# Arithmetic Sequences

Sequence, arithmetic sequence, common divisor

# Organizational Information

• Lesson Number: Arithmetic Sequences
• Time Required: 40 minutes
• Original Author: Marion Blanchard

# Big Ideas

## Essential Question(s) for this Lesson

• What patterns exist in Math?
• How can you use Math to make problems simpler?

## NYS Standards Addressed for this Lesson

• A2.A.29 Identify an arithmetic or geometric sequence and find the formula for its nth term
• A2.A.30 Determine the common difference in an arithmetic sequence
• A2.A.32 Determine the specified term of an arithmetic or geometric sequence.
• G.PS.2 Observe and explain patterns to formulate generalizations and conjectures
• Math B – Modeling / Multiple Representations
-- 4A Represent problem situations symbolically …
-- Use symbolic form to represent an explicit rule for a sequence.

## Evidence of Student Understanding (Assessment) for this Lesson

• Participation
• Worksheet in class
• Closure work

# Lesson Preparation

## Student Preparation Prior to this Lesson

• None, this is the first class in the unit

## Materials Required

• Number and Patterns worksheet – I per student
• Challenge problem – 1 per student
• Unit tile blocks for manipulative approach to sequences if necessary

## Specific Purpose(s) or Objective(s)

• Use of mathematical notation
• See that tedious and repetitive math problems may have a simpler pattern / solution
• Pattern recognition
• Arithmetic sequence notation and usage

# Lesson Sequence

## Hook

• A rich relative is leaving you money. You get a penny today, 2 pennies tomorrow, 4 pennies the next day, this continues. On which day do you get \$10.01?

## Step by Step Explanation of Activities/Strategies

1. Do Now – (3 minutes)
• Put these sequences on the board. Students to find the next 3 numbers:
• 2, 4, 6, 8, … (arithmetic)
• 1, 4, 9, 16, …
• 15, 13, 17, 15, 19, ….
• 5, 10, 20, 40, …. (geometric)
• 1, 1, 2, 3, 5, … (Fibonacci)
• 1, 7, 13, …

2. Talk about each sequence. (6 minutes)
• Any pattern you can explain is valid, as long as you can explain it to me.
• Mention Fibonacci sequence, geometric sequence.
• Our lesson will be on arithmetic sequences”
• What is the first term? What is the next term? What did you do to get there?

3. Hand out number patterns worksheet and challenge worksheet (5 minutes)
• Class start working on them. Stop after 5 minutes. Save this worksheet. You will need it again

4. Introduce notation: a1, an, d (common difference) (15 minutes)
• backwards terminology! – we are adding and the word refers to subtraction
• Write out the a1, a2, a3,.. of some of the series from the worksheet. Go forward 2 terms, what is answer? Two more. Then ask for the 105th term.
• Look for general definition of nth term: an = a1 + (n-1)d
• So back to the starting problem – How do we set this up?
• an = a1 + (n-1)d;
• an = \$10.01,
• a1 =\$.01,
• d = .02
• 10.01 =.01 + (n-1) * .02,
• n = 501.
• It takes more than 500 days to get to \$10.00. When we start doing summations of sequences in a few days I will show you something interesting about this problem.

5. Define homework (2 minutes)
• Homework will be to finish worksheet. Write these instructions on the worksheet. Next to each sequence write a big “A” if it is an arithmetic sequence. If it is - write the a1 term and the d term. Write that on the work sheet now so you don’t forget. The challenge problem is optional.

## Closure

• In class – everyone make up an arithmetic sequence. Give it to your partner to solve. The person who is solving: be sure to define a1, d. Find the 243rd term of your sequence. You will hand in to me as you leave. Put both of your names on the sheet.
• I will use some of these as tomorrow morning’s do-nows.----

# Accommodations for Students with Disabilities or Diverse Learning Styles

• Auditory learning – repeat key words – Common difference, nth term, sequence
• Manipulative learning – have small unit square tiles available. Students can create piles of blocks, each pile a term in a sequence.
• Challenge problem for advanced students