Background Information

Content/Course

Algebra 2

Unit

Unit 4 – Inferences and Conclusions from Data

Essential Questions/Enduring Understandings Addressed in the Lesson

Enduring Understandings

Mathematics can be used to solve real world problems and can be used to communicate solutions to stakeholders.

It is important to be well informed on the correct ways to interpret data and make sound decisions.

Relationships between quantities can be represented symbolically, numerically, graphically and verbally in the exploration of real world situations

The context of a question will determine the data that needs to be collected and will provide insight on the best method for collecting and analyzing the data.

Recognize when to apply simulations to model real world situations.

The results of statistical analysis must be interpreted and analyzed to determine if there is a significant evidence to justify conclusions about real world situations.

Statistics needs to be applied to make inferences and justify conclusions

Recognize possible sources of error and types of error in context of the real world.
Essential Questions

When is mathematics an appropriate tool to use in problem solving?

What types of situations correspond with the different methods of data collection?

What is the most effective method of data analysis in order to draw conclusions?

What characteristics of problems determine how to model a situation and develop a problem solving strategy?

What characteristics of a problem influence the choice of representation and analysis of the data?

How can mathematical representations be used to communicate information effectively?

How can data be represented to best communicate important information about a problem?

What characteristics of problems would determine how to model the situation and develop a problem solving strategy?

How can statistical analysis be used to decide what conclusions need to be drawn?

What role does probability have in decision making?

How can the process of data analysis justify a conclusion?

Standards Addressed in This Lesson

S.ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution
and to estimate population percentages. Recognize that there are data sets for
which such a procedure is not appropriate. Use calculators, spreadsheets, and
tables to estimate areas under the normal curve.
Note: This lesson addresses only the first sentence of this standard.

Lesson Title

Let’s Roll the Dice: Introduction of the Normal Distribution

Relevance/Connections

How does this lesson connect to prior learning/future learning and/or other content areas?
Students will build on the understanding of key ideas for describing distributionsshape, center, spread that were developed in the Grades 68 Statistics and Probability Progression. This lesson will allow students to add a key measure of variation to their toolkits. Prior to this lesson students should have had a lesson on how to calculate standard deviation by hand so they have a sense of where this statistical measure comes from. Students will be able to use standard deviation as a means of describing data distributions that are approximately normal in shape.
Additional lessons will be needed to explore all aspects of this S.ID.4. A next lesson might include data sets for which the use of standard deviation is not appropriate. Estimating areas under normal curves would also need to be developed in future lessons.

Student Outcomes

The student will:

Find the mean and standard deviation of a data set.

Find and interpret the standardized score for an observation in a data set.

Determine if a data set fits an approximately Normal distribution.

Estimate population percentages using the mean and standard deviation of a Normal distribution.

Estimate the areas under a Normal curve.

Summative Assessment
(Assessment of Learning)

How will this “Student Outcome” be assessed on a Summative Assessment? What evidence of student learning would a student be expected to produce to demonstrate attainment of this outcome?
Teacher should collect worksheets from Let’s Roll the Dice lesson for evidence of ability to: Find the mean and standard deviation of a data set, find and interpret the standardized score for an observation in a data set, determine if a data set fits an approximately Normal distribution, estimate population percentages using the mean and standard deviation of a Normal distribution and estimate the areas under a Normal curve.
A subsequent summative assessment would involve a new context for which a similar process of analysis is appropriate. Given a data set, calculate the mean and standard deviation, then determine if the data set fits an approximately Normal distribution. The student should also be assessed on the ability to find and interpret the standardized score for an observation in a data set.

Prior Knowledge Needed to Support This Learning
(Vertical Alignment)

Students need an understanding of interpreting differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (Algebra I Unit 3Descriptive Statistics S.ID.3). They also need an understanding of standard deviation to complete the activities in this lesson.

Method for determining student readiness for the lesson

How will evidence of student prior knowledge be determined?
A warmup exercise that requires students to determine the mean and standard deviation of a data set. The students will also need to construct a histogram from the data set and describe its center, shape and spread (variability) later in the lesson.
What will be done for students who are not ready for the lesson?
Students will be paired for support; teacher will need to assess what skills or concepts are barriers to student unable to engage in this lesson.

