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.

(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.

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)Given situations, the student looks for patterns and represents generalizations algebraically.

  1. 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.

  1. 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

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.

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.

SATEC/Algebra I/Foundations of F’ns/1.01 Identifying Patterns.doc/Rev. 07-01 Page 1/15

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

Identifying Patterns #1

  1. Write your pattern rule here and have your teacher check it for accuracy.

______

  1. Use your rule to determine the number of blocks needed to build the house

for term number:

  1. 25______Show your process here:______
  1. 90______Show your process here: ______
  1. 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.
  1. 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.
  1. 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.

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

Identifying Patterns #2

  1. Write your pattern rule here and have your teacher check it for accuracy.

______

  1. Use your rule to determine the number of blocks needed to build the pattern for term number:
  1. 25______Show your process here:______
  1. 90______Show your process here: ______
  1. 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.
  1. 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.
  1. 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.

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

Identifying Patterns #3

  1. Write your pattern rule here and have your teacher check it for accuracy.

______

  1. Use your rule to determine the number of blocks needed to build the pattern for term number:
  1. 25______Show your process here:______
  1. 90______Show your process here: ______
  1. 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.
  1. 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.
  1. 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.

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

Identifying Patterns #4

  1. Write your pattern rule here and have your teacher check it for accuracy.

______

  1. Use your rule to determine the number of blocks needed to build the pattern for term number:
  1. 25______Show your process here:______
  1. 90______Show your process here: ______
  1. 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.
  1. 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.
  1. 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 calledLogger Pro.

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

Identifying Patterns #5

  1. Write your pattern rule here and have your teacher check it for accuracy.

______

  1. Use your rule to determine the number of blocks needed to build the pattern for term number:
  1. 25______Show your process here:______
  1. 90______Show your process here: ______
  1. 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.
  1. 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.
  1. 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 calledLogger Pro.

SATEC/Algebra I/Foundations of F’ns/1.01 Identifying Patterns.doc/Rev. 07-01 Page 1/15