Saitama Public High School Entrance Examination - Mathematics study plan

Expand and simplify polynomial products including (a+b)(a−b), (a±b)², and (x+a)(x+b) without a formula sheet, scoring at least 18 of 20 on a self-made drill.

Factorise quadratic expressions by choosing between common factor, difference of squares, and sum-product patterns, and justify each choice in one written line.

Solve linear equations and simultaneous equations in two variables, then verify each solution by substitution before checking an answer key.

Convert between decimals and fractions and simplify expressions containing square roots, including rationalising a denominator with a single surd term.

Solve linear inequalities and state the answer on a number line, correctly handling multiplication or division by a negative number.

Solve quadratic equations by factorisation and by completing the square, and decide in advance which method is faster for a given equation.

Apply algebraic expressions to word problems about speed, concentration, and price, defining the variable in writing before solving.

Reproduce from memory the expansion and factorisation identities and the quadratic formula, then check them against a textbook.

Sketch linear, quadratic, and inverse-proportion graphs from their equations, marking intercepts and the vertex or asymptotes.

Translate a graph left, right, up, and down and write the new equation for each shift.

Determine the equation of a line from two points, from a point and a gradient, and from a graph read off a grid.

Solve problems that combine a linear graph with a quadratic graph, finding intersection points algebraically and confirming them on the sketch.

Interpret rate-of-change as gradient in real contexts such as distance–time and volume–time graphs, and describe the meaning of a flat section or a steeper section in words.

Compute mean, median, mode, and range from a frequency table, and explain which measure best represents a skewed data set.

Read and construct histograms and relative-frequency tables, choosing appropriate class intervals.

Calculate the probability of simple and compound events using the addition and multiplication rules, distinguishing mutually exclusive from independent events.

State and apply the angle properties of parallel lines, triangles, and polygons, including the interior and exterior angle sums, in multi-step angle chases.

Prove two triangles congruent using the three congruence conditions and name the condition used at each step.

Apply the Pythagorean theorem and similarity ratios to find unknown lengths, including problems with two similar triangles sharing a vertex.

Compute areas and volumes of composite figures made from triangles, circles, prisms, and pyramids.

Write a two-column or paragraph proof of a triangle or circle property, starting from the given and ending at the conclusion, with every claim supported.

Identify the converse, contrapositive, and negation of a statement and judge whether each is true.

Distinguish a conjecture from a proof, and produce a counterexample to disprove a false general statement.

Use inscribed-angle and tangent properties to justify angle and length results in circle problems.

Complete a full past-paper-style paper under strict time limits, allocating time per section before starting.

Classify every wrong answer by domain and by cause, separating careless slips from gaps in understanding.

Re-solve all gap-type errors from memory the following day without looking at the original solution.

Speed up routine algebra and graph sketching so that time saved is reallocated to geometry and proof questions.

Re-drill the five weakest skills identified in week 7 until each is solved correctly twice in a row without notes.

Practise explaining a proof aloud in under two minutes, covering the given, the construction, and the conclusion.

Review the error logs from weeks 1–7 and rewrite any rule that was misstated.

Sit one final timed paper and confirm that section timings match the plan.