Sequence, arithmetic sequence, common divisor

Organizational Information

  • Grades/Course: 11, Algebra 2
  • 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


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