TITLE: Applying Measures of Central Tendency and Variation to Reaction Time
TASK DEVELOPER: Mr. Vasic
GRADE AND CONTENT AREA: 10th grade Application Class
SCOPE AND SEQUENCE: Section in Cluster 3 of the New Jersey HSPA
TARGET TEACHING DATE: March 14, 2007
SCHOOL: John F. Kennedy High School

STANDARDS:

DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS - GRADE 10

STANDARD 4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS: All students will develop an understanding of the concepts and techniques of data analysis, probability, and discrete mathematics, and will use them to model situations, solve problems, and analyze and draw appropriate inferences from data.

Strand A. Data Analysis: Building upon knowledge and skills gained in preceding grades, by the end of Grade 10, students will:

1. Use surveys and sampling techniques to generate data and draw conclusions about large groups.

  • Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling)

2. Evaluate the use of data in real-world contexts.

  • Accuracy and reasonableness of conclusions drawn
  • Bias in conclusions drawn (e.g., influence of how data is displayed)
  • Statistical claims based on sampling

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PERFORMANCES:

The students will analyze the reaction times of the whole class by finding the  mean, median, mode, range, lower quartile, upper quartile, interquartile range, and outlier of the data.  They will then graph their results and determine which measures best represent the reaction times of the entire class.  Finally, they will discuss how the mean, median, mode, range, lower quartile, upper quartile, interquartile range, and outlier help paint a representation of the entire class.

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SETTING:

Real World Setting: Recreation and sports

You are a coach for the 100m sprint team in track and field. You are faced with picking the best runner for the event. All of your runners run equally fast so you must pick the runner with the fastest reaction time. Once you have completed your test of all your runners' reaction time, you will analyze the data in order find where most of your runners stand and also to pick the runner with the best reaction time.

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SMARTSKILLS:

Level I: Acquiring Data - Data students will acquire in this standards-based task:

Numbers: The students' reaction times will be used as the data

Level II: Visualizing Information - Data from Level I that are visualized as information in this standards-based task:

Arranging: The data will be arranged in increasing order so that its analysis will be easier

Level III: Applying Knowledge - Visualized information from Level II that is applied knowledge in this standards-based task:

Making decisions: After students analyze and scrutinize the data they will determine which indicator represents all the runners best, as well as which runner has the fastest reaction time.

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PREFERENCES:

Student Involvement - The students will complete the task as a cooperative group as well as in a whole class group setting

Instruction - Activities will be organized and delivered as a teacher-facilitated set of hands-on activities.  Then the students will work on their own or in small groups to analyze the data.  The teacher will provide calculators and yard sticks.

Special Education Accommodations - Students with special needs will require the following electronic devices: Calculator

Use of Resources - The students will provide classroom materials such as pencils, paper, notebooks and homework time

Customer for Student Work - The student will present their work as evidence of task completion to their peers and to the teacher.

Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task: The student's teacher and Peers

Assessment of Student Work - The following forms of assessment will be used to determine progress and results: Performance assessment based on completion of the various tasks

Reporting Results - The students will hand in their completed analysis of the reaction time data.  This will then be scored according to the rubric and further discussed in class.  

Timeline - The estimated time needed to plan, teach, and score this task is two class periods.

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ACTIVITIES:

Teaching for Understanding in Mathematics

Activity 1: The teacher reviews yesterday's lesson (Estimated time: 1 minute)

  • Step 1: Review measures of central tendency
    • mean - the sum of the numbers of data divided by the number of items
    • median - the middle number...
    • mode - the number or numbers that appear most often in a set of data

  • Step 2: Review measures of variation
    • Range - the difference between the greatest and the least values of a set of data.
    • Lower Quartile - divides the lower half of the data into two equal parts.
    • Upper Quartile - divides the upper half of the data into two equal parts.
    • Interquartile Range - the difference between the upper and lower quartiles of a set of data.
    • Outlier - a value in a set of data that is much less or much greater than the rest of the data
      • occurs if a set of data is at least 1.5 interquartile ranges
        • less than the lower quartile
        • greater than the upper quartile

Technology: Calculator
Materials: Student workbook (Pre-HSPA Success)
Student product or performance: Answer questions to jog their memory
Links or connections between different parts of the lesson: This lesson is a continuation of the material presented for the past three days.
Scoring: None

Activity 2: Measure students' reaction time with a yard stick
(Estimated time: 10 minutes)

  • Step 1: The instructor shows the students how to measure the reaction time.
    • Person A holds the index finger and thumb one inch apart
    • Person B holds the yard stick vertically with the measure of 0 inches just above the one inch gap.
    • Person B drops the yard stick without cue and Person A must grab the yard stick as fast as possible.
    • The top of the measure is recorded as the data. (Note: faster reaction times will have a lower number)

  • Step 2: Repeat this two more times and record the data.

  • Step 3: Then Person A drops for Person B three times.
  • Note: The teacher will also take his or her reaction time.  The teacher will make one of his or her reaction times as slow as possible (30 to 36 inches), so as to create an outlier.

Technology: None
Materials: Yard stick and notebook
Student product or performance: Students will help each other take data of their reaction times.
Links or connections between different parts of the lesson: Measurement - reading a ruler
Scoring: None.  Reaction times are recorded, not scored.

Activity 3: The data is analyzed (Estimated time: 10 minutes)

  • Step 1: All data is written on the board

  • Step 2: The students calculate the measures of central tendency and variation

  • Step 3: The students create a box and whisker plot.

Technology: Calculator
Materials: Data, paper, pencil
Student product or performance: Students may work in small groups of about 3 or 4 to fulfill the above steps.
Links or connections between different parts of the lesson: Students have done this with data so it should be familiar to them.
Scoring: The data calculations will be reviewed with the entire class.  At the end of the class they will be collected.

