# Cluster 1: Investigate patterns of association in bivariate data. (Supporting Cluster)Archived

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.8.SP.1
Title: Investigate patterns of association in bivariate data. (Supporting Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Statistics & Probability

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.8.SP.1.AP.1a
Graph data using line graphs, histograms or box plots.
MAFS.8.SP.1.AP.1b
Graph bivariate data using scatter plots and identify possible associations between the variables.
MAFS.8.SP.1.AP.1c
Using box plots and scatter plots, identify data points that appear to be outliers.
MAFS.8.SP.1.AP.2a
Draw the line of best fit on a scatter plot.
MAFS.8.SP.1.AP.3a
Interpret the slope and the y-intercept of a line in the context of data plotted from a real-world situation.
MAFS.8.SP.1.AP.4a
Analyze displays of bivariate data to develop or select appropriate claims about those data.
MAFS.8.SP.1.AP.2b
Identify outliers on a scatter plot given the line of best fit.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

## 3D Modeling

Wind Farm Design Challenge:

In this engineering design challenge, students are asked to create the most efficient wind turbine while balancing cost constraints. Students will apply their knowledge of surface area and graphing while testing 3D-printed wind farm blades. In the end, students are challenged to design and test their own wind farm blades, using Tinkercad to model a 3D-printable blade.

Type: 3D Modeling

## Formative Assessments

Two-Way Relative Frequency Table:

Students are asked to convert raw data to relative frequencies by both rows and columns given a two-way frequency table.

Type: Formative Assessment

Tuition:

Students are asked to use a linear model to make a prediction about the value of one of the variables.

Type: Formative Assessment

Stretching Statistics:

Students are asked to interpret a specific solution and the y-intercept of a linear equation that describes a context.

Type: Formative Assessment

Sleepy Statistics:

Students are given a scatter plot in a real-world context and asked to describe the association between the variables.

Type: Formative Assessment

School Start Time:

Students are asked to describe an association between two variables given a table of relative frequencies by column.

Type: Formative Assessment

Population Density:

Students are given a scatterplot in a real-world context and asked to describe the association between the variables.

Type: Formative Assessment

Music and Sports:

Students are asked to construct a two-way frequency table given a set of raw data.

Type: Formative Assessment

Infectious Statistics:

Students are given a scatterplot in a real-world context and asked to describe the association between the variables.

Type: Formative Assessment

Foot Length:

Students are asked to interpret the line of best fit, slope, and y-intercept of a linear model.

Type: Formative Assessment

Developmental Data:

Students are asked to interpret the slope of a linear model.

Type: Formative Assessment

Cheesy Statistics:

Students are given a scatterplot in a real-world context and asked to describe the association between the variables.

Type: Formative Assessment

Bungee Cord Data:

Students are asked to construct a scatterplot corresponding to a given set of data.

Type: Formative Assessment

Siblings and Pets:

Students are asked to describe an association between two variables given a table of relative frequencies by row.

Type: Formative Assessment

Two Scatterplots:

Students are asked to compare two lines fitted to data to determine which fit is better.

Type: Formative Assessment

Three Scatterplots:

Students are asked to informally assess three lines fitted to data to determine which fit is the best.

Type: Formative Assessment

Line of Good Fit - 2:

Students are asked to informally fit a line to model the relationship between two quantitative variables and to assess how well that line fits the data.

Type: Formative Assessment

Line of Good Fit - 1:

Students are asked to informally fit a line to model the relationship between two quantitative variables and to assess how well that line fits the data.

Type: Formative Assessment

## Lesson Plans

Why Correlations?:

This lesson is an introductory lesson to correlation coefficients. Students will engage in research prior to the teacher giving any direct instruction. The teacher will provide instruction on how to find the correlation coefficient by hand and using Excel.

Type: Lesson Plan

Why Correlations?:

This lesson is an introductory lesson to correlation coefficients. Students will engage in research prior to the teacher giving any direct instruction. The teacher will provide instruction on how to find the correlation coefficient by hand and using Excel.

Type: Lesson Plan

Where Should We Move? STEM Lesson Plan:

Students will collect data to identify planet composition, average temperature, and the distance of some planets within the Milky Way Galaxy from the Sun. Students will complete two-way tables to make comparisons. Students will then analyze and interpret their data. Students will make inferences and justify their reasoning.

Type: Lesson Plan

Sea Turtle Nesting Associations:

In this lesson, students preview a Sea Turtle Research Perspectives Video to introduce the idea that sea turtle gender hatching rates and temperature are related. Students will use scatterplots to visualize the relationships and draw conclusions about associations and how it helps in interpreting two variable data sets.

