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# The Normal Distribution

Lesson 4 of 6

## Objective: Students will be able to estimate population percentages on a normal distribution using the mean and standard deviation.

*50 minutes*

#### Warm Up and Homework Check

*10 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. This lesson’s Warm Up- Normal Distribution asks students to explain the mistakes made in finding the mean of a frequency table. My goal for this lesson is to give students some practice in evaluating mistakes as well as ensuring that they understand how to read a frequency table.

I also use this time to correct and record the previous day's Homework.

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#### Normal Distribution

*35 min*

I begin this lesson by looking at an example of the normal distribution using SAT scores. SAT scores are my lesson hook because they are important to these students as many of them are juniors and facing this test at some point this year. We discuss the idea of a normal distribution and then look at an example of skewed data to provide a comparison.

Using the short below, I then show the students the importance of standard deviation. Please watch my Video Narrative explaining my thoughts on the use of videos in a mathematics classroom.

I pass out a copy of the normal distribution that will pasted into their notes. We then discuss the empirical rule as well as the fact that the area under the curve is equal to 1 or 100%.

The students to talk to their partner about how the shape of a normal distribution will change as the standard deviation gets larger (**Math Practice 1**). This is a great opportunity for students to deepen their understanding of the normal distribution.

We then look at the SAT distribution again and add the markings for each standard deviation. I ask the students a variety of questions surrounding this distribution.

*What score range did the middle 68% of students have?*

*What is the range of scores within 2 standard deviations? *

*What range of scores did the top 0.1% get?*

*What range of scores did the bottom 2.3% get?*

*What range of scores did the middle 99.7% get?*

These are just samples. More can be added to ensure that the students are grasping the concept. I may pull out white boards for this activity depending on my students. White boards ensure that all of the students are putting in some good effort.

The next task gives students the following information with no visual model: *The average height of adult males is 70 inches (5’10”) with a standard deviation of 4 inches.* Again I ask students a variety of questions based on this information.

*What heights are within one standard deviation from the average?*

*What heights are more than 3 standard deviations from the average?*

*93.3% of males are shorter than what height?*

*93.3% of males are taller than what height? *

The final task turns the question creation back to the students (**Math Practice 4**). Here are the instructions:

*The average score on a test was 78 and there was a standard deviation of 4. Write three questions that can be answered with the normal curve.*

Once they have created their questions, there are a variety of activities they can be used for. If we are short on time, I may collect them and ask the class some of my favorites. If we have more time, I may shuffle and pass them back out for pairs of students to complete. I would then have each pair share their favorite. They could also be added to the homework assignment.

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#### Exit Ticket

*5 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

Today's Exit Ticket asks students to use the standard deviation on a normal distribution to determine the interval for 95% of a population.

#### Resources

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This assignment allows the students to practice finding intervals and percentages using a variety of real life contexts (**Math Practice 4**). My goal is to the deepen their knowledge of the empirical rule on a normal curve which will help them in the next lesson.

#### Resources

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review