Download 2, 4, 6 ,8, What is the Sequence that We Make?

Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Patterns, Relationships, and Algebraic Thinking
Activity:
2, 4, 6 ,8, What is the Sequence that We Make?
TEKS:
(6.4) Patterns, relationships, and algebraic thinking. The student uses
letters as variables in mathematical expressions to describe how one
quantity changes when a related quantity changes.
The student is expected to:
(A) use tables and symbols to represent and describe proportional
and other relationships such as those involving conversions,
arithmetic sequences (with a constant rate of change),
perimeter and area;
(6.12) Underlying processes and mathematical tools. The student
communicates about Grade 6 mathematics through informal and
mathematical language, representations, and models.
The student is expected to:
(A) communicate mathematical ideas using language, efficient
tools, appropriate units, and graphical, numerical, physical, or
algebraic mathematical models; and
(6.13) Underlying processes and mathematical tools. The student
uses logical reasoning to make conjectures and verify conclusions.
The student is expected to:
(A) make conjectures from patterns or sets of examples and nonexamples; and
(B) validate his/her conclusions using mathematical properties and
relationships.
Overview:
Students will learn to determine whether or not a series of numbers is an
arithmetic sequence by finding the difference between consecutive terms
in the sequence (the constant rate of change). Students will create
artifacts in the form of notes that may be used to document participation
and understanding. Use of both oral and written communication skills is
also required as students justify their solutions.
Materials:
Overhead or board
Understanding Arithmetic Sequences handout
Sequence Exploration Questions handout
Grouping:
Whole group and small groups of 2-3
Time:
60-90 minutes
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Lesson:
1.
Procedures
Write the following vocabulary terms on the
board/overhead. Ask students if they have
ever heard these terms before. Then using
class discussion, provide either a definition
or example of the use of each term.
sequence
constant
element
one-to-one correspondence
term
Notes
Focus on key vocabulary terms
and have students take notes
during the discussion component
of the lesson.
Look out for definitions of terms
that demonstrate an
understanding of the word, but not
the technical meaning that is
needed for its application in
mathematics. For example, a
student may say that a term is the
length in office of a political
official. Although this is correct, it
is not the mathematical meaning
that is needed for this lesson. If
these types of definitions arise,
validate their use, but also discuss
with students that there is another
meaning associated with the word
and provide definitions that are
more aligned with the use in
mathematics.
Suggest definitions and examples
to clarify meaning and include in
student notes. For example:
A sequence is a set of numbers in
a specific order. What this means
is that the set of numbers can be
put into a one-to-one
correspondence with the Counting
Numbers (1, 2, 3, 4, ...). You can
talk about the 1st element (or
term) in a sequence or the 10th
element in a sequence or the
101st element in a sequence.
2.
Write this sequence on the board/overhead:
1, 4, 7, 10, 13, 16, . .
Ask students “What is the fourth element or
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Use a Think-Pair-Share strategy
with this part of the lesson.
Students have a minute or so to
study the sequence and think
about a response. Then students
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Procedures
term in this sequence? What would be the
next 3 terms in this sequence? Explain how
you determined those terms.”
Notes
are directed to share their ideas in
pairs or groups of 3. Then call
upon several students to orally
describe the pattern they see and
how the next number in the
sequence is produced. Look for
consensus of explanations.
After some discussion, students
should summarize that the
sequence that begins 1, 4, 7, 10,
13, 16, . . . is extended by adding
3 each time. Explain to students
that this is an arithmetic
sequence since the difference
between consecutive terms is
always 3. This difference of 3
may also be described as the
constant rate of change. Direct
them to add this example and
definition to their notes.
Note: Adding three to the previous
term is recursive thinking.
Students have used recursive
thinking throughout their
mathematics explorations up to 6th
grade. In 6th grade the goal is to
move students to an input/output
way of thinking.
3.
Term #
Sequence
1
1
2
4
3
7
4
10
5
13
Help students generalize this
sequence by pointing out that the
repeated add 3 can be expressed
as multiplication.
1X3=3
2X3=6
3X3=9
However, these outputs do not
match the given sequence.
Ask, “Do the two sets of output
have anything in common?” Yes,
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Procedures
Tarleton State University
Notes
both have a common difference of
three between terms.