Type: Lesson Plan

Clean Up, Collect Data, and Conserve the Environment!:

Students will participate in collecting trash either on campus or another location. They will compare the distance traveled and the weight of the trash bag collected. Students will explore the use of mean and median in finding the ratios of the data set. They will discuss the use of mean and median in finding the relationship between the independent and dependent variables. Students will examine their scatter plot and determine if any patterns of association exist. They will compare their data to a coastal cleanup report. Finally, students will use the data to help determine interventions at the local, state and national level regarding environmental issues.

Type: Lesson Plan

The changing climate is an important topic for both scientific analysis and worldly knowledge. This lesson uses data collected by the National Snow and Ice Data Center to create and use mathematical models as a predictive tool and do critical analysis of sea ice loss.

Type: Lesson Plan

Sensoring Data:

In this follow up lesson, students will explore data collection using the weather station sensor and perform statistical analysis of the data. Students will use a scientific method of inquiry to plan an investigation of their own. This activity is meant to allow students to use a variety of skills they have acquired throughout a statistics unit in a personally meaningful way.

Type: Lesson Plan

In this design challenge, students will explore electrical circuits. Students will use their skills in science, math, and technology to determine how many light bulbs can be powered off of one circuit. Students will build circuits, measure luminosity, graph data, analyze the data and then report their findings to Kiser construction.

Type: Lesson Plan

Shipwrecked Pirates:

In this lesson, students will take the role of shipwrecked pirates. Working in groups, they will have to use the concepts of force, speed, scatter plots, and literal equations to come up with a way of getting one student to a nearby sister island so that they will both have enough food to survive.

Type: Lesson Plan

Mixtures and Solutions Uncovered:

This lesson is a hands-on approach to SC.8.P.8.9 that the students enjoy and are engaged in. The main activities cover making anchor charts (teacher lead) that will assist them in completing activities that cover vocabulary and a break down of characteristics for mixtures. There are four group activities that will guide the students to an understanding of the standard outlined. This is a two-day lab that adds teacher demonstration and allows for collaborative group and student-talk sessions.

Type: Lesson Plan

Sensoring Data:

In this follow up lesson, students will explore data collection using the weather station sensor and perform statistical analysis of the data. Students will use a scientific method of inquiry to plan an investigation of their own. This activity is meant to allow students to use a variety of skills they have acquired throughout a statistics unit in a personally meaningful way.

Type: Lesson Plan

Steel vs. Wooden Roller Coaster Lab:

This lesson is a Follow Up Activity to the Algebra Institute and allows students to apply their skills on analyzing bivariate data. This STEM lesson allows students the opportunity to investigate if there is a linear relationship between a coaster's height and speed. Using technology the students can determine the line of best fit, correlation coefficient and use the line for interpolation. This lesson also uses prior knowledge and has students solve systems of equations graphically to determine which type of coaster is faster.

Type: Lesson Plan

Height Arm Juxtaposition:

This lesson is a Follow Up Activity to the Algebra Institute and allows students to apply their skills on analyzing bivariate data. This STEM lesson allows students the opportunity to investigate if there is a linear relationship between a person's height and arm length. Using technology the students will explore in-depth how to perform a least square regression as a procedure for determining the line of best fit.

Type: Lesson Plan

Height Scatterplot Lab:

This lesson is a Follow Up Activity to the Algebra Institute and allows students to apply their skills on analyzing bivariate data. This STEM lesson allows students the opportunity to investigate if there is a linear relationship between a person's height and foot length. Using technology the students can determine the line of best fit, correlation coefficient and use the line for interpolation.

Type: Lesson Plan

Star Scatter Plots:

In this lesson, students plot temperature and luminosity data from a provided star table to create a scatter plot. They will analyze the data to sequence the colors of stars from hottest to coolest and to describe the relationship between temperature and luminosity. This lesson does not address differentiation between absolute and apparent magnitude.

Type: Lesson Plan

Scattering Plots:

Students will find the slope of the line of regression in a scatter plot. Students will interpret the line of regression in context to a real-world situation involving bivariate data.

Type: Lesson Plan

Two-Way Math Survey:

In this lesson, students will construct a two-way table based on data given. They will interpret the data to find if there is a correlation between the two variables from the same subject. As the lesson progresses, students will create their ownÂ testable questions that they will collect data on using the survey method. Students will show their mastery of this math concept with the presentation of theÂ data in a two-way table and interpretation of the data to draw inferences.

