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Home Up Scale and Mapping Invasion rates Quadrat sampling Graphing 1 Graphing 2

Effective Graphing:  Using Graphs to Present Data (Part 1)

 Grade Level: 6-8

Time Frame: 40-45 minutes

Subject:  Math

Introduction to Lesson: 

Almost all scientific and technical reports, as well as more "mainstream" media like newspapers and magazines, contain tables and figures. The purpose of these tables and figures is to display data and illustrate ideas and concepts in an organized fashion, so that it is easier for others to understand. There are many different types of graphs. Depending on the nature of the data, some graphs are more appropriate than others.  Knowing the appropriate type of graph to use to illustrate a given data set is a crucial skill that students need to master.  This and the subsequent lesson in the sequence will work with students to review / learn how to pick the appropriate kind of graph to display different kinds of data sets.  Students then will practice creating different kinds of graphs using the data on the Phragmites  invasion that they’ve collected in previous classes.

NJ- New Jersey Core Curriculum Content Standards

STANDARD 4.1 (Number and numerical operations) All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.

A. Number Sense

1.        Use real-life experiences, physical materials, and technology to construct meanings for numbers 

·          Explore the use of ratios and proportions in a variety of situations.

C.  Estimation

1.        Use a variety of strategies for estimating both quantities and the results of computations.

2.        Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.

3.        Determine the reasonableness of an answer by estimating the result of operations.

4.        Determine whether a given estimate is an overestimate or an underestimate.

STANDARD 4.2 (Geometry and measurement) All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.

D. Units of Measurement

5.        Use measurements and estimates to describe and compare phenomena

E. Measuring Geometric Objects

5. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one's foot).

STANDARD 4.3 (Patterns and algebra) All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

C. Modeling

2.        Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.

1.        Changes over time

2.        Relations between quantities

3.        Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

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.

A. Data Analysis (or Statistics)

1.        Collect, generate, organize, and display data.

2.        Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data.

o        Bar graph, line graph, circle graph, table, histogram

o         Range, median, and mean

o        Calculators and computers used to record and process information

3.        Respond to questions about data, generate their own questions and hypotheses, and formulate strategies for answering their questions and testing their hypotheses.

STANDARD 3.1 (Reading) All students will understand and apply the knowledge of sounds, letters, and words in written English to become independent and fluent readers, and will read a variety of materials and texts with fluency and comprehension.

A. Concepts About Print/Text

1.        Use a … glossary independently and appropriately.

C. Decoding and Word Recognition

3.  Apply knowledge of new words correctly

STANDARD 1.2 (Creation and Performance) All students will utilize those skills, media, methods, and technologies appropriate to each art form in the creation, performance, and presentation of dance, music, theater, and visual art.

D. Visual Art

1.        Individually or collaboratively create two and three-dimensional works of art employing the elements and principles of art.

3.   Recognize and use various media and materials to create different works of art

 Materials and Resources:

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Powepoint review: (pdf version)  What type of graph should be used when?  (Skip this if this information has been covered previously in the curriculum).

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Student worksheet for Powerpoint review  (MSword format) (pdf format

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Powerpoints for the two games Discrete or Continuous (or in pdf format) and What’s wrong with this picture? (pdf format)

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Graphic organizer worksheets for the two games  (Discrete or continuous worksheet (.doc format; pdf format);   "What's wrong with this picture?" worksheet (.doc format; .pdf format)

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Data collected in previous classes

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Worksheet for graphing exercise

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Computers running excel or similar software, or graph paper, rulers and pencils (if bar charts are to be made by hand) and compasses and protractors (if pie charts are to be made by hand)

Objectives: 

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Students will review the vocabulary of graphing to ensure that they remember which axis is the X and which is the Y, and the idea that the X axis is used to quantify the independent (controlled) variable, while the Y axis is used to record values of the dependent (response, or measured) variable.

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Students will practice determining whether an X-axis (independent) variable is continuous or categorical

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Students will review the common types of graph available for describing data sets and the criteria that make one graph type better suited for describing on type of data set than another

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Students will practice what they’ve learned by critiquing a set of graphs shown to them by the teacher.

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Students will apply their knowledge through generation their own graphs to represent data generated in previous lessons within this sequence.

Vocabulary

Continuous variable:  when a variable can have any possible value (fractions, decimals etc.) within the scale of the measurement, then that variable is said to be continuous.  If a variable is continuous, it is always possible to find an intermediate value between any two values on the scale e.g.  Length, weight, nutrient concentration. 

Discrete variable:  one which has specific values for which no meaningful intermediate exists.  Discrete variables can be categorical (nominal) or ordinal.   Categorical variables are those which exist in discrete units to which there is no obvious order (e.g. “male, female” or “red, blue, green”).  Ordinal variables are still discrete but can be arranged a logical order  (e.g. “poor, middle class, wealthy” or “private, sergeant-major, general”).  

Graph:  A diagram that provides a visual representation of a data set (sometimes also called chart or plot).

X / Y Axis (plural = axes):   In two-dimensional plots the vertical scale is usually the y-axis, and the horizontal scale is the x-axis

Parts of a graph

A graph showing the y-axis, the origin and x-axis

Common types of graph

Graph type

Description

Used for

Nature of X axis (independent) variable

Line graph

Displays a series of dots connected by lines.  When time is on the horizontal axis the plot is called a time-series graph.  In such graphs the slope of the line between two points represents the rate of change in that variable  over time

Line graphs are used to show patterns or trends in the measured variable (Y) as a result of changes in the size of the independent variable (X). 

Continuous

Scatterplot

(X,Y) points are plotted without a connecting line.  A best fit line may or may not be fitted to the data

Used to display or assess the relationship between two variables

Continuous

Vertical bar graph

Rectangular bars are plotted with vertical height of each bar representing the value of the measured (dependent) variable.  Since X represents discrete data, there should be spaces between bars on the X axis. Generally preferred over horizontal bar graphs since the assignment of independent and dependent variables to the X and Y axes then match those used with line graphs.

Used to display the values of two or more different types / levels of the independent variable. 

Discrete

Horizontal bar graph

Similar to vertical bar charts except the categories of the independent variable are assigned to the Y axis and the value of each measured variable is represented as the horizontal length of the bar. Useful when category names are too long to fit at the base of column

Used to display the amounts / levels scores for different types independent variable.

Discrete

Histogram

Looks similar to a vertical bar chart but Y axis data is always frequency. Also no spaces should be left between bars unless a category has zero frequency. 

Used to display count (frequency, number of times something occurs) of data points falling within various ranges within a data set

Continuous, but X axis variable has then been broken into discrete “buckets” or ranges

Pie chart

Circular chart divided into segments (slices) with the size of each segment being proportional to the size (percent) of the quantity being represented within the overall sample. If values are not markedly different, differences may not be easy to decipher. Labeling pie segments with their actual values overcomes this problem.

Compares a small number of categories.

Discrete

 

 Anticipatory Set: 

Teacher will provide the students with handouts with the vocabulary words and table with characters of the different graphs.  If desired, review the vocabulary of graphing and information on types of graph with the students.  Alternately, teacher can jump straight into playing the “categorical or continuous” and / or “What’s wrong with this picture?” games with students using the PowerPoint presentations provided and allow the students to “learn through application”.

 Sequence Instruction:

  1. “Discrete or Continuous?” Game.  In this first game, a data set will be described (PowerPoint provided) and students will be asked to identify which variable should be plotted on the X axis, and whether that variable is continuous or discrete, and what that means for what kind of graph should be used in displaying that data set.

To keep students on task, a graphic organizer is provided that students can use to keep track of the different data sets, and note for each which the independent (X) and which was the dependent or measured variable (Y).  Further, for each X variable the students will note which are continuous or categorical

  1. “What’s wrong with this picture?” Game.  In this game, the students will be asked to identify errors in a variety of graphs.  Errors may include reversed independent and dependent variables (so measured (response) variable is depicted on X and independent variable is on the Y), use of a bar chart when a line chart would be called for (or visa versa), etc.  To keep students on their toes, some graphs will have no errors.  For each graph, teacher will engage students in a discussion of WHY whatever they have identified as an error was wrong, and what needs to be done to fix it. 

Again, to keep students on task, a graphic organizer is provided that students can use to keep track of the graphs they look at, the errors they detected, the reason that the graphs were wrong and what they would need to do to present the data correctly.

  1. You want to compare the change in average densities of the four most abundant native plant species as well as invasive Phragmites in the marsh over time and by presenting that data in a graph.  What should be the X variable and what will be the Y variable in this graph?  Why?
  2. What is the nature of the X axis variable? (i.e. is it discrete or continuous?)
  3. What does that mean for the type of graph that you need to use to represent this data correctly?
  4. Create that graph using the data that you calculated in the quadrat exercise.  Be sure to label the axes appropriately and to give your graph a good title.
  5. What kind of graph would you use to represent the average relative proportions (percent) of the various species in the marsh before the Phragmites invasion?  Why?

  1. Create that graph using the data that you calculated in the quadrat exercise.  Be sure to label the axes appropriately and to give your graph a good title.

Accommodations and Modification:                                                                        

  1. Group students such that each group has a mix of students with different learning styles and strengths.
  2. Provide students with visual impairment with alternative visual media (printouts of PowerPoint slides, graphs made with puffy pens or embossing) to allow them to participate in the games
  3. If needed, pair students so that one student can read directions aloud to another
  4. Provide handouts and worksheets with larger font sizes, raised or embossed writing, or Braille for visually impaired students

Assessment / Homework / Anchor Activity (if lesson goes shorter than planned)

Have students complete similar graphs for data from the marsh at different stages in the invasion.

Closure  

Each student will create a graph with a deliberate mistake (reversed axes, missing legends, wrong type of graph for the type of data being presented etc.).  This graph could make use of data collected in previous classes or, alternately, the student could create a totally fictional data set.  Once the graph is made, have students pair up and exchange graphs and challenge one another to find the error and write that error on the sketch / print out.  Each student will then provide feedback to their partners (acknowledging and congratulating them on correctly diagnosing errors, providing hints if the error is not recognized by the partner).   Students will turn in their graphs, and their partner’s diagnoses of the error(s) to the teacher as they leave for the day.

Alternately, teacher could ask a lightning round of questions re: the vocabulary for the lesson (continuous, variable, etc.)  (S)he could use graphs to ask the questions or refer to the ones the students have done.

 

TEACHER FEEDBACK REQUEST:  We are always to working to improve these lesson plans. If you use this lesson plan, we'd love to hear from you with your thoughts, comments and suggestions for future improvements.  Please take the time to fill in our survey at http://www.zoomerang.com/Survey/?p=WEB229JA3BEWBD .  Thanks!

 

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© 2009.  Louise Wootton.    Edited by Claire Gallagher

 Although the information in this document has been funded wholly or in part by the United States Environmental Protection Agency under assistance agreement NE97262206  to Georgian Court   University, it has not gone through the Agency's publications review process and, therefore, may not necessarily reflect the views of the Agency and no official endorsement should be inferred.

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