TExES Math 4-8 (115) cheat sheet

Every rational number can be written as a/b where a and b are integers and b ≠ 0; irrational numbers cannot.

Place value: each position to the left is 10 times the value to its right; to the right, each position is 1/10 of the value to its left.

For fractions a/b and c/d: add or subtract with common denominators; multiply numerators and denominators; divide by multiplying by the reciprocal d/c.

Percent means per 100: part = percent × whole; percent change = (new − old)/old × 100%.

Order of operations: grouping symbols, exponents and roots, multiplication and division left to right, addition and subtraction left to right.

GCF is the product of shared prime factors; LCM is the product of the greatest powers of all prime factors appearing in the numbers.

Integer operations: same signs add and keep the sign; different signs subtract and take the sign of the larger absolute value; negative × negative = positive.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts exponents.

Real numbers include rational and irrational numbers; integers are whole numbers and their opposites; whole numbers are 0, 1, 2, 3, ...

Slope m = (y2 − y1)/(x2 − x1); slope-intercept form y = mx + b; point-slope form y − y1 = m(x − x1).

A linear function has a constant rate of change; its graph is a straight line; Ax + By = C is standard form.

Solve systems by substitution, elimination, or graphing; the solution is the point where the graphs intersect.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a) for ax² + bx + c = 0.

Discriminant b² − 4ac: positive gives two real roots, zero gives one real root, negative gives two complex roots.

To factor ax² + bx + c, find two numbers whose product is ac and whose sum is b; check by expanding.

Arithmetic sequence: a_n = a_1 + (n−1)d; geometric sequence: a_n = a_1 r^(n−1).

Exponential growth or decay: y = a(1 ± r)^t, where r is the rate per period and t is the number of periods.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.

Area formulas: rectangle A = lw; triangle A = ½bh; parallelogram A = bh; trapezoid A = ½(b1 + b2)h; circle A = πr².

Circumference C = 2πr = πd; volume of a rectangular prism V = lwh; volume of a cylinder V = πr²h.

Pythagorean theorem: a² + b² = c² for right triangles; its converse identifies right triangles.

Similar figures have equal corresponding angles and proportional side lengths; scale factor k multiplies lengths by k, areas by k², and volumes by k³.

Angle relationships: vertical angles are equal; complementary angles sum to 90°; supplementary angles sum to 180°; triangle angles sum to 180°.

Distance formula √[(x2−x1)² + (y2−y1)²]; midpoint formula ((x1+x2)/2, (y1+y2)/2).

Transformations: translation slides, rotation turns, reflection flips, dilation scales; rigid motions preserve size and shape.

Surface area of a rectangular prism SA = 2lw + 2lh + 2wh; surface area of a cube SA = 6s².

Unit conversions: multiply by a conversion factor with the desired unit in the numerator; 1 in = 2.54 cm and 1 km = 1000 m.

Probability of an event = favorable outcomes / total equally likely outcomes; 0 ≤ P(E) ≤ 1.

Complement rule: P(not E) = 1 − P(E).

Independent events: P(A and B) = P(A)P(B); dependent events use P(A and B) = P(A)P(B|A).

Mutually exclusive events: P(A or B) = P(A) + P(B); otherwise subtract P(A and B).

Mean = sum / count; median = middle value in ordered data; mode = most frequent value; range = maximum − minimum.

Interquartile range IQR = Q3 − Q1; outliers are often defined as values below Q1 − 1.5IQR or above Q3 + 1.5IQR.

Correlation coefficient r ranges from −1 to 1; near ±1 indicates strong linear association; near 0 indicates weak or no linear association.

Empirical rule for normal distributions: about 68% of values within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Random sampling reduces bias; larger samples generally produce more stable estimates of population characteristics.

Problem solving: understand the problem, devise a plan, carry out the plan, and look back to verify the result.

Multiple representations include verbal, numeric, algebraic, graphical, and geometric forms of the same mathematical idea.

Mathematical modeling translates a real situation into mathematics, solves the model, interprets the result, and validates it in context.

Reasoning and proof include inductive pattern finding and deductive argument; a single counterexample disproves a general statement.

Communication requires precise mathematical language, symbols, definitions, and justification of each step.

Connections link mathematics topics to one another, to other disciplines, and to real-world contexts.

Procedural fluency and conceptual understanding reinforce each other; memorized steps without meaning are fragile.

Technology and tools can support exploration, but results require estimation and reasonableness checks.

Piaget described sensorimotor, preoperational, concrete operational, and formal operational stages; instruction should match cognitive readiness.

Vygotsky defined the zone of proximal development as the gap between independent performance and performance with guidance; scaffolding supports learning in that zone.

Bruner's concrete-representational-abstract sequence moves from manipulatives to drawings to symbolic notation.

Formative assessment monitors learning during instruction; summative assessment evaluates learning at the end of instruction.

Diagnostic assessment identifies prior knowledge and misconceptions before instruction begins.

Error analysis examines patterns in student mistakes to reveal misconceptions and guide reteaching.

Differentiation adjusts content, process, or product based on readiness, interests, and learning profiles.

Manipulatives and multiple representations build conceptual understanding before procedural fluency.

Standards-based instruction aligns objectives, activities, and assessments with defined learning expectations.