9.1 Patterns, Sequences & Functions

Overview & Exam Relevance

Competency 003 of the TExES Core Subjects EC-6 (391) Mathematics subtest (902) assesses the candidate's understanding of patterns, relations, functions, and algebraic reasoning. In elementary education, algebraic thinking does not begin with secondary-level quadratic equations or formal abstract proofs. Instead, early algebra represents a coherent habit of mind: noticing regularity, analyzing mathematical structure, generalizing computational properties, exploring functional covariation, and conceptualizing equivalence.

The Texas Essential Knowledge and Skills (TEKS) systematically progress from concrete pattern exploration in Pre-Kindergarten and Kindergarten (identifying, replicating, and extending repeating sensory patterns) to analyzing growing additive patterns in Grades 1–3, formalizing input-output tables in Grades 4–5, and graphing ordered pairs from patterns and tables in the first quadrant in Grade 5 and in all four quadrants in Grade 6. This section covers patterns, sequences, and functions; equality, equations, inequalities, and graphing follow in the next section. Exam candidates must know how to diagnose common student misconceptions—particularly regarding the equals sign and functional tables—and select developmentally appropriate representations that bridge concrete models to symbolic notation.

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Classification of Mathematical Patterns

Elementary mathematics organizes patterns into distinct structural categories based on their generative rules: repeating patterns, growing (arithmetic) patterns, shrinking patterns, and geometric (multiplicative) patterns.

MATHEMATICAL PATTERNS IN ELEMENTARY GRADES
│
├── Repeating Patterns (Fixed Core Unit)
│   ├── AB, ABB, AAB, ABC, AABB
│   └── Predict term n via Modular Arithmetic: n ÷ (core length) → Remainder
│
├── Growing & Shrinking Patterns (Arithmetic Sequences)
│   ├── Constant Additive/Subtractive Difference (d)
│   ├── Recursive View: Next = Current + d
│   └── Explicit/Functional View: y = d · n + c
│
└── Geometric Patterns (Exponential Sequences)
    ├── Constant Multiplicative Ratio (r)
    └── Exponential Growth: y = a · r^(n-1)

1. Repeating Patterns

A repeating pattern contains a distinct, unchanging sequence of elements called the core (the shortest repeating block) that recurs cyclically without variation. Repeating patterns are typically introduced using tactile manipulatives (colored counters, linking cubes, attribute blocks) and auditory rhythms before transitioning to symbolic letter representations:

• AB Pattern: Red, Blue, Red, Blue... (Core length: 2)
• AAB Pattern: Circle, Circle, Triangle, Circle, Circle, Triangle... (Core length: 3)
• ABC Pattern: Clap, Snap, Stomp, Clap, Snap, Stomp... (Core length: 3)
• AABB Pattern: Dog, Dog, Cat, Cat, Dog, Dog, Cat, Cat... (Core length: 4)

Determining Distant Terms (Modular Arithmetic)

A foundational pedagogical objective is weaning students from drawing or building patterns step-by-step to find distant terms. Instead, students learn to use division with remainders:

• Problem: What is the 58th shape in an ABC pattern where A = Square, B = Circle, and C = Triangle?
• Step 1: Identify the core length ($L = 3$).
• Step 2: Divide the target position by the core length: $58 \div 3 = 19$ with a remainder of 1 ($58 = 19 \times 3 + 1$).
• Step 3: Interpret the remainder: The core repeats 19 complete times. The remainder of 1 points to the 1st element in the core, which is the Square. If the remainder were 2, it would be the Circle; if the remainder were 0 (no remainder), the term would correspond to the final element of the core (Triangle).

2. Growing and Shrinking Patterns (Arithmetic Sequences)

A growing pattern (or shrinking pattern) increases or decreases systematically according to an arithmetic rule characterized by a constant difference ($d$):

• Numeric Example: 4, 7, 10, 13, 16... (Constant difference: $+3$)
• Visual / Figural Patterns: Tile trains, toothpick perimeter frames, or staircase arrays where each consecutive stage adds a fixed number of geometric elements.

