AQA Decision 1 Algorithms. Section 1: Communicating an algorithm

Transcription

1 AQA Decision 1 Algorithms Section 1: Communicating an algorithm Notes and Examples These notes contain subsections on Flow charts Pseudo code Loops in algorithms Programs for the TI-83 graphical calculator Flow charts Although flow charts are not specifically required by the syllabus, it is useful to understand how they work. Drawing a flow chart first can help you to write instructions for an algorithm. It s useful to be aware of the box convention used for drawing flowcharts. A flowchart may have circular boxes to start and end the process it describes. Rectangular boxes contain instructions to be carried out on the variables in the problem Diamond shaped boxes are used for questions/decisions. The route followed from such boxes depends on the outcome. Boxes with curved corners are used for inputs and outputs. Example 1 uses a flow chart to express an algorithm. Example 1 Draw a flowchart to divide one number by another using repeated subtraction. The output should be the quotient and the remainder. 1 of 7 10/01/13 MEI

2 Start Read x, y N = 0 x < y? No x = x y N = N + 1 Yes Output N, x N is being used to count the number of subtractions you have done. End When x < y you cannot subtract any more, and the value of x is the remainder. Pseudo code (pseudo English) An algorithm is simply a set of precise instructions. Although in this chapter (and throughout Decision Maths 1) you will be working through algorithms by hand, in real life algorithms are usually programmed into computers. Pseudo code (sometimes called pseudo English) is a kind of halfway step between instructions given in English and a computer program. If you have had any experience with programming computers (you may have used LOGO or another turtle graphics program to draw patterns in late primary or early secondary school, or you may have programmed a graphic calculator) you will know that there are many different programming languages and that these are constantly evolving. However, there are common elements to most programming languages: values are assigned to variables; loops allow you to repeat the same instruction more than once; the values of variables are compared. Pseudo code allows you to write instructions in the form similar to a computer program without having to worry too much about using the correct language and syntax. Two different people would probably write different versions of pseudo code for the same task, but would both be able to understand what the other has written. It is a bit like text messaging: you want to express things as concisely as possible but clearly enough that anyone can understand what you are communicating. 2 of 7 10/01/13 MEI

3 Line numbers are often given when using pseudo code: this can be useful when using the GOTO statement. Line numbers are often given as multiples of 10: this means that if you need to add an extra instruction between existing lines you do not need to renumber everything! Some useful instructions for psuedo code: INPUT Indicates that the user must provide a number that the algorithm is to process. PRINT The algorithm here produces a result (the output of the algorithm) LABEL This provides a reference point (e.g. LABEL X means that you are labelling this point in the algorithm X. You can then use GOTO X). GOTO This makes the algorithm jump to a different point. You can just use a line number here (e.g. GOTO 40), or you can label a point (X, say) and then use GOTO X this approach means that if you change line numbers then you do not need to change the GOTO instruction. LET This assigns a number to a variable (e.g. LET N = 1). You can also use LET to add to existing variables (e.g. LET N = N + 1 adds 1 to the existing value of N, and LET N = N + X adds the current value of X to the current value of N, and assigns the new value to N). IF THEN This means just what it does in normal English. After the IF is a statement (e.g. IF X = Y), and after the THEN is an instruction (e.g. THEN GOTO A). If the statement is true then the algorithm follows the instruction after the THEN; if it is not true then the algorithm proceeds to the next line.) STOP This signals the end of the algorithm. END Sometimes used instead of STOP. Example 2 below shows how the algorithm in Example 1 can be expressed in pseudo code; remember, however, that there are many other ways it could be written. Example 2 (i) Use pseudo code to write instructions to divide one number by another using repeated subtraction. The output should be the quotient and the remainder. (ii) Use the pseudo code to produce a trace table to divide 134 by 12, and give the output. (i) 10 INPUT X 20 INPUT Y 30 LET N = 0 40 LABEL A 50 IF X < Y THEN GOTO B 60 LET X = X Y 70 LET N = N + 1 X is the dividend (the number you are dividing) and Y is the divisor (the number that you are dividing by) When X < Y you cannot subtract any more, and the value of X is the remainder. 3 of 7 10/01/13 MEI

