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Home arrow Mathematics arrow Modeling mathematical ideas: developing strategic competence in elementary and middle school


One problem we have had teachers try is the following rabbit digging burrows problem:

Six rabbits can dig six burrows in 6 minutes. Assuming each rabbit works at the same rate how long will it take eight rabbits to dig eight burrows?

When the teachers were not allowed too much time to think and were asked to tell what they thought the answer was, majority of them admitted that the answer they thought was 8 minutes. It was helpful at that point to employ a technology tool that was developed in-house as a part of this project to clarify their misconception using a pictorial demonstration of reasoning up and reasoning down strategy. First one can illustrate the given input in the problem by the following representation.

Not only is this methodical process helpful to understand the proportional reasoning process but also provides the opportunity for the students to learn conceptually.


1. Solve the following problems:

a. 10/13 = n/26

b. 5/13 = 25/n

2. A can of fruits cost $2. How many cans can you buy with $14?

  • 3. Five-eighths of a room is now painted. If John did two-fifths of the painting, then how much of the room did he paint?
  • 4. A painting was originally 32 ft tall and 40 ft wide. What is the new height if the painting needs to be reduced to a width of 2.5 ft?
  • 5. Two triangles are similar and the length of their base ratio is 1:2. The smaller triangle has a base that equals 5 and an area that is 110. What is the area of the larger triangle?
  • 6. On a map 7, 72 miles represents 12 centimeters length. How many miles does a 17-centimeter length represent?
  • 7. When I drove up to the gas station, I notice that my gas tank was one-eighth full. I purchased 7.5 gallons of gasoline for $10. With this additional gasoline, my tank now is five-eighths full. How many gallons of gasoline can my tank hold when it is full?
  • 8. A worm climbs a 40-ft tree. Each day he climbs 5 feet, and each night he slides down 2 feet. In how many days will the worm reach the top of the tree?
  • 9. If 2 people can paint 2 rooms in 2 hours then, then how many people can paint 18 rooms in 6 hours?
  • 10. If there are 15 total bicycles and tricycles in a store and there is a total of 37 wheels. How many tricycles are there?

The Task

The task is to paint your bedroom, using whatever color you choose but done in the most time and cost effective way. (We only want to spend what we have to in order to get the bedroom fully painted, and we would hope for this project to not take too long)

What do we want to know? What questions should we ask ourselves, to make sure we complete this task in the most time and cost effective way?

How many people will it require to complete the job in half the time?

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