Activity 4: Data is scrutinized (Estimated time: 10 minutes)

  • Step 1: All students should have the same number and graphical analysis of the data
  • Step 2: Students work in pairs or small groups to discuss
    the ways the coach can interpret the data. This information is then recorded.
    • Which measures of central tendency and variation best summarize the data?
  • Step 3: Different students or group leaders are called on to give their opinion on what the coach should do with the data.

Technology: none
Materials: results from Activity 3
Student product or performance: Students are to actively participate.
Links or connections between different parts of the lesson: This connects the setting with the results from Activity 3.
Scoring: The class will comment on the conclusions made by others.

Activity 5: The teacher sums up the lesson, data, and different ways data can be interpreted.
Note: No homework is typical, unless a student needs some extra time.
(Estimated time: 10 minutes)

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BENCHMARKING:

Student Performance One:

Assessment Benchmarking Example: Students will show all work and solve the problems. They will also draw and label a box-and-whisker plot.  The mean, median, mode, range, lower quartile, upper quartile, interquartile range, and outlier will depend on the data obtained through experimentation in class.  The students will also have to answer questions pertaining to the setting problem, including which calculation best represent all the reaction times.

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SCORING:

Maryland High School Mathematics Rubric: Extended Constructed Response
Level 4 The response indicates application of a reasonable strategy that leads to a correct solution in the context of the problem. The representations are correct. The explanation and/or justification is logically sound, clearly presented, fully developed, supports the solution, and does not contain significant mathematical errors. The response demonstrates a complete understanding and analysis of the problem.
Level 3

The response indicates application of a reasonable strategy that may or may not lead to a correct solution. The representations are essentially correct. The explanation and/or justification is generally well developed, feasible, and supports the solution. The response demonstrates a clear understanding and analysis of the problem.
Level 2

The response indicates an incomplete application of a reasonable strategy that may or may not lead to a correct solution. The representations are fundamentally correct. The explanation and/or justification supports the solution and is plausible, although it may not be well developed or complete. The response demonstrates a conceptual understanding and analysis of the problem.
Level 1

The response indicates little or no application of a reasonable strategy. It may or may not have the correct answer. The representations are incomplete or missing. The explanation and/or justification reveals serious flaws in reasoning. The explanation and/or justification may be incomplete or missing. The response demonstrates a minimal understanding and analysis of the problem.
Level 0 The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know."
Notes: Explanation refers to the student using the language of mathematics to communicate how the student arrived at the solution.

Justification refers to the student using mathematical principles to support the reasoning used to solve the problem or to demonstrate that the solution is correct. This could include the appropriate definitions, postulates and theorems.

Essentially correct representations may contain a few minor errors such as missing labels, reversed axes, or scales that are not uniform.

Fundamentally correct representations may contain several minor errors such as missing labels, reversed axes, or scales that are not uniform.
Source: http://www.mdk12.org/mspp/high_school/structure/algebra/index.html

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METACOGNITION:

Cognitive Information: I will collect the following information verbally by asking questions near the end of the period.

  1. Describe what skills you needed to complete this task.
  2. Explain how you solved the goal, problem, or issue in this task.
  3. What are the different applications of this activity to real life? Explain.

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RESULTS:

Reflect: I was surprised at how well the students dealt with the abundance of data.  Each group wrote the reaction times for each member on the left side of the board.  There were 15 students present that day and I made up the data for the 16th individual. All together we had 48 reaction times listed.  In the previous days the students mostly worked with around 10 number values. From the suggestion of a student, the class then went about writing the data in order from least to greatest from the left to the right of the entire board.  From there the groups went on to determine the median, mean, etc.  Some preferred to work from their seat to find the median, but those that needed to physically count each number took advantage of the fact that all numbers were in order.

It was great to see that some groups were checking with others to make sure they were calculating similar results.  They were very engaged with the lesson and enjoyed seeing where their reaction times stood with the rest of the class.  Some students even wanted to take more reaction times, but I took the ruler and allowed them to get them back when they finished.  This actually did motivate some to work harder.

The connection to the real world setting was an easy sell, but I realized that I wasn't painting the whole reaction time picture to the students.  The students measured their reaction distance, which is actually related to the square root of the time.  This is something I could revisit at the end of the year and let the students actually calculate the reaction times.

As far as task completion goes, the students returned assignment of a high caliber.  Four students in the class had a Level 4 response as they fully answered the question and organized the data very well.  Nine students were at Level 3 since they were missing a few things, and two students handed in Level 2 assignments.  They usually have difficulty in math, but working in groups helped them understand a little better.  I noticed that these two students did better on their test than in previous test, showing about a 13% test score increase.  The rest of the class, also had excellent test scores, especially on the open-ended questions, showing that their understanding was an average 9% higher than the class that I didn't do this activity with.  However, that is the score difference on all the other test.

 Many students commented that they enjoyed finding out their reaction time and how it affects their performance in sports and want to do more activities like this in the future.

Action Plan: I will complete the following TaskBuilder Figure 8 Strategy Action Plan to prepare for my next standards-based task.

1. Plan - My next standards-based task will focus on: stem-and-leaf plots

  • Title: Introduction to Stem-and-Leaf Plots
  • Content Area: Mathematics, App. Math, Statistics
  • Learning Standard(s): 4.4.A.2
  • Intent: (Instruction or Assessment) Students will be introduced to stem and leaf plots.  Then they will apply today's data to make a stem-and-leaf plot of the reaction times.  From this, they will look for similarities between the stem-and-leaf plot and the boxplot of the same data.  They will report on any additional information, they can interpret

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