Lesson Plan: 8.SP.A.1-2 Associations in Bivariate Data
(This lesson should be adapted, including instructional time, to meet the needs of your students.)
Background Information /Content/Grade Level / Statistics and Probability/Grade 8
Unit / Understand and interpret patterns of association between two sets of data
Essential Questions/Enduring Understandings Addressed in the Lesson / Essential Questions:
How does analysis of patterns in data inform and influence decisions?
To what extent can we predict the future?
Enduring Understandings:
A difference exists between causation and correlation.
Modeling data can help us understand patterns.
Standards Addressed in This Lesson / 8.SP.A.1: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.A.2: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
It is critical that the Standards for Mathematical Practice be incorporated in ALL lesson activities throughout each unit as appropriate. It is not the expectation that all eight Mathematical Practices will be evident in every lesson. The Standards for Mathematical Practice make an excellent framework on which to plan instruction. Look for the infusion of the Mathematical Practices throughout this unit.
Lesson Topic / Constructing and interpreting scatter plots for bivariate measurement; determining a line of best fit, as appropriate, to analyze the relationship between two data sets.
Relevance/Connections / 8.SP.B.3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
8.EE.B.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
Student Outcomes / · Ability to relate the scenarios to authentic, student-centered situations, and to integrate technology, if possible.
· Ability to keep paired data organized in relation to one another within two sets of quantities.
· Ability to informally fit a straight line for scatter plots that suggest a linear association; and informally assess the model fit by analyzing the distribution and closeness of the data points relative to the line.
Prior Knowledge Needed to Support This Learning / · 5.G.A.1 and 5.G.A.2: Graph points on the coordinate plane to solve real-world and mathematical problems.
Method for determining student readiness for the lesson / Battleship on Grid Paper (Illustrative Mathematics)
· Materials – Each student will need:
o One copy of Attachment #1: Battleship Grid Paper
o Two different color pencils: One color (for example red) to mark ordered pairs on the grid that represent explosions on the student’s ships and the opponent’s ships; a different color (for example black) to mark ordered pairs that the student calls out but are not a “hit” on one of the opponent’s battleships.
· Setup
o Students begin by folding their copy of Attachment #1 in half along the fold line. The half with the coordinate plan should be flat on the desk with the other half opening up so that it makes a wall/barrier so students’ opponents cannot see each other’s grids. Students will need to plot ordered pairs to represent five battleships:
§ Two PT boats that are sitting across 5 ordered pairs,
§ One Submarine that is sitting across 6 ordered pairs.
§ One Destroyer that is sitting across 7 ordered pairs, and
§ One Air Craft Carrier that is sitting across 8 ordered pairs
o Each battleship should be drawn horizontally or vertically on the coordinate plane, not diagonally. See Attachment #2: Sample Battleship Grid.
· Actions
o Students play in pairs, sitting opposite each other and making certain to keep the folded half of the grid paper in an “up” position so they cannot see each other’s coordinate planes. Students take turns calling out ordered pairs. Each student should mark the ordered pair they call out on their coordinate plane. They should mark in black if they “missed” one of the opponent’s ships and red if they “hit” a ship. Another option is to list in the margin of the grid paper the ordered pairs that represent a “miss” (in black) versus a “hit” (in red).
o Since students sometimes inadvertently transpose the numbers in a coordinate pair, players should keep a list of the ordered pairs they call out written in(x, y)form on a piece of paper that both players can see. This strategy will minimize later disagreements.
o The first player to make at least one hit on all five of the opponent’s ships and sink the opponent’s fleet is the winner.
Learning Experience /
Component / Details / Which Standards for Mathematical Practice does this activity address? How is each Practice used to help students develop proficiency?
Warm Up/Motivation for Activity 1 / Share with students a likeness of Charlemagne, along with the following legend:
Some historical accounts imply that the left foot of Emperor Charlemagne, who ruled much of Europe in the late 8th century, became the standard of length for the measure we use today and refer to as a “foot.” The width of Charlemagne’s thumb – a space just one twelfth the length of his foot – was used as an “inch.”
In the 18th century, Charlemagne's skeleton was measured, and his height was estimated to have been 6'4". According to ancient records, Charlemagne's height was seven times the length of his foot.
Discuss with students:
· Based on this information, was Charlemagne’s foot truly a foot long, according to today’s standard measurement? Why or why not?
· Was the width of his thumb truly an inch wide, according to today’s standard measurement? Why or why not?
Activity 1
UDL Components
· Multiple Means of Representation
· Multiple Means for Action and Expression
· Multiple Means for
Engagement
Key Questions
Formative Assessment
Summary / UDL Components:
· Principle I: Representation is present in the activity. Prior knowledge is activated through the Warm-up and Motivation. Options perception are provided through discussion of Charlemagne’s legend and the photographs of modeled “yard” and “span” measurements using the human body.
· Principle II: Expression is present in the activity. The grouping of students for different tasks provides options for physical activity. Similarly, students interact with each other and with various physical materials as they measure their “yard” and “span” lengths; as they record their self-generated data and review the entire class’s data on the overhead or the document camera; and as they collaborate while fitting “lines” of linguini on the sample scatter plot patterns.