Learning Experience

How will this experience help students to develop proficiency with one or more of the Standards for Mathematical Practice?
Which practice(s) does this experience address?

Component

Details


Warm Up/Drill

Given the data set: Test scores are 85, 84, 87, 92, 75, 86, 94, 99, 100

Determine the mean and standard deviation and describe its meaning in the context of the situation.

Construct a histogram and describe its center, shape and spread (variability).

Make sense of problems and persevere in solving them.
Teacher should pose the question and as much as possible have students brainstorm as to how to investigate

Motivation

Teacher poses question, “Can you predict what you are going to roll?” Students may say that there is an equal chance for each outcome or be able to recognize that some outcomes may occur more often than others when rolling two dice (sum of 7 is more likely than 2 or 12).

Model with mathematics.
Students will reflect on whether the results make sense. They should recognize that the individual data may not necessarily represent the entire population as well as the class data.
Construct viable arguments and critique the reasoning of others.
Teacher can pose a question as to why the individual histograms may appear different than the class histogram.
Use appropriate tools strategically.
Students will be using knowledge of mean, standard deviation(variability), and histograms to create a visual representation of the data.
Attend to precision.
Students will be careful to label axes to clarify the quantities appropriate for a single 6sided number cube and a pair of 6sided number cubes.
Look for and make use of structure.
Teacher will prompt class to discuss differences and similarities between the Warm Up exercise and Activity 1. (Rolling the 6sided number cubes.)

Activity 1

This activity can be used as a reinforcement of S.ID.3 (Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).)
Materials Needed

“Let’s Roll the Dice” handout

6 sided number cubes (2 per student or student pair)

Calculator

Power Point Slides
Implementation

Give each student a copy of the “Let’s Roll the Dice” handout.

Give each student one 6sided number cube.

Tell students that they will be generating some data by rolling a number cube 18 times and recording the results on their “Let’s Roll the Dice” handout.

As indicated on the handout, students should make a bar graph /histogram using their data and should also determine the values of the indicated summary statistics.

Combine all of the student data into one table and display the results.

Instruct students to record this data on the provided table on their handout.

Instruct students to make a bar graph/histogram and to determine the values of the summary statistics for the class data.

Facilitate a discussion of the results using questions such as those shown below.
UDL Connections
This activity adheres to UDL Principle III Provide Multiple Means of Engagement Guideline 7, in that the rolling of dice to collect data is a different way of engaging in the learning process. Such an activity will be engaging for many learners.
Guideline 7: Provide options for recruiting interest
Information that is not attended to, that does not engage learners’ cognition, is in fact inaccessible. It is inaccessible both in the moment and in the future, because relevant information goes unnoticed and unprocessed. As a result, teachers devote considerable effort to recruiting learner attention and engagement. But learners differ significantly in what attracts their attention and engages their interest. Even the same learner will differ over time and circumstance; their “interests” change as they develop and gain new knowledge and skills, as their biological environments change, and as they develop into selfdetermined adolescents and adults. It is, therefore, important to have alternative ways to recruit learner interest, ways that reflect the important inter and intraindividual differences amongst learners.
Key Questions:

What do you notice about your histogram? Class distribution? (For the individual histograms, students may notice that one outcome is more likely than another. Students may notice that their distribution is different from a peer’s distribution. For the class distribution, students should notice a uniform shape for Activity 1 and symmetric distribution centered around 7 for Activity 2.)

Why are all the heights of the class frequency similar for Activity 1? (Students may say that each outcome is equally likely. The 6sided number cube is fair.)


Activity 2

Materials Needed

Students will need to use their second die.

“Let’s Roll the Dice” handout (problems 7 12)
Implementation

Distribute a second number cube to each student.

Instruct students to roll their pair of number cubes 18 times and record the sum of the numbers from each roll on the provided table

As indicated on the handout, students should make a bar graph /histogram using their data and should also determine the values of the indicated summary statistics.

Combine all of the student data into one table and display the results.

Instruct students to record this data on the provided table on their handout.

Instruct students to make a bar graph/histogram and to determine the values of the summary statistics for the class data.

Facilitate a discussion of the results using questions such as those shown below.
Key Questions

How would you describe the shape of the distribution for Activity 2?

Why do you think it has the shape that it does?

Model with mathematics.
Students will construct their own normal curves based on mean and standard deviations given.
Use appropriate tools strategically.
Students will apply their newly acquired knowledge of Normal Distributions to construct and interpret a given data value using mean and standard deviation.
Attend to precision.
Students will carefully construct their curves using appropriate labeling and calculations given the context of the exercises.
Look for and make use of structure.
Teacher will prompt class to discuss differences and similarities between the standard Normal Curve and the example exercises.
Reason abstractly and quantitatively.
Students will determine a zvalue for a given data value and estimate the area under the Normal curve that corresponds to that data value.

Activity 3

This activity can be used as an introduction to the Normal Distribution.
Materials Needed

“What is Normal?” handout (one for each student)

Power Point Slides
Implementation

Distribute a “What is Normal?” handout to each student.

Discuss different distributions, including uniform, symmetric, and skewed distributions. (use power point slides)

Discuss differences between a symmetric bellshaped curve and a normal curve. (Normal curves are always bell curves, but bell curves are not always Normal curves.)

Introduce the Empirical Rule (68%95%99.7% Rule). (Students may need clarification that “within 1σ” means in the positive and negative direction.)

Label sections of the normal curve, as students do the same on their note page.

Present the scenario of Example 1: Heights of Women (Students may need to be told that normal curves occur in many reallife scenarios. Many aspects of humans are approximately normal, such as height, weight, length of appendages.)

Instruct students to label the provided curve with the mean and standard deviation.

Ask “What is the probability that a randomly selected woman would be 67” tall or taller?” (The reason for the inequality component of the question is to identify the likelihood of the event or more extreme possibilities)

Instruct students to identify and mark where the desired outcomes are located on the graph.

Explain that it is necessary to sum the probabilities for each region that is part of the solution. (The answer should look like: .136 +.021 + .001 = .158. So 15.8% of all women will be 67” or taller.)

Continue with Example 2: Walleye Fish.
(The answer to “What percentage of the fish population can you keep from this lake,” students should mark 1 standard deviation below the mean (to the left) and sum the probabilities .34 +.34 + .136 +.021 + .001. The final answer is .838, which can be interpreted as 83.8% of the walleye fish population can be kept from this lake.)
Key Questions

Do the percentages in each region stay the same for different normal curves? Yes, the percentages are based on standardized positions.

What is the total area of any normal curve? (100%.)

Name a difference between two curves that have a similar mean, but different standard deviations. (The normal curves will be centered at the same value, but will be spread across the number line differently.)


Closure
How will evidence of student attainment of the lesson outcomes be determined?
 
Ask students to complete the provided summary questions: Ask students to provide an example of a population that may also follow a normal distribution. (Some examples may include age, test scores, computer sales, song lengths, movie lengths, etc.)

Assign Normal Sketch Practice WS.

Supporting Information


Details

Interventions/Enrichments

Special Education/Struggling Learners

ELL

Gifted and Talented

Gifted and Talented:
Students will be able to use normal probabilities to perform statistical significance tests and develop appropriate confidence intervals.
Special Education/Struggling Learners:
Teacher will have prepared data collection sheets with data/axes/histogram, normal calculations completed prior to lesson, based on student needs.

Materials

Student: Sixsided number cubes, “Let’s Roll the Dice” Worksheet, What is Normal Note Page, , graph paper, ruler , Normal Sketch Practice WS
Teacher: What is Normal PowerPoint (or .pdf)

Technology

Calculators

Warm Up
Given the data set:
Test scores: 85, 84, 87, 92, 75, 86, 94, 99,100

Determine the mean and standard deviation and describe
its meaning in the context of the situation.

Construct a histogram and describe its center, shape and spread (variability).
Warm UpAnswer Key
Given the data set:
Test scores: 85, 84, 87, 92, 75, 86, 94, 99,100

Determine the mean and standard deviation and describe
its meaning in the context of the situation.
Mean
Standard Deviation
Add the squares of the deviations from the mean.
Divide the sum of the deviations from the mean by a number that is one less than the number of data values. (This is the formula used when the data set is a sample and not a population.)
Take the square root of the quotient from the previous step
The standard deviation is a measure of spread or variance. In this example the standard deviation tells us that that the distribution of test scores is somewhat clustered around the mean. In other words most of the student’s test scores fell within approximately 8 points of 89%.

