Trigonometry — ҚМЖ, Математика, 11 сынып
Алдын ала қарау
Материалды ашу
Қысқамерзімді жоспар
| Бөлім |
Trigonometric Functions and Equations
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| Сабақтың тақырыбы |
Trigonometry
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| Сыныбы |
11
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| Оқу бағдарламасына сәйкес оқыту мақсаттары |
Classics
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| Сабақтың мақсаты |
Students know the definitions and key properties of sine, cosine, and tangent on the unit circle.
Students know how to apply fundamental trigonometric identities to simplify expressions and solve equations.
Students know how to solve standard trigonometric equations and verify solutions within a specified interval.
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| Құндылықтар |
Accuracy and intellectual honesty are fostered through showing complete solution steps and checking results. Perseverance is encouraged as students compare strategies for solving equations and correct errors using mathematical reasoning.
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Сабақтың барысы
| Сабақтың кезеңі/ уақыт | Педагогтің іс-әрекеті | Оқушының іс-әрекеті | Бағалау | Ресурстар |
|---|---|---|---|---|
| Organisational Moment 3 minutes |
Organisational moment:
Welcome students, check attendance and readiness, and ensure that notebooks and calculators are available.
Display the lesson topic, “Trigonometry,” and outline the lesson sequence.
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Students — prepare notebooks and calculators and listen to the lesson outline. | Readiness checklist: materials prepared — 1 point; follows the opening instructions — 1 point. Stage maximum — 2 points. | board, notebooks, calculators |
| Concept Review Presentation 7 minutes |
New material explanation:
Present the unit-circle definitions of sine and cosine and explain that tangent is the ratio of sine to cosine when cosine is nonzero.
Review the identity sin²x + cos²x = 1 and the period and symmetry properties used in solving equations.
Ask: “What are sin x and cos x represented by on the unit circle?” and “When is tan x undefined?”.
Answers:
sin x is the y-coordinate and cos x is the x-coordinate of the point on the unit circle.
tan x is undefined when cos x = 0.
Слайд 1 — Trigonometric Functions
Слайд 2 — A point moves around a wheel, and its horizontal and vertical positions change as it goes.
Слайд 3 — By the end of this lesson you will be able to…
Слайд 4 — Coordinates on the Circle
Слайд 5 — Tangent and Its Domain
Слайд 6 — What matters more when deciding whether tangent exists: the angle’s position in a full ro…
Слайд 7 — A Fundamental Identity
Слайд 8 — Period, Symmetry, and Equations
Слайд 9 — Answering the Opening Question
Слайд 10 — Recall and Apply
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Students — follow the explanation and record the definitions and identity. Students — answer that sine is the y-coordinate, cosine is the x-coordinate, and tangent is undefined when cosine equals zero. | Correct unit-circle coordinate definitions — 1 point each, max 2 points. Correct condition for undefined tangent — 1 point. Stage maximum — 3 points. | |
| Equation Strategy Game 8 minutes |
Instructions:
Organize students into small teams and explain that each team will solve a trigonometric equation card and explain the chosen method.
Give teams the equations sin x = 1/2, cos x = −√2/2, and tan x = 1, with solutions requested on 0 ≤ x < 2π.
Ask teams to justify their solutions using the unit circle and to check that each answer lies in the interval.
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Students — work in teams to solve the assigned equation, identify all solutions in 0 ≤ x < 2π, and explain their reasoning. Students — report the relevant unit-circle angles: for sin x = 1/2, x = π/6, 5π/6; for cos x = −√2/2, x = 3π/4, 5π/4; for tan x = 1, x = π/4, 5π/4. | Correct solutions for the assigned equation — 2 points; valid interval and completeness check — 1 point; clear mathematical justification — 1 point. Stage maximum — 4 points. | |
| Guided Problem Solving 9 minutes |
Questions to the class:
Ask: “Solve 2sin²x − 1 = 0 for 0 ≤ x < 2π. Which identity or equation form helps?”.
Ask: “What values of sin x satisfy the equation, and which angles in the interval have those values?”.
Answers:
Using sin²x = 1/2 gives sin x = ±√2/2.
The solutions are x = π/4, 3π/4, 5π/4, and 7π/4.
Model how to substitute the solutions back into the original equation to verify them.
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Students — solve the equation independently, then compare steps with a partner. Students — state that sin x = ±√2/2 and give the four solutions π/4, 3π/4, 5π/4, and 7π/4. Students — check the solutions in the original equation. | Correct transformation to sin²x = 1/2 — 1 point; correct values of sine — 1 point; all four interval solutions — 2 points; verification — 1 point. Stage maximum — 5 points. | textbook, board, notebooks |
| Independent Assignment 8 minutes |
Instructions:
Distribute the assignment: solve sin(2x) = 0 for 0 ≤ x < π, and simplify (1 − cos²x)/sin x, stating any restriction on x.
Remind students to show each algebraic step, list solutions in the required interval, and check restrictions before simplifying.
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Students — solve sin(2x) = 0 on the stated interval and simplify the expression, recording any restriction. Students — show their reasoning and check their answers. | Complete solution set for sin(2x) = 0 — 2 points; correct use of the double-angle equation — 1 point; simplification to sin x with the restriction sin x ≠ 0 — 2 points. Stage maximum — 5 points. | |
| Knowledge Check Quiz 6 minutes |
Instructions:
Give a short individual quiz: state the fundamental identity; give the period of sin x; solve cos x = 1/2 for 0 ≤ x < 2π.
Collect responses and review the answer criteria after students finish.
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Students — complete the quiz independently. Students — identify sin²x + cos²x = 1, state the sine period as 2π, and solve cos x = 1/2 with x = π/3 and 5π/3. | Correct fundamental identity — 1 point; correct sine period — 1 point; both solutions in the interval — 2 points. Stage maximum — 4 points. | |
| Reflection 4 minutes |
Runs the "Reflection" reflection. End the lesson with quick student feedback — emoji vote and share the results.
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Students complete the "Reflection" reflection and assess their understanding of the topic. | Names one accurate concept — 1 point; identifies a relevant checking or practice step — 1 point. Stage maximum — 2 points. |
Денсаулық және қауіпсіздік техникасын сақтау
| Денсаулық және қауіпсіздік техникасын сақтау | Use calculators responsibly and keep them on the desk; do not use electrical devices with damaged cables or near liquids. Keep bags and materials clear of walkways to prevent trips. |
Материал туралы
- Материал түрі
- ҚМЖ (ҚМЖ)
- Пән
- Математика
- Сынып
- 11 сынып
- Тіл
- Ағылшын
- Оқыту бағыты
- Физмат
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