B.E.S.T. Math: Algebraic Reasoning K-12

B.E.S.T. Math: Algebraic Reasoning Domain (K-12)

Overview

Algebraic Reasoning builds from understanding patterns in early grades to manipulating symbolic expressions in high school.

Critical Benchmarks by Grade Band

Grades K-2: MA.2.AR.1.1

Benchmark: Solve addition problems with two two-digit numbers and subtraction problems involving two-digit numbers and one-digit whole numbers.

Plain English: Students use place value knowledge to add and subtract within 100, building foundation for seeing addition and subtraction as inverse operations.

Grades 3-5: MA.3.AR.1.1

Benchmark: Apply the distributive property to multiply a one-digit number and a two-digit number. Apply properties of multiplication to find a product of one-digit whole numbers.

Plain English: Students break apart harder multiplication (like 6 x 15) into easier ones (6 x 10 and 6 x 5) and combine results. Cornerstone for mental math and future algebra.

Grades 6-8: MA.7.AR.2.1

Benchmark: Write and solve two-step equations in one variable within a mathematical or real-world context.

Plain English: Students translate word problems into algebraic equations (like 2x + 5 = 17) and solve them, moving from arithmetic to symbolic problem-solving.

Grades 9-12: MA.912.AR.1.1

Benchmark: Interpret and rewrite algebraic expressions and equations in equivalent forms.

Plain English: Students manipulate expressions (factoring, expanding) to reveal different properties or simplify problem-solving. Key skill for all higher-level math.

Vertical Progression

Properties of Operations (Teach Explicitly)

Commutative Property: - Addition: a + b = b + a - Multiplication: a × b = b × a

Associative Property: - Addition: (a + b) + c = a + (b + c) - Multiplication: (a × b) × c = a × (b × c)

Distributive Property: - a(b + c) = ab + ac - Critical for mental math and algebra

Identity Property: - Addition: a + 0 = a - Multiplication: a × 1 = a

Zero Property: - a × 0 = 0

Instructional Strategies

Number Talks for Algebraic Thinking: - Pose problems that invite property use - Example: "How would you solve 9 × 6?" - "I did 10 × 6 minus 6" (distributive) - "I did 9 × 5 plus 9" (distributive) - "I know 6 × 6 is 36, plus 18 more" (building facts)

Error Analysis: Common error: 3(x + 2) = 21 solved as 3x + 2 = 21 - Students must identify: didn't distribute to both terms - Deepens understanding through diagnosing mistakes

Which One Doesn't Belong? Display: 2(x + 3), 2x + 6, 2x + 3, x + x + 6 - Promotes critical thinking about equivalent expressions

FAST Item Types for Algebraic Reasoning

Equation Editor: Type equations/expressions (30% of items) - Students must know on-screen keypad - Common trap: forgetting to simplify

Multiselect: Select ALL that apply - Example: "Which expressions equal 2(x + 3)?" - Must get all correct for credit

Common Distractors: - Misapplying distributive property - Confusing inverse operations - Order of operations errors

Intervention Approach

If Grade 4 student struggles with multiplication properties: 1. Go back to Grade 2/3 addition properties 2. Build concrete understanding with manipulatives 3. Use arrays to visualize distributive property 4. Connect to real-world contexts

Key: The intervention point is often earlier than the current grade level - reinforce foundations.