Type: Lesson Plan

Text Count and Homework Minutes:

Text Count and Homework Minutes

The Students will engage in counting Text minutes and Homework minutes. They will keep a journal to be kept over a four day period, Monday thru Thursday. The assignment can be done on Friday. The students will record the number of text messages received and sent out during the hours of 3:00-11:00. On the second journal the students will also record the number of minutes spent on homework between the hours of 3:00-11:00. They will compile this information into graphs on that 5th day Friday. With this data the students will be able to plot a scatter plot graph and learn how to recognize patterns on a scatter plot graph. Students will see the relevance of a correlation even if the graph is not linear or a straight line. Students will identify trends and articulate the reason for the trends.

Type: Lesson Plan

Scattered Time:

Through a slide show presentation, students will be led through data in one variable, bivariate data, scatter plots, lines of best fit, outliers, and correlations. They will be able to analyze bivariate data to determine correlations, associations, and the impact of outliers.

Type: Lesson Plan

Tackling 2 Way Tables:

This highly engaging and interactive lesson will have students constructing two-way tables, calculating relative frequency and analyzing the bivariate data to determine a possible association between the two variable categories.

Type: Lesson Plan

If the line fits, where's it?:

In this lesson students learn how to informally determine a "best fit" line for a scatter plot by considering the idea of closeness.

Type: Lesson Plan

Scatter Plots at Arm's Reach:

This lesson is an introductory lesson to scatter plots and line of best fit (trend lines). Students will be using m&m's to represent different associations in scatter plots, and measure each other's height and arm span to create their own bivariate data to analyze. Students will be describing the association of the data, patterns of the data, informally draw a line of best fit (trend line), write the equation of the trend line, interpret the slope and y-intercept, and make predictions.

Type: Lesson Plan

Help, My Data is Scattered!!!:

This lesson provides activities for students to conceptually understand how to gather, organize, and interpret data using a scatter plot. They will be required to work cooperatively to complete certain tasks. They will use estimation strategies to complete a experiment that compares the the length of their hand to the number of centimeter cubes they can grab in order to make predictions about data with a larger sample size. Students will be assessed formally and informally throughout lesson via slide notes and data collection tool.

Type: Lesson Plan

Where do I fit in?:

Students will do an exploration activity with paper airplanes. They will be up and moving around. Students will be able to understand and analyze bivariate data that they will be able to plot from the exploration activity. They will be able to describe patterns, and also find facts and come up with conclusions based on the data they gather.

Type: Lesson Plan

Creating a Linear Model:

Students will analyze data to create scatter plots. They will draw the line of best fit to determine linear models. The teacher will use PowerPoint and activities included to guide the students into finding the line of best fit.

Type: Lesson Plan

What's Your Association: Scatter Plots and Bivariate Data:

In this lesson, students will learn how to graph, describe, and determine patterns and associations for bivariate data. This lesson incorporates vocabulary development and collaboration.

Type: Lesson Plan

Linear Statistical Models:

In this lesson, students will learn how to analyze data and find the equation of the line of best fit. Students will then find the slope and intercept of the best fit line and interpret the meaning in the context of the data.

Type: Lesson Plan

This lesson will get your student up to speed with constructing and interpreting scatter plots using bivariate data. With this lesson, patterns of association are investigated between two quantities and positive or negative association as well as linear and non-linear association is determined through the use of data from the NFL as reported by ESPN. Data from Consumer Reports is used to determine associations between shoe price and quality.

Type: Lesson Plan

Chemical or Physical Change? That is the Question!:

Students will conduct an investigation on the effect of laundry detergent on water temperature, use technology to graph their data, and determine whether a physical or chemical change occurred. Students will also read articles to gather evidence to write an evidence-based claim using the CLEVER method.

Type: Lesson Plan

How Fast Can You Go:

Students will apply skills (making a scatter plot, finding Line of Best Fit, finding an equation and predicting the y-value of a point on the line given its x-coordinate) to a fuel efficiency problem and then consider other factors such as color, style, and horsepower when designing a new coupe vehicle.

Type: Lesson Plan

Scattered Data:

This lesson allows students to use real-life problem-solving skills to construct and interpret scatter plots by generating and recording their own data. Students will investigate patterns between bivariate measurement data. They will model linear associations with a line of best fit.

Type: Lesson Plan

Spaghetti Bridges:

Students use data collection from their spaghetti bridge activity to write linear equations, graph the data, and interpret the data.

