Lesson Title: ______________________________________________ Course: ___________________________
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the
integers.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Students will pair up and exchange the data sets that
they developed for homework. Each student will write
a recursive and explicit formula for his or her partner’s
data set. Students will then compare solutions.
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and provide
a foreshadowing of tomorrow? List the questions.
An individual exit ticket will close the lesson.
When you were born, your grandmother opened a savings
account in your name for college. She deposited $250 in a
Bank of America account that had a yearly interest rate
and made you promise not to touch the money until you
graduated from high school.



Age in Years
11
12
13
14
Account
Balance
$474.57 $503.05 $533.23 $565.23
Write a recursive and explicit formula to model the
situation.
How much money will you have in your account
when you graduate? Explain how you got your answer
using either formula.
Is one formula better to use than the other? Why?
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson?
1. Introduce the investigation by asking, “What will happen if a certain ball is dropped from a specific height?
What will happen with the rebound height? How will the rebound height change over time?” (Note: This is an
introductory discussion to access students’ prior knowledge, not to delve into mathematical solutions.)
2. Form groups of 3-4. Provide each group with a motion sensor, a different ball, and graph paper. (Note: Use
different balls for different groups to generate different rebound heights. Consider also giving everyone or some
groups the same ball, so that all or some rebound heights and, thereby, recursive formulas would be the same.)
3. Have each group collect data in order to write a recursive formula to model the rebound height of the ball over
time. Make sure to have students record their starting height. The students should notice that the data does not
have a common difference, and they may begin to look for a common ratio. Teachers should facilitate this
process using guiding questions. If the pattern is not a common difference, what other operations could you
consider for vertical change? What is the best method for finding a common ratio? How should your group best
generalize the common ratio if the data is slightly different? (Note: Notation for geometric recursive is: a0 =
starting point and an+1 = common ratio (an), where n  0. In this case, a0 will be the initial height the ball was
dropped from and the common ratio will be the mean of the rebound ratios from different points.) If groups are
able to write the recursive formula, direct them to write the explicit from the data as well.
4. As a class, discuss what similarities and differences were apparent in this activity as compared to yesterday’s
arithmetic activity. Give group a Venn Diagram and have the small groups brainstorm for about 5 minutes. Have
students share ideas, noting that the rebound height was a geometric sequence and had a common ratio as
opposed to arithmetic with a common difference.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: ______________________________________________ Course: ___________________________
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
5. Have each group present its recursive formula to the class and explain how members arrived at that conclusion.
Be sure to have students include a0. Record the formulas on the board making sure to label the type of ball used.
Discuss the differences among all groups’ formulas and how the type of ball could have affected them. If all or
some of the balls used were the same, focus on calculating and generalizing the rebound ratio for the data found.
6. Ask students to write explicit formulas from the recursive. Allow students to drive discussion and eventually
rewrite each group’s recursive. Have students prove that these formulas yield the same data values. Lead
students to generalize the connection between the recursive and explicit formulas for geometric sequences.
(Note: As an extension, explore exponential decay, rebound ratio, and the rate of decay.)
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student mastery? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened (and conceptual) understanding.
Students will write a recursive and an explicit formula given a geometric data set or sequence. They will translate
and make connections between the two formulas and recognize the limitations of the recursive.
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Using different balls for the activity will yield different rebound heights and thus different recursive formulas for
each group. When the class discusses what causes the difference in formulas, have students discuss the difference in
the balls and how that is shown in the formulas. To simplify, consider giving everyone the same ball or giving some
groups the same ball, so that all or some rebound heights and, thereby, recursive formulas would be the same. The
discussion following would be less rich, but could focus more on gaining the recursive formula if the students were
having trouble mastering that skill.
A common mistake in geometric recursive formulas is to use a1 instead of a0. If a0 is not given, students must divide
a1 by the common ratio to generate a0. This is necessary to derive the explicit formula.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
A different type of ball for each group
Motion Sensors/CBLs for each group
Calculators
Graph paper
Venn Diagrams
Create your own real-life arithmetic or geometric data set
and write recursive and explicit formulas for each. Bring
this to class and be ready to share. (Note: These can be
used to have students determine which are arithmetic and
which are geometric in a following lesson.)
Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you use to
reflect on the effectiveness of this lesson?
Can the students distinguish between recursive and explicit formulas?
Can students calculate the common ratio and find a0 from a set of geometric data?
Can the student write explicit and recursive formulas given a geometric data set?
Can the students determine and justify the appropriate formula to be used for different input values?
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.