MathBuilders - Unit 8
Data Collection, Graphing, and Probability/Statistics
 
Catalog No. 7-03565-01
Copyright © 2009,
American Printing House for the Blind, Inc.
Louisville, KY 40206-0085
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Louisville, KY 40206-0085
Catalog Number 7-03565-01
Standards are reprinted with permission from Principles and Standards for School Mathematics, copyright 2000 by the National Council of Teachers of Mathematics. All rights reserved.
Credits

Project Leader:
Jeanette Wicker

Mathematics Consultant:
Derrick W. Smith, Ed.D., COMS

Contributors:
Katherine Corcoran, Darlene Donhoff, Frank Hayden, Dr. William E. Leibfritz, Eleanor Pester, Tom Poppe, Erica Rucker, and Anita Rutledge

Advisory Committee:
Anthony Evancic, Dr. Phillip H. Hatlen, Dr. William E. Leibfritz, Ted Lennox, Mary Mitchell, Andrew S. Papineau, Dr. LaRhea Sanford, and Merry Vahala

Field Testers:
Ginger Barnes, Cynthia Carter, Janet Chlapek, Melinda Demaris, Jane Herder, Roxanne Lawrence, Christine May, Renee Toy, and Kelly van Dyk
Special Acknowledgement goes to Dr. Phillip H. Hatlen, Superintendent of the Texas School for the Blind and advisory committee member for this project, and to the following members of his staff for providing APH with the conceptual framework upon which much of this program is based: Nancy Levack, Robin Washburn, and Brigitte Starkey.

Table of Contents

 MathBuilders - Unit 8
 Credits
 Introduction
 Bibliography
 Learning Objectives
 Kindergarten
 

 Learning Objectives
 Materials Suggested for Lessons
 D K-1 Data Collection, Graphing, and Probability/Statistics
 D K-2 Data Collection, Graphing, and Probability/Statistics
 D K-3 Data Collection, Graphing, and Probability/Statistics
 D K-4 Data Collection, Graphing, and Probability/Statistics
 

 First Grade
 Learning Objectives
 Materials Suggested for Lessons
 D 1-1 Data Collection, Graphing, and Probability/Statistics
 D 1-2 Data Collection, Graphing, and Probability/Statistics
 D 1-3 Data Collection, Graphing, and Probability/Statistics
 D 1-4 Data Collection, Graphing, and Probability/Statistics
 D 1-5 Data Collection, Graphing, and Probability/Statistics
 

 Second Grade
 Learning Objectives
 Materials Suggested for Lessons
 D 2-1 Data Collection, Graphing, and Probability/Statistics
 D 2-2 Data Collection, Graphing, and Probability/Statistics
 D 2-3 Data Collection, Graphing, and Probability/Statistics
 D 2-4 Data Collection, Graphing, and Probability/Statistics
 D 2-5 Data Collection, Graphing, and Probability/Statistics
 D 2-6 Data Collection, Graphing, and Probability/Statistics
 D 2-7 Data Collection, Graphing, and Probability/Statistics
 

 Third Grade
 Learning Objectives
 Materials Suggested for Lessons
 D 3-1 Data Collection, Graphing, and Probability/Statistics
 D 3-2 Data Collection, Graphing, and Probability/Statistics
 D 3-3 Data Collection, Graphing, and Probability/Statistics
 D 3-4 Data Collection, Graphing, and Probability/Statistics
 D 3-5 Data Collection, Graphing, and Probability/Statistics
 D 3-6 Data Collection, Graphing, and Probability/Statistics
 D 3-7 Data Collection, Graphing, and Probability/Statistics
 D 3-8 Data Collection, Graphing, and Probability/Statistics
 D 3-9 Data Collection, Graphing, and Probability/Statistics
 D 3-10 Data Collection, Graphing, and Probability/Statistics
 

 Assessment Check List
 

 Appendix A
 Copies of Student Worksheets

Introduction
One need only read a newspaper or magazine to see the abundant use of graphs to display statistical information. Graphs not only capture the reader's attention, but also can often convey a message more directly and clearly. Graphs enable the reader to make comparisons, see and study relationships, and draw conclusions much more easily.

Since more and more information (data) is presented in graphical form, young children need to develop skills that allow them to read, interpret and analyze such information. Graphical representations should be included in the early mathematics curriculum because of the appeal and excitement they generate with students. Graphs are incorporated in most all areas of mathematics to communicate information and provide many problem-solving opportunities. Graphs of all types are most meaningful when generated by data actually collected by the students. Graphing can be even more valuable when the statistics represented are useful in solving a real problem.

