Finding the nth Term

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Objective

Students will be able to use inductive reasoning to find the rule for a linear pattern.

Big Idea

By playing the Silent Board Game, students will take risks to try to find the rule to represent a linear pattern.

Silent Board Game

15 minutes

Since we use both inductive and deductive reasoning extensively throughout this year in geometry, I use the Silent Board Game to introduce students to inductive reasoning while activating some of their pattern recognition skills.

I like to introduce Inductive Reasoning through the Silent Board Game.  In the Silent Board Game, there are two rules:

  1. Everyone must stay silent.
  2. If I hand a student a marker, the student must come forward to put down a guess.  If the answer is correct, it will stay on the board.  If the answer is incorrect, I will erase it.

The Silent Board Game allows students to problem solve by looking for patterns, taking risks, and ultimately, finding a general rule (MP1).  Additionally, it helps us to create a classroom culture of making mistakes and learning from them, since incorrect guesses often lead us to a better understanding of how x and y relate to each other.

 

y = 2x+10

 

x

0

2

4

1

-1

10

 

y

10

14

18

12

8

30

 

 

y = x2+1

 

x

0

2

4

1

-1

10

 

y

1

5

17

2

2

101

 

 

 

 

Investigation: Finding the nth Term

10 minutes

In the Finding the nth Term Investigation, I ask students to look for patterns in the table and to algebraically represent how points on a line divide the line into segments and non-overlapping rays.  After students check in with me about how they expressed the general rule, I give them four different linear patterns for which they will complete a table, and find a general rule that represents the situation.  As students get more exposure to linear patterns, they notice the features of linear equations: a constant rate of change and some kind of initial value--in this sense, students express regularity in repeated reasoning (MP8) and look for and make use of structure (MP7).

 

Debrief

10 minutes

I debrief the lesson by telling students that rules that generate a sequence with a constant difference are called linear functions.  I tell them that inductive reasoning is the process of observing data, looking for patterns, and making generalizations about those patterns.

 

Exit Ticket:  Generalize the pattern to find the expression for the nth term.

Term

1

2

3

4

5

n

Value

20

27

34

41

48