Illuminations: Patterns That Grow

# Patterns That Grow

## Exploring Other Number Patterns

 Students analyze numeric patterns, including Fibonacci numbers. They also describe numeric patterns and then record them in table form.

### Learning Objectives

 Students will: analyze numeric patterns record numeric patterns use a table to record and analyze patterns

### Materials

 Pattern blocks Calculators (optional)

### Instructional Plan

Give each student several pattern blocks and call on volunteers to make pattern cores with blocks in at least two different shapes (for example, square, square, trapezoid, hexagon). Then ask them to identify the pattern core. Ask each student to make a copy of the pattern core. Now place the students in groups of six and ask them to make the pattern using all six repeats. Then record on a table the number of shapes that they used in all. (An example of how four repeats of the sample pattern would be recorded is shown on the table below.)

Then ask what the entries would be if there were eight or ten people. Encourage the students to skip count to find the answers.

Number of Shapes (located within the cells of the table)

Repeats of the Pattern Core (1, 2, 3, 4, 5, or 6)

 Shape 1 2 3 4 5 6 Triangle 0 Hexagon 4 Square 8 Rhombus 0 Trapezoid 4

Then ask what the entries would be if there were eight or ten people. Encourage the students to skip count to find the answers.

Next, display the pattern 1, 1, 2, 3, 5, 8, 13, 21. Tell the students that this special pattern is named after a man who lived in Italy many years ago, and that the Fibonacci pattern is different from those they have studied before. Ask the children to speculate on the rule for this pattern [each pair of numbers is added to get the next number in the series] and what the next 3 numbers in it will be [34, 55, 89]. Ask the students to make a list of the first ten Fibonacci numbers, and then record the difference between each pair of numbers. [The differences will be 0, 1, 2, 3, 5, 8, 13, and so on.]

### Questions for Students

 If the pattern core is triangle, square, trapezoid, trapezoid, how many squares and how many trapezoids would you use in five repeats of the pattern? How many triangles would you use in six repeats? [5 squares, 10 trapezoids; 6 triangles.] What would come next in this pattern? 5, 10, 15, 20, 25, 30 … [35, 40, 45, etc.] My pattern is 0, 4, 8, 12, 16. What are the next three numbers in my pattern? How do you know? [20, 24, 28; add 4 to the previous number.] My pattern is 1, 4, 7, 10. What are the next five numbers in my pattern? What is the rule for my pattern? How would you make a table to show that? [13, 16, 19, 22, 25; add 3 to the previous number; student responses may vary.] What would the pattern be if you started at five and added two each time? [5, 7, 9, 11, etc.] Here is how the Fibonacci pattern begins: 1, 1, 2, 3, 5, 8. What comes next? Then what? How do you know? Can you state the rule in words? [13, 21, 34, etc.; each # is the sum of the previous two numbers.]

### Assessment Options

 At this stage of the unit, it is important for students to know how to: analyze numeric patterns record numeric patterns use a table to record and analyze patterns

### Extensions

 If it is appropriate for your class, you may wish to give the students calculators and have them find the quotient of each pair of numbers, dividing the smaller number by the larger in each pair. [The quotients will be 1, 0.5, 0.667, 0.6, 0.625, 0.615, 0.619, 0.617, 0.618] Ask the students if they notice anything about the quotients. [After the first 2, they are all close to 0.62, and they follow a pattern of over, then under, 0.62.]

### NCTM Standards and Expectations

 Algebra 3-5Represent and analyze patterns and functions, using words, tables, and graphs. Describe, extend, and make generalizations about geometric and numeric patterns.
 This lesson prepared by Grace M. Burton.

1 period

### NCTM Resources

Principles and Standards for School Mathematics

 More and Better Mathematics for All Students
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