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Linear Regressions and Making Predictions (The Wave)

Objectives: Students will be able to:

· Review linear equation vocabulary.

· Create a best-fit line after plotting data points.

· Find the equation for the best-fit line created.

· Make a prediction based on the equation found.

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Instructional Plan

“THE WAVE”

Mathematics Governor’s Institute 2005

Names of group members: Lawrence Lee, Brian Long,

Bernadette Louden,

Sandy Madden, Jennifer Taylor,

Topic/Theme: Linear Regression and Making Predictions

Level: Pre-Algebra or Algebra

Time Element: 2 days

NCTM Standards Addressed:

· Understand patterns, relations and functions.

· Represent and analyze mathematical situations and structures using algebraic symbols.

· Use mathematical models to represent and understand quantitative relationships.

· Compute fluently and make reasonable estimates.

PA Math Standards Addressed:

· 2.8.8.B: Discover, describe and generalize patterns, including linear, exponential and simple quadratic relationships.

· 2.8.8.G: Represent relationships with tables or graphs in the coordinate place and verbal or symbolic rules.

· 2.8.11.A: Analyze a given set of data for the existence of a pattern and represent the pattern algebraically and graphically.

· 2.8.11.L: Write the equation of a line when given the graph of the line, two points on the line, or the slope of the line and a point on the line.

· 2.8.11.M: Given a set of data points, write an equation for a line of best fit.

Math Assessment Anchors Addressed:

· M8.D.1.1: Analyze, extend or develop descriptions of patterns or functions.

· M8.D.2.2: Create and/or interpret expressions, equations or inequalities that model problem situations.

· M8.D.4.1: Represent relationships with tables or graphs on the coordinate plane.

· M8.A.2.2: Represent or solve problems using rates, ratios, proportions, and/or percents.

Reading Assessment Anchors Addressed:

1.1.8. B: Identify and use common organizational structures and graphic features to comprehend information.

1.1.8. F: Understand the meaning of and apply key vocabulary across the various subject areas.

1.4.8. B: Write multi-paragraph informational pieces (e.g. letters descriptions, reports, instructions, essays, articles, interviews).

1.4.8. D: Maintain a written record of activities, course work, experience, honors and interests.

1.5.8. B: Write paragraphs that have details and information specific to the topic and relevant to the focus.

1.6.8. A: Listen to others.

· Ask probing questions.

· Analyze information, ideas, and opinions to determine relevancy.

· Take notes when needed.

1.6.8. C: Speak using skills appropriate to formal speech situations.

· Use complete sentences.

· Pronounce words correctly.

· Adjust volume to purpose and audience.

1.6.8. D: Contribute to discussions.

· Ask relevant, probing questions.

· Respond with relevant information, ideas or reasons in support of opinions expressed.

· Listen to and acknowledge the contributions of others.

· Adjust tone and involvement to encourage equitable participation.

· Clarify, illustrate or expand on a response when asked.

· Present support for opinions.

· Paraphrase and summarize, when prompted.

1.6.8. E: Participate in small and large group discussions and representations.

· Initiate everyday conversation.

· Select a topic and present an oral reading.

· Organize and participate in informal debates.

1.8.8. A: Select and refine a topic for research.

1.8.8. B: Locate information using appropriate sources and strategies.

· Determine valid resources for researching the topic, including primary and secondary sources.

· Evaluate the importance and quality of sources.

· Select essential sources (e.g. dictionaries, encyclopedias, other reference materials, interviews, observations, and computer databases.

· Use tables of contents, indices, key words, cross-references and appendices. Use traditional and electronic search tools.

Objectives: Students will be able to:

· Review linear equation vocabulary.

· Create a best-fit line after plotting data points.

· Find the equation for the best-fit line created.

· Make a prediction based on the equation found.

Instructional Strategies and Plan (include strategies used to help different types of learners, i.e. auditory, visual, etc):

· Use a vocabulary development web to review vocabulary related to linear equations.

· Students form a circle of chairs and are instructed in how to do the “WAVE.” Also, provide students with a brief history of the wave.

· Practice doing the “WAVE” a few times so that students feel comfortable and the data is more reliable.

· Do the “WAVE” between ten and fifteen times, using a different number of students for each trial.

· Record the data with “NUMBER OF PEOPLE” in the first column and “TIME IN SECONDS” in the second column. Round to the nearest half second. Go up to about 40 people.

· Discuss which information is dependent and which is independent so students know how to label axes.

· Monitor students as they plot the ordered pairs from the data gathered.

· After checking to see that all students have plotted correctly, instruct students to draw what they feel would be the best-fit line through the data points.

· Instruct students to find the equation for the line they have drawn, expressing the slope and the y-intercept as decimals.

· Have students come to the board and record their equation. The slope and y-intercept expressed as decimals will make these important facts easy to compare.

· Students will observe that everyone has a similar slope and y-intercept. Take time to help students who do not have similar answers.

