# SmartGraphs

In this project, we're creating and disseminating "SmartGraphs," digital objects that "know" about themselves and that provide scaffolding to students to help them learn about graphs and the concepts graphs convey, in mathematics, science and other subjects.

## SmartGraphs: Algebra App

Our new app helps you learn algebra using 19 graph-related activities on four key topics: linear equations, quadratic equations, transformations of functions and exponential functions.

SmartGraphs runs directly in a Web browser. It is free, and there is nothing to download or install (except Java is needed on a user's computer in order to run activities that use a Vernier GoMotion! Sensor).

Activities lasting one class period or less are available in a variety of subjects, including physical science, algebra, and social science. Click on the Curriculum "tab" above to find and use the existing activities with your students.

The Research "tab" above provides information about research on the use of SmartGraphs conducted with thousands of students and dozens of teachers. Students in a large randomized experimental group outperformed students in the control group at a statistically significant level.

In addition, an authoring system allows teachers, curriculum developers, or others to create and disseminate new SmartGraphs activities. No special programming knowledge is required; instead, authors fill in online forms. When an activity is ready, it can be shared easily by providing students or other users with a URL. Click on the Authoring "tab" above to find out more about and to use the authoring system.

A two-sided, tri-fold brochure describing SmartGraphs can be downloaded by clicking here

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## Activity Spotlight

### Inverses of Functions

Learn to graph the inverses of a series of discrete and continuous functions.

Learn MoreIn the fall of 2011, SmartGraphs conducted a randomized experimental trial in 29 Pennsylvania schools. In 14 schools, 17 experimental teachers used SmartGraphs activities with 49 8th and 9th grade Physical Science classes studying the motion of objects (such as position-time and velocity-time graphs). In the other 15 schools, 18 control teachers taught the same topic using the same textbooks, with 42 Physical Science classes. This study sought to answer two research questions. A summary of findings is provided below. In October 2013 the *Journal of Science Education and Technology* published an article about our research on its website; for additional information besides the summary, see the accepted manuscript, Helping Students Make Sense of Graphs: An Experimental Trial of SmartGraphs Software. (Note: the final publication is available online at link.springer.com.)

The first research question was: What do teachers using SmartGraphs physical science activities believe about the software, including how well the software matches important learning goals for the motion unit of study? Based on the 203 sessions when experimental teachers used a SmartGraphs activity with a class, teachers agreed or strongly agreed that the SmartGraphs activities address important learning goals and help students meet those goals. Also, in 99% of the sessions teachers reported they would use the activity again, either exactly as is (63%) or with only minor changes (36%).

The second research question was: Do students who use SmartGraphs activities learn more than comparison students studying the same topic from the same textbooks, but who do not use SmartGraphs activities? To answer this question, nearly 1,700 students completed a pre- and a post-test that included multiple-choice and constructed-response items assessing students' understanding of about a dozen learning goals. The experimental students showed greater gains than the control students on all three measures: multiple-choice items, constructed-response items, and total score (see table below). In each case, the results are statistically significant.

*Gain Scores from Pre to Post Test*

Experimental | Control | Significance (p) | |
---|---|---|---|

Total | 5.07 | 4.30 | .008 |

Multiple-Choice | 1.16 | 1.07 | .049 |

Constructed-Response | 4.19 | 3.64 | .043 |

Answers to both research questions were positive, which is encouraging. In addition, we are continuing to do additional research in Pennsylvania in the fall of 2012.

Note: The SmartGraphs project has made use of existing research about students' understanding of graphs and graphing. Two articles that we have found especially useful are "Making sense of graphs: Critical factors influencing comprehension and instructional implications" (Friel, Curcio, & Bright, 2001) and "Representation in school mathematics: Learning to graph and graphing to learn" (Monk, 2003).

### SmartGraphs Activities

Click on one of the topics below, and then a sub-topic (if any), to see a list of available activities. Then click on the activity name to start the activity.

Activities (below) are available for Physical Science, Biology, Algebra, and Social Science and Other. In each activity, data persist during the browser session but are not otherwise saved. If you wish there were activities covering additional topics, you may want to create new activities of your own using the authoring system (see the authoring tab). A brief guide to using activities is available here.

**Maria's Run**

Shows that the motion of an object can be described by its position, direction of motion, and speed. (The activity requires a Vernier Go!Motion sensor, an inexpensive device that plugs into your USB port.)

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Motion Toward and Away**

Explores different ways of describing motion on a graph. (The activity requires a Vernier Go!Motion sensor, an inexpensive device that plugs into your USB port.)

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**How Fast Am I Moving?**

Uses the position of an object at several times to determine the direction and velocity traveled during different time intervals. (The activity requires a Vernier Go!Motion sensor, an inexpensive device that plugs into your USB port.)

