Look at the toothpick pattern and complete Problems B1-B8.
Groups: Work in pairs. There are two common ways of thinking about the toothpick pattern, and one of the reasons it's useful to work in pairs or small groups is to observe this variety.
Some people see the pattern as groups of 4 toothpicks, with 2 toothpicks added on to the right-most triangle, and they come up with the following rule: Take the number of triangles, multiply by 4, and add 2. Others see the pattern as groups of 6 toothpicks, with 2 toothpicks removed from all of the triangles except the right-most triangle. The rule they come up with is: Take the number of triangles, multiply by 6, and then subtract 2 times 1 less than the number of triangles.
When using a variable, the first rule turns out to be 4n + 2; the second becomes 6n - 2(n-1). The second rule may be harder to represent‹"subtract 2 times 1 less than the number of triangles" can be tricky to express. Consider going through a derivation of this rule, starting with: If the number of triangles is 5, what¹s 1 less? So we subtract 2 times that. If the number of triangles is 10, what¹s 1 less? Then subtract 2 times that number. Finally, if the number of triangles is n, what¹s 1 less? It is n - 1.
The second difficulty may be seeing the need for the parentheses when multiplying this quantity by 2. Explore the expressions 2(n - 1) and 2n - 1, testing some particular values for n to decide which accurately describes "2 times 1 less than n."
Those who come up with a symbolic rule right away, without even looking at the drawing, should think about where the "4-ness" or the "6-ness" comes from in the picture. Make sure to make the link between what's going on in the equation and what's happening in the picture.
Groups: No matter how people come up with them, explore the two different rules as a full group. Do they continue the table in the same way? Are the rules equivalent, even though they look different? Or are they genuinely different?
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