Identifying Patterns:
Exploring Functional Relationships
Introduction – About the Mathematics
A. Student Performance Objectives - EOC and Related TEKS
Correlation
Objective 2: The student will graph problems involving real-world and
mathematical situations.
(b)(1) Foundations for functions. The student understands that a function
represents a dependence of one quantity on another and can be described in
a variety of ways.
(D)
The student represents relationships among quantities using [concrete
models,] tables, graphs, diagrams, verbal description, equations, and
inequalities.
Objective 3: The student will write linear functions (equations of lines) to
model problems involving real-world and mathematical situations.
(c)(1) Linear functions. The students understands that linear functions can be
represented in different ways and translates among their various
representations.
(C)
The student translates among and uses algebraic, tabular, graphical,
or verbal descriptions of linear functions.
Objective 4: The students will formulate or solve linear equations/inequalities and
systems of linear equations that describe real-world and mathematical situations.
(b)(1) Foundations of functions. The student understands that a function
represents a dependence of one quantity on another and can be described in
a variety of ways.
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(C)
The student describes functional relationships for given problem
situations and writes equations or inequalities to answer questions
arising from the situations.
(b)(3) Foundations of functions. The student understands how algebra can be used
to express generalizations and recognizes and uses the power of symbols to
represent situations.
(B)
Given situations, the student looks for patterns and represents
generalizations algebraically.
Objective 7: The student will use problem-solving strategies to analyze,
solve, and/or justify solutions to real-world and mathematical problems
involving exponents, quadratic situations, or right triangles.
(c)(3) Linear functions. The student formulates equations and inequalities based on
linear functions, uses a variety of methods to solve them, and analyzes the
solutions in terms of the situation.
(A)
The student analyzes situations involving linear functions and
formulates linear equations or inequalities to solve problems.
(B)
The student investigates methods for solving linear equations
and inequalities using [concrete models,] graphs, and the
properties of equality, selects a method, and solves the
equations and inequalities.
Objective 8: The student will use problem-solving strategies to analyze,
solve, and/or justify solutions to real-world and mathematical problems
involving one-variable or two-variable situations.
(b)(1) Foundations of functions. The student understands that a function
represents a dependence of one quantity on another and can be described in
a variety of ways.
(E)
The student interprets and makes inferences from functional
relationships.
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Objective 9: The student will use problem-solving strategies to analyze, solve
and/or justify solutions to real-world and mathematical problems involving
probability, ratio and proportion, or graphical and tabular data.
(b)(1) Foundations of functions. The student understands that a function
represents a dependence of one quantity on another and can be
described in a variety of ways.
(B)
The student gathers and records data, or uses data sets, to
determine functional (systematic) relationships between quantities.
(b)(3) Foundations of functions. The student understands how algebra can be used
to express generalizations and recognizes and uses the power of symbols to
represent situations.
(B)
A.
Given situations, the student looks for patterns and represents
generalizations algebraically.
Critical Mathematics Explored
In developing mathematical models it is important to recognize patterns. In
Algebra I, the most common patterns studied are the linear pattern and the
quadratic pattern. The patterns are most easily recognized by looking at a graph
of the pairs within the pattern. Patterns can also be recognized by looking at a
sequence of numbers. The Pattern Block activity is simply a collection of linear
sequences. All of the rules for the sequences fit the pattern mx + b.
B.
How Students Will Encounter the Concepts
The sequences in this activity are represented by pattern block designs. This gives
the student the opportunity to visualize the numbers involved in a concrete way.
In each activity the student must represent a number concretely with a block
design, verbally with a written description, and by mathematical rule by describing
the arithmetic needed to arrive at the sequence term. We are not trying to teach
them about sequences nor do we want to give any discussion on the mx + b form.
We in fact do not care whether or not the student uses these letter designations.
We want to have them start recognizing the (0,b) pair. We also want them to see
that many patterns are in the form:
constant * variable + constant
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Setting Up--Preparation
List of Materials
Cubes or squares - at least 24 for each group of students
Optional technology
Graphing Calculator or Logger Pro Program
Identifying Patterns Activity Sheets
What The Teacher Should Do To Prepare
You should separate blocks or cubes into baggies with about 24 to a bag. There
should be at least two colors of blocks in the bag since some of the patterns
involve blocks of contrasting colors.
