### The Primary and Junior Programs, in greater detail

The Primary and Junior Programs were developed a number of years ago by Dr Hoffman. The Primary Program is for mainly Year 6 students and the Junior Program is for mainly Year 7 students. Each program focuses on systematic approaches to problem solving. This is done in the context of problems that involve systematic counting (combinatorics), chance (random) processes, geometric relationships, and prime factorisation. A consistent theme is that a problem may have no solutions, one solution, several solutions or many solutions. Solving a problem means finding all solutions, not merely one of them (if there is more than one).

In the Primary and Junior Programs, the problems are explored over an extended period, typically several months. First a simple version of the problem is examined, then a simple extension, then more complex extensions. Each problem is explored to sufficient depth for students to appreciate the solution strategy involved and the process by which it is obtained. The problems span the range of arithmetic, geometric and symbolic contexts.

The nature of the programs is probably best illustrated by some examples of the problems tackled by students. The solutions to the problems, together with some discussion, are given after the problems. The reader is urged to attempt the problems before examining the solutions to problems 1 to 2 here and the solutions to problems 3 to 6 here.

### Sample Primary and Junior Problems

**Problem 1:**

List all the different ways in which **$2.35** change can be given in **exactly seven coins.**

**Problem 2:**

526 is a 3-digit number. The product of its digits is 60 (5 x 2 x 6 = 60).

List all the 3-digit numbers for which the product of the digits is **24**.

**Problem 3:**

List all of the different 3-letter arrangements that can be made from the letters of the word “bent”.

**Problem 4:**

List all the different 3-member subsets that can be made from the set consisting of the letters

b, e, n, and t, that is { b , e , n , t }

**Problem 5:**

the diagram on the right. Two selections are NOT different if one can be rotated to

match the other.

**Problem 6**

(Chance Processes) Famous Footballer cards are produced by Kolleggs, the makers of the breakfast cereal, Brekky Bites. There are 100 cards in the whole series. Kolleggs put 5 cards in each large packet of Brekky Bites. You can also buy the cards in packs of 50. One million cards of each footballer were printed. The whole supply of cards was then thoroughly mixed before the cards were put in the packs and into the packets of Brekky Bites.

Suppose you buy a 50-pack of the cards. On average, for how many footballers would you get 0 cards?