Mathematics
What this domain tests
The mathematics domain tests the number, operations, algebraic thinking, geometry, measurement, and data knowledge needed to teach K–5 students. Items frequently show student work or a representation and ask you to diagnose the misconception and select a mathematically valid response. The focus is on meaning, not just procedures.
Core rules and concepts
- Place value is base-ten: multiplying by 10 shifts digits one place left; dividing by 10 shifts them right. This explains decimal algorithms.
- Fractions can be understood as part-whole, as locations on a number line, and as division. Equivalent fractions come from multiplying by a form of 1.
- Dividing by a fraction asks how many of that fraction fit in the dividend. Invert-and-multiply is a shortcut derived from common denominators or multiplicative inverses.
- Properties-commutative, associative, distributive-justify mental math and algebraic moves. Order of operations is a convention, not a property.
- Area measures square units covering a region; perimeter measures distance around it. They are not interchangeable.
Worked example
Compute 3/4 ÷ 1/2. Ask: how many halves are in three-fourths? Two halves make 1, which is more than 3/4, so the answer is between 1 and 2. Using common denominators: 3/4 ÷ 2/4 = 3 ÷ 2 = 1 1/2. The invert-and-multiply form gives 3/4 × 2/1 = 6/4 = 1 1/2. Check by multiplying: 1/2 × 1 1/2 = 3/4. The quotient is 1 1/2, not 3/8; multiplying instead of dividing is the common error.
Traps that cost marks
- Applying “keep, change, flip” to addition or subtraction.
- Assuming multiplication always makes a number larger; multiplying by a fraction less than 1 makes it smaller.
- Placing the decimal point by counting digits without estimating magnitude (0.3 × 0.4 = 0.12, not 1.2).
- Confusing area and perimeter formulas.
- Evaluating left to right while ignoring grouping symbols and precedence.