· Principle III: Engagement is present in the activity. Sources of information are in an authentic, student-centered context. The task allows for active participation, exploration and experimentation, and also invites personal response, evaluation and reflection. Also, depending on resources available to teacher, the activities can be explored using graphing calculators and computerized statistics programs.
Directions:
Present the students with the following scenario and use photographs to demonstrate how to measure a “yard” and a “span”:
In former times, arms and hands also have been used to measure lengths. A “yard” was commonly considered the length from the tip of one’s nose to the tip of one’s longest finger. A “span” was the length of one’s open hand, from the tip of the pinky finger to the tip of the thumb.
Yard Span
· Ask students to think about Charlemagne’s “foot” and “inch,” and predict whether a relationship exists between the length of a person’s “yard” and “span.” Students should identify/describe on a piece of paper or in a journal entry what the relationship might be should one exist, or why none exist. Note to Teacher: In all likelihood, the data should provide a positive correlation.
· Place students in small groups of 3 or 4.
· Give each group a measuring tape.
· Have students measure each other’s “yard” and “span” lengths, being careful to match each set of measures with its owner. Note to teacher: a decision must be made re: use of inches versus centimeters.
· Provide each group with a transparency copied from attached graphic: Attachment #3: Yard and Span Scatter Plot. Remind the students that in this case it doesn’t matter which variable is the dependent or independent variable.
· Have the class define the scale for the horizontal and vertical axes, including the unit of measure they are using (inches vs. centimeters). Ask group members to copy the class’s chosen scale on their transparencies.
· After student groups complete their measuring task, have them plot their points on the group’s transparency. The coordinates of the points should match each student’s “yard” measure and “span” measure.
· Ask students to place their transparencies on top of one another on an overhead (or document camera), taking care to line up the axes and scales.
· Direct students to respond to the following directions on their papers or in their journal entries.
· Look at the collection of points that represent data collected from your class:
o Was your prediction correct? Why or why not?
o Do the points on the scatter plot create a pattern? Write a brief description of a pattern you observe.
o Do the points correlate in any particular direction?
o Are the points close together or far apart?
o What relationship between the “yard” data and “span” data do you think the points represent?
· Call on students to share their various responses with the class as a means of discussing and defining important vocabulary such as bivariate data, correlate, correlation, clustering, outliers, positive versus negative associations, linear association, and nonlinear association. These terms should be posted in the room for student reference.
· Distribute copies of the attached activity sheets, Attachment #4: Degree of Correlation and Attachment #5: Example Scatter Plots and use during the class discussion to provide visual representations of vocabulary terms noted above.
· Provide each student with a piece of linguini. Inform students that straight lines are widely used to model relationships, or correlations, between two quantitative variables, such as the bivariate data set they have collected. Teacher should (1) use linguini to demonstrate on the overhead the linearity of the “yard”/”span” data from the transparencies, and (2) ask students to describe/identify the slope for that line as positive, negative, slope of 0, of no slope (undefined).
· Place students back in their small groups and direct them to consider which scatter plots (from the activity sheets) suggest a linear association. Students should work collaboratively to:
o Informally fit a straight line with their linguini.
o Informally assess each line’s fit by (1) judging the closeness of the data points to the line, and (2) counting the number of points on each side of the line – should be an approximately equal number on each side.
o Describe/identify slope as positive, negative, 0, or undefined.
NOTE:
If technology is available, this activity would lend itself well to exploration using the statistics functions on a graphing calculator or a statistics computer program (e.g., TinkerPlots©). Students would have an alternative opportunity to plot their data points, work with lines of fit, and investigate relationships between bivariate data. / Make sense of problems and persevere in solving them by requiring students to gather data on “span” and “hand;” to determine the appropriate scale for the scatter plot; and to use the scatter plot to display their data.
(SMP # 1)
Reason abstractly and quantitatively by requiring students to perceive and analyze relationships among values in bivariate data; to decontextualize these quantitative relationships by defining a pattern(s) within the displayed data; and to explore options for a line of fit.
(SMP #2)
Construct viable arguments and critique the reasoning of others by requiring students to justify their conclusions with mathematical ideas in their group discussions and personal journals.
(SMP #3)
Attend to precision by expecting students to plot their points accurately; use exact mathematical language when expressing their ideas to describe what the data is showing them; and understand the meaning of the informally fit line in the context of the problem.
(SMP #6)
Closure / · Project the coordinate grid, Attachment #6: Closure Scatter Plot on a sheet of poster paper stuck to a flat surface (chalkboard, wall, door, etc.).
· Give each student a sticker to use as the point that represents the coordinates of his/her birthday (by month and by day).
· Have all students plot their points on the grid for Attachment #6. Note to Teacher: In all likelihood, the plotted data for these two variables will have no correlation.
· Ask students to analyze the bivarate data represented on the scatter plot. Lead students in a discussion in which they review correlation, non-correlation, clustering, outliers, positive versus negative associations, linear association, and nonlinear association, etc.
Warm-up/Motivation for Activity 2 / Display the scatter plot for Attachment #7: The Flu.
Present students with the following scenario:
· A nurse at the local medical clinic has noticed a correlation, or relationship, between the number of people who have come to the clinic complaining of the flu and the number of people to whom she gave flu shots this year. Describe this bivariate correlation.
· Possible responses:
o The greater the number of people with the flu, the fewer the number of shots/jabs given.