As every one knows, the word problems supplied by textbooks that attempt to appear realistic are mostly artificial, silly, and elicit from students, if not disgust, then a heartfelt jeer. Teachers who are obligated to follow a curriculum and a prescribed text find themselves in a bind: they often share students' perception of those problems. There is an absorbing discussion at Dan Meyer's blog. Many teachers and teacher educators believe that students must be given problems they can relate to, something realistic and pertinent to their everyday experiences. This of course leads to the kind of problems that give bad name to an Algebra I course.
As there is very little mathematics in children's experience the effort should be concentrated elsewhere, beside a search for relevance. The sooner children are led to experience the characteristic abstraction of mathematics the better. One way to achieve this goal is to ask students for a problem modification that would end up with the same answer. As every formulation brings along a different context, abstractedly same solutions may have meaning in one but no meaning in another. This, too, could provide food for thought, both to teachers and students.
For example, let's consider the subject of discussion at Dan's blog:
A youth group with 26 members is going to the beach. There will also be 5 chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed?
First of all I suggest to do away with the chaperones. Their presence is irrelevant and distractive:
A youth group with 26 members is going to the beach. 5 vehicles are available, vans and cars. Each van seats 6 students. Each car seats 4 students. How many vans and cars will be needed?
In my post at Dan's blog I suggested several problems that lead to exactly same system of equations such that, at the end of the exercise, students would have solved several problems in a single go. Here I'll expand the list:
- 26 astronauts are to man 5 cruisers, some with 4 and some with 6 seats. How many of each were there?
- There are 5 pizzas. The smaller ones cut into 4 pieces, while the big ones are cut into 6. There are 26 kids to feed. Ask the question.
- An art teacher brought 5 packs of 4 and 6 brushes to distribute between 26 students. If every student got a brush and none was left over, how many of each kind of packs were there?
- At an animal farm, 5 residents have ordered warm socks for the coming winter. Horses ordered 2 pairs of socks each, but pigs each ordered an extra pair. They received the total of 26 socks. How many pigs were there?
- At a soccer tournament, the best 5 players have scored the total of 26 goals: some 4 and some 6 goals each. How many have scored 6 goals.
- 5 pirates had to divide a booty of 26 precious stones. The senior pirates took 6 stones each, the junior pirates took 4 stones each. How many junior pirates were there?
- Some of genetically engineered tulips sprout 4 flowers out of a single bulb, some 6. 5 tulip bulbs that have been planted in a flower bed showed 26 flowers for the effort. How many 4 sprout bulbs have been planted?
- A fishing trawler, brought up a net with 5 sharks, some big, some small. When they cut up the sharks they found in their stomachs old car license plates, 4 plates in each of the small sharks, 6 plates in each of the big sharks. If they found the total of 26 plates, how many big sharks did they catch?
- During a baseball game, 5 players spewed the tobacco cuds the total of 26 times. Some performed the disgusting act 4 times, others 6 times. How many of the more gross players were there?
After investing into the effort of coming up with different formulations, students may be less surprised that so many problems may be presented and solved with the same system of equations. However, besides being an engaging activity, the exercise shows a clear way to the ultimate reformulation, i.e., the introduction of variables and the equations.