Pretest items

The pre-equation problem presented a situation where four friends commute to school. They were at different distances from their school, but the total distance that was travelled was given. There were four unknowns in this problem; however they were related to one another, meaning three of the unknowns could be written in terms of the fourth. The task in this problem was to find the distance travelled by each person. The mathematical concept, which was depicted in this problem, is solving first degree equations. Students are familiar with this concept; however, many students find it hard to deal with several unknowns that are related to one another. The first step in the solving process of this word problem is to properly organize the given data in a way that would provide a clearer picture of the situation. There are four friends: Lara, Mohammad, Karim and Sam. Students can refer to the distances travelled by each as L, M, K and S. They each travel a different distance to school.

M travels L/2; K travels M+L; S travels 3L

All together they travel 36 Km, so L+M+K+S=36

Noticing that M, K and S can be written in terms of L, students should replace each unknown with its value in terms of L: L+L/2+M+L+3L=36; but M=L/2 so substituting again should give us the following equation: L+L/2+L/2+L+3L=36. After simplifying and solving the equation, the distance travelled by Lara can be calculated and it turns out to be 6 Km. The other distances can be calculated by substituting the value for L in the given

45 relations. So, the distance travelled by Mohammad is 3 Km, by Karim 9 km, and by Sam 18 Km. It is important so check whether the calculated distances are correct by verifying that they add up to 36 Km. The answers should be presented in full sentences which gives real-life meaning to the calculated numbers. The solution presented above is considered a sophisticated solution strategy in which a student should receive a full mark. A less sophisticated strategy would be to use guess and check. Students might start by trying out different distances, starting with the distance Lara travels. For example, if Lara travels 8 Km, then Mohammad travels 4, Karim 12 and Sam 24 Km, but adding all distances gives more than 36, so a smaller distance needs to be chosen.

The pre-pattern problem presented a situation where six teams were going to play with one another, each team would play with the other team only once, disregarding who won, and the task was to find out how many games will be played. The mathematical concept that is presented in this word problem is combinations. Students are not familiar with this concept or with the general formulas that can be used to find a fast answer. Thus, students should rely purely on their problem solving skills in order to come up with a solution. They might refer to each team with a letter or a number, and start matching who plays with whom. For example, T1 represents the first team and T2 represents the second team and so on. Thus, the following list shows the games to be played.

T1-T2; T1-T3; T1-T4; T1-T5; T1-T6 5 games that first team will play in.

T2-T3; T2-T4; T2-T5; T2-T6 4 games that second team will play in. T2 can not play with T1 again and of course can not play with itself.

46 T4- T5; T4-T6  2 games.

T5-T6  1 game.

Adding up all the games gives a total of 15 games.

An advanced problem solver might not need to do the whole list, and is satisfied with only the first two or three rows before such a solver notices that there is a pattern here, one less game for each row. Students could use various representation strategies to solve the problem, but mostly they are similar to the above list. They might use a table or matching up among two columns.

The pre-reasoning problem presented a situation that would require mathematical reasoning, specifically numerical reasoning, to complete the task, which was to fill a tank, of maximum capacity of 10 Liters, with 6 Liters of water using only 5 Liter container and 8 Liter container. The word problem provided an extra given data which is the maximum capacity of the drinking tank. Students were required to check if there was extra information in the given data. This was used to check whether students use given information appropriately. A simple diagram would help students get a clearer idea about the situation at hand. The solution which requires the smallest number of steps is to fill the 8 Liter container with water and then pour this water into the 5 Liter container. This leaves 3 liters in the 8 Liter container. This water should be poured into the tank, and the procedure is repeated another time for the remaining 3 liters. A solution with more steps is to fill the 5 Liter container and pour the water into the 8 Liter container. Fill the 5 Liter container again, and pour the water into the 8 Liter container. Two liters would be left in the 5 Liter

47 container, since 3 liters were poured into the 8 Liter container. Pour these 2 liters into the tank, and repeat the above steps three times to fill the tank with 6 liters.