A BIT MORE DIFFICULT 3rd year of secondary education

 1. FINDING THE RIGHT FORMULA In this window we can see the first five pictures in a sequence containing a certain number of counters (represented by the blue dots) which are laid out in a certain way. Change the value of the parameter "place" to see each layout. 1.- Look carefully at the five layouts and copy them into your exercise book. Write down the number of counters used in each case. 2.- Use this information to try and work out the number of counters needed for the arrangement of counters in the following positions in the series: 8, 12 and 15. Draw each layout. 3.- How many dots are there in the layout of the nth term in the series?

 2. CALCULATE AND OBSERVE There are five rows of numbers in this window. All the numbers in any one column are related to each other. The numbers can be changed by altering the corresponding parameter in the window. (n1 gives us the first row and so on). 4.- Change the value of each parameter and copy the numbers that appear into a table in your exercise book. 5.- There is a very straight forward relationship between the first three numbers in each row. Do you know what it is? What's the connection between the first three numbers and the fourth and fifth terms? 6.- Look carefully at the values that are given in the last column. What can you say about them? Compare them to the last two values given in each row. Show that the pattern you have found is true for any three consecutive numbers.

3. SURPRISE SURPRISE!

In the following window there are four identical little machines. A certain number is entered into each one which then undergoes a series of operations, as illustrated in the window, before giving us a final result when it comes out of the machine.

 7.- Change each of the parameters and look carefully at the changes in the window. Draw a table in your exercise book and complete it with the values that are given in each machine. What is special about the numbers which are put into each machine? 8.- What can you say about the numbers that come out of each machine? What can you say about the remainder after each division which is carried out by the machine? Try and explain why the machines produce the final results that are given in the window.

 Josep Mª Navarro Canut Spanish Ministry of Education. Year 2001