» Lesson Plan and Student Assessment documents are also available. Standards Addressed

Three activities are designed for use in a standard motion unit in Physical Science classes. These activities focus on position-time graphs, velocity-time graphs, and acceleration due to gravity. Each takes 30-40 minutes for students to complete. Portions of these activities require use of a motion sensor.

**Describing Velocity**

Connects the motion of an object to the corresponding position- time and velocity-time graphs to determine the velocity during different intervals.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Was Galileo Right?**

Explores the effects of gravity on light and heavy objects during free fall.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed

Two activities are designed for use in a standard motion unit in Physical Science classes. These activities focus on position-time graphs, velocity-time graphs, and acceleration due to gravity. Each takes 30-40 minutes for students to complete. These activities do not require use of a motion sensor.

**Boiling Water**

This demo illustrates users' interactions with graphs in a SmartGraphs activity.

**African Lions: Modeling Populations**

Students explore exponential and logistic growth models to analyze population data for African lions and identify carrying capacity.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**4.1 Population Curve: Significance of Breakpoints**

The logistic curve, or S-curve, is important in understanding how populations change over time. It's a complex curve that can be analyzed with a linear fit to help identify the breakpoints where changes of growth rate occur. This activity brings together many graph interpretation skills, and puts them to use to design and analyze a population experiment using the ubiquitous aquatic plant, duckweed.

» A Lesson Plan is also available.

**Linear Equations: Ski Slope**

This activity introduces students to the numeric representation of the slope of a line, including using "rise" and "run" to graph lines with a given slope. This is the first of seven activities about linear functions.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**y=mx+b**

This activity teaches students to graph lines using a slope and a y-intercept, or identify the slope and y-intercept given an equation in slope-intercept form. This is the second of seven activities about linear functions.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Linear Equations: Points, Intercepts, and Slopes, Oh My!**

In this activity students learn to graph lines using the x- and y- intercepts of equations, and how to graph a line given an equation in point-slope form. This is the third of seven activities about linear functions.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Linear Equations Word Problems**

Students graph linear equations to solve real world problems. This is the fourth of seven activities about linear functions.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Solving Systems of Linear Equations by Graphing**

In this activity students use systems of two linear equations to model real world situations with graphs. This is the fifth of seven activities about linear functions.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Systems of Linear Equations Word Problems Part 1**

Students explore two real life scenarios (mixing chemicals; a school bake sale) that require them to solve systems of linear equations. This is the sixth of seven activities about linear functions.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Systems of Linear Equations Word Problems Part 2**

Students explore two more real life scenarios (speed of a plane; attendees at a school play) requiring them to solve systems of linear equations. This is the seventh of seven activities about linear functions.

»Lesson Plan and Student Assessment documents are also available. Standards Addressed

This sequence of activities is designed to supplement normal classroom instruction focused on graphing linear equations.

**Graphing Quadratic Equations**

This activity teaches students to graph a parabola using the coordinates of the vertex and the x-intercepts. It is appropriate for use in Algebra I or Algebra II.

» Lesson Plan and Student Assessment documents are also available.**Quadratic Word Problems Part 1**

Students solve two problems involving the motion of projectile objects. This is the second of a series of activities about quadratic equations.

» Lesson Plan and Student Assessment documents are also available.**Quadratic Word Problems Part 2**

Students solve two problems modeled with quadratic equations. This is the third of a series of activities about quadratic equations.

» Lesson Plan and Student Assessment documents are also available.**Quadratic Functions in Vertex Form**

Students learn to graph quadratic equations that are given in vertex form. This is the fourth in a series of activities about quadratic equations.

» Lesson Plan and Student Assessment documents are also available.

**Transformations of Functions 1: Translations**

Students learn to translate functions: change the equation to move the graph of a function up, down, left, or right, without changing its shape.

» Lesson Plan and Student Assessment documents are also available.**Transformations of Functions 2: Dilations**

Students learn to dilate functions: change the equation to stretch or compress the graph of the equation either vertically or horizontally.

» Lesson Plan and Student Assessment documents are also available.**Transformations of Functions 3: Reflections**

Students learn to translate functions by changing the equation, and graph equations that include a reflection.

» Lesson Plan and Student Assessment documents are also available.**Transformations of Functions 4: All Transformations**

Students use translations, dilations, and reflections to change equations, and also graph equations that include these transformations.

» Lesson Plan and Student Assessment documents are also available.**Inverses of Functions**

Given a function, discrete or continuous, students learn to graph the function's inverse.

» Lesson Plan and Student Assessment documents are also available.

**Graphing Exponential Equations**

Students graph exponential growth and decay functions by connecting ordered pairs. This is the first of three activities about Exponential Functions.