Type: Lesson Plan

Finding the Hottest Trend:

In this lesson, students will graph a scatter plot and learn how to recognize patterns. The students will learn that correlation may still exist even though the points are not in a perfectly straight line (linear function). Students will be able to identify outliers, describe associations, and justify their reasoning.

Type: Lesson Plan

Why Correlations?:

This lesson is an introductory lesson to correlation coefficients. Students will engage in research prior to the teacher giving any direct instruction. The teacher will provide instruction on how to find the correlation coefficient by hand and using Excel.

Type: Lesson Plan

Guess the Celebrities' Heights!:

In this activity, students use scatter plots to compare the estimated and actual heights of familiar celebrities and athletes. They will determine how their answers impact the correlation of their data, including the influence of outliers. Finally, they will compare their correlation to that provided in a scatter plot with a larger data sample.

Type: Lesson Plan

Graphing Equations on the Cartesian Plane: Slope:

The lesson teaches students about an important characteristic of lines: their slope. Slope can be determined either in graphical or algebraic form. Slope can also be described as positive, negative, zero, or undefined. Students get an explanation of when and how these different types of slope occur. Finally, students learn how slope relates to parallel and perpendicular lines. When two lines are parallel, they have the same slope and when they are perpendicular their slopes are negative reciprocals of one another. Prerequisite knowledge: Students must know how to graph points on the Cartesian plane. They must be familiar with the x- and y- axes on the plane in both the positive and negative directions.

Type: Lesson Plan

Constructing and Calibrating a Hydrometer:

Students construct and calibrate a simple hydrometer using different salt solutions. They then graph their data and determine the density and salinity of an unknown solution using their hydrometer and graphical analysis.

Type: Lesson Plan

## Original Student Tutorials

The Notion of Motion, Part 3 - Average Velocity:

Describe the average velocity of a dune buggy using kinematics in this interactive tutorial. You'll calculate displacement and average velocity, create and analyze a velocity vs. time scatterplot, and relate average velocity to the slope of position vs. time scatterplots.Â

This is part 3 of 3 in a series that mirrors inquiry-based, hands-on activities from our popular workshops.

• ClickÂ  to open The Notion of Motion, Part 1 - Time Measurements
• Click HERE to open The Notion of Motion, Part 2 - Position vs Time

Type: Original Student Tutorial

The Notion of Motion, Part 2 - Position vs Time:

Continue an exploration of kinematics to describe linear motion by focusing on position-time measurements from the motion trial in part 1. In this interactive tutorial, you'll identify position measurements from the spark tape, analyze a scatterplot of the position-time data, calculate and interpret slope on the position-time graph, and make inferences about the dune buggy’s average speed

Type: Original Student Tutorial

Scatterplots Part 6: Using Linear Models :

Learn how to use the equation of a linear trend line to interpolate and extrapolate bivariate data plotted in a scatterplot. You will see the usefulness of trend lines and how they are used in this interactive tutorial.

This is part 6 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 5: Interpreting the Equation of the Trend Line :

Explore how to interpret the slope and y-intercept of a linear trend line when bivariate data is graphed on a scatterplot in this interactive tutorial.

This is part 5 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 4: Equation of the Trend Line:

Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this interactive tutorial.

This is part 4 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 3: Trend Lines:

Explore informally fitting a trend line to data graphed in a scatter plot in this interactive online tutorial.

This is part 3 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 2: Patterns, Associations and Correlations:

Explore the different types of associations that can exist between bivariate data in this interactive tutorial.

This is part 2 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 1: Graphing:

Learn how to graph bivariate data in a scatterplot in this interactive tutorial.

This is part 1 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

## Perspectives Video: Expert

Birdsong Series: Statistical Analysis of Birdsong:

Wei Wu discusses his statistical contributions to the Birdsong project which help to quantify the differences in the changes of the zebra finch's song.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Determining Strengths of Shark Models based on Scatterplots and Regression:

Chip Cotton, fishery biologist, discusses his use of mathematical regression modeling and how well the data fits his models based on his deep sea shark research.

Type: Perspectives Video: Professional/Enthusiast

Asymptotic Behavior in Shark Growth Research:

Fishery Scientist from Florida State University discusses his new research in deep sea sharks and the unusual behavior that is found when the data is graphed.

Type: Perspectives Video: Professional/Enthusiast

Slope and Deep Sea Sharks:

Shark researcher, Chip Cotton, discusses the use of regression lines, slope, and determining the strength of the models he uses in his research.