Graphing is a natural way to integrate mathematics with other content areas. In social studies, students can survey and graph characteristics of their families, communities, cities, etc., or describe the population of their classroom or school. In science, students can graph the growth of plants or record daily weather characteristics. A teacher can use graphs to get a better understanding of his/her students by surveying their likes or dislikes such as foods, music, sports teams, TV shows, movies, etc. These items can be graphed, then displayed for students to learn more about each other.
Graphing can be used as a school wide project, with each classroom gathering, recording, and graphing characteristics appropriate to that grade. These graphs can be displayed with a short paragraph describing the procedures, findings, conclusions, and predictions posted nearby. Collecting and recording data in graphical form increases the value of any project and allows more information to be communicated than text is capable of providing. If the project answers a question or solves a problem, then it is even more worthwhile.

Along with their importance in real-world applications, graphing aids in the development of problem solving and thinking skills. Many problems can be generated from a graph. Students can even develop new problems or questions from a graph they have produced. Graphs combine numerical, sensory (both visual and tactile), and written information. Many skills are needed to fully understand a graph and still others are needed to construct one. A student must be able to compare and contrast numbers, associate written and numerical labels with various components, draw conclusions, and communicate the information presented. When constructing a graph, a student must have such skills as counting, classifying, sorting, corresponding, labeling, and organizing. Time spent on graphing is good use of mathematical time.

Young students should be led systematically through their experiences with graphing. Initially they should have many experiences with concrete objects to represent data. This should then lead to using materials such as tiles to represent the actual information. Symbols can then be used to introduce more abstract graphs including bar graphs, pictographs, and line graphs. Research indicates that children should construct their first graphs themselves with concrete materials. This will allow them to better understand the process of collecting information, organizing data, and communicating the findings they have discovered. Like all areas of mathematics, personal involvement increases the likelihood of student achievement.
Probability is closely related to graphing and data collection. Graphs can be used to estimate or predict the likelihood or the probability of an event. Probability is typically represented by a number from 0 to 1. For young students, it is better to use terms with which they are most familiar, such as likely, not likely, more likely, and less likely.

While much of mathematics is exact, probability involves uncertainty. Since we cannot always know for sure what event will occur, we must make decisions based on what we know about their probability. Through probability activities students can use the data they have collected to predict future outcomes.
Young students have many misconceptions about probability. For example, they may make decisions about an outcome solely on preference or expect all outcomes to be equally likely. It is important for students to realize that while some aspects of mathematics may require exactness and precision, probability deals with choices based on the likelihood of an event. Students can learn that making decisions in real life often involves selecting the best possible choice based on the information available.

Graphing and probability are basic experiences that all children need to have at an early age. They need to learn how to analyze information in graphs, which would then lead to the ability to make decisions based on this analysis. Graphing should be a tool for solving problems and act as a source of problem solving activities. Graphing is important in developing and extending mathematical concepts. Graphs can result from lessons that focus on such things as classifying, counting, addition, subtraction, etc.

As teachers bring the mathematics of the real world into the classroom, students will collect and display real information and describe what they have found. These types of activities will help students better understand and appreciate the value of graphing data as well as the basic ideas and needs of probability.

The objectives for each lesson were identified by an APH focus group and aligned to the Principles and Standards for School Mathematics (NCTM 2000). Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or the validity of these alignments.

In addition, some objectives are noted as being a NCTM Curriculum Focal Point. In 2006 the National Council of Teachers of Mathematics published Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. In this document NCTM provides a "description of the most significant mathematical concepts and skills at each grade level" (NCTM, 2006, p. 1).

Included at the end of this unit is an Assessment Checklist that includes all of the objectives for the unit. This checklist is divided by grade levels. Prior to beginning the activities, review the checklist to determine which objectives your student has mastered and which objectives you will need to teach or review. Because some students have gaps in their skills, you may want to review the objectives for the preceding grade levels. Additionally, your school or district may align the math curriculum differently. You may find that you will need to teach lessons that are included at a different grade level or delay some objectives until the next school year.

Some children will need more time and experience with making graphs and interpreting data than others. The following lessons may only provide one worksheet per lesson. If your student needs more practice, less difficult activities or more challenging activities, use products like Tangible Graphs, Graphic Aid for Mathematics, Hundreds Board, and Graph Sheets that will allow you to create problems and activities to meet your child's specific needs.
Literature plays an important role in the learning process. There are many good children's books that reinforce math concepts. Following is a brief list of books you may want to use to introduce or reinforce the concepts of Data Collection, Graphing, and Probability/Statistics. Not all are available in braille and some are dependent on pictures to convey meaning.

*   Aber, Linda. Who's got spots? New York: Kane Press Paperback, 2000. 
*   Axelrod, Amy. Pigs at odds: fun with math and games. New York: Simon & Schuster, 2000. 
*   Baylor, Byrd. Guess who my favorite person is. New York: Scribner, 1977. 
*   Baer, Edith. This is the way we go to school. New York: Scholastic, 1990. [braille] 
*   Carlson, Nancy. Harriet's Halloween candy. Minneapolis: Carolrhoda Books, 1982. 
*   Dorros, Arthur. Alligator shoes. New York: Dutton Press, 1982. 
*   Jenkins, Emily. Five creatures. New York: Farrar, Straus & Giroux, 2001. 
*   Kellogg, Steven. The mysterious tadpole. New York: Dial Press, 1977. [braille] 
*   Knapp, Brian. Tables and charts. Danbury, CT: Grolier, 1999. 
*   McGrath, Barbara. The M & M's brand counting book. Watertown, MA: Charlesbridge, 1994. 
*   Morris, Ann. Hats! Hats! Hats! Boston: Houghton Mifflin, 1996. [braille] 
*   Morris, Ann. Bread! Bread! Bread! New York: Mulberry Books, 1993. [braille] 
*   Murphy, Stuart J. Lemonade for sale. New York: HarperCollins Publishers, 1998. [braille] 
*   Murphy, Stuart J. Pepper's journey: a kitten's first year. New York: HarperCollins, 2000. 
*   Murphy, Stuart J. Probably pistachio. New York: HarperCollins, 2001. 
*   Nagda, Ann Whitehead. Tiger math: learning to graph from a baby tiger. New York: Henry Holt, 2000. 
*   Ochiltree, Dianne. Bart's amazing charts. New York: Scholastic, 2001. 
*   Smith, David J. If the world were a village: a book about the world's people. Toronto, ON; Tonawanda, NY: Kids Can Press, 2002. 
*   Stineson, Kathy. Red is best. Toronto: Annick Press, 1982. 
*   Wright, Suzette. Thingamajig. Louisville, KY: American Printing House for the Blind, 1996. [braille] 

Bibliography
National Council of Teachers of Mathematics. Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: National Councils of Teachers of Mathematics, 2006.

National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 2000.

Learning Objectives

Kindergarten
D K-1   Count class members and real objects to determine answers to real life questions.
D K-2   Arrange objects on a floor or table graph according to attributes.
D K-3   Construct pictographs using people or real objects in lines.
D K-4   Look for largest group, the smallest group, and groups with the same amount.

First Grade
D 1-1   Collect data from class members, observations, etc. to answer real life questions.
D 1-2*   Record data on simple charts or tables.
D 1-3*   Construct pictographs or bar graphs with blocks, cubes, or symbols in stacks or lines to represent individual people, objects, etc.
D 1-4   Look for the largest grouping and the extremes - smallest to largest amount of the data (range).
D 1-5*   Answer questions related to data displayed in graphs (e.g., how many more in a group compared to another group, how many in two groups combined, etc.).

Second Grade
D 2-1   Collect and organize data from observations, samplings, surveys, polls, etc.
D 2-2   Record simple tally/frequency counts of the data.
D 2-3*   Record data on simple charts or tables.
D 2-4   Construct bar graphs with stacks or lines of blocks or cubes and then change these to lines of single, simple characters, letters, or stickers to make bar graphs on paper.
D 2-5*   Identify the mode (the number that occurs the most often) and the range (the smallest to largest) of the data.
D 2-6*   Locate information on simple charts or tables and bar graphs.
D 2-7   Use data to describe events as likely or unlikely to occur.

Third Grade
D 3-1   Collect data from observations, samplings, surveys, polls, etc.
D 3-2   Record simply tally/frequency counts of data.
D 3-3*   Record data on charts or tables.
D 3-4*   Construct bar graphs with a single block, cube, simple character, letter, or sticker corresponding to a single item being studied.
D 3-5*   Construct bar graphs with intervals greater than one.
D 3-6*   Read and label points on a coordinate graph.
D 3-7*   Construct simple line graphs.
D 3-8   Identify the mode, median, mean, and the range of the data.
D 3-9*   Read and interpret information on line graphs and bar graphs.
D 3-10   List some of the possible outcomes based on data and predict whether given outcomes are more, less, or equally likely to occur.
    * Worksheet included

Kindergarten
 
Learning Objectives
Kindergarten
D K-1   Count class members and real objects to determine answers to real life questions.
D K-2   Arrange objects on a floor or table graph according to attributes.
D K-3   Construct pictographs using people or real objects in lines.
D K-4   Look for largest group, the smallest group, and groups with the same amount.

Materials Suggested for Lessons

A limited number of manipulatives are included in the kit that accompanies this unit. Additionally, you may want to use the following items to add variety and to give the child more experiences with objects from his or her environment.

Floor graph D K-2
Shapes from Hundreds Board D K-2, D K-3, D K-4
Shapes from Wheatley Tactile Diagramming Kit
D K-2, D K-3, D K-4 
Tactile objects D K-1, D K-2, D K-3, D K-4
D K-1
Data Collection, Graphing, and Probability/Statistics

Objective: The student will count class members and real objects to determine answers to real life questions.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should pose questions and gather data about themselves and their surroundings.

Curriculum Focal Point

Suggested Materials:
*   Tactile objects 

Materials: Listed materials are only a suggestion. Feel free to substitute other appropriate materials including those from other math programs.

Strategies: This lesson focuses on the student counting class members or real objects. The teacher should look for opportunities to have the student count things in the environment and ask the student questions. D K-1, D K-2, and D K-3 are closely related. D K-2 and D K-3 will present a more formal process for graphing information. Save the data from the exercises in this lesson to use in developing graphs that are more formal in the following lessons.

Class Activity: Giving the student an opportunity to count class members is important. Since a student with visual impairments cannot do this in typical ways, the teacher needs to present an activity in a game that allows the student to count other class members. An example would be using a modified version of "Duck-Duck-Goose" and having all the students count on the first trip around, possibly saying "one duck, two ducks, three ducks, etc." This would give the students a fun opportunity to count class members.

Another option is to have the students line up against the wall. Have students take turns doing the following:
1.   Count the total number of students. 
2.   Count the boys. 
3.   Count the girls 
4.   Ask questions such as, "Are there more girls than boys?" 

Other class activities might include having the students move to designated areas of the rooms to respond to specific questions.
How many people in our class have pets at home?
What is the favorite ice cream in our class? Is it chocolate or strawberry?
What is the favorite pizza in our class? Is it pepperoni or cheese?
What is the favorite fruit in our class? Is it apple or banana?

Have the students move to a designated area in the classroom to vote for their favorite or to answer the question. Once all students are in place, have the students count off, count in unison, or have one student count the number of students in a particular line or group. Help the students arrive at the answer to the initial question by comparing the number of students in each line or group.

Individual Activity: Present the student with tactile objects and ask her questions related to the number. For example, have the student count the number of worksheets for an activity and then ask the student if there are enough for the work group or the entire class. Provide the student with other opportunities to use data to answer real-world questions.

Other individual activities might include:
Use a calendar to count the number of days until a specific event such as a holiday, school event, or the weekend.
Use the class weather calendar to count the number of rainy days and sunny days.
Determine and count the number of items needed for a small group activity, pencils for the math group, scissors for an art activity, or snacks for a celebration.

Connections: These activities are excellent ways to have the student with visual impairments develop interpersonal skills by providing her opportunities to talk to all the students and learn information about them.
Counting is a foundational component of mathematics as well as life. Taking every opportunity to reinforce counting in games and play is important to the development of real-world math skills.

Read the book This Is the Way We Go to School by Edith Baer. This book, available in braille, contains many fun filled rhymes. The student learns how children around the world get to school. Discuss with the student the different types of ways that children get to school and what groupings she would use to sort the children.

D K-2
Data Collection, Graphing, and Probability/Statistics

Objective: The student will arrange objects on a floor or table graph according to attributes.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should sort and classify objects according to their attributes and organize data about the objects.

Suggested Materials:
*   Floor graph 
*   Small Tactile Table Graph from the kit 
*   Shapes from the Wheatley Tactile Diagramming Kit 
*   Shapes from the Hundreds Board 
*   Tactile objects 
*   Feel 'n Peel Stickers from the kit 

Strategies: The table graph and floor graph are the foundational graphs in mathematics and should be presented to the student as early as possible. Table graphs and floor graphs come in a variety of sizes starting with a 2×2 seen below. Use a 2×2 table/floor graph when comparing two things, and count occurrences with tactile objects or tally marks.

Label A   Label B
x x x x   x x x x x x x x x x x

Class activity: Many floor graphs are commercially available. However, the floor graph will have to be made tactile by using different textures on the graph. Make sure that the dividing line between the sections is clearly defined (use a broomstick, yardstick, or a divider that makes the sections easy to find). Using the floor graph, have the students place different types of toys on each side of the graph arranged by their attributes (e.g., things with wheels on one side, dolls or stuffed animals on the other side). Then have the students count to see how many things are on each side. Provide the student multiple opportunities to use the floor graph with the class.

Individual activity: Provide the student with the Small Tactile Table Graph and an assortment of tactile objects such as the shapes from the Hundreds Board or from the Wheatley Tactile Diagramming Kit. Have the student explore the Table Graph noting the area for the label and the individual compartments for the tactile objects. Show the student how to position the table in both the horizontal and the vertical positions.

