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KS3 Year 9 / Mathematics / Curriculum

Graphs and functions

Graphs and functions: structured theory, worked examples, answered practice, and a mastery checklist for KS3 Year 9.

Цели обучения

Recognise, explain, and apply “Linear graphs”.
Recognise, explain, and apply “Gradient and intercept”.
Recognise, explain, and apply “Quadratic graphs introduction”.
Recognise, explain, and apply “Reciprocal graphs introduction”.
Recognise, explain, and apply “Real-life graphs”.

Ключевые идеи и проверки

I move between formula, table, and graph.
I interpret slope, intercepts, and key features.
Every solution or explanation for “Graphs and functions” should include a method, justification, and final check.

Раздел

Основная теория

Ключевые идеи главы в понятной последовательности перед практикой.

01

Input-output relationship

A function maps each allowed input to one output. Formula, table, and graph are different representations of the same relationship.

02

Parameters and graph

Parameters change slope, position, amplitude, or rate of change. Read the key features before plotting individual points.

03

Domain and interpretation

The domain states which inputs are allowed. In a real model, also check whether the output is physically meaningful.

04

Linear graphs

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

05

Gradient and intercept

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

06

Quadratic graphs introduction

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

07

Reciprocal graphs introduction

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

08

Real-life graphs

Use a suitable representation, state the rule, and keep every numerical or algebraic step equivalent. Finish with estimation, an inverse operation, or substitution as a check.

Mathematics

Разобранные примеры

Следуйте методу шаг за шагом и проверяйте обоснование каждого шага.

Worked function example

For f(x) = 3x - 6, find f(5) and the x-intercept.

1Substitute x = 5: f(5) = 3·5 - 6.
2For the intercept set f(x) = 0: 3x - 6 = 0.
3Solve for x and check the point on the graph.

f(5) = 9, x-intercept = 6/3

Reasoning example

Before calculating, explain the key idea from “Linear graphs” and which conditions must be checked.

1Define the idea in one clear sentence.
2Connect it to a representation, law, or formula.
3State a restriction, unit, or final check that makes the solution valid.

The answer should show not only which rule is used for “Linear graphs”, but also why it is valid here.

Graphs and functions

Практика с ответами

Восемь заданий от базовых навыков до экзаменационного уровня. Сначала решите самостоятельно.

Базовый навык4 минут

Solve 9x + 5 = 50.

Подсказка

Subtract 5 from both sides, then divide by 9.

Ответ

x = 5.

Базовый навык5 минут

Expand and simplify 5(x + 2) - 2x.

Подсказка

Multiply every term inside the bracket.

Ответ

5x + 10 - 2x = 3x + 10.

Применение6 минут

Factorise 8x + 32.

Подсказка

Take out the common factor 8.

Ответ

8(x + 4).

Применение7 минут

For f(x) = 4x + 6, find f(10) and solve f(x) = 30.

Подсказка

Substitute, then solve the equation.

Ответ

f(10) = 46; x = 6.

Обоснование8 минут

A sequence starts 3, 10, 17. Find its nth term and 10th term.

Подсказка

The common difference is 7.

Ответ

7n + -4; term 10 = 66.

Обоснование9 минут

Solve 3x + 5 > 20.

Подсказка

Use equation steps; the coefficient of x is positive.

Ответ

x > 5.

Экзаменационный10 минут

Solve x + y = 8, x - y = 4.

Подсказка

Add the two equations.

Ответ

x = 6, y = 2.

Экзаменационный12 минут

A learner writes 9(x + 4) = 9x + 4. Identify and correct the error.

Подсказка

The outside factor multiplies every term.

Ответ

9(x + 4) = 9x + 36.

Частые ошибки

Confusing f(x) with the product f times x.
Plotting without a scale or key points.
Ignoring domain and model restrictions.

Проверка понимания

I move between formula, table, and graph.
I interpret slope, intercepts, and key features.
I state domain and relevant restrictions.

Graphs and functions

Домашняя работа по главе

Шесть разных заданий от базового уровня до сложного с временем, подсказками и ответами.

Базовый8 минут

Homework 1: Solve 5x + 6 = 36.

Подсказка

Subtract 6 from both sides, then divide by 5.

Ответ

x = 6.

Базовый10 минут

Homework 2: Expand and simplify 8(x + 3) - 3x.

Подсказка

Multiply every term inside the bracket.

Ответ

8x + 24 - 3x = 5x + 24.

