100+ Free Tochigi Public High School Entrance Examination — Mathematics Practice Questions
Prepare for the Tochigi Prefectural Public High School Entrance Examination — Mathematics (栃木県立高等学校入学者選抜 学力検査 数学) exam with instant access — no signup required.
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Key Facts: Tochigi Public High School Entrance Examination — Mathematics Exam
Practise for the Tochigi Prefectural Public High School Entrance Exam in Mathematics (数学) with 100 four-option questions covering 数と式, 関数, 図形 and 資料の活用, including genuine worked calculations, weighted to the official 100-point, 50-minute paper sat second on 2月24日 in 令和9年度. The official paper is handwritten and requires 作図, a full 図形の証明 and 途中の計算過程, so this English-language MCQ bank drills the underlying methods — it is not a format simulation, and you still need to practise writing proofs out.
Sample Tochigi Public High School Entrance Examination — Mathematics Practice Questions
Try these sample questions to test your Tochigi Public High School Entrance Examination — Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.
1Compute: -8 + 3 - (-2).
2Compute: (-3)^2 - (-2)^3.
3Compute 1/4 + 2/3 and write the answer as a reduced fraction.
4Compute 7/12 - 1/3 and write the answer as a reduced fraction.
5What is the prime factorisation of 180?
6Solve for x: 3x - 7 = 11.
7Solve for x: 2x + 5 = 3x - 2.
8Solve the simultaneous equations (連立方程式): x + y = 8, x - y = 2.
9Solve the simultaneous equations: 2x + 3y = 13, x + y = 5.
10Solve: x^2 - 5x + 6 = 0.
About the Tochigi Public High School Entrance Examination — Mathematics Exam
The Tochigi Prefectural Public High School Entrance Examination in Mathematics (数学) tests lower-secondary graduates on numbers and expressions, functions, geometry, and data and probability. The general academic test (学力検査) covers five subjects (国語, 数学, 外国語[英語, リスニング含む], 社会, 理科), each worth 100 points (total 500) and each 50 minutes, all sat on one day. From 令和9年度 (2027) Tochigi runs a single 本検査: every 全日制 applicant applies to the 一般選抜 and sits the 学力検査 on 2月24日(水), with 数学 second at 10:50-11:40; 学校独自検査 for 特色選抜 applicants follows on 2月25日・26日, 追検査 is 3月8日(月) and 合格者発表 is 3月12日(金). The 数学 paper is worked by hand in Japanese across about 6 大問 and requires compass-and-ruler 作図, a written 図形の証明 and 途中の計算過程 with partial credit, so it is not a multiple-choice paper; a 関数のグラフと図形の融合問題 appears regularly and is among the hardest items. Selection combines the 調査書(内申) with the 学力検査 score under each school's chosen weighting; 小山高校 数理科学科 weights 数学 at 1.5×. The paper is aligned to the MEXT junior-high Course of Study (平成29年 中学校学習指導要領 数学編), calculators are not permitted, and it tests 知識・技能 plus 思考力・判断力・表現力.
Assessment
100-point paper aligned to the MEXT 数学 Course of Study, covering 数と式 (25%), 関数 (25%), 図形 (30%) and 資料の活用 (20%) across about 6 大問: a 小問集合, then 文字式・方程式, 平面図形 with 作図, データの活用と確率, 関数, and a 総合/規則性 problem. Answers are handwritten, including compass-and-ruler 作図, a full 図形の証明, and 途中の計算過程 that earns partial credit. The five academic-test subjects each carry 100 points (max 500), all sat on a single day. Problems are at junior-high graduation level; calculators are not permitted; 小山高校 数理科学科 weights 数学 at 1.5× (150点). The paper tests 知識・技能 plus 思考力・判断力・表現力.
Time Limit
Sat as part of the one-day five-subject 学力検査: each subject is a 100-point paper of 50 minutes, 500 points in total. For 令和9年度 (2027) the 本検査 is 2月24日(水), with 数学 second at 10:50-11:40; 学校独自検査 for 特色選抜 applicants runs 2月25日(木)・26日(金), 追検査 is 3月8日(月) and 合格者発表 is 3月12日(金). 出願 is 1月29日(金)〜2月8日(月) and 志願先変更 is 2月10日(水)〜12日(金). Candidates bring a compass and ruler; calculators are not permitted. Source: 令和9年度 学力検査時間割及び出題方針等について / 関係諸日程, 栃木県教育委員会.