Have the student sort the objects into two categories based on shape (round or square), size (large or small), or texture (rough or smooth). Initially you may want to use a Work-Play Tray to help the student sort the objects. As the student becomes more skilled, he may be able to place the objects directly onto the Tactile Table Graph. Help the student arrange the shapes on the Tactile Table Graph. Help the student place a label on the graph for each column. Labels can consist of a tactile object or shape, an initial consonant in braille using Feel 'n Peel Alphabet Stickers, or the whole word if appropriate. Secure the labels to the graph. Provide the student with opportunities to graph objects in both horizontal and vertical arrangements.

Vertical Table:
Label A   Label B
Data   Data
Data   Data
Data   Data

Horizontal Table:
Label A   Data   Data   Data
Label B   Data   Data   Data

Connections: Tables are a common way of organizing data and are found in many places including newspapers such as Weekly Reader, magazines such as Squid, and textbooks. It is important that the student begins to formulate how tables "look" in order to be able to read tables that are more complex in the future.

Read the book Five Creatures by Emily Jenkins. Discuss with the student what attributes he would use to sort the five creatures. What labels would he put on the table or floor graph? How would he represent the five creatures on the graph?

D K-3
Data Collection, Graphing, and Probability/Statistics

Objective: The student will construct pictographs using people or real objects in lines.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should represent data using concrete objects, pictures, and graphs.

Suggested Materials:
*   Small Tactile Table Graph from the kit 
*   Shapes from the Wheatley Tactile Diagramming Kit 
*   Shapes from the Hundreds Board 
*   Tactile objects 
*   Fell 'n Peel Stickers from the kit 

Strategies: A pictograph is a graph that uses pictures to represent real-world objects. Simple pictographs have one picture representing one object while pictographs that are more complex have pictures representing multiple objects. At this age, a one-to-one ratio should be used.
 
Class activity: A class project could involve creating a monthly weatherboard that tracks the weather each day. Every morning have the students determine the weather for the day and place a tactile picture (sun, clouds, clouds with rain, etc.) on a table graph. At the end of the month, ask the students questions regarding the month's weather. How many days were sunny, rainy, or cloudy? Were there more rainy or sunny days?

Individual activity: Use the Small Tactile Table Graph from the kit and tactile shapes to create a pictograph. Refer to lesson D K-1 and use the data you collected and saved from that lesson. The first activity in lesson D K-1 had the class determine the number of boys and girls in the room. Determine which tactile shape will represent the boys and which tactile shape will represent the girls. Have the student count out the number of tactile shapes needed to represent the boys and place in the Tactile Table Graph. Have the student count out the number of tactile shapes needed to represent the girls and place in the Tactile Table Graph. Label the graph. Labels can consist of a tactile object or shape, an initial consonant in braille using Feel 'n Peel Alphabet Stickers, or the whole word if appropriate. Have the student explore the pictograph, noting the columns, the labels, and the symbolic representation of data. Provide the student with opportunities to make additional pictographs of other activities completed in lesson D K-1.

Connections: Students with visual impairments are often presented with tactile graphics that represent something else. The pictograph activities can reinforce the concept of a symbol representing something else. This can connect to tactile maps in orientation and mobility lessons.

Read the book Harriet's Halloween Candy by Nancy Carlson. In this book Harriet learns that sharing her Halloween candy makes her feel much better than eating it all herself. Harriet also sorts her candy by different types. Discuss with the student the different ways for sorting the candy and graphing the candy.

D K-4
Data Collection, Graphing, and Probability/Statistics

Objective: The student will look for largest group, the smallest group, and groups with the same amount.

NCTM Standard: Data Analysis for Pre-K - 2 - Select and use appropriate statistical methods to analyze data. All students should describe parts of the data and the set of data as a whole to determine what the data show.

Curriculum Focal Point

Suggested Materials:
*   Small Tactile Table Graph from the kit 
*   Shapes from the Wheatley Tactile Diagramming Kit 
*   Shapes from the Hundreds Board 
*   Tactile objects 

Strategies: Provide the student with the Small Tactile Table Graph and an assortment of tactile objects to use for graphing. Have the student build pictographs. Some suggestions for developing a pictograph are:

Mary is having a birthday party. Four of her friends want to go swimming. Five of her friends want to go skating. Make a pictograph to show what her friends want to do at her birthday party.

John went trick-or-treating. He has six Milky Way® candy bars and six Twix® bars in his bag. Make a pictograph to show what kinds of candy bars he collected.

Ms. Smith's class travels to school in different ways. Four students come to school in cars. Six students ride the bus. Make a pictograph to show how the class travels to school.

Help the student determine what information he will be graphing and what tactile object will represent each group. Have the student count out the number of objects he will need to make the graph and place the tactile objects in the appropriate columns. Ask the student to determine which group is the largest, which group is the smallest, or if either group is the same.

