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- 1 Introduction to algorithms with the Algobox software.
- 2 Algorithmic exercises: calculator, spreadsheet and algobox.
- 2.1 Algorithm for solving second degree equations.
- 2.2 Dichotomy algorithm.
- 2.3 Simulate a randomized experiment.
- 2.4 Program a function.
- 2.5 The root of 7.
- 2.6 Distance between reals.
- 2.7 The inequalities.
- 2.8 Compare two numbers.
- 2.9 Define a function.
- 2.10 Compare two real numbers.
- 2.11 Conjecture on an algorithm.

## Introduction to algorithms with the Algobox software.

**Algobox** is an easy-to-use educational software for the introduction to **algorithms**.

With the help of a mini algorithmic language in French and a simple interface, this software allows to design mini programs and to test simple algorithms, which can be encountered in the teaching of mathematics in secondary school (seconde, première and terminale).

An **algorithm** is a program that contains a sequence of actions to achieve a result in a finite number of steps.

The word **algorithm** comes from the Arab-Muslim mathematician Al-Khawarizmi, nicknamed the father of algebra. The field that studies algorithms is called **algorithmics**.

Some tests of the mathematics baccalaureate S involve algorithms created by Algobox.

The code of the algorithms is built step by step with the help of commands already predefined in the software (read, display, affect, if-then statement, loops for from and as long as ): this allows the beginner to concentrate mainly on algorithmic logic, programming and loop creation rather than on learning a complex syntax.

Once the algorithm is finished, it is enough to execute the program to see its rendering. **AlgoBox and algorithms** also allows to understand predefined algorithms.

In order to be able to approach the various mathematical situations that can be encountered in high school,

This software also includes the possibility of using a numerical function as well as drawing points and segments in a predefined frame.

## Algorithmic exercises: calculator, spreadsheet and algobox.

### Algorithm for solving second degree equations.

Here is an algorithm for solving the second degree equation:

Entries :

Enter a,b,c (a not null)

Processing:

D takes the value

Show D

If D>0 then

X takes the value

Y takes the value

Display X,Y

FinSi

If D=0 then

X takes the value

Show X

FinSi

If D<0 then

Show “No solutions”

FinSi

1. Code this algorithm in a programming language with your calculator or algobox .

2. Execute the obtained program with a=1, b= – 5 and c= 6.

### Dichotomy algorithm.

f is the function on [0;1] by :

1. Draw the representative curve of f on the screen of your calculator.

2. We consider the algorithm below:

Initializations :

a takes the value 0

b takes the value 1

Processing:

As long as b-a > 0.01

m takes the value

If f(m)>0 then

b takes the value m

or

a takes the value m

FinSi

FinTantque

Outings :

Display a,b

1. Explain the role of this algorithm.

2. Can the condition b-a>0,01 which manages the loop be modified? Explain.

3.a. Translate this algorithm into a programming language with algobox or your calculator.

b. Enter the resulting program into the calculator or computer.

c. Check the operation of the program.

### Simulate a randomized experiment.

An urn contains 3 white balls and 2 black balls.

We draw a ball at random from the urn and note its color.

The following program, written with the Algobox language, performs a simulation of this random experiment.

1. Explain the test performed in this program.

2. Finalize this program so that it matches the statement.

3.a. Enter this program into the computer.

b. Check the operation of the program.

### Program a function.

Let f be the polynomial function of degree 2, of canonical form

We propose to write a program that computes the image by the function f

of a given real number.

1. Calculate the image for x=0; x=3 .

2. Write this program with algobox and check your results.

Indication:

### The root of 7.

Consider the following algorithm.

1. Run this algorithm: we can perform four iterations.

We will note the values of a and b obtained after each iteration.

2. What is the role of this algorithm?

3. Build the associated program with the spreadsheet, the calculator or algobox.

4. Indicate the framework given by the program.

Initializations :

a takes the value 2.

b takes the value 3.

Processing:

As long as

m takes the value

If then

a takes the value m

or

b takes the value m

FinSi

FinTantque

Outings :

Display a,b .

### Distance between reals.

Write an algorithm

which reads two real numbers x and y and

which displays the distance between these two numbers.

### The inequalities.

a. Consider the inequation .

Solve this inequation by following the instructions of the following algorithm step by step:

- Cut 7 in both members.
- Divide the two members by 6.
- Write the set of solutions .

b. Write an algorithm to solve the inequation :

### Compare two numbers.

Algorithm to compare and .

and are real numbers.

1. We consider the algorithm :

Entries:

Enter x,y

Processing:

takes the value .

takes the value

Output:

View .

Questions:

a. Write the corresponding program with the calculator.

b. Conjecture the comparison of and according to the values of and .

2. Demonstration:

a. Develop and reduce .

b. Deduce the comparison of the square of the sum of two reals with the sum of their squares.

### Define a function.

Here is an algorithm:

1. Read ( non-zero number).

2. Give the value .

3. Give the value .

What is the function defined by this algorithm?

### Compare two real numbers.

An algorithm to compare two reals:

Write an algorithm that reads a non-zero number and

which displays, depending on the values of x, the larger of the two numbers and .

### Conjecture on an algorithm.

Here is a screenshot of the Algobox software.

1. Test this algorithm with n = 4, then n = 7.

2. A student has entered n = – 3. What is going on?

3. Make a conjecture about the result provided by this algorithm.

4. Prove algebraically this conjecture.

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