Passing Score
No fixed prefecture-wide pass mark. The 数学 paper is scored out of 100 points (500 across the five 学力検査 subjects, each 100 points). Under the 令和8年度 一般選抜 rules the 調査書(内申) 135点 (中1-中3 each 9教科×5段階 = 45点×3) was converted to a 500-point scale and combined with 学力検査 500点, with each school/department choosing a weighting between 1:9 and 9:1, then selecting through a two-stage 第1次審議 (top 80% of the intake) + 第2次審議 comprehensive process; 小山高校 数理科学科 multiplies the 数学 score by 1.5 (150点). Admission is competitive and relative to other applicants, not against a published pass line. The 令和9年度 実施細則 was still 暫定版 as of June 2026, with the 確定版 due late August 2026 — confirm the weighting rules there. Source: 栃木県立高等学校入学者選抜実施細則, 栃木県教育委員会.
Exam Fee
The 入学考査料 is ¥2,200 for a 全日制 course and ¥950 for a 定時制 course, set by 栃木県立学校の授業料等に関する条例 第1条. It is paid by electronic payment (電子納付) via the 栃木県電子申請システム (credit card, PayPay, Pay-easy and other methods). 栃木県収入証紙 were also accepted for 令和8年度 but sales stopped at the end of March 2026, so confirm the current method in the year's 実施細則 on the 栃木県教育委員会 高校教育課 selection page. The fee is not refunded once paid. (Tochigi Prefectural Board of Education (栃木県教育委員会), 高校教育課, 〒320-8501 宇都宮市塙田1-1-20 県庁舎南別館3階)
Tochigi Public High School Entrance Examination — Mathematics Exam Content Outline
Numbers & Expressions (数と式)
Integers, prime factorisation, linear equations, simultaneous equations (連立方程式), quadratic equations, inequalities, algebraic expressions and identities, polynomials, exponents, and square roots.
Functions (関数)
Linear functions, proportional and inverse proportional relationships, quadratic functions y=ax² (二次関数), parabolas, vertex and axis of symmetry, intersections of parabolas with lines, and function–geometry fusion problems.
Geometry (図形)
Congruence and similarity, triangles, quadrilaterals, circles, arcs and chords, inscribed and central angles (円周角の定理), the Pythagorean theorem (ピタゴラスの定理), parallel-line proportionality, surface area and volume of solids, constructions and 記述式 proof.
Data & Probability (資料の活用)
Mean, median, mode and range, histograms, frequency distributions, relative frequency, quartiles and box-and-whisker plots, variance, sampling surveys, and probability.
How to Pass the Tochigi Public High School Entrance Examination — Mathematics Exam
What You Need to Know
- Passing score: No fixed prefecture-wide pass mark. The 数学 paper is scored out of 100 points (500 across the five 学力検査 subjects, each 100 points). Under the 令和8年度 一般選抜 rules the 調査書(内申) 135点 (中1-中3 each 9教科×5段階 = 45点×3) was converted to a 500-point scale and combined with 学力検査 500点, with each school/department choosing a weighting between 1:9 and 9:1, then selecting through a two-stage 第1次審議 (top 80% of the intake) + 第2次審議 comprehensive process; 小山高校 数理科学科 multiplies the 数学 score by 1.5 (150点). Admission is competitive and relative to other applicants, not against a published pass line. The 令和9年度 実施細則 was still 暫定版 as of June 2026, with the 確定版 due late August 2026 — confirm the weighting rules there. Source: 栃木県立高等学校入学者選抜実施細則, 栃木県教育委員会.
- Assessment: 100-point paper aligned to the MEXT 数学 Course of Study, covering 数と式 (25%), 関数 (25%), 図形 (30%) and 資料の活用 (20%) across about 6 大問: a 小問集合, then 文字式・方程式, 平面図形 with 作図, データの活用と確率, 関数, and a 総合/規則性 problem. Answers are handwritten, including compass-and-ruler 作図, a full 図形の証明, and 途中の計算過程 that earns partial credit. The five academic-test subjects each carry 100 points (max 500), all sat on a single day. Problems are at junior-high graduation level; calculators are not permitted; 小山高校 数理科学科 weights 数学 at 1.5× (150点). The paper tests 知識・技能 plus 思考力・判断力・表現力.
- Time limit: Sat as part of the one-day five-subject 学力検査: each subject is a 100-point paper of 50 minutes, 500 points in total. For 令和9年度 (2027) the 本検査 is 2月24日(水), with 数学 second at 10:50-11:40; 学校独自検査 for 特色選抜 applicants runs 2月25日(木)・26日(金), 追検査 is 3月8日(月) and 合格者発表 is 3月12日(金). 出願 is 1月29日(金)〜2月8日(月) and 志願先変更 is 2月10日(水)〜12日(金). Candidates bring a compass and ruler; calculators are not permitted. Source: 令和9年度 学力検査時間割及び出題方針等について / 関係諸日程, 栃木県教育委員会.
- Exam fee: The 入学考査料 is ¥2,200 for a 全日制 course and ¥950 for a 定時制 course, set by 栃木県立学校の授業料等に関する条例 第1条. It is paid by electronic payment (電子納付) via the 栃木県電子申請システム (credit card, PayPay, Pay-easy and other methods). 栃木県収入証紙 were also accepted for 令和8年度 but sales stopped at the end of March 2026, so confirm the current method in the year's 実施細則 on the 栃木県教育委員会 高校教育課 selection page. The fee is not refunded once paid.
Keys to Passing
- Complete 500+ practice questions
- Score 80%+ consistently before scheduling
- Focus on highest-weighted sections
- Use our AI tutor for tough concepts
Tochigi Public High School Entrance Examination — Mathematics Study Tips from Top Performers
Frequently Asked Questions
Is this question bank the same as the real Tochigi 数学 exam?
No. The official paper is worked by hand in Japanese and requires compass-and-ruler 作図, a written 図形の証明 and 途中の計算過程; its exact item count is not published and the prefecture does not publish past papers online. This bank is an English-language multiple-choice study adaptation of the same junior-high syllabus (数と式, 関数 including y=ax², 図形 with 相似・円周角・三平方の定理, and 資料の活用) and cannot replace writing out your own proofs and calculations in Japanese under timed conditions.
How long is the Tochigi 数学 paper and what is it worth?
It is scored out of 100 points, one of five subjects totalling 500 points (each subject 100 points), and each paper runs 50 minutes. 数学 is the second paper of the day — 10:50-11:40 on 2月24日(水) for 令和9年度 — and 合格者発表 is 3月12日(金). 小山高校 数理科学科 multiplies the 数学 score by 1.5, giving it a 150-point weighting.
Are calculators allowed?
No. The mathematics paper is sat without a calculator, in line with standard prefectural academic-test rules; numerical reasoning and exact computation are expected.
How is the selection structured?
Tochigi combines the 調査書(内申) with the 学力検査 500点. Under the 令和8年度 rules the 内申 135点 (中1-中3 each 9教科×5段階 = 45点×3) was converted to a 500-point scale, each school chose a weighting between 1:9 and 9:1, and selection ran in two stages — 第1次審議 for the top 80% of the intake, then 第2次審議 for the rest. 小山高校 数理科学科 weights 数学 at 1.5×, and 宇都宮, 宇都宮女子 and 栃木 adjust the internal mark allocation on 国語・数学・英語. From 令和9年度 特色選抜 and 一般選抜 share one 本検査: 出願 is 1月29日(金)〜2月8日(月), 志願先変更 2月10日〜12日, 学力検査 2月24日(水), 学校独自検査 2月25日・26日 and 合格者発表 3月12日(金).
What kinds of problems appear on the 数学 paper?
The paper runs to about 6 大問 covering 数と式, 関数, 図形 and 資料の活用, opening with a 小問集合 and closing with a 総合/規則性 problem. Everything is handwritten: there is compass-and-ruler 作図, a full 図形の証明, and 途中の計算過程 that earns partial credit. A 関数のグラフと図形の融合問題 appears regularly and is among the hardest items. The paper tests 知識・技能 plus 思考力・判断力・表現力.
How much is the examination fee?
The 入学考査料 for a 全日制 course is ¥2,200, paid by electronic payment (電子納付) via the 栃木県電子申請システム (credit card, PayPay, Pay-easy, etc.) or by 栃木県収入証紙; 定時制 courses pay less. Confirm the current method in the year's 実施細則 on the 栃木県教育委員会 高校教育課 page.