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# Z-Scores and Normal Distribution

Lesson 5 of 6

## Objective: Students will be able to estimate percentages from normally distributed data using a z-score.

## Big Idea: Get beyond the empirical rule... students will be able to find any percentage or probability for normally distributed data.

*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- Z-Score asks students list all of the percentages they can find on a normal distribution using the empirical rule. Please watch my Video Narrative for more information on this warm up.

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

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#### Z-Scores

*35 min*

This lesson begins with a task that shows students the need for finding a population percentage other than those found through the empirical rule. *The average height of women in the U.S. is 65 inches (5’5”) with a standard deviation of 3.5 inches.* First I ask the students to identify the percent of women between 58” and 72” which are two standard deviations in either direction. I then ask my students to find the percent of women between 60” and 70” which lie between one and two standard deviations in either direction. They quickly find that they do not have to tools to do a problem like this.

I then give them an easier problem designed to push them towards intuitively deriving the formula for finding z-score (**Math Practice 1**). *The number of miles walked by a group of student last week had a mean of 20 with a standard variation on 4.* This problem is nice because all miles lie on quarter deviations which are easy for students to visualize. I ask them a variety of questions that build to this idea. We then name and formalize this formula.

The remainder of the lesson focuses on the women’s height example. I ask a variety of questions that highlight the many problems that can be solved with a z-score. Each student will receive a copy of the z-score percentages table at this time. I found this z-score table on the website for the University of Texas in Dallas.

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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 find the probability that a woman will be 6'2" given the statistical information from the lesson.

#### Resources

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This assignment gives students the opportunity to find the z-score and use the z-score table in a variety of contexts such as bird watching, basketball wins, and test scores(**Math Practice 4**).

Some of the problems from this homework were inspire by this assignment.

#### 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