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Learning Math: Patterns, Functions, and Algebra

Patterns in Context Part B: Patterns in Situations (45 minutes)

Session 2, Part B

In This Part

  • Toothpick Triangles
  • More Toothpick Triangles

Problem A5 describes several different ways to extend the pattern given in Problem A1. When a sequence of numbers is not tied to a context, there is no way to determine which extension of a pattern is “correct.” Note 5

Describing patterns is most useful when the regularity is embedded in some well-understood situation. For example, examine the toothpick pattern given here:

 


Problem B1

Describe the pattern in as many ways as you can.


Problem B2

Imagine you keep adding triangles. Complete the following table:

Triangles Toothpicks
1
2
3
4
10

 


Problem B3

If someone tells you the number of triangles, explain how to tell them the number of toothpicks they would need. How do you know your description will always give the right number of toothpicks?


Problem B4

Use your explanation to complete this table:

Triangles Toothpicks
2
3
4
10
100
26
46
102

 

Video Segment

In this segment, participants describe and explain different patterns they found in the toothpick triangles problem, and Professor Cossey discusses the importance of understanding where a pattern comes from. Watch the segment after you have completed the toothpick triangles problems, Problems B1-B4, and consider the different patterns that the onscreen participants identified.

Do Sue-Anne and Frederick’s rules extend the table in the same way? Is there only one correct way to extend the table in Problem B2?

You can find this segment on the session video, approximately 14 minutes and 10 seconds after the Annenberg Media logo.

More Toothpick Triangles

Problem B5

How does the table in Problem B4 compare to the table from Problem A1? Is there more than one way to extend the triangle and toothpick table? Explain.
Note 6

Table from Problem B4

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

Table from Problem A1

Input Output
1 6
2 10
3 14
4 18
5 22
6 26
? ?
? ?

Problem B6

Which of the descriptions below from Problem A5 are valid for the toothpick triangles? Why are the others invalid?

  1. As the input increases by 1, the output increases by 4.
  2. If you add 2 to 1 and double it, you get 6. If you add 3 to 2 and double it, you get 10. If you add 4 to 3 and double it, you get 14. Or, if you add the input to the next input, double that, you get the output.
  3. The units digits are in the sequence 6, 0, 4, 8, 2, so the next number would be 26, then 30, 34, 38, and 42, and then 46, 50, 54, 58, 62, etc.
  4. To get the output, multiply the input by 4 and add 2.
  5. To get the output, triple the input, then add 2 more than the input.
  6. After 6 as an input, the output numbers repeat over again: 6, 10, 14, 18, 22, 26, etc.
  7. After 6, the output n

 


Problem B7

Explain your rule for calculating the number of triangles if you are given the number of toothpicks. Why does it work?
Note 7


Problem B8

Will every number of toothpicks correspond to a number of triangles? What does this tell you about your rule?

 

Notes

Note 5

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, what’s 1 less? So we subtract 2 times that. If the number of triangles is 10, what’s 1 less? Then subtract 2 times that number. Finally, if the number of triangles is n, what’s 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?


Note 6

Groups: Go over Problem B5 as a full group.

In this situation there is only one way to fill in the table because the data is linked to a situation. Whatever the number of triangles, we can visualize how many toothpicks are needed.


Note 7

Be sure to go over Problem B7. Here, we are given the number of toothpicks and are asked to work backwards to find the number of triangles. Working backwards is a hallmark of algebraic thinking; we will revisit this idea in a later session.

Solutions

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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