Is there anything we could do to
the second sequence (3, 6, 9, …)
to turn it into the first sequence?
Yes, subtract two from each term.
1X3-2=3-2=1
2X3-2=6–2=4
3X3-2=9–2=7
Now we are ready to generalize.
The nth term then would be
3n – 2
An alternative approach is to use
the table and back up to the zeroth
term.
4.
Write this sequence on the board/overhead:
8, 6, 4, 2, 0, -2, -4, . . .
5.
This is an arithmetic sequence
since the difference between
consecutive terms is always -2.
Describe how you would determine the next
3 terms in the sequence. Is this an example
of an arithmetic sequence? Explain why/why
not.
Some students may have difficulty
with this sequence since it
involves integers. You may wish
to have several students come to
the board to justify that this is an
arithmetic sequence by finding the
difference between each term.
Now write this sequence on the
board/overhead:
This is not an arithmetic
sequence. After the first two
terms, each term is found by
adding the previous two terms.
2, 2, 4, 6, 10, 16, 26, . . .
Describe how you would find the next three
terms in the sequence. Is this an example of
an arithmetic sequence? Explain why or why
not.
Students should say that there is
not a constant rate of change or a
common difference between
terms.
If students are still unsure about
this, use additional non-arithmetic
sequences for practice.
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Procedures
Tarleton State University
Notes
1, 3, 6, 10, 15, 21, . . .
This sequence is quadratic.
2, 4, 8, 16, 32, . . .
This sequence is exponential.
It is very important for students to
work with both examples and nonexamples. Do not leave out this
step. Any non-linear sequence
can be used if the ones here are
not appropriate for your students.
6.
Distribute Sequence Exploration Questions
handout.
After you have explored the
sequences from the previous
component of the lesson, have
students work in groups to answer
the questions from the Sequence
Exploration Questions handout.
Some students may have difficulty
with sequences that involve
integers. Have the groups work
together to discuss and answer
the questions. Then conduct a
brief class discussion after about 5
minutes of small group discussion.
7.
Consider the sequence:
Remind students that in order to
identify whether a pattern is an
arithmetic sequence you must
examine consecutive terms. If all
consecutive terms have a
common difference, you can
conclude that the sequence is
arithmetic.
4, 11, 18, 25, 32, . . .
Since
11 - 4 = 7
18 - 11 = 7
25 - 18 = 7
32 - 25 = 7
the sequence is arithmetic.
This sequence is arithmetic. We
can continue to find subsequent
terms by adding 7. Therefore, the
sequence continues:
39, 46, 53, etc.
Ask student to find an expression
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
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Mathematics TEKS Refinement 2006 – 6-8
Procedures
8.
Distribute Understanding Arithmetic
Sequences handout.
Make a copy for overhead and place on
overhead.
Ask students to consider the sequence:
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Notes
for the general term of the
sequence (nth term).
Have students now look at a
different representation of this
sequence by organizing a table.
Discuss the phrase “nth term.”
This can represent any term in the
sequence. Have students add this
to their notes.
4, 8, 12, 16, . . .
Direct students that they will now create a
table that can be used to also show the
arithmetic sequence and identify each term
in the sequence, including the nth term.
Have students come to overhead to
complete the table for the first 2 terms.
Direct the class to pay attention to
how the table is to be read. To
this point, we have looked for a
common difference to determine
an arithmetic sequence. What is
the common difference with our
new sequence? It is 4. But now,
we are going to place our data in a
table and look for a mathematical
way of showing how to produce
the next term in the sequence. As
we enter the information in the
table, be aware of common
misunderstandings that may
occur.
Error Alerts:
We have found common
differences, and some students
may wish to express the nth term
as n + 4 rather than 4n.
Ask students how this part of the
lesson differs from what they have
done in the previous exercises.
Note: In the previous exercises,
only the common differences were
the focus. Now we look at the
common differences and a table
that will show a different
representation as we organize the
data and complete the process
column.
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Procedures
Tarleton State University
Notes
Point out to students that the idea
behind arithmetic sequences is to
determine a common difference
as well as any add-on values used
to produce the sequence.
Compare the two answer keys
provided. Look first at the key
which provides the add-on
process in the process column,
and then compare that to the
multiplication in the second key.
Help students make connections
among the two representations,
but remind them that if it cannot
be shown as an add-on of the
common difference, then it is not
an arithmetic sequence.
