6.2 Operations & Algebraic Thinking

Addition and Subtraction: Problem Types

The GACE tests the CGI (Cognitively Guided Instruction) problem-type framework. A Georgia K-2 teacher should recognize four broad classes, each split into subtypes by what is unknown.

Joining, Separating, Part-Part-Whole, Comparison

The cognitive load increases as the unknown moves from the result to the change/start/part/difference. Students who can solve result-unknown but not change-unknown have memorized "add the numbers" without modeling the action.

Worked Example: Change-Unknown Join

"Maya had 6 stickers. Her teacher gave her some more. Now Maya has 14. How many did the teacher give her?"

A child who adds 6 + 14 = 20 has misread the structure. The correct model is 6 + ? = 14, solved by counting on from 6 up to 14 — that is 8. Equivalently, 14 − 6 = 8. The same problem can be modeled by addition or subtraction; this flexibility is exactly what "operations and algebraic thinking" cultivates.

Multiplication and Division: From Equal Groups to Algorithms

The GACE expects you to know multiple representations of multiplication and division and the connections among them.

Representations of 3 × 4 = 12

• Equal groups: 3 groups of 4 counters
• Repeated addition: 4 + 4 + 4 = 12
• Array: 3 rows of 4 (rows × columns)
• Area model: a 3-by-4 rectangle, area 12 square units
• Number line: 3 jumps of 4, landing on 12
• Skip counting: 4, 8, 12

Each representation reveals a different facet. The array makes the commutative property visible (rotate it: 4 rows of 3 is also 12). The area model later generalizes to fractions and polynomials.

Division as the Inverse

Division has two meanings:

• Partitive (fair-sharing): 12 cookies shared equally among 3 friends — how many does each get? (12 ÷ 3 = 4)
• Quotitive (measurement): 12 cookies, 3 per bag — how many bags? (12 ÷ 3 = 4)

Same answer, different story. The GACE may ask which interpretation a word problem requires.

Properties of Operations

The distributive property is the bridge from arrays to the standard multiplication algorithm and to algebra. A 4th grader computing 7 × 12 mentally uses the distributive property: 7 × (10 + 2) = 70 + 14 = 84.

Order of Operations (5.OA.1)

By 5th grade, students evaluate expressions with parentheses, brackets, and braces:

Evaluate 3 + 4 × (6 − 2)

1. Parentheses first: 6 − 2 = 4
2. Multiply: 4 × 4 = 16
3. Add: 3 + 16 = 19

A common error is to add 3 + 4 first to get 7, then multiply by 4 to get 28. The GACE checks that you know multiplication has higher precedence than addition unless parentheses override it.

Patterns, Relationships, and Expressions (5.OA)

Three Kinds of Patterns

Kindergarten emphasizes repeating patterns; 3rd-5th shifts to growing patterns, especially arithmetic sequences described by an explicit rule.

Function Tables and Rules

A 4th-grade function table might show:

The rule is "multiply by 2, then add 3", or y = 2x + 3. The GACE expects you to identify the rule from a table and predict the next output (input 5 → 13).

Writing and Evaluating Expressions

Students move from words to symbols. "Add 4 to a number, then multiply by 3" becomes 3(n + 4) — note the parentheses force the addition first. Evaluating for n = 5: 3 × (5 + 4) = 3 × 9 = 27.

A 5th-grade anchor task: "Write an expression for 'the sum of 8 and a number, divided by 2.'" Answer: (8 + n) ÷ 2. Without parentheses, 8 + n ÷ 2 would divide only n by 2 before adding — a different value. The parentheses carry meaning, not decoration. A student who can read 3(n + 4) and explain why the parentheses matter is already thinking algebraically, which is the bridge from 5.OA to the middle-grades Expressions and Equations domain.