4 80 GOTO A 90 LABEL B 100 PRINT N 110 PRINT X 120 STOP N is the quotient and X is the remainder. (ii) X Y N The output is 2, 11 (i.e. the quotient is 11 and the remainder is 2). Compare the pseudo code in Example 2 with the flowchart in Example 1: you will see that the process is the same. Loops in algorithms In Example 2, an IF THEN command and GOTO commands were used to produce a loop. An alternative approach is a FOR NEXT loop. FOR NEXT loops are useful tools in many algorithms, so it is important that you understand how they work. Below are a couple of examples with some notes to explain how they work. Example 3 shows a simple FOR NEXT loop. Example 3 Write a set of instructions to print the first three square numbers. 10 FOR I = 1 to PRINT I 30 NEXT I This program will print out 1 on the first circuit of the loop, 4 on the second circuit and 9 on the third and final circuit. 4 of 7 10/01/13 MEI

5 The instruction FOR I = 1 to 3 means go through the loop between FOR and NEXT with I = 1, then I = 2, then I = 3. The instruction PRINT I 2 is within the loop and so is carried out each time the loop is followed, so 1, 4 and 9 get printed out. Example 4 shows a nested FOR NEXT loop, in which one loop is inside another. Example 4 (i) Write a set of instructions to calculate the sum of (ii) Give a trace table for the algorithm. 2 3 r r r for r = 1 to r = 3. (i) 10 LET X = 0 20 FOR I = 1 to 3 30 FOR J = 1 to 3 40 J LET A = I 50 LET X = X + A 60 NEXT J 70 NEXT I 80 PRINT X (ii) X I J A The J loop is nested within the I loop. Every time the I loop is followed, the J loop is followed 3 times. The instruction PRINT X is outside the loops and so only occurs once all of the loops have been completed. Programs for the TI-83 graphical calculator Programming a graphical calculator to carry out simple algorithms like the ones in this chapter is quite easy. The programming language used varies from one make of calculator to another, but they are all quite similar to pseudo code so are easy to learn. It is well worth trying some programming, as this will help your understanding of pseudo code and flowcharts. 5 of 7 10/01/13 MEI

6 If you would like to try some programming, here are some ideas. 1. Program your calculator to find the roots of a quadratic equation. Write the program so that if there are no real solutions, the calculator displays the message No real solutions. 2. Zeller s algorithms finds the day of the week for any date (it is the subject of Exercise 9A Question 5 in the Oxford textbook). Here are the instructions for Zeller s algorithm: Let day number = D, month number = M, year = Y. If M is 1 or 2 add 12 to M and subtract 1 from Y. Let C be the first two digits of Y and let Y be the last two digits of Y. Add together the integer parts of (2.6M 5.39), (Y /4) and (C/4), then add on D and Y and subtract 2C. Find the remainder when this quantity is divided by 7. If the remainder is 0 the day was a Sunday, if it is 1 a Monday, etc. Try Zeller s algorithm yourself for a few dates, such as today s date or your date of birth. Then try programming your calculator to perform the algorithm. On the following pages are programs for the TI-83 for the algorithms above. Try programming them yourself before looking at the programs. If you have a different make of graphical calculator, you should be able to adapt the programs by referring to your calculator manual. Quadratic Equation Formula A program for the TI-83 graphical calculator. QUADRATIC EQUATION SOLVER Displays the roots of a quadratic equation or suitable message if the discriminant is negative. ClrHome Disp "QUADRATIC" Disp " " Prompt A,B,C B 2-4AC D If D>0 Then Disp (-B+ (D))/(2A) Disp (-B- (D))/(2A) Else Disp "NO REAL ROOTS" End 6 of 7 10/01/13 MEI

7 Examples If A = 1, B = 3 and C = -2, then the answers displayed are and If A = 1, B = 1 and C = 1, then the message displayed is NO REAL ROOTS. Zeller s Algorithm A program for the TI-83 graphical calculator. ZELLER S ALGORITHM Enter a date in the form day:month:year and display the day of the week. ClrHome Disp "ZELLER" Disp "------" Disp "DAY OF THE WEEK" Prompt D,M,Y If M<3 Then M+12 M Y-1 Y End int((y/100) C Y-100C X int((2.6m- 5.39)+int((X/4)+int((C/4)+ D+X-2C S S-7int(S/7) Z If Z=0:Disp "SUNDAY" If Z=1:Disp "MONDAY" If Z=2:Disp "TUESDAY" If Z=3:Disp "WEDNESDAY" If Z=4:Disp "THURSDAY" If Z=5:Disp "FRIDAY" If Z=6:Disp "SATURDAY" Example If D = 21, M = 3 and Y = 1950, then the day is TUESDAY. 7 of 7 10/01/13 MEI