If a bus can hold 36 If a bus can hold 36 passengers, and there passengers, and there are 2 adults per bus…are 2 adults per bus…

a.a. If there are 232 students, how If there are 232 students, how many buses are needed?many buses are needed?

b.b. If there are 20 busses, how many If there are 20 busses, how many students are on the busses?students are on the busses?

c.c. Is number of students vs. number Is number of students vs. number of buses a function? Explain.of buses a function? Explain.

Agenda• Go over warm-up problem

• Begin Generalizing Patterns

• Exploration 2.7

Find a Generalized Formula

• A formula is an equation that can be used to find a particular amount (e.g., length, temperature, area, time) if we are given a particular piece of information.

• The formula only works if we know what the variables are standing for, and if we are given enough information.

Examples of formulas• A = L • w

– A = area; L = length of rectangle; w = width of rectangle

• P = # desired outcomes/# total outcomes– P = probability of event

• C = (5/9)(F - 32) – C = ˚Celsuis; F = ˚Fahrenheit

Translating to Algebra• Some basic vocabulary.• Suppose we have a pattern:• 1, 4, 9, 16, 25, 36, 49, …, n• Term means each of the elements in the

sequence. So, 4 is a term, 16 is a term, 49 is a term, and n is a term in the sequence.

• nth term refers to where in the sequence the term is: 1 is the first term, 4 is the second term, 36 is the 6th term, and n is the nth term.

Two easy ones.• Consider the sequence:• 4, 5, 6, 7, 8, …, n• Let’s make a table:• n: 1 2 3 4 …n• nth term: 4 5 6 7 ?• So, when n = 1, the 1st term is 4. When

n = 3, the 3rd term is 6. When n = 4, the 4th term is 7. • What is the 8th term?• What is the nth term?

Another easy one• Consider the sequence• 10, 15, 20, 25, 30, 35, 40, …, n• n: 1 2 3 4 5• nth term: 10 15 20 25 30• What is the 6th term? What is the 10th term?• What is the nth term?

Find a general formula• First, make sure you understand the

pattern: draw the next three terms.

Finding a general formula• Describe in words what is happening:

• “Each time, you are adding two more.”

• “The sides get bigger each time.”

• “The sides are the same length.”

• “The first term has no sides.”

• Experiment with finding a pattern.

Generalized formula• Let’s make a table to help us.• Term: 1 2 3 4 5 6 7• nth term: 1 3 5 7 9 11 13• We know that each term increases by

2. So, try 2n: 2, 4, 6, 8, 10, 12, 14, …• Close: Each term is too big by 1, so try

2n - 1: 1, 3, 5, 7, 9, … √

Another thing• If that doesn’t jump out at you… think it through.• 1, 1 + 2, 1 + 2 + 2, 1 + 2 + 2 + 2, …• This can be simplified:• 1, 1 + 2, 1 + 2 • 2, 1 + 2 • 3, 1 + 2 • 4.• Now relate it to your terms:• Term: 1 2 3 4 5• nth term: 1 1 + 2 1 + 2 • 2 1 + 2 • 3 1 + 2 • 4• 1 + n 1 + 2(n-1) 1 + 2(n-1), • 1 + 2(n - 1) = 1 + 2n - 2, or 2n - 1. Check this.

Three more• Arrange pattern blocks in a straight line:• (1) Find the perimeter of n orange

squares.

• (2) Find the perimeter of n yellow hexagons.

• (3) Find the perimeter of n red trapezoids.

• (1) Find the perimeter of n orange squares. 2n + 2

• (2) Find the perimeter of n yellow hexagons. 4n + 2

• (3) Find the perimeter of n red trapezoids.

3n + 2

Exploration 2.7• p. 32

• Part 1 #1 a, b, c; 2a, b, c; 3, 6.

• Note for 6: if you cannot find the general formula, write in words how to get to the next term.

http://mathforum.org/library/drmath/view/56941.html

Term # dots

1 5

2 5 + 7

3 12 + 10

4 22 + 13

5 35 + 16

n n(n - 3)/2 or (n + 1)(3n + 2)/2