Nara Public High School Entrance Examination - Mathematics study plan
Nara Public High School Entrance Examination - Mathematics · Give each week a clear focus.
8 weeks · About 7 hours per week
Numbers and Algebraic Expressions (数と式) & Geometry and Figures (図形)
Expand and simplify polynomial products and factorisations, including the difference of two squares and perfect-square patterns, until you can complete ten mixed items in fifteen minutes with no sign errors.
Rationalise denominators containing square roots and simplify expressions such as 6 over root 3 plus root 2, stating each intermediate step.
Apply the Pythagorean theorem and the special right-triangle ratios (1:2:root 3 and 1:1:root 2) to compute unknown sides in composite figures.
Convert between the three standard forms of the quadratic function (general, factored, completed-square) and read off the vertex from each.
Your checkpoint
A one-page formula sheet in your own handwriting covering expansion, factorisation, radical simplification, and the Pythagorean ratios, plus a scored set of 20 mixed computation items with all errors corrected in red.
Geometry and Figures (図形) & Mathematical Reasoning and Proof (数学的な見方・考え方・証明)
Write full congruence proofs for triangles using the three congruence conditions, naming corresponding vertices in the correct order every time.
Use properties of isosceles triangles, parallelograms, and the midsegment theorem to derive equal angles and equal lengths within a single figure.
Distinguish clearly between a statement, its converse, and a counterexample, and produce one counterexample for a false converse.
Set out a proof in the standard Japanese format: hypothesis, conclusion, construction line, then the argument with reasons cited for each step.
Your checkpoint
Three complete written proofs (triangle congruence, isosceles-triangle angle property, parallelogram diagonal bisection) each marked against a checklist of hypothesis, construction, reasons, and conclusion.
Numbers and Algebraic Expressions (数と式) & Equations and Inequalities (方程式・不等式)
Solve linear equations with fractional and decimal coefficients by clearing denominators first, and verify each solution by substitution.
Solve simultaneous linear equations in two variables by both elimination and substitution, choosing the faster method per problem.
Solve quadratic equations by factoring, by completing the square, and by the quadratic formula, and identify when the discriminant is negative and no real solution exists.
Translate two short word problems (age and mixture type) into equations and solve them, defining the variable explicitly.
Your checkpoint
A 25-item timed equation set (linear, simultaneous, quadratic) completed in 40 minutes with a written error log classifying each mistake as arithmetic, method, or misreading.
Functions and Proportional/Inverse Relationships (関数・比例・反比例) & Equations and Inequalities (方程式・不等式)
Sketch and interpret graphs of direct and inverse proportion, stating the constant of proportionality and the domain restriction for inverse proportion.
Determine the equation of a linear function from two points or from one point and a gradient, and from a graph read intercepts accurately.
Solve linear inequalities and represent the solution set on a number line, handling multiplication by a negative correctly.
Solve simultaneous inequalities and express the combined solution as a single interval.
Your checkpoint
A graph-and-inequality portfolio: six accurate hand-drawn graphs (two direct proportion, two inverse proportion, two linear) with equations derived from the drawings, plus five inequality solution sets shown on number lines.
Geometry and Figures (図形) & Integrated and Applied Problems (総合問題・応用)
Compute lengths, angles, and areas in circles using the inscribed-angle theorem, the tangent-radius perpendicularity, and arc and sector formulas.
Solve solid-geometry problems involving the volume and surface area of prisms, pyramids, cylinders, cones, and spheres.
Apply similarity ratios to relate lengths, areas, and volumes between two similar figures, and use them to find an unknown dimension.
Break a multi-step applied figure problem into sub-goals and write the solution as an ordered chain of justified steps.
Your checkpoint
Two fully worked integrated geometry problems (one circle-plus-triangle, one solid-plus-similarity), each with a labelled diagram, a step list, and a self-check of units and plausibility.
Data Analysis and Probability (資料の活用・確率) & Functions and Proportional/Inverse Relationships (関数・比例・反比例)
Construct and read histograms, box plots, and frequency tables, and compute mean, median, mode, and range from raw and grouped data.
Interpret relative frequency as an estimate of probability and distinguish it from theoretical probability.
Compute probabilities of combined events using the addition rule and the multiplication rule, and use tree diagrams for two-stage experiments.
Use a scatter plot to judge correlation and read a trend line to make a prediction, noting the limits of extrapolation.
Your checkpoint
A short data investigation: collect or use 20 or more data values, present them as a histogram and box plot, compute all summary statistics, and answer three probability questions about the same data set with working shown.
Mathematical Reasoning and Proof (数学的な見方・考え方・証明) & Integrated and Applied Problems (総合問題・応用)
Write a complete proof of the inscribed-angle theorem and of one similarity criterion, with every reason named.
Identify the logical gap in a flawed proof supplied by yourself or a study partner and rewrite the proof correctly.
Solve at least three past-paper-style integrated problems that combine functions with geometry, showing the reasoning chain rather than only the answer.
Justify why a result holds using a general argument (letters) rather than a single numerical example.
Your checkpoint
A proof portfolio of four proofs (two written from scratch, two corrected from flawed drafts) with a short reflection on which logical habit caused each original error.
Integrated and Applied Problems (総合問題・応用) & Numbers and Algebraic Expressions (数と式) & Equations and Inequalities (方程式・不等式) & Functions and Proportional/Inverse Relationships (関数・比例・反比例) & Geometry and Figures (図形) & Data Analysis and Probability (資料の活用・確率) & Mathematical Reasoning and Proof (数学的な見方・考え方・証明)
Complete two full-length mock papers under the real time limit, with no notes and no pauses.
Mark each mock against the answer key and tag every lost mark by domain and by error type.
Re-solve every missed item from scratch within 48 hours and confirm the fix on a fresh similar item.
Finalise a one-page pre-exam card of the formulas and proof patterns you most often forgot.
Your checkpoint
Two marked mock papers with a domain-by-domain accuracy table, a corrected-error list, and a final one-page revision card.
Milestones
Week 2: You can write a complete triangle congruence proof unaided in under eight minutes, with hypothesis, construction, and reasons all present.
Week 4: You score at least 80 percent on a self-made 20-item mixed set covering expansion, factorisation, linear and quadratic equations, and inequalities, with no sign errors.
Week 6: You can compute mean, median, mode, and range from a grouped frequency table and answer a two-stage probability question using a tree diagram, both without reference material.
Week 8: Both mock papers are marked, every error is tagged by domain, and your weakest domain shows measurable improvement between mock one and mock two.