Connections: This activity can be completed with multiple objects but coins may be the most obvious objects to sort. Have the student sort different kinds of coins and then ask the same type of questions. Which group is the largest? Which group is the smallest? Are any groups the same? This will aid in teaching the student to distinguish the coins by touch.
First Grade
 
Learning Objectives

First Grade
D 1-1   Collect data from class members, observations, etc. to answer real life questions.
D 1-2*   Record data on simple charts or tables.
D 1-3*   Construct pictographs or bar graphs with blocks, cubes, or symbols in stacks or lines to represent individual people, objects, etc.
D 1-4   Look for the largest grouping and the extremes - smallest to largest amount of the data (range).
D 1-5*   Answer questions related to data displayed in graphs (e.g., how many more in a group compared to another group, how many in two groups combined, etc.).
    * Worksheet included

Materials Suggested for Lessons
A limited number of manipulatives are included in the kit that accompanies this unit. Additionally, you may want to use the following items to add variety and to give the child more experiences with objects from his or her environment.

Bi-fold felt board D 1-2
Shapes from the Hundreds Board D 1-3, D 1-4
Shapes from the Wheatley Tactile Diagramming Kit D 1-2, D 1-3, D 1-4
Wikki Stix D 1-2

D 1-1
Data Collection, Graphing, and Probability/Statistics

Objective: The student will collect data from class members, observations, etc. to answer real life questions.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should pose questions and gather data about themselves and their surroundings.

Suggested Materials:

Strategies: Lessons D 1-1 through D 1-3 are closely related. The student will plan a survey, collect and record the data on simple charts or tables, create a pictograph and/or a bar graph, and answer questions about the information. Select one or more of the following suggestions to use through the next two lessons. Alternatively, you may choose to create a survey question that more closely meets the interest of your student. Limit the categories or the options to two choices for this lesson.

What is the most popular pet? Is it a cat or a dog? 
Do you like the food in the cafeteria?
What is your favorite subject? Is it reading or math?
What is your favorite cake? Is it chocolate or white?
Select questions that meet the interest and the skill level of your student(s). Help the student determine how many people she will survey, the people she will survey, what question she will ask, and how she will collect and record the data. Some students may need to practice the question with you before starting the survey. You may want to limit the number of respondents to the survey for ease in collecting and graphing the data.

Connections: Allow the student with visual impairments to add tactile stickers to class attendance chart. All students can place their own stickers, but allow the student with visual impairments to explore the chart. Ask the student who has been absent the most this month, who has not been absent at all, how many days they have missed, etc.

Read the book Red is Best by Kathy Stineson. This is the tale of one toddler's obstinate color preference. Have the student name her favorite color. Does she think this is the preference of most students in her class? Have the student determine the favorite color of her friends or her classmates.

D 1-2
Data Collection, Graphing, and Probability/Statistics

Objective: The student will record data on simple charts or tables.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should sort and classify objects according to their attributes and organize data about the objects.

Suggested Materials:
*   Bi-fold felt board 
*   Shapes from the Wheatley Tactile Diagramming Kit 
*   Feel 'n Peel Stickers from the kit 
*   Wikki Stix (Cut in 1" lengths) 
Worksheet D 1-2

Materials: Listed materials are only a suggestion. Feel free to substitute other appropriate materials including those from other math programs.

Strategies: Have the student record the data for the question selected in lesson D 1-1 on a simple chart or table using various tools. The first part of this activity is to teach tallying, the process of counting with marks or symbols. Listed below are three options that you may choose from based on your student's needs and abilities.

Using the Wheatley Tactile Graphing Kit and textured shapes have the student place objects on either division of the bi-fold board to represent the data. Discuss with the student which side of the board will represent the different responses. Help the student make a one-to-one correspondence between each answer to the survey and each tactile object placed on the board. Ask the student to count the total number of tactile objects in each group to determine the answer to the survey question.

Included with this lesson is a simple chart, Worksheet D 1-2. Explore the worksheet with the student noting the areas for a heading and the areas for the data. Place braille headings in the appropriate columns for the student or use Feel 'n Peel Alphabet Stickers to note the beginning letter of the survey items. Provide the student two different types of Feel 'n Peel Stickers to represent the responses to the survey. Help the student make a one-to-one correspondence between each answer to the survey and each tactile sticker placed on the chart. Ask the student to count the total number in each group to determine the answer to the survey question.

Use the worksheet, D 1-2, and one-inch pieces of Wikki Stix to record the data on the chart. Place braille headings in the appropriate columns for the student or use Feel 'n Peel Alphabet Stickers to note the beginning letter of the survey items. Help the student make a one-to-one correspondence between each answer to the survey and each Wikki Stix placed on the chart. Ask the student to count the total number of Wikki Stix in each group to determine the answer to the survey question.

For more advanced students, have the student count the stickers or Wikki Stix and then add the total number to the worksheet as a final step to this process. Braille the number on a sticker or use Feel 'n Peel Number Stickers. This is a major step as the student is moving from real objects to the more abstract concepts of numbers representing a group.

Connections: Read the book Thingamajig by Suzette Wright with the student. As you read this tactile book, available in braille, discuss with the student how the main character sorts his Thingamajig collection into different categories. Ask the student to decide on categories he would use to sort the items on page two and page thirty of the book.

D 1-3
Data Collection, Graphing, and Probability/Statistics

Objective: The student will construct pictographs or bar graphs with blocks, cubes, or symbols in stacks or lines to represent individual people, objects, etc.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should represent data using concrete objects, pictures, and graphs.

Curriculum Focal Point

Suggested Materials:
*   Small Tactile Table Graph from the kit 
*   Shapes from the Wheatley Tactile Diagramming Kit 
*   Shapes from the Hundreds Board 
*   Large Tactile Table Graph from the kit 
*   Graphing Squares from the kit 
*   Feel 'n Peel Stickers from the kit 
Worksheets D 1-3 (a), D 1-3 (b), and D 1-3 (c)

Strategies: This activity introduces the student to the concept of a bar graph, which is a graph that uses bars to show an amount instead of pictures or tallies. The important difference is that bar graphs are labeled on two axes or sides.

Example of a bar graph:
 
Pictograph activity: Provide the student with the Small Tactile Table Graph and an assortment of tactile objects to use for graphing. Help the student build a pictograph based on the data collected and charted in the previous lesson. Help the student determine which tactile object will represent the data, how many of each tactile objects she will need, and what information she will place on the labels.

Bar Graph activity: Provide the student with the Large Tactile Table Graph from the kit. Have the student explore the graph noting the differences between the two graphs (e.g., more columns and rows and two areas for labeling). Explain that she will not use pictures or tactile objects on this graph but squares of equal sizes that will make a bar. She will also place labels on two axes (sides) of the graph. Working with the same data used to make the pictograph have the student make a bar graph. Help the student determine the quantity of squares needed for each response. Help the student arrange the squares on the graph and label the graph.

Have the student compare the three ways she has displayed the data: the chart from the previous lesson, the pictograph, and the bar graph. How are they different? How are they alike? Which one does she prefer?
Provide the student with the worksheets D 1-3 (a), D 1-3 (b), and D 1-3 (c). These worksheets provide a bar graph and two pictographs of the same information, "Pets in My Neighborhood." Worksheet D 1-3 (c) includes a legend to the pictograph. Explain the purpose of a legend and demonstrate how to use a legend to read a graph.

Help the student explore the embossed graphs. What information do the graphs provide? Which graph is easiest to read? Are there more dogs or cats in the neighborhood? Are there more cats or fish in the neighborhood? Which is the largest group? Which is the smallest group?

Connections: The pictograph activity provides an opportunity to reinforce the basic shapes of the circle, square, and triangle.
Use the APH Tactile Demonstration Thermometer to reinforce the concept of a bar graph. This product allows the teacher to teach temperature as well as reading a simple bar graph.

D 1-4
Data Collection, Graphing, and Probability/Statistics

Objective: The student will look for the largest grouping and the extremes - smallest to largest amount of the data (range).

NCTM Standard: Data Analysis for Pre-K - 2 - Select and use appropriate statistical methods to analyze data. All students should describe parts of the data and the set of data as a whole to determine what the data show.

Suggested Materials:
*   Shapes from the Wheatley Tactile Diagramming Kit 
*   Shapes from the Hundreds Board 
*   Large Tactile Table Graph from the kit 
*   Graphing Squares from the kit 
*   Feel 'n Peel Stickers from the kit 

Strategies: Part I. Provide the student with an assortment of four types of objects and have him sort them into groups. Explain that a group is a collection of things that are the same. After the student has sorted the objects, have the student place the objects on the Large Tactile Table Graph and label the columns and rows by the type of shape and by number. Ask the student questions such as:
1.   How many items are in each group? (It is advisable to label each group such as the "circle group" or "the smooth stone" group.) 
2.   Which group has the most items? 
3.   Which group has the least amount of items? 
4.   If applicable, have the student determine the difference between the largest group and the smallest group. If he cannot yet do simple subtraction, have the student make pairs of items from each group until the smaller group has no items left. Have the student count the number of objects that remain in the larger group. 

