Mathematical Processes and Number Sense
Mathematical Processes and Number Sense
This domain tests how you solve unfamiliar problems, justify reasoning, connect representations, and work flexibly with rational and irrational numbers. Expect proportional reasoning, integer and fraction operations, number theory, estimation, and precise communication of steps.
Core rules and concepts
- Follow order of operations: grouping, exponents and roots, multiplication and division left to right, addition and subtraction left to right.
- Use properties of operations to rewrite expressions: distributive, associative, commutative, identity, and inverse.
- Convert among fractions, decimals, and percents using equivalent forms; compare by benchmarks or common denominators.
- A ratio compares quantities; a rate compares different units. Set up proportions with matching units in the same positions.
- Prime factorization supports GCF, LCM, simplifying radicals, and divisibility checks.
- Absolute value gives distance from zero; square roots of non-perfect squares are irrational.
Worked example
A recipe uses 2 1/3 cups of flour to make 12 muffins. How much flour is needed for 30 muffins?
The scale factor is 30/12 = 2.5. Convert 2 1/3 to 7/3. Then (7/3) × (5/2) = 35/6 = 5 5/6 cups. Reasoning check: 12 muffins use about 2.33 cups, so 30 muffins (2.5 times as many) need about 5.83 cups; 5 5/6 is reasonable.
Traps that cost marks
- Treating a part-to-part ratio as a part-to-whole fraction.
- Dropping negative signs when subtracting integers or distributing a negative.
- Confusing “percent of” with “percent increase”; a 20% increase multiplies by 1.20, not 0.20.
- Mixing units in a proportion, such as cups with muffins in the same ratio position.
- Rounding too early; keep exact fractions until the final step.