Углубленный12 минут

Homework 3: Factorise 4x + 20.

Подсказка

Take out the common factor 4.

Ответ

4(x + 5).

Углубленный15 минут

Homework 4: For f(x) = 7x + 2, find f(9) and solve f(x) = 16.

Подсказка

Substitute, then solve the equation.

Ответ

f(9) = 65; x = 2.

Сложный18 минут

Homework 5: A sequence starts 4, 7, 10. Find its nth term and 10th term.

Подсказка

The common difference is 3.

Ответ

3n + 1; term 10 = 31.

Сложный20 минут

Homework 6: Solve 6x + 6 > 42.

Подсказка

Use equation steps; the coefficient of x is positive.

Ответ

x > 6.

50 минут / 40 баллов

Полный тест по главе

Полный тест с таймером, баллами и образцами ответов.

Start the timer when ready, work without notes, show every step, and open model answers only after finishing.

1. Solve 3x + 6 = 24.

2 баллов

Ответ

x = 6.

2. Expand and simplify 6(x + 3) - 3x.

3 баллов

Ответ

6x + 18 - 3x = 3x + 18.

3. Factorise 9x + 45.

3 баллов

Ответ

9(x + 5).

4. For f(x) = 5x + 2, find f(7) and solve f(x) = 12.

4 баллов

Ответ

f(7) = 37; x = 2.

5. A sequence starts 4, 12, 20. Find its nth term and 10th term.

4 баллов

Ответ

8n + -4; term 10 = 76.

6. Solve 4x + 6 > 30.

4 баллов

Ответ

x > 6.

7. Solve x + y = 10, x - y = 4.

5 баллов

Ответ

x = 7, y = 3.

8. A learner writes 3(x + 5) = 3x + 5. Identify and correct the error.

5 баллов

Ответ

3(x + 5) = 3x + 15.

9. Model this: 2 tickets cost x euros each plus a fixed 6 euros. Write the cost and find it for x = 8.

5 баллов

Ответ

C = 2x + 6; C(8) = 22.

10. Represent “Quadratic graphs introduction” as an equation, a graph, and a verbal description of the same model.

5 баллов

Ответ

Example y = 9x + 4: a line with gradient 9, intercept 4, meaning “starts at 4 and rises by 9 per unit”.

Раздел

Официальные источники

Источники программы; порядок тем уточняйте в школе и у преподавателя.

UK Department for Education - MathematicsOfficial national curriculum progression and attainment guidance.

Mathematics

KS3 Year 9: структурированный маршрут по математике и физике с главами, практикой и следующими шагами.

Назад к предмету

Что покрывает глава

Структура следует учебнику и помогает организовать изучение темы.

Linear graphs

Gradient and intercept

Quadratic graphs introduction

Reciprocal graphs introduction

Real-life graphs

Graphical solutions

Function notation introduction

Где сфокусироваться

Темы, где чаще возникают ошибки или нужна дополнительная практика.

I move between formula, table, and graph.

I interpret slope, intercepts, and key features.

I state domain and relevant restrictions.

Источники, ежедневные материалы и ресурсы

С чего начать: учебник, daily material, PDFs, videos и worked examples.

Start from the official textbook or specification referenced on the subject page.

Use the notes and examples as support, not as a replacement for the official syllabus.

Check the current syllabus version before exam preparation.

Практика по подтемам

Целевая практика перед full tests, чтобы coverage было понятным.

Solve 9x + 5 = 50.

Expand and simplify 5(x + 2) - 2x.

Factorise 8x + 32.

For f(x) = 4x + 6, find f(10) and solve f(x) = 30.

A sequence starts 3, 10, 17. Find its nth term and 10th term.

Solve 3x + 5 > 20.

Solve x + y = 8, x - y = 4.

A learner writes 9(x + 4) = 9x + 4. Identify and correct the error.

Mocks и проверка прогресса

Как измерять прогресс по главе и когда включать ее в общий mock.

Start with untimed practice by subtopic.

Move to a short timed checkpoint only after completing the mastery checklist.

Record each error with the correct method and revisit it after 48 hours.

Следующий шаг

Что делать после главы и как она связана со следующим разделом.

Complete the practice without support.

Explain the core method aloud in under two minutes.

Continue to the next chapter or request targeted tutor support.

Примечание: официальную экзаменационную программу каждого года всегда уточняйте в школе, у преподавателя и по текущим официальным источникам.