• Observe/evaluate the class discussion. Each student’s contribution meets a standard of quality if it is original, relevant to the mathematical content in a specific way, and adds to the collective knowledge of the class. The discussions may also be advanced by student questions and responses to them.
• Evaluate student performance on the
  • research into various types of numbers, the connection of triangular and square numbers (Are both pattern and calculations correct?).
  • illustration of the Golden Ratio found within the Fibonacci Sequence.
  • article written on the prevalence and uses of the Golden Ratio (Is the approximation to 1.6 appropriate?).
  • investigation into patterns found within Pascal’s Triangle.
  • connection of Pascal’s Triangle to the Fibonacci Sequence.
  • illustration of arithmetic and geometric sequences (Did any students point out that some mathematical patterns relate topics that appear unrelated , such as geometry and algebra?).

“These notations are used interchangeably. Notice that we can represent the Golden Ratio with illustrations, words, and symbols.”

Activity 3

Have students support the following statement, using a detailed and clear illustration and explanation: If you take the ratio of any two successive Fibonacci terms, the ratio approaches the Golden Ratio, as n gets larger and approaches infinity.

Activity 4

Have students, in pairs, write a short article highlighting the prevalence of the Golden Ratio, [IS.10 - Struggling Learners] as seen in the world around us. Choose one specific example and illustrate how that object/structure represents the Golden Ratio. At least one illustration/drawing is required. Students can use tables, symbols, and any other necessary representations. They need to clearly describe and illustrate how the Golden Ratio is demonstrated by the real-world example.

Ask each pair to share their articles and drawings with the class. They can create a class collage, portraying all the different representations of the Golden Ratio. For example, illustrations might include polygons, rectangular pyramids, seeds, petals, shells, other pyramids, etc. Student articles should clearly illustrate the Golden Ratio found within the examples.

Activity 5: Pascal’s Triangle

Demonstrate fractal geometry illustrations to students, such as http://en.wikipedia.org/wiki/File:Von_Koch_curve.gif. This link shows the animation of the Koch snowflake. Distribute the Fractals handout, showing Sierpinski’s Triangle and four versions of the Koch snowflake (M-A1-2-3_Fractals.docx).

“Has anyone heard of Pascal’s Triangle? How is Pascal’s Triangle created? What patterns are shown?” Give students an opportunity to draw the triangle. “Pascal’s Triangle is a triangular formation of patterns.”

Divide students into groups of three or four. Have students undertake an investigation into patterns found within Pascal’s Triangle (M-A1-2-3_Pascal Triangle.xlsx). They should answer the following questions:

• What number sets or patterns can you find within the triangle? (Example: The natural number set can be found within the second diagonal.)
• Can you find the triangular numbers within the triangle? Any other numbers?
• How can you describe the sum of the numbers in each row of the triangle?

Lead students toward the discovery that the sum of the numbers in each row is a power of 2. Use a table with headings for row number, sum of numbers, and alternate way to write the sum. After students create the table, ask them to discover the rule for the sum of the numbers in the nth row. (The sum for row n is ). The pattern is

1 + 2 + 4 + 8 + 16 + ….

Note: Pascal’s Triangle illustrates the fact that the sum of the diagonals equals the number in the next row. Refer to the Pascal Triangle handout (M-A1-2-3_Pascal Triangle.xlsx) to see the progression of sums in successive rows. For example, if we add 1, 4, 10, and 20, we get 35. This number is found below and to the left of the 20, in the next row!

Activity 6

Divide students into groups of three or four. Each group should examine ways to connect Pascal’s Triangle to the Fibonacci sequence. In other words, they can find the Fibonacci sequence within Pascal’s Triangle. They should specifically identify the Golden Ratio, as it can be drawn from the triangle.

Arithmetic Sequences

“Suppose we have the following sequence illustrated on a graph.”

“Let’s write this sequence on the board: −9, −6, −3, 0, 3, 6,… Now let’s write it using a table:

What is going on here? How can we describe what is happening from term to term? Can we find the next term using the previous term? Can we find the nth term without knowing the previous term? If so, how can we write those rules?”

“Considering this example, we can see that each term is 3 more than the previous term. In other words, we add 3 to the previous term to get the successive term. Since the terms increase by a constant amount, the sequence is an arithmetic sequence.”

“If we were to write the sequence as a recursive rule, we would write:

If we were to write the sequence as an explicit or close-form rule, we would write:

Do you know how this last formula was derived? Is there a general rule/form for arithmetic sequences? If so, what is it?” Students should work in groups on finding this general form. Hand out the Arithmetic Sequences resource (M-A1-2-3_Arithmetic Sequences.docx).

“Let’s add another facet to our discussion of this arithmetic sequence. Consider these questions:

• Does the sequence converge or diverge? If it converges, what does it approach?
• Does the sum of the sequence converge or diverge? If the sum converges, what does it approach?
• Do all arithmetic sequences follow the same type of behavior? Provide an example.”

Activity 7

Have students identify an arithmetic sequence and illustrate the behavior as n increases, using an Excel spreadsheet. [IS.11 - Struggling Learners] Their sequence may be connected to real-world phenomenon or not.

Geometric Sequences

“Suppose we have the following sequence illustrated on the number line:

Let’s write this sequence on the board: ”

“Now let’s write it using a table:

N	a(n)
1	1
2
3
4
5
6
N

What is going on here? How can we describe what is happening from term to term? Can we find the next term using the previous term? Can we find the nth term without knowing the previous term? If so, how can we write those rules?”

“Considering this example, we can see that each term is  the previous term. In other words, we either multiply the previous term by  or divide by 2. Since the terms increase by a constant multiplier, r = , the sequence is a geometric sequence.”

“If we were to write the sequence as a recursive rule, we would write:

If we were to write the sequence as an explicit or close-form rule, we would write:

Do you know how this last formula was derived? Is there a general rule/form for arithmetic sequences? If so, what is it?” (Students should work toward finding this general form.)

“Let’s add another facet to our discussion of geometric sequence. Consider these questions:

• Does the sequence converge or diverge? If it converges, what does it approach?
• Does the sum of the sequence converge or diverge? If the sum converges, what does it approach?
• Do all geometric sequences follow the same type of behavior? Provide an example.”

Activity 8

Direct students to identify at least one sequence of importance. They should present the sequence using a variety of representations and descriptors. For example, you might discuss the idea of convergence/divergence and relate it to the context of the sequence. [IS.12 - Struggling Learners]

Geometric Connections

Provide the following geometric connection to sequences:

Ask students to complete the following table, write the sequence for the diagonals drawn in the given polygons, and develop a conjecture for a rule to describe the sequence.

Description of Pattern: _________________

Rule: _______________________________

“What if you examined the position number n in relation to the number of diagonals? Would that change the rule? How?”

To review the lesson, hold a class discussion. Have each student answer the following question: How is the study of patterns and sequences related to the study of notations? Students should provide an example.

Extension:

• Once students understand the concept of a geometric illustration of a sequence, ask each student to create a geometric representation of a sequence, identifying at least one sequence from the representation. (For example, students might create a pool activity/representation or illustrate growing triangles.)