebra What does this mean?
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Boolean logic Boolean algebra deals with the values 0 and 1 or the values false and true ‘the algebra of truth’ It is ideally suited for use in a computer: WHY? Operations using just 0 and 1 can be easily replicated using simple electronic circuits AND OR NOT XOR
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Truth Tables Boolean Operations are represented in Truth tables
Input (s) Input A Input B Output The operations are carried out by electronic ‘gates’ INPUT OUTPUT
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The simplest function - NOT
The NOT gate takes an input and reverses it. also known as an inverter If the input is 1, the output is 0 If the input is 0, the output is 1 (very useful for 2s complement when we flip the bits) INPUT OUTPUT A NOT A 1 Symbol (used on circuit diagrams) Truth table
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Task Find symbols and meanings for: AND OR NOT XOR
Can you also write a truth table for each?
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AND A 1 is required on both inputs to give a 1 at the output
Only returns a high value if both inputs are high. INPUT OUTPUT A B A AND B 1 Symbol Truth table
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OR A 1 on either input gives a 1 at the output.
Gives a high output if at least one of the inputs is high INPUT OUTPUT A B A OR B 1 Symbol Truth table
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Exclusive OR (EXOR XOR)
You only get a 1 at the output when the two inputs are different Returns a high output if either (but not both) of the inputs is high. INPUT OUTPUT A B A XOR B 1 Symbol Truth table
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Other gates There are other gates:
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How do Boolean operations help?
Searches: Help to refine searches: Find ‘this term’ AND ‘this term’ Find either ‘this term’ or ‘this term’ (or both) Find ‘this term’ but not ‘this term’ Etc
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Programming Can be used with conditional statements:
E.g. ‘if’ statements
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Addition So how can we use logic gates to add up?
Need to know the rules of binary arithmetic: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 and carry 1
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