Math lesson plan · 9th grade (Year 10 in England) · 60 minutes
Average rate of change over an interval
Algebra
CCSS HSF.IF.B.6England: GCSE Higher Algebra
Objective and success criteria
Students will: Average rate of change over an interval.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Function notation; Slope (gradient) of a line
Leads to: Estimate the gradient of a curve with chords
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 6 min
- Quick retrieval from earlier learning (Function notation; Slope (gradient) of a line). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- f(x) = x² − 5x − 4. Find f(−2). → 10Prompt: Replace every x with (−2).
- f(x) = −4x + 8. For what value of x is f(x) = −16? → x = 6Prompt: Set the expression equal to the value.
- f(x) = 3x² + x + 3. Find f(−3). → 27Prompt: Replace every x with (−3).
Hook · 6 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Average rate of change over an interval. Read the success criteria together.
I do: model it · 12 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: x: 0, 1, 2, 3, 4 f(x): 2, 8, 20, 38, 62 Find the average rate of change of f from x = 0 to x = 4.
- Step 1: (62 − 2) ÷ (4 − 0)
- Step 2: = 15
- Answer: 15
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 12 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- x: 0, 1, 2, 3, 4 f(x): −4, −5, −2, 5, 16 Find the average rate of change of f from x = 1 to x = 4. → 7Prompt: Average rate = change in f(x) ÷ change in x.
- f(x) = x² + 5x + 4. Find the average rate of change of f between x = −3 and x = −1. → 1Prompt: Evaluate f at both ends of the interval.
You do: independent practice · 17 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 7 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- f(x) = 2x² + 5x + 5. Find the average rate of change of f between x = 1 and x = 5. → 17Prompt: Evaluate f at both ends of the interval.
- A ball is thrown upwards. Its height in metres after t seconds is h(t) = 17t − 5t². Find its average speed upwards (rate of change of height, in m/s) from t = 2 to t = 3. A negative answer means it is coming down. → -8Prompt: Find the height at each time.
Watch out for
- Doing an operation to one side of an equation but not the other.
- Reading 3x as 3 + x, or thinking x always stands for one particular number.
Support and stretch
Support
- Review first: Function notation.
- Use concrete materials: balance scales or a balance drawing.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Estimate the gradient of a curve with chords.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-9.
Free lesson plan from eurekami.com/lesson-plan-builder