» Lesson Plan and Student Assessment documents are also available.**Exponential Growth**

Learn how equations for several exponential functions are used to model population growth and compound interest. This is the second of three activities about exponential functions.

» Lesson Plan and Student Assessment documents are also available.**Exponential Decay**

In this activity students learn to use two different equations for modeling real-life situations that involve exponential decay. This is the third of three activities about Exponential Functions.

» Lesson Plan and Student Assessment documents are also available.

**Reflections 1**

In this lesson, students reflect line segments over the x-axis or the y-axis and learn how coordinates of points on the plane change when the points are reflected.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Fence Painting**

Students investigate the effect of a worker's work rate on job completion time. Linear graphs are used to display rates over time.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Biker Betty**

Students use their knowledge of linear equations to interpret a graph, with a focus on rates and slopes.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Driving to Grandma's House**

This activity was used by the SmartGraphs team to conduct research about student's understanding of slope, based on three graphs of increasing difficulty.**Translations of Functions!**

In this activity students find out what happens to the graph of a function when a constant is added to it, or subtracted from it.

» Lesson Plan and Student Assessment documents are also available. Standards Addressed**Reflections 2 - Functions**

In this lesson students compare the graph of a function, f(x), to graphs of f(-x) and -f(x).

» Lesson Plan and Student Assessment documents are also available. Standards Addressed

**College Debt**

This activity explores the impact of borrowing varying amounts of money to pay for college and helps students understand financial implications of their plans.

» Lesson Plan and Student Assessment documents are also available.

Activities can focus on graphs used in social science courses, for example in a Psychology course. College Debt, which focuses on financial literacy, takes about 45 minutes to complete.

**1.2 Equivalent Graphs**

Students investigate graphs that are equivalent, in the sense that they represent the same data, though they look different because they employ different scales. (Addresses Graph Literacy Objective 1.2: Understand how zooming, panning, stretching, and shrinking do not change the data within a graph.)

» A Lesson Plan is also available.**1.3 Interpolation**

In this activity students interpolate between data points in the context of questions involving a train timetable. (Addresses Graph Literacy Objective 1.3: Interpolate between points on a graph.)

» A Lesson Plan is also available.**1.4 Independent and Dependent Variables**

For various scenarios students select which of two variables should be considered independent and which dependent, and to explain their choice. (Addresses Graph Literacy Objective 1.4: Determine the dependent and independent variables for display on a graph.)

» A Lesson Plan is also available.**2.1 Graphs Tell a Story**

Students match a word story to the correct set of graphs involving temperature change over time. (Addresses Graph Literacy Objective 2.1: Identify the overall shape and direction of a line graph, and connect the shape with the real-world meaning.)

» A Lesson Plan is also available.**2.2 Hurricane Katrina**

Students are given graphs relating to Katrina, the hurricane that devastated New Orleans and much of the gulf coast in 2005, and are asked to identify various events, such as the moment when the hurricane made its closest approach to New Orleans. (Addresses Graph Literacy Objective 2.2: Identify the maxima and minima of a graph and interpret their meaning.)

» A Lesson Plan is also available.**2.3 Growing Up**

This activity asks students to interpret the slope of sections of a line graph of the height of U.S. girls and boys from ages two to twenty. (Addresses Graph Literacy Objective 2.3: Estimate the slope of a line and describe its real-world meaning.)

» A Lesson Plan is also available.

SmartGraphs activities require the use of an up-to-date Web browser, such as Google Chrome, Apple Safari 4 or above, Mozilla Firefox 3 or above, or Microsoft Internet Explorer 7 and above.

## Activity Spotlight

### Inverses of Functions

Learn to graph the inverses of a series of discrete and continuous functions.

Learn MoreThe SmartGraphs Authoring System was built using software called Hobo. Just as is the case for the use of a SmartGraphs activity, there is nothing for the user to download or install. The authoring system works directly in a Web browser.

The authoring system allows anyone to create a new SmartGraphs activity consisting of as many pages as the author wants. Creating an activity requires filling in online forms that specify, for example, text that should appear on each page, the graphs that should be displayed when a student runs the activity, the questions to be asked, the correct answers to those questions, and hints that appear if students provide an incorrect answer. Sharing the finished activity with students or colleagues is achieved simply by providing them with the URL for that activity.

A Smartgraphs activity called Authoring Demo demonstrates the features available to authors who want to create new activities, or revise existing ones. By running this activity you will see each of the "building blocks" that can be combined to build an activity.

An online SmartGraphs Authoring Manual provides instructions on using the authoring system. New users can become familiar with the system by copying and modifying an existing activity, allowing them to see how an activity looks "under the hood." No special programming knowledge is required.

A brief guide to using activities is available here.