Item: tVerbalRuleExplicitModelGeometric3a

Stem:

Katie is a biochemist. During her last trip to the Amazon rainforest, she brought back some leaves from an exotic plant. The substance she extracted from the leaves has some amazing properties: one drop on a given cell causes it to grow and divide, so that it generates an additional cell of the same size, along with a smaller bonus cell. That is, at the first division, there are three cells, made up of two large cells and one small cell. At the next division, the number of large cells and small cells doubles, so there are a total of six cells (4 large and 2 small cells) at the second division. The same pattern continues: as a result of each division, the number of large cells doubles; the number of small cells also doubles, so there are always half as many small cells as large cells.

Your task is to find the expression for the number of total cells present as a result of the nth division.

Options:

1. 3 * 2n-1

2. 2 * 3n-1

3. 3n

4. 2n

Key:

• Observable: 1. Score: 1; Evidence Value: 1.
  • Student:Correct, good job!
  • Teacher:The student has successfully modeled a real world problem by finding an explicit formula for a geometric sequence.
• Observable: 2. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's not right. Try plugging in some numbers into the formula you chose, starting with the first division: n = 1. Then 2 * 3n-1 = 2 * 30 = 2 * 1 = 2. But we know there are three cells, not two, after the first division, so your answer is not right. The geometric sequence in this case is: 3, 6, 12, 24... for each division. The correct answer is: 3 * 2n-1.
  • Teacher:The student may have memorized the formula for a geometric sequence, and confused some of the terms.
• Observable: 3. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's not right. Try plugging in some numbers. Suppose n = 2. Then 3n = 32 = 9. But we know that there are six cells, not nine, after the second division. The geometric sequence in this case is: 3, 6, 12, 24... for each division. The correct answer is: 3 * 2n-1.
  • Teacher:In finding the formula, the student may have only looked at the number of cells following the first division, and may not have verified the pattern.
• Observable: 4. Score: 0; Evidence Value: 0.
  • Student:Sorry, that's not right. But you are close. You did find the correct formula for the number of large cells as a result of the nth division. Try plugging in some numbers. Suppose n = 1. Then 2n = 21 = 2. But we know that there are three cells, not two, after the first division. The geometric sequence in this case is: 3, 6, 12, 24.... for each division. The correct answer is: 3 * 2n-1.
  • Teacher:The student may have observed that the number of large cells and small cells both double, and may have inferred that a power of two was involved. In fact this is the correct formula for the number of large cells.

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