Type: Perspectives Video: Professional/Enthusiast

You and Michael:

In this problem solving task, students will test Marcus Vitruvius"s theory that a person"s height is approximately equal to their arm span (wingspan). Students will test this theory via collection, recording, graphing and analysis of data.

Bear Hugs:

In this problem solving activity, students are tasked with measuring the arm lengths of fellow students. Students will record the data and use it to construct a boxplot and scatterplot to help draw conclusions.

Modeling Linear Relationships:

In this lesson, which is part of a unit on bivariate data and analysis, students use data collected comparing height and arm span and create a scatter plot from the bivariate data. In the first section, How Square Can You Be?, students look at measurement comparisons between the height and arm span of 24 people. The second section, Analyzing the Differences, asks students to look at the data again to answer questions about proportion of height to arm span, such as who is square and who is rectangular. The third section, Using a Scatter Plot, has students analyze the same data on a scatter plot with the line representing Height = Arm Span. There are questions for student to answer independently, but hints are provided.

How Is the Weather?:

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Music and Sports:

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.

Students are asked to examine a scatter plot and then interpret its meaning. Students should identify the form of the relationship (linear, curved, etc.), the direction or correlation (positive or negative), any specific outliers, the strength of the relationship between the two variables, and any other relevant observations.

US Airports:

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Teaching Idea

Birds & Eggs:

In this task, students start by examining a scatter plot about the size of various bird eggs from a collection of measurements. In particular, students are asked to identify a correlation, sketch an approximation for the line of best fit, determine the equation of that line, use the equation of the line and/or the graph to make interpolative predictions, and draw conclusions about the properties of specific eggs by using the graphical presentation of the data.

Type: Teaching Idea

## Tutorial

Scatter Plots:

Scatterplots are used to visualize the relationship between two quantitative variables in a binary relation. As an example, trends in the relationship between the height and weight of a group of people could be graphed and analyzed using a scatter plot.

Type: Tutorial

## Unit/Lesson Sequences

Linear Functions and Slope:

This session on linear function and slope contains five parts, multiple problems and videos, and interactive activities geared to help students recognize and understand linear relationships, explore slope and dependent and independent variables in graphs of linear relationships, and develop an understanding of rates and how they are related to slopes and equations. Throughout the session, students use spreadsheets to complete the work, and are encouraged to think about the ways technology can aid in teaching and understanding. The solutions for all problems are given, and many allow students to have a hint or tip as they solve. There is even a homework assignment with four problems for students after they have finished all five parts of the session.

Type: Unit/Lesson Sequence

Direct and Inverse Variation:

"Lesson 1 of two lessons teaches students about direct variation by allowing them to explore a simulated oil spill using toilet paper tissues (to represent land) and drops of vegetable oil (to simulate a volume of oil). Lesson 2 teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases." from Insights into Algebra 1 - Annenberg Foundation.

Type: Unit/Lesson Sequence

## Video/Audio/Animations

Interpreting Scatter Plots - Study Time, Shoe Size, and Test Score:

This 3-minute video provides an example of how to solve a problem involving scatter plots.

Type: Video/Audio/Animation

Trend Lines (Smoking in 1945):

This 5-minute video provides an example of how to solve a problem using a trend line to estimate data through a problem called, "Smoking in 1945."

Type: Video/Audio/Animation

Linear Equations in the Real World:

Linear equations can be used to solve many types of real-word problems. In this episode, the water depth of a pool is shown to be a linear function of time and an equation is developed to model its behavior. Unfortunately, ace Algebra student A. V. Geekman ends up in hot water anyway.

Type: Video/Audio/Animation

Fitting a Line to Data:

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.

Type: Video/Audio/Animation

## Virtual Manipulatives

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Data Flyer:

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Univariate and Bivariate Data:

This lesson is designed to introduce students to the difference between univariate and bivariate data, and how the two can be represented graphically. This lesson provides links to model discussions and online graphing applets, as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Virtual Manipulative

This is an online graphing utility that can be used to create box plots, bubble graphs, scatterplots, histograms, and stem-and-leaf plots.

Type: Virtual Manipulative

Curve Fitting:

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

Line of Best Fit:

This manipulative allows the user to enter multiple coordinates on a grid, estimate a line of best fit, and then determine the equation for a line of best fit.

Type: Virtual Manipulative

KidsZone: Create a Graph:

Create bar, line, pie, area, and xy graphs.