If students have trouble with this
last part, go back to the previous
arithmetic sequences and build a
table for each of them.
9.
Student Reflection
Give an example of an arithmetic sequence.
You must provide mathematical proof that it
is an arithmetic sequence by identifying that
all consecutive terms have a common
difference. Provide a non-example of an
arithmetic sequence. Again, you must use
mathematics to prove that it is not an
arithmetic sequence.
Homework:
Consider the following sequence: 2, 4, 6, 8, 10, …
1. Write a verbal explanation of how to find the next 5 terms in the
sequence.
2. Create and complete a table for the sequence. Use the following
format for the table:
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Term
Tarleton State University
Process
Number
3. Identify the next 5 terms in the sequence.
4. Write an algebraic representation that shows how to determine
the nth term in the sequence.
Suppose a sequence has a common difference of 5 and starts with
– 13. Generate the first 5 terms of the sequence. Put the term
number and the sequence value in a table. Show the process you
used to generate the terms.
Assessment:
Place the following words on the board/overhead. Ask students to
write a paragraph that summarizes today’s lesson. Each of the
following words must be used in the paragraph in a mathematically
correct way. Students should also provide examples and nonexamples as part of their writing.
sequence
constant
element
one-to-one correspondence
term
constant rate of change
arithmetic sequence
common difference
Extensions:
1. Pick one of the activities from the lesson. Have students change a
term in the sequence and describe how it affects the sequence.
2. Consider the sequence 4, 11, 18, 25, 32 . . . once again.
Use the following tabular representation and look for a pattern to
complete the table:
Term
Term 1
Term 2
Term 3
Process 1
4
4+7
11 + 7
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Process 2
= 4 + (1 X 7)
= 4 + 7 + 7 = 4 + (2 X 7)
Number
= 11
= 18
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Note: You may not wish to have students go as far as coming up with
the algebraic representation, but they will need to know that
sometimes the arithmetic sequence may not be simply put in
terms of multiplication.
Key:
Term 1
Term 2
Term 3
Term 4
.
.
.
Term 12
Term n
4
4+7
11 + 7
18 + 7
= 4 + (1 X 7)
= 4 + 7 + 7 = 4 + (2 X 7)
= 4 + 7 + 7 + 7 = 4 + (3 X 7)
.
.
.
74 + 7
= 11
= 18
= 25
.
.
.
= 4 + (11 X 7)
= 4 + [(n - 1) X 7]
.
.
.
= 81
The same strategy can be used with any arithmetic sequence. If the
first term is designated by the letter a and the common difference is
designated by the letter d, then the value of the nth term, an, can be
described by:
an = a + (n – 1)*d
Modifications: 1. Distribute the Sequence Exploration Questions student handout.
After you have explored the sequences, have students work in
groups to answer the questions from the Sequence Exploration
Questions handout. Some students may have difficulty with
sequences that involve integers. Allow groups to work together to
discuss and answer the questions. Conduct a brief class
discussion after about 5 minutes of group discussion.
2. Counters may be used to model the sequences which do not
include negative integers.
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Sequence Exploration Questions
Student Handout
1.
What effect does a negative starting number have
on the sequence?
2.
What effect does a large negative starting number
have on the sequence?
3.
What effect does a positive starting number have
on the sequence?
4.
What effect does a large positive number have on
the sequence?
5.
What effect does a negative add-on have on the
sequence?
6.
What effect does a positive add-on have on the
sequence?
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Understanding Arithmetic Sequences
Term
Process
Number
n
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Understanding Arithmetic Sequences
Keys
Term
Process
Number
1
1x4
4
2
2x4
8
3
3x4
12
4
4x4
16
5
5x4
20
6
6x4
24
7
7x4
28
n
n x 4
4n
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Term
Process
Number
1
4
4
2
4+4
8
3
4 + (4 + 4)
12
4
4 + (4 + 4 + 4)
16
5
4 + (4 + 4 + 4 + 4)
20
6
4 + (4 + 4 + 4 + 4 + 4)
24
7
4 + (4 + 4 + 4 + 4 + 4 + 4)
28
n
Patterns, Relationships, and Algebraic Thinking
2, 4, 6, 8, What is the Sequence that We Make?
4n
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