· Discuss information about PNC Park. Teacher has timed the “WAVE” previously. Students will receive a copy of the park and will discuss the sections that should be used to make a prediction for how long the wave takes to go around the park once.

· Discuss how the “WAVE” looks at any sports event. Guide students to understand that more than one person stands up at a time in a stadium as opposed to the way we conducted the “WAVE” in class. Ask students to guess how many people stand up at one time. I have observed that approximately 5 people stand at a time so guide students to realize that one student in the classroom represents 5 people in PNC Park.

· Guide students to the map of PNC Park, asking them which sections should be used as an “average”. Guide students to using the 100 level. There are 40 sections, each with approximately 20 seats per section. Have students figure out the number to substitute in for ‘x’ (the number of people) to get the time in seconds.

· Average the predictions together and find the approximate time the wave takes to go around PNC Park.

· Follow up with technology by creating a scatter plot and figuring out the line of regression on the graphing calculator. Compare the equation to student equations and compare predictions.

Materials/Resources:

· Worksheet with a brief history of the wave and a chart to record data.

· Map of PNC Park.http://pittsburgh.pirates.mlb.com/NASApp/mlb/pit/ballpark/seating_pricing.jsp

· Graph paper labeled appropriately.

· Graphing calculator and overhead calculator.

· Stopwatch

For students with ADD/ADHD, helpful strategies for this group project would include:

1. To have student work in a mixed ability group utilizing group reward reinforcements

2. To encourage peer assistance and collaborative learning

3. Clarify the rules when setting up the procedure

4. Allow the student to have a responsible job within the sub-group

For students who have specific learning disabilities:

1. Provide direct instruction while in the “group processes”

2. Prepare the sub-group members to include and help the student

3. Provide extra structure within the group by stating the goal(s), objectives and tasks, and listing the necessary steps to accomplish this goal

For students who suffer the effects of autism:

1. Require the student to stay in and perform the group activity if it is meaningful

2. Provide this type of student with a transitional warning before group activities begin. This can be performed with the use of a special transitional object (i.e. ringing of a special bell, verbal five minute warning prior to the start of the group activity, etc.)

3. Allow student to leave the activity if frustration sets in

4. Provide this particular type of student with a sequence of the activities.

For students who display emotional disturbances:

1. Have set proactive strategies to prevent behavior problems related to group activities (point/token system, time-out, etc.)

2. Reinforce appropriate classroom behavior prior to activity.

3. Pair student with socially competent student for modeling purposes.

4. Use non-verbal cues to gauge when to intervene

For students with Asperger’s Syndrome:

1. Acknowledge when appropriate social communication skills are used.

2. Encourage use of informal language with peers.

3. Model use of formal language appropriate with adults.

Pair student with socially competent group, and explain to other students about typical AS behavior.

Interdisciplinary Connections:

· Reading

Vocabulary Development Map on “Linear Equations”

· Technology

Creating a scatterplot on a graphing calculator and graphing the line of best fit as a way to verify results.

Wave simulator at http://angel.elte.hu/wave/

· Other

Social Studies – wave originated in Mexico

Assessment Strategies:

· Formative Evaluation (checking student understanding during the lesson):

o Students will be working in groups and will be able to check each other’s work.

o Teacher will check each group’s work at crucial points of the activity.

o Students will put their equations on the board

o Corrections can be made if needed.

· Summative Evaluation (How will it be determined that the objectives were achieved?):

o Use height and wingspan information collected previously from 3 or 4 of the teacher’s classes and ask students to create a scatter plot, find an equation, notice patterns and make predictions. Students will also use the graphing calculator to verify their answers. All of these components will be turned in and evaluated. In addition, students will write a short paragraph summarizing the process. Results will be discussed in class. Projects will be graded using a performance assessment rubric. http://teach-nology.com/web_tools/rubrics/math/ (Scroll down to create your math rubric.)

Correctives/Remediation:

· Small group instruction/re-teaching.

Extensions/Enrichment:

· Figure out the time it takes the “WAVE” to go around levels closer to the field or further from the field.

· Figure out the time it takes the “WAVE” to go around other stadiums.

Special Accommodations (special needs students)

· Description of the Special Needs student selected:

o Our selected student in a male that does not attend classes regularly. He comes to school, but often cuts his classes. He loves reading and writing but the teachers are unable to assess his abilities because of his poor attendance.

Accommodations to use with this student:

· Provide the student with the responsibility in class that involves writing.

1. Writing the warm-up

2. Writing the objectives

3. Writing the examples

· Pair the student with others who have trouble reading/writing.

· Make journalizing a part of this activity. As well, make sure this student is asked to share responses.

· While participating in the group work, this student could be the recorder. Could possibly pair the student with other students who need help with the reading/writing but also with a student who will hold everyone accountable.

· Student could possibly read problem/lessons to the class.

· As a reward, this student could possibly read to students in the lower grades.

· Ask student to write/keep notes.

· Keep summary of information for students who are absent.

· Have student keep a special journal for the class.