Type: Virtual Manipulative

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorials

The Notion of Motion, Part 3 - Average Velocity:

Describe the average velocity of a dune buggy using kinematics in this interactive tutorial. You'll calculate displacement and average velocity, create and analyze a velocity vs. time scatterplot, and relate average velocity to the slope of position vs. time scatterplots.Â

This is part 3 of 3 in a series that mirrors inquiry-based, hands-on activities from our popular workshops.

• ClickÂ  to open The Notion of Motion, Part 1 - Time Measurements
• Click HERE to open The Notion of Motion, Part 2 - Position vs Time

Type: Original Student Tutorial

The Notion of Motion, Part 2 - Position vs Time:

Continue an exploration of kinematics to describe linear motion by focusing on position-time measurements from the motion trial in part 1. In this interactive tutorial, you'll identify position measurements from the spark tape, analyze a scatterplot of the position-time data, calculate and interpret slope on the position-time graph, and make inferences about the dune buggy’s average speed

Type: Original Student Tutorial

Scatterplots Part 6: Using Linear Models :

Learn how to use the equation of a linear trend line to interpolate and extrapolate bivariate data plotted in a scatterplot. You will see the usefulness of trend lines and how they are used in this interactive tutorial.

This is part 6 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 5: Interpreting the Equation of the Trend Line :

Explore how to interpret the slope and y-intercept of a linear trend line when bivariate data is graphed on a scatterplot in this interactive tutorial.

This is part 5 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 4: Equation of the Trend Line:

Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this interactive tutorial.

This is part 4 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 3: Trend Lines:

Explore informally fitting a trend line to data graphed in a scatter plot in this interactive online tutorial.

This is part 3 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 2: Patterns, Associations and Correlations:

Explore the different types of associations that can exist between bivariate data in this interactive tutorial.

This is part 2 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

Scatterplots Part 1: Graphing:

Learn how to graph bivariate data in a scatterplot in this interactive tutorial.

This is part 1 in 6-part series. Click below to open the other tutorials in the series.

Type: Original Student Tutorial

How Is the Weather?:

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Music and Sports:

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.

Students are asked to examine a scatter plot and then interpret its meaning. Students should identify the form of the relationship (linear, curved, etc.), the direction or correlation (positive or negative), any specific outliers, the strength of the relationship between the two variables, and any other relevant observations.

US Airports:

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

## Student Center Activity

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

## Tutorial

Scatter Plots:

Scatterplots are used to visualize the relationship between two quantitative variables in a binary relation. As an example, trends in the relationship between the height and weight of a group of people could be graphed and analyzed using a scatter plot.

Type: Tutorial

## Video/Audio/Animations

Trend Lines (Smoking in 1945):

This 5-minute video provides an example of how to solve a problem using a trend line to estimate data through a problem called, "Smoking in 1945."

Type: Video/Audio/Animation

Linear Equations in the Real World:

Linear equations can be used to solve many types of real-word problems. In this episode, the water depth of a pool is shown to be a linear function of time and an equation is developed to model its behavior. Unfortunately, ace Algebra student A. V. Geekman ends up in hot water anyway.

Type: Video/Audio/Animation

Fitting a Line to Data:

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.

Type: Video/Audio/Animation

## Virtual Manipulatives

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Data Flyer:

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

This is an online graphing utility that can be used to create box plots, bubble graphs, scatterplots, histograms, and stem-and-leaf plots.

Type: Virtual Manipulative

Curve Fitting:

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

Line of Best Fit:

This manipulative allows the user to enter multiple coordinates on a grid, estimate a line of best fit, and then determine the equation for a line of best fit.

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

How Is the Weather?:

This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.

Music and Sports:

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.

Students are asked to examine a scatter plot and then interpret its meaning. Students should identify the form of the relationship (linear, curved, etc.), the direction or correlation (positive or negative), any specific outliers, the strength of the relationship between the two variables, and any other relevant observations.

US Airports:

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

## Teaching Idea

Birds & Eggs:

In this task, students start by examining a scatter plot about the size of various bird eggs from a collection of measurements. In particular, students are asked to identify a correlation, sketch an approximation for the line of best fit, determine the equation of that line, use the equation of the line and/or the graph to make interpolative predictions, and draw conclusions about the properties of specific eggs by using the graphical presentation of the data.

Type: Teaching Idea

## Video/Audio/Animation

Fitting a Line to Data:

Khan Academy tutorial video that demonstrates with real-world data the use of Excel spreadsheet to fit a line to data and make predictions using that line.

Type: Video/Audio/Animation

## Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative