Nagasaki High School Entrance Exam - Mathematics study plan

Expand and factorize quadratic and cubic-style expressions accurately without a calculator

Simplify radical expressions including rationalizing denominators and combining like surds

State and apply the Pythagorean theorem and its converse to find unknown lengths in right triangles

Compute areas and volumes of composite figures built from triangles, circles, and prisms

Solve 20 mixed calculation problems under a 30-minute timer with at least 90% accuracy

Prove two triangles congruent using SSS, SAS, ASA, and the right-triangle HL condition

Identify corresponding and alternate angles in parallel-line configurations and justify each step

Use congruence to prove equal segments and equal angles in a written two-column proof

Manipulate algebraic expressions inside geometric proofs (e.g., expressing a length as x+3)

Rewrite three flawed proofs, correcting each missing justification

Apply the inscribed-angle theorem and the tangent-radius perpendicularity property to find unknown angles

Distinguish central angles, inscribed angles, and angles in the same segment in a single diagram

Plot and read linear functions y=ax+b, identifying slope and intercept from a graph

Find the equation of a line from two points and from a point plus slope

Solve intersection-point problems between two lines algebraically and confirm graphically

Interpret proportional and inverse-proportional relationships y=ax and y=a/x, including domain restrictions

Convert between function tables, equations, and graphs for the same relationship

Solve systems of two linear equations by substitution and by elimination

Solve quadratic equations by factoring, completing the square, and the quadratic formula

Model a word problem as an equation and verify the answer against the original conditions

Solve linear inequalities and represent solution sets on a number line, including compound conditions

Determine how many solutions a quadratic equation has from its discriminant

Apply the relationship between roots and coefficients to find unknown coefficients

Solve simultaneous equations where one equation is quadratic (line and parabola intersection)

Check solutions for extraneous values introduced by squaring or by denominator restrictions

Use similarity of triangles to set up and solve proportion equations for unknown lengths

Apply the midpoint theorem and the property that a line parallel to one side divides the other two proportionally

Compute rates of change from a graph, including average speed over an interval

Find the area of a triangle on a coordinate plane using base and height or the surrounding-rectangle method

Combine a parabola and a line in one coordinate problem to find intersection and enclosed area

Organize raw data into a frequency table and compute mean, median, and mode, choosing the appropriate representative value

Read and construct histograms and box plots, and compare two data sets using quartiles and range

Compute theoretical probability for dice, cards, and bag-drawing experiments, using the complement rule where efficient

Explain a solution strategy in writing so another student could follow it without asking questions

Justify why a given generalization fails by producing a counterexample

Complete two full-length mock exams under strict time limits, one at the start of the week and one at the end

Classify every error from the first mock as knowledge gap, calculation slip, misread, or time pressure

Re-solve every missed problem from scratch without looking at notes

Reduce repeated error types by at least half between the first and second mock

Finalize a one-page formula and theorem reference sheet written from memory