You do not have to give all 10 of the activities at once. You may wish to hold back a
few of them for an assessment activity.
Please do not give a lecture on linear patterns. Also do not tell the students how
to arrive at the pattern rules. Give them time to work on the patterns and
discover the rules for themselves. As you check on each group, look to see how
they are doing in writing the general rule. They may need individual help on using a
variable in a rule or formula. This would be the appropriate time to explain how the
variable may be used to represent any particular term in the sequence.
You may use the vocabulary word sequence, since that is what we are actually using,
but refrain from giving kind of formal definition.
The Teacher's Perspective
Objective: The student will recognize patterns and write a model to represent the
pattern.
Be sure while you are discussing the general rule for each activity that you use the
word model. Each activity actually contains several kinds of models:
Concrete: The picture of the blocks or the actual pattern made with the
blocks.
Verbal: The written description of the pattern.
Table: The columns called term number and numerical value of the term
Function Rule or Formula: The last entry in the numerical value column.
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Answers and Notes
In this answer key, we are providing only the mathematical rule for the pattern.
Pattern 1: n + 6
Pattern 2: 2n + 6
Pattern 3: 4n + 1
Pattern 4: 3n + 3
Pattern 5: 2n + 3
From Identifying Patterns R&A: Pattern 6: 2n + 1
From More Patterns Practice:
Pattern 7: 4n + 3
Pattern 8: 3n + 2
Pattern 9: 5n + 0 (We have included the 0 in order to match the other patterns;
Pattern 10: 3n + 1
however, 5n is an acceptable response.)
Extensions
Internet Research
Search for mathematical sites that have sequence puzzles.
Project Ideas
Give students other mathematical rules such as 3n + 5 and have them make block
designs that match the rule.
Have the students create a block design sequence and then write the rule for the
design sequence.
Students could create a graph of each block pattern. This project would give good
practice with ordered pairs and have the added bonus of letting see that the
patterns all have the shape of a line.
Assessment
To give an assessment on this activity, reserve a few of the block activities/
worksheets. You could also create additional pattern blocks activities to use.
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Name:____________________________Date:____________Period:___
IDENTIFYING PATTERNS #1
Term
Number
Visual
Form
Written
Description
Thought
Process
Numerical
Value
0
1
A house with a 1 block
chimney takes 7
blocks to build
2
A house with a 2
block chimney takes 8
blocks to build
3
A house with a 3
block chimney takes 9
blocks to build
4
5
10
58
100
n
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Identifying Patterns #1
1. Write your pattern rule here and have your teacher check it for accuracy.
_____________________________
2. Use your rule to determine the number of blocks needed to build the house
for term number:
a. 25______ Show your process here:_________________________
b. 90______ Show your process here: ________________________
3. Suppose that it takes 206 blocks to build the house. What would be the
term number of the house? Show how you get your answer.
4. Suppose that it takes 316 blocks to build the house. What would be the
term number of the house? Show how you get your answer.
5. The power of algebra is that it lets us determine formula (symbolic) answers
to questions when we don’t know all the numbers. This exercise is a good
place to see how that works. Think of the words you could use to describe
how you get the answers to numbers 3 and 4. You could say: “subtract 6
from the number of blocks in the house. Mathematically that looks like this:
206-6 and 316-6. With this clue, write an expression that answers this
question:
Suppose that it takes B blocks to build the house. What
Would be the term number of the house?
____________________________________________
This is a good opportunity to learn to use the graphing calculator to create scatter
plots. Your teacher may also want to let you learn to use the computer program
called Logger Pro.
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Identifying Patterns #2
Term
Number
Visual
Form
Written
Description
Thought
Process
Numerical
Value
0
1
8 blocks are needed to
surround 1 empty space
2
10 blocks are needed to
surround 2 empty spaces
3
12 blocks are needed to
surround 3 empty spaces
4
5
10
58
100
n
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Identifying Patterns #2
1.
Write your pattern rule here and have your teacher check it for accuracy.