Construct a histogram and describe its center, shape and spread (variability).

3







2







1








7075

7680

8185

8690

9195

96100

Let’s Roll the Dice
Activity 1

Roll one 6sided number cube 18 times. Record a tally mark next to the appropriate number after each roll. After rolling the cube 18 times, determine the frequency for each number by counting the tally marks.
Individual Data

Number Rolled

Tally Marks

Frequency

1



2



3



4



5



6




Create a histogram or bar graph for your individual data.

Determine the value of each of the following for your data set.


Mean

Median

Mode

Range

Interquartile Range

Standard Deviation

Record the combined class totals in the table below.
Number Rolled

Frequency

1


2


3


4


5


6



Create a histogram or bar graph for the class data.

Determine the value of each of the following of the Class data set.


Mean

Median

Mode

Range

Interquartile Range

Standard Deviation
Activity 2
Individual Data

Sum of the Numbers Rolled

Tally Marks

Frequency

2



3



4



5



6



7



8



9



10



11



12



Roll two 6sided number cubes 18 times. Record a tally mark next to the appropriate number after each roll. After rolling the cube 18 times, determine the frequency for each number by counting the tally marks.

Create a histogram or bar graph for your Individual Data for Activity 2.

Determine the value of each of the following for your data set.


Mean

Median

Mode

Range

Interquartile Range

Standard Deviation

Record the combined class data for Activity 2 in the table below.
Sum of the Numbers Rolled

Frequency

2


3


4


5


6


7


8


9


10


11


12



Create a histogram or bar graph for the Class Data for Activity 2.

Determine the value of each of the following for the class data set.


Mean

Median

Mode

Range

Interquartile Range

Standard Deviation
What is Normal?

What terms can be used to describe the shape of a data distribution?

Make a sketch to illustrate each shape.

Provide a real world example of a data set that might produce each type of distribution.

What is a symmetric bellshaped curve?

What is a normal curve?

What is the difference between a symmetric bellshaped curve and a normal curve?

What is the Empirical Rule?

Label the provided Normal Curve to illustrate the Empirical Rule.

Assume that the data which reflects the “Heights of Women” is normally distributed.
Heights of Women
Mean = µ = 64.5 in.
Standard Deviation =σ = 2.5 in.


Label the provided normal curve to reflect this information.

What is the probability that a randomly selected woman would be 67 inches tall or taller? Explain how you determined your answer.

Shade the graph to reflect this information.

The population of walleye fish in a local lake is normally distributed.
Population of Walleye Fish
Mean = µ = 8 in.
Standard Deviation =σ = 0.6 in.
The game commission will only allow you to keep fish that are at least 7.4 inches long.
What percentage of the fish population can you keep from this lake? Label the provided normal curve to illustrate how you determined your answer.
Closure Questions

Do the percentages in each region stay the same on all normal curves? Explain your answer.

What is the total area under any normal curve?

Describe the differences in the shapes of normal curves used to represent two normally distributed data sets which have the same mean but different standard deviations.

Provide an example of a population which would probably be normally distributed. Justify your answer.
Normal Sketch Practice
In each problem: sketch, label, shade the region of the curve, and answer the question. The information in each question may not be accurate in reallife.
1. Music video artists are not the only ones that get to play around with money. The average salary for a music video producer is normally distributed with a mean of $134,000 with a standard deviation of $4,200. What percentage of music video producers make more than $125,600?
2. A real estate agent, working entirely on commission, makes an average of $850 with a standard deviation of $260 weekly selling property in the city. If we assume this distribution is roughly normal, what is the probability that a real estate agent will make between $1110 and $1,630 selling property in the city?
3. Mr. Cerutti likes to collect random things, such as hotel bar soap. At one hotel, he found a 3.5oz bar of soap. Wholesale distributors report that the weight of hotel bar soap is normally distributed with a mean of 2.3oz and standard deviation of 0.4oz. How rare was this bar of soap? Hint: Find the area above that standard deviation!
4. The length it takes a teenager to respond to a Facebook update is (surprisingly) normally distributed with a mean of 4 hours and standard deviation of 23 minutes. What is the probability that a recorded time will not be in between 217 and 286 minutes?
DRAFT Maryland Common Core State Curriculum Lesson Plan for Algebra II August 2012 Page of
