6.3 Patterns, Relationships & Algebraic Thinking

Patterns, Relationships, and Algebraic Thinking

Patterns/Relationships/Algebra is one of the four Mathematics Problem Solving content clusters (6 of 48 items on a real Intermediate 2 report). Pearson keeps the cluster names the same across levels, while the content grows with the grade: early levels ask students to continue patterns and find missing numbers, middle levels use input-output tables and expressions, and upper levels and the TASK Mathematics subtest include equations and graphing.

---

Sequence Analysis: Arithmetic, Geometric, and Visual Patterns

A mathematical sequence is an ordered list of elements governed by a defined rule:

1. Arithmetic Sequences

An arithmetic sequence changes by adding or subtracting a constant numerical value known as the common difference ($d$) between consecutive terms ($d = a_{n} - a_{n-1}$).

• Recursive Form: $a_n = a_{n-1} + d$
• Explicit Formula: $a_n = a_1 + (n - 1)d$

Worked Example: Find the 25th term of the sequence: $4, 11, 18, 25, \dots$

• First term: $a_1 = 4$
• Common difference: $d = 11 - 4 = 7$
• Apply explicit formula: $a_{25} = 4 + (25 - 1)(7) = 4 + (24)(7) = 4 + 168 = 172$.

2. Geometric Sequences

A geometric sequence changes by multiplying or dividing by a constant non-zero value known as the common ratio ($r$) between consecutive terms ($r = \frac{a_n}{a_{n-1}}$).

• Recursive Form: $a_n = a_{n-1} \times r$
• Explicit Formula: $a_n = a_1 \times r^{n-1}$
• Example: For the sequence $3, 6, 12, 24, \dots$, the ratio is $r = 2$. The 6th term is $a_6 = 3 \times 2^{6-1} = 3 \times 32 = 96$.

3. Visual and Figural Patterns

Pattern items often display geometric arrays—such as growing tile borders or toothpick figures—and prompt students to identify the mathematical rule governing the $n$-th figure. Students should convert the visual figures into a numerical sequence table representing figure number ($n$) versus total components ($T$), then identify the governing sequence formula.

---

Input-Output Function Tables and Rule Discovery

Function tables represent mathematical relations where every input value ($x$) corresponds to exactly one output value ($y$). Discovering the governing algebraic rule requires a methodical 4-step diagnostic protocol:

Step-by-Step Rule Derivation

1. Determine the Rate of Change (Slope $m$): Divide the change in output by the change in input: $m = \frac{\Delta y}{\Delta x} = \frac{11 - 7}{2 - 1} = \frac{4}{1} = 4$ Notice that between $x = 4$ and $x = 6$, $\Delta y = +8$ and $\Delta x = +2$, yielding $\frac{8}{2} = 4$, confirming a constant rate.
2. Isolate the Constant Offset ($y$-intercept $b$): Substitute the rate $m = 4$ and any known $(x, y)$ coordinate into the linear model $y = mx + b$: $7 = 4(1) + b \implies 7 = 4 + b \implies b = 3$
3. Formulate the Rule: $y = 4x + 3$ (or in functional notation, $f(x) = 4x + 3$).
4. Verify Across Rows: Test with $x = 6$: $4(6) + 3 = 24 + 3 = 27$, matching the table output perfectly.

Translating Verbal Phrases into Algebraic Expressions

Translating linguistic statements into symbolic expressions requires decoding operational signal vocabulary. Subtraction and division phrasing require strict attention to order.

Exam Trap: Multiple-choice items often include the reversed expression as a distractor. For "9 less than twice a number $n$," the correct algebraic translation is $2n - 9$. The distractor $9 - 2n$ is a classic error trap.

---

Writing and Solving Linear Equations

Solving an algebraic equation means finding the specific numerical value that preserves mathematical balance across the equal sign. This requires applying inverse operations symmetrically:

• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.

Two-Step Linear Equation Protocol

1. Initial Equation: $5x - 8 = 37$
2. Step 1 (Undo Subtraction): Add 8 to both sides: $5x - 8 + 8 = 37 + 8 \implies 5x = 45$
3. Step 2 (Undo Multiplication): Divide both sides by 5: $\frac{5x}{5} = \frac{45}{5} \implies x = 9$

Always verify by substituting $x = 9$ into original equation: $5(9) - 8 = 45 - 8 = 37$, confirming validity.

---

Coordinate Graphing and the Cartesian Plane

The Cartesian coordinate plane is constructed from two perpendicular real number lines intersecting at the origin $(0, 0)$:

• $x$-axis: Horizontal number line ($x > 0$ to the right, $x < 0$ to the left).
• $y$-axis: Vertical number line ($y > 0$ upward, $y < 0$ downward).

Points are specified by ordered pairs $(x, y)$, where the first coordinate designates horizontal displacement and the second designates vertical displacement.

Calculating Distances on the Coordinate Grid

For two points aligned on a horizontal or vertical grid line, distance equals the absolute value of coordinate differences:

• Horizontal Distance (same $y$): $d = |x_2 - x_1|$. Distance between $(-4, 5)$ and $(3, 5)$ is $|3 - (-4)| = |3 + 4| = 7$ units.
• Vertical Distance (same $x$): $d = |y_2 - y_1|$. Distance between $(2, -3)$ and $(2, 6)$ is $|6 - (-3)| = |6 + 3| = 9$ units.