Related Images
Find the total number of triangles (of all sizes) in the image
About
The Mathematical Problem Solving Program provides year-long after school courses for mathematically able and gifted school students aged 10 to 15. In operation since 1992, the program runs at ECU’s Mount Lawley Campus and is directed by Dr Norm Hoffman.
Each year, more than 200 students enrol from primary and secondary schools, government and non-government, throughout the Perth metropolitan area to participate in the national program, Mathematics Challenge for Young Australians. This program is produced by the Australian Mathematics Trust and comes as a complete package with student notes and problems, and teacher guides. ECU offers three levels of the program: Euler (Year 8), Gauss (Year 9), and Noether (Year 10).
Activities
At the primary level there are two programs, Primary 1 (mainly Year 6 students) and Primary 2 (mainly Year 7 students). These have been specially developed and refined over a number of years by Dr Hoffman. They focus on systematic approaches to problem solving, explored through arithmetic, symbolic and geometric situations. There is also a strand that focuses on chance processes and the nature of randomness. The scope of activities in the primary programs is best illustrated by the following examples. Solutions are given on the bottom of the page.
Arithmetic:
List all the 3-digit numbers for which the sum of the digits is 4.
Symbolic:
List all the 3-letter arrangements that can be made from the letters of the word “page”.
Geometric:
Find the total number of triangles (of all sizes) in the image “geometric problem” shown right.
Chance Processes:
Fruit Drop lollies are made in ten different flavours: Apricot, Banana, Cherry, Date, Elderberry, Fig, Grape, Kiwi fruit, Lemon and Mandarin. In each production run, ten million lollies are made, one million of each flavour. They are then mixed thoroughly and put into packets of ten. On average, how many different flavours might you expect to get in a packet?
In the primary classes, the problems are explored over an extended period, typically 2 or 3 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 inter-relatedness of many of the problems is drawn out.
At all levels of the program (primary and secondary) there is a consistent focus on clear and effective presentation of solutions. Students not only improve their problem solving skills but they also develop skills in presenting a solution, once it has been obtained. The marking of the solutions also provides a means of monitoring student progress.
Achievement
Student participation and achievement is recognised through certificates presented at presentation functions held each August and November.
Solutions
Arithmetic problem:
0+0+4 = 4, 3-digit numbers: 400
0+1+3 = 4, 3-digit numbers: 103 130 301 310
0+2+2 = 4, 3-digit numbers: 202 220
1+1+2 = 4, 3-digit numbers: 112 121 211
Algebraic problem:
pag pga apg agp gpa gap
pae pea ape aep epa eap
pge peg gpe gep epg egp
age aeg gae gea eag ega
Geometric problem:
There are 69 triangles altogether.
Chance processes problem:
By Monte Carlo simulation, the expected number of different flavours is approximately 6.5.€