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Math lesson plan · 11th grade (Year 12 in England) · 30 minutes

Combine regions of a normal curve with the 68–95–99.7 rule

Statistics and probability

CCSS HSS.ID.A.4England: A-level Statistics

Objective and success criteria

Students will: Combine regions of a normal curve with the 68–95–99.7 rule.

  • 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: The normal distribution (68–95–99.7 rule); Calculate and use z-scores

Leads to: Find normal probabilities with z-scores and technology

Materials: Squared paper for charts, Class survey data, Sticky notes for a live chart, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 4 min

  • Quick retrieval from earlier learning (The normal distribution (68–95–99.7 rule); Calculate and use z-scores). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. A variable is normally distributed with mean 103 and standard deviation 6. About what percentage of values lie between 85 and 121? → 99.7Prompt: How many standard deviations from the mean are these limits?
  2. A test has mean 44 and standard deviation 7. Find the z-score of a mark of 58. → 2Prompt: z = (x − mean) ÷ standard deviation.
  3. Scores have mean 45 and standard deviation 2. What is the z-score of 47? → 1Prompt: z = (x − mean) ÷ standard deviation.

Hook · 3 min

  • Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
  • Share the objective: Combine regions of a normal curve with the 68–95–99.7 rule. Read the success criteria together.

I do: model it · 7 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: A quantity is normally distributed with mean 21 and standard deviation 9. What percentage of values lie between 3 and 30?
  • Step 1: z from −2 to 1
  • Step 2: 81.5%
  • Answer: 81.5
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 6 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. A quantity is normally distributed with mean 63 and standard deviation 2. What percentage of values lie between 63 and 67? → 47.5Prompt: Convert each end to 'number of standard deviations from the mean'.
  2. A factory fills 200 bottles. Volumes are normally distributed with mean 320 ml and standard deviation 40 ml. About how many bottles hold more than 400 ml? → 5Prompt: That value is 2 standard deviations from the mean.

You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.

Exit ticket · 3 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. A factory fills 2000 bottles. Volumes are normally distributed with mean 450 ml and standard deviation 11 ml. About how many bottles hold less than 428 ml? → 50Prompt: That value is 2 standard deviations from the mean.
  2. A quantity is normally distributed with mean 75 and standard deviation 7. What percentage of values lie between 54 and 82? → 83.85Prompt: Convert each end to 'number of standard deviations from the mean'.

Watch out for

  • Forgetting to put data in order before finding the median.
  • Reading a bar's height wrongly when the scale goes up in 2s or 5s.

Support and stretch

Support

  • Review first: The normal distribution (68–95–99.7 rule).
  • Use concrete materials: squared paper for charts.
  • 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: Find normal probabilities with z-scores and technology.
  • 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-11.

Free lesson plan from eurekami.com/lesson-plan-builder

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Combine regions of a normal curve with the 68–95–99.7 rule

Name

Date

Score/ 6

  1. Question 1

    A quantity is normally distributed with mean 48 and standard deviation 5. What percentage of values lie between 53 and 58?

    Answer

  2. Question 2

    A factory fills 2000 bottles. Volumes are normally distributed with mean 300 ml and standard deviation 50 ml. About how many bottles hold less than 250 ml?

    Answer

  3. Question 3

    A quantity is normally distributed with mean 73 and standard deviation 8. What percentage of values lie between 65 and 89?

    Answer

  4. Question 4

    A factory fills 2000 bottles. Volumes are normally distributed with mean 112 ml and standard deviation 25 ml. About how many bottles hold more than 162 ml?

    Answer

  5. Question 5

    A quantity is normally distributed with mean 76 and standard deviation 7. What percentage of values lie between 83 and 90?

    Answer

  6. Question 6

    A factory fills 1000 bottles. Volumes are normally distributed with mean 392 ml and standard deviation 11 ml. About how many bottles hold more than 403 ml?

    Answer

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Answer key: Combine regions of a normal curve with the 68–95–99.7 rule

  1. 1. 13.5

  2. 2. 320

  3. 3. 81.5

  4. 4. 50

  5. 5. 13.5

  6. 6. 160