At a glance. One lesson of about 60 minutes. It assumes the unit circle from lesson 1 and the Year 10 stretches and reflections. Students sketch y = a sin bx and y = a cos bx, read amplitude and period, and write an equation from a description. A GDC is optional, to check a sketch.
Calculator: GDC optional. The GDC checks a sketch and shows the peaks. Degree mode is the one setting that matters; the sketch is done by hand.
What students will learn
Know
- Amplitude is the distance from the middle line to a peak.
- The period of y = a sin bx is 360° ÷ b.
- The cosine graph starts at its highest point.
Understand
- a is a vertical stretch and b a horizontal stretch, as in Year 10.
- Cosine is sine moved 90° to the left.
- The tangent has period 180° and asymptotes.
Do
- Read the amplitude and period from an equation.
- Write an equation from a description.
- Evaluate a model at a given angle.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: amplitude, period and the sine and cosine shapes (check and questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to predict how the graph changes before moving a slider.
- Criterion C. Communicating: sketching with labelled axes, amplitude and period marked.
- Criterion D. Applying mathematics in real-life contexts: a sound wave modelled by a sine.
The one thing to do with the figure
Change one slider at a time. Ask “What will move if I change only b?” and take predictions before it moves.
In the second figure, ask where the cosine curve crosses the axis, and compare with the sine.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the tuning forks. Collect predictions. |
| 10 min | First figure and key idea: amplitude and period. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose C, ask what 360 ÷ 12 is. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | Second quadrant; y = 3f(x); −1 and 0. |
|---|---|
| Hook | A larger a raises the peaks and leaves their spacing alone; a larger b brings them closer together. |
| Worked examples | Amplitude 4, period 120°; y = 5 sin 4x. |
| Faded example | y = 2.5 cos 3x. |
| 1. What changing b does | B. A changes the amplitude; C multiplies instead of dividing; D is a translation. |
| 2. Period from the equation | 24. 360 ÷ 15. |
| 3. Amplitude from the lowest value | 7. The lowest value is −a. |
| 4. b from the length of a wave | 8. 360 ÷ 45. |
| 5. Evaluating a model | −3. 3 × cos 180°. |
| 6. b from a peak position | 3. b × 30 = 90. |
| 7. Finding b, then using it | 3.54. 5 × sin 45°. |
| 8. Writing an equation from a description | C. A swaps the 1.5 and the 10; B uses the period as b; D uses the range. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 (the check) | Changes the amplitude, multiplies instead of divides, or reads it as a shift. | Ask: is the 12 outside the sine or inside it? |
| 5400, 15 or 60 in question 2 | Multiplies 360 by b, reads b as the period, or uses the amplitude. | Say: b is the number of waves in 360°. |
| −7, 14 or 3.5 in question 3 | Reads off the value, the range, or halves it. | Sketch the graph with its highest and lowest values. |
| 0.125, 45 or 16 200 in question 4 | Inverts, gives the period, or multiplies. | Ask: how many 45° waves fit into 360°? |
| 2.12, 3 or 8.49 in question 5 | Leaves out b, takes cos 180° as 1, or multiplies b into the amplitude. | Write the angle 4 × 45 on its own line. |
| 12, 0.33 or 6 in question 6 | Treats 30° as a period or half a period, or inverts. | Sketch one wave and mark where the first peak falls. |
| 0.44, −1.71 or 0.05 in question 7 | Leaves out b, uses the period as b, or inverts. | Find b first and write it down. |
| A, B or D in question 8 | Swaps a and b, uses the period as b, or uses the range. | Read the highest value off first, then ask how many waves of 36° fit into 360°. |
Questions worth asking
- Which transformation of Year 10 does a stand for? And b? a is a vertical stretch of factor a; b is a horizontal stretch of factor 1 divided by b.
- How do you know a sine graph has a peak at a quarter of its period? sin is 1 at 90°, and 90° is a quarter of 360°.
- How are sine and cosine graphs related? The cosine graph is the sine graph moved 90° to the left.
- What does a negative a do? It turns the graph upside down. The amplitude is still the size of a.
Differentiation
A way in for anyone stuck
- Give a grid with x marked in steps of 90° and ask for a sketch of one wave only.
- Use a two-column table: number in front (amplitude) and number multiplying x (waves in 360°).
- Provide the five key points of sine: 0, 1, 0, −1, 0 at 0°, 90°, 180°, 270°, 360°.
Stretch
- Write an equation for the graph of y = 2 sin 3x reflected in the x-axis, and say what happens to its period.
- Show that y = cos x and y = sin (x + 90°) give the same value at x = 0, 30°, 60° and 90°.
- What if angles were measured in radians, with a full turn equal to 2π? Say what the period of y = sin x would be, and how the period of y = sin 3x would be written.
Students learning in English as an additional language
“Amplitude” is the height of a wave from its middle line. “Period” is the length of one complete wave. “Peak” is the highest point. “Midline” is the horizontal line halfway between the highest and lowest values.
| Word | Plain meaning | Example |
|---|---|---|
| amplitude | the distance from the middle line of a wave to its highest point | the amplitude of y = 2 sin x is 2 |
| period | the length of one complete wave on the horizontal axis | 360° for y = sin x |
| periodic | repeating exactly after a fixed interval | a wave that repeats |
| midline | the horizontal line halfway between the highest and lowest values | the x-axis for y = sin x |
| asymptote | a line that a graph gets closer to but never reaches | the x-axis for y = 1 ÷ x, far from the origin |
Sentence starters for explaining aloud or in writing:
- The amplitude is … because ….
- There are … waves in 360°, so the period is ….
- The graph of cosine is the graph of sine ….
Two things not to say
“Sine goes from 0 to 1 and back.” It goes below the axis too, to −1.
“The bigger b is, the longer the wave.” The wave is shorter.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- AmplitudeState the amplitude of y = 4 cos 2x.
4. - PeriodState the period of y = sin 6x.
60°. 360 ÷ 6.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA sound wave is y = 0.8 sin 9x with x in degrees. How long is one wave?
40°. 360 ÷ 9. - Earlier work (Lesson 1)State the sign of cos 250°.
Negative: third quadrant.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan y = 3 sin 2x ever equal 4? Explain.
No: its highest value is 3. - Working backwardsA graph has a period of 72° and a lowest value of −1.5. Write an equation using sine.
y = 1.5 sin 5x.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Give each pair a sheet of squared paper and ask for y = sin x at 30° steps, then y = 2 sin x and y = sin 2x on the same axes.