_____________________________
2. Use your rule to determine the number of blocks needed to build the
pattern for term number:
a. 25______ Show your process here:_________________________
b. 90______ Show your process here: ________________________
3. Suppose that it takes 86 blocks to build the pattern. What would be the
term number of the pattern? Show how you get your answer.
4. Suppose that it takes 316 blocks to build the pattern. What would be the
term number of the pattern? Show how you get your answer.
5. Think of the words you could use to describe how you get the answers to
numbers 3 and 4. Write an expression that answers this question: Suppose
that it takes B blocks to build the pattern. What would be the term number
of the pattern?
_______________________________________________
This is a good opportunity to learn to use the graphing calculator to create scatter
plots. Your teacher may also want to let you learn to use the computer program
called Logger Pro.
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Identifying Patterns #3
Term
Number
Visual
Form
Written
Description
Thought
Process
Numerical
Value
0
1
1st windmill takes 5
blocks to build
2
2nd windmill takes 9
blocks to build
3
3rd windmill takes 13
blocks to build
4
5
10
58
100
n
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Identifying Patterns #3
1.
Write your pattern rule here and have your teacher check it for accuracy.
_____________________________
2. Use your rule to determine the number of blocks needed to build the
pattern for term number:
a. 25______ Show your process here:_________________________
b. 90______ Show your process here: ________________________
3. Suppose that it takes 89 blocks to build the pattern. What would be the
term number of the pattern? Show how you get your answer.
4. Suppose that it takes 321 blocks to build the pattern. What would be the
term number of the pattern? Show how you get your answer.
5. Think of the words you could use to describe how you get the answers to
numbers 3 and 4. Write an expression that answers this question: Suppose
that it takes B blocks to build the pattern. What would be the term number
of the pattern?
_____________________________________________
This is a good opportunity to learn to use the graphing calculator to create scatter
plots. Your teacher may also want to let you learn to use the computer program
called Logger Pro.
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Identifying Patterns #4
Term
Number
Visual
Form
Written
Description
Thought
Process
Numerical
Value
0
1
A chair with 1 block high
legs and back uses 6
blocks
2
A chair with 2 block high
legs and back uses 9
blocks
3
A chair with 3 block high
legs and back uses 12
blocks
4
5
10
58
100
n
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Identifying Patterns #4
1.
Write your pattern rule here and have your teacher check it for accuracy.
_____________________________
2. Use your rule to determine the number of blocks needed to build the
pattern for term number:
a. 25______ Show your process here:_________________________
b. 90______ Show your process here: ________________________
3. Suppose that it takes 102 blocks to build the pattern. What would be the
term number of the pattern? Show how you get your answer.
4. Suppose that it takes 321 blocks to build the pattern. What would be the
term number of the pattern? Show how you get your answer.
5. Think of the words you could use to describe how you get the answers to
numbers 3 and 4. Write an expression that answers this question: Suppose
that it takes B blocks to build the pattern. What would be the term number
of the pattern?
_____________________________________________
This is a good opportunity to learn to use the graphing calculator to create scatter
plots. Your teacher may also want to let you learn to use the computer program
called Logger Pro.
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Identifying Patterns #5
Term
Number
Visual
Form
Written
Description
Thought
Process
Numerical
Value
0
1
A cup 1 space deep
takes 5 blocks to build
2
A cup 2 spaces deep
takes 7 blocks to build
3
A cup 3 spaces deep
takes 9 blocks to build
4
5
10
58
100
n
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Identifying Patterns #5
1.
Write your pattern rule here and have your teacher check it for accuracy.
_____________________________
2. Use your rule to determine the number of blocks needed to build the
pattern for term number:
a. 25______ Show your process here:_________________________
b. 90______ Show your process here: ________________________
3. Suppose that it takes 105 blocks to build the pattern. What would be the
term number of the pattern? Show how you get your answer.
4. Suppose that it takes 321 blocks to build the pattern. What would be the
term number of the pattern? Show how you get your answer.
5. Think of the words you could use to describe how you get the answers to
numbers 3 and 4. Write an expression that answers this question: Suppose
that it takes B blocks to build the pattern. What would be the term number
of the pattern?
_____________________________________________
This is a good opportunity to learn to use the graphing calculator to create scatter
plots. Your teacher may also want to let you learn to use the computer program
called Logger Pro.
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