At a glance. One lesson of about 60 minutes. It assumes lesson 2 and translations from Year 10. Students read the midline, amplitude, period and first maximum from an equation, build a model from the highest value, the lowest value and the period, and use it at a given time. A GDC is optional, to check a model against a table.
Calculator: GDC optional. The GDC graphs a model and tabulates it, so students can check a hand value. Degree mode is the one practical point. Building the model from the data is by hand.
What students will learn
Know
- The midline is halfway between the highest and lowest values.
- The amplitude is half the range.
- x − c moves the graph c to the right.
Understand
- The same family fits many periodic situations.
- A cosine is a sine moved a quarter of a period.
- The shift sits inside the bracket with b.
Do
- Build a model from the highest value, the lowest value and the period.
- Use a model at a given time.
- Find when a model reaches a given level.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: midline, amplitude, period and shift (check and questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to predict the first maximum before moving the shift slider.
- Criterion C. Communicating: stating the parameters before writing the model, and giving units.
- Criterion D. Applying mathematics in real-life contexts: tides and a wheel, where the model is built from measurements and used to answer a question.
The one thing to do with the figure
Predict, then move. In the first figure, ask “Where will the first maximum be with c = 60°?” and take values before the slider moves.
In the second figure, ask for the height at t = 15 before showing it, and ask why the graph starts at its lowest point.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the harbour mouth. Collect which of the items students would need. |
| 10 min | First figure and key idea: the family of graphs. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose B, ask where a sine peaks when there is no shift. |
| 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 | Amplitude 3 and period 90°; y = f(x − 5); 7. |
|---|---|
| Hook | The highest depth, the lowest depth, the time between two high tides, and the time of one high tide. The ship's colour is not needed. |
| Worked examples | Midline 5.4 m and amplitude 3.2 m; d = 5.4 + 3.2 cos(30t), 7.0 m at t = 2. |
| Faded example | h = 7 − 6 cos(15t). |
| 1. Reading a shifted sine | A. B is the shift only; C moves the wrong way; D is the amplitude. |
| 2. Highest value of a model | 15. 12 + 3. |
| 3. Midline from highest and lowest | 6.4. 12.8 ÷ 2. |
| 4. Evaluating a shifted model | 26. 20 + 6 sin 90°. |
| 5. Time of the first maximum | 5. t − 2 = 3. |
| 6. Building and using a wheel model | 22.5. h = 14 − 12 cos(9t) at t = 15. |
| 7. How long above a level | 4. t from 1 to 5. |
| 8. Choosing a model | D. A starts at high tide; B starts by falling; C uses the range. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 (the check) | Uses the shift alone, moves the peak the wrong way, or reads the amplitude as the highest value. | Ask: where does y = sin x peak? Then shift it. |
| 3, 36 or 9 in question 2 | Reads the amplitude, multiplies, or finds the lowest value. | Ask: what is the biggest sin can be? |
| 5.6, 2.8 or 12.8 in question 3 | Gives the range, the amplitude or the sum. | Write: midline = (highest + lowest) ÷ 2. |
| 23, 23.4 or 14 in question 4 | Leaves out the shift, subtracts it from the angle, or takes sin 90° as −1. | Do the bracket on its own line first. |
| 2, 3 or 1 in question 5 | Reads the shift, forgets it, or subtracts it. | Write: the bracket must equal 3. |
| 5.5, 20 or 2.4 in question 6 | Uses + cos (a rider starting at the top), uses the period as b, or leaves out b. | Sketch one turn first, then write the model, the angle and its cosine on separate lines. |
| 1, 6 or 8 in question 7 | Gives the first time, half the period, or the hours below. | Sketch the sine with a horizontal line at 0.5. |
| A, B or C in question 8 | Uses a cosine, starts the wave falling, or uses the range. | Find the midline and amplitude first, then ask what the depth does at t = 0. |
Questions worth asking
- Why is the midline the average of the highest and lowest values? The wave goes the same distance above the midline as below it.
- Why does a model that starts at its lowest value use a negative cosine? A cosine starts at its highest value; turned upside down it starts at its lowest.
- How does y = f(x − c) relate to Year 10? It is a translation of c to the right, now of a sine.
- How would you find the first time the tide is at its highest if it is not at t = 0? Set the angle inside the bracket to 90° (for a sine) and solve for t.
Differentiation
A way in for anyone stuck
- Give a labelled frame: highest, lowest, midline, amplitude, period, b.
- Provide the graph of one model and ask for the parameters before asking for the equation.
- Allow the GDC graph as a check once the hand model is written.
Stretch
- Write the model for a wheel of the same size that starts at the top of its turn instead of the bottom.
- Find the first two times the seat is at 14 m, for h = 10 − 8 cos(6t). Say what the answers should add up to over one period.
Students learning in English as an additional language
“Midline” is the level halfway between the highest and the lowest values. “Shift” is a movement of the whole graph without changing its shape. “Model” is an equation that describes a real situation. “Tide” is the rise and fall of the sea.
| Word | Plain meaning | Example |
|---|---|---|
| midline | the level halfway between the highest and lowest values | the line y = 5 for a wave between 3 and 7 |
| amplitude | half the distance between the highest and lowest values | 2 for a wave between 3 and 7 |
| period | the time or angle after which the pattern repeats | 12 hours for a tide |
| phase shift | a movement of a periodic graph to the right or left | the 70° in sin (x − 70°) |
| model | an equation that describes a real situation | the depth of water against time |
Sentence starters for explaining aloud or in writing:
- The highest value is … and the lowest is …, so the midline is ….
- The period is …, so b = ….
- It starts at its … value, so I use ….
Two things not to say
“Minus means move left.” x − c moves the graph right.
“The midline is always the x-axis.” It is d, which can be anything.
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.
- MidlineA wave has a highest value of 9 and a lowest value of 1. State the midline and amplitude.
Midline 5, amplitude 4. - ShiftWhere is the first maximum of y = sin (x − 20°)?
At x = 110°. 90 + 20.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA swing's height above the ground is h = 1.4 + 0.6 sin(90t), in metres. Find the lowest height.
0.8 m. 1.4 − 0.6. - Earlier work (Lesson 2)State the period of y = 2 sin 5x.
72°. 360 ÷ 5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs y = 3 + 2 sin(x − 30°) ever below 0? Explain.
No: its lowest value is 3 − 2 = 1. - Working backwardsA model has midline 6, amplitude 2, period 8 and starts at its lowest value. Write it with t.
h = 6 − 2 cos(45t).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders 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 drawing of a wheel and ask them to measure the height of a seat above the ground at 30° intervals and plot them against time.