Recursive versus Explicit Thinking

Elementary educators must understand the critical cognitive distinction between recursive and explicit reasoning:

• Recursive Reasoning (Local / Iterative): Observing how a term relates to the immediately preceding term ("add 3 to the previous number"). While intuitive for young learners, recursive reasoning has severe limitations: finding the 100th term requires calculating all 99 preceding terms.
• Explicit / Functional Reasoning (Global / Structural): Formulating a mathematical rule that directly connects the term position ($n$ or input $x$) to the term value ($y$). For the sequence 4, 7, 10, 13... ($n = 1, 2, 3, 4$), the explicit rule is $y = 3n + 1$. Finding the 100th term is calculated directly: $y = 3(100) + 1 = 301$.

3. Geometric Patterns (Exponential Sequences)

A geometric pattern changes by a constant ratio ($r$) through repeated multiplication or division, rather than repeated addition:

• Numeric Example: 3, 6, 12, 24, 48... (Constant ratio: $\times 2$)
• Classroom Models: Biological cell division, doubling lily pads on a pond, or successive paper-folding layers ($1, 2, 4, 8, 16, 32$). The $n$-th term follows the exponential rule $y = a \cdot r^{n-1}$.

Functions, Relations & Input-Output Tables

In Grades 3–5, the TEKS introduce functional relationships through function machines and input-output tables (T-charts).

Core Functional Definitions

• Relation: Any mathematical pairing between a set of inputs and a set of outputs.
• Function: A specialized relation where each input value ($x$, domain) corresponds to exactly one unique output value ($y$, range).
• Independent Variable: The input variable ($x$), which can be chosen freely.
• Dependent Variable: The output variable ($y$), whose value depends entirely on the input and the functional rule.

Input-Output Table Comparison Matrix

The "Vertical Analysis" Trap in Function Tables

One of the most heavily tested diagnostic errors on the TExES 391 is the vertical difference trap. Consider the following table:

When asked to identify the rule, elementary students frequently look down the output column, see that numbers increase by 3 ($5 + 3 = 8; 8 + 3 = 11$), and declare: "The rule is add 3." When writing the algebraic equation, they incorrectly write $y = x + 3$.

Pedagogical Intervention: The teacher must prompt the student to test their rule horizontally against the first pair: "If $x = 1$, does $1 + 3$ equal 5?" ($1 + 3 = 4 \neq 5$). The teacher then scaffolds the structural relationship: the vertical increment of $+3$ represents the multiplicative rate of change (coefficient), meaning the rule involves $3x$. Multiplying input 1 by 3 yields 3; to reach the output of 5, one must add 2. The explicit functional equation is therefore $y = 3x + 2$.

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Classroom Scenario Application

Classroom Context: Mr. Henderson is teaching a 4th-grade mathematics unit on algebraic expressions. He presents students with a table showing the relationship between the number of picnic tables ($t$) and the maximum number of chairs ($c$) that can be arranged around them:

| Tables ($t$) | 1 | 2 | 3 | 4 | | Chairs ($c$) | 6 | 10 | 14 | 18 |

Observed Challenge: When asked to write a rule connecting tables to chairs, several students state: "The rule is add 4." When asked how many chairs are needed for 20 tables, students attempt to repeatedly add 4 twenty times, making frequent addition errors.

Diagnostic Analysis: Students are analyzing the table recursively down the output column rather than functionally across from input to output. They confuse the rate of change (+4) with an additive rule ($c = t + 4$), which fails when tested ($1 + 4 = 5 \neq 6$).

Targeted Instructional Plan:

1. Concrete Visual Modeling: Mr. Henderson has students arrange square tiles with counters representing chairs. Students observe that each table contributes 4 chairs on the sides, plus 2 permanent chairs at the two ends.
2. Decompose the Output: The teacher creates an intermediate column in the table showing: $1 \times 4 + 2 = 6$; $2 \times 4 + 2 = 10$; $3 \times 4 + 2 = 14$.
3. Horizontal Functional Generalization: Students generalize that the total number of chairs equals 4 times the number of tables plus 2 ($c = 4t + 2$).
4. Efficient Computation: Students compute the chairs for 20 tables directly: $c = 4(20) + 2 = 82$ chairs.