Solutions for Session 2, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6| B7 | B8

Problem B1

Some possible descriptions:

 • As the number of triangles increases by 1, the number of toothpicks increases by 4. • To get the number of toothpicks, multiply the number of triangles by 4 and add 2. • To get the number of toothpicks, triple the number of triangles, then add 2 more than the number of triangles. • To get the number of toothpicks, take the number of triangles, multiply by 6, and then subtract 2 times 1 less than the number of triangles

Problem B2

Here's the completed table:

 Input Output 1 6 2 10 3 14 4 18 10 42

Look familiar? It should -- it's the table from Problem A1.

 Problem B3 The number of toothpicks is 4 times the number of triangles, plus 2. Look at what happens when each new shape is built. Specifically, imagine that you have the shape for 1 triangle already built, and you need to build the shape for 2 triangles. You would only need to add 4 toothpicks to the left of the existing shape. Sure enough, these 4 are what you'd need to build the 3rd shape on to the 2nd, and so on. So each new triangle means 4 new toothpicks. The "+ 2" comes from the lower right corner of the original triangle, which is not part of the shape added on each time. You can be sure that the description always gives the correct number of toothpicks because there is a specific context to the problem. The method of creating new triangles never changes, so you can be sure that each triangle will have 4 more toothpicks than the last. Compare this to Problem A5; without a context, there is no guaranteed pattern.

Problem B4

Some predictions:

 Triangles Toothpicks 1 6 2 10 3 14 4 18 10 42 100 402 6 26 11 46 25 102

 Problem B5 One subtle change between this table and the table in Part A is that we can be sure the pattern in this table will continue as long as necessary.

 Problem B6 Patterns like the ones listed in parts (f) and (g) in Problem A5 will not happen in this context. The others (parts (a) through (e)) are valid.

 Problem B7 One way to calculate the number of triangles is to "undo" the way we turn triangles into toothpicks. To get toothpicks from triangles, we multiply by 4, then add 2. So to "undo" these operations, we have to subtract 2, then divide by 4. So the number of triangles can be determined by this process, or it can be determined by the rule n = (T - 2) / 4.

 Problem B8 No. For example, there would never be a number of triangles for 1004 toothpicks. This means that the rule developed in Problem B7 only works for numbers that produce a positive whole number as the output.

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