Now have the student replace each tactile object with a Graphing Square from the kit. Help the student explore the new bar graph he has made. Help him note the labels of the groups and the labels of numbers. Repeat the above questions with the new bar graph. Help the student understand that both graphs show the same information but that the graphs are different in the way they present the information.
Part II. If the student can read and write numbers, provide the student with a series of numbers on cards that are not in order.
1.   The first series of numbers should have all single numbers, no repeats. 
2.   Future series should have multiples (such as 1, 3, 5, 6, 3, 5, 8) 
Have the student put the numbers in order (e.g., 1, 2, 3...). After he has mastered this skill and moved to series with multiples, have the student determine which number in the series is repeated the most. Next, have the student determine the largest and smallest numbers and if able, subtract the two to determine the range.

Connections: Visit the "Read Write Think" web site (http://www.readwritethink.org) for a lesson that integrates social studies and math. The lesson entitled, Mail Time! An Integrated Postcard and Geography Study, also incorporates writing skills. Students write to friends and family asking them to send postcards. This activity provides motivation for writing and reading and provides a wonderful opportunity to learn about maps as students discover where their family members and friends live. Construct graphs to record the different towns, states, or countries from which the postcards originate.

D 1-5
Data Collection, Graphing, and Probability/Statistics

Objective: The student will answer questions related to data displayed in graphs (e.g., how many more in a group compared to another group, how many in two groups combined, etc.).

NCTM Standard: Data Analysis for Pre-K - 2 - Select and use appropriate statistical methods to analyze data. All students should describe parts of the data and the set of data as a whole to determine what the data show.
Curriculum Focal Point

Suggested Materials:
Worksheets D 1-5 (a), D 1-5 (b), and D 1-5 (c)

Strategies: Provide the student with the worksheets. Help the student explore the graph and answer questions regarding the graphs.

Possible questions:
1.   What type of graph is this? Is it a pictograph or a bar graph? 
2.   What information does this graph give us? 
3.   Which group is the largest? 
4.   Which group is the smallest? 
5.   How many items are in each group? 
6.   How many more items are there in the first group than in the second group? 
7.   How many items would there be if we combined the two groups? 
8.   How are the two bar graphs alike? 
9.   How are the two bar graphs different? 

Connections: Visit the NCTM Illuminations web site (http://illuminations.nctm.org) for a lesson that ties science and math. In the lesson, Graphing Trash Material, the student will gain experience gathering and categorizing data, making and interpreting graphs, and developing an action plan for recycling.

Second Grade
 
Learning Objectives

Second Grade
D 2-1   Collect and organize data from observations, samplings, surveys, polls, etc. 
D 2-2   Record simple tally/frequency counts of the data.
D 2-3*   Record data on simple charts or tables.
D 2-4   Construct bar graphs with stacks or lines of blocks or cubes and then change these to lines of single, simple characters, letters, or stickers to make bar graphs on paper.
D 2-5*   Identify the mode (the number that occurs most often) and the range (the smallest to largest) of the data.
D 2-6*   Locate information on simple charts or tables and bar graphs.
D 2-7   Use data to describe events as likely or unlikely to occur.
    * Worksheet included

Materials Suggested for Lessons
A limited number of manipulatives are included in the kit that accompanies this unit. Additionally, you may want to use the following items to add variety and to give the child more experiences with objects from his or her environment.
Braille paper D 2-2
Counters D 2-2
DRAFTSMAN Tactile Drawing Board D 2-2
Paper bag D 2-7
Perkins Brailler D 2-2
Pipe cleaner D 2-7
Plastic eggs (6) D 2-7
Wheatley Tactile Diagramming Kit D 2-2
Wikki Stix D 2-2
Work-Play Tray D 2-2

D 2-1
Data Collection, Graphing, and Probability/Statistics

Objective: The student will collect and organize data from observations, samplings, surveys, polls, etc.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should pose questions and gather data about themselves and their surroundings.

Suggested Materials:

Strategies:
The teacher should introduce the student to new terminology including:

Observations: Noting or recording information.

Survey: To gather information or data. 

Polls: Another term for a survey.

Provide examples of each from newspapers and magazines such as Weekly Reader and Squid.
Discuss with the student the terms observations, surveys, and polls. Ask the student why people conduct surveys. Help the student select a survey question that interests her. Determine the number of students she will survey. You may want to limit the number of students surveyed in order to limit the amount of data the student will need to manipulate.

Lessons D 2-1 through D 2-4 are closely related. The student will select a survey question, collect data, record the data on simple charts or tables, and create a bar graph to answer questions about the survey. You may select one or more of the following suggestions to use through the next three lessons; or, you may want the student to create a survey question that more closely meets her interests or those of her class. Limit the categories or options to three choices for this lesson.

Sample survey questions:
What is your favorite sandwich? Is it a hamburger, a hot dog, or a chicken sandwich?
Where should we go on a class trip? Should we go to the zoo, a play, or the fire station?
What is your favorite way to spend time after school? Is it playing with friends, watching TV, or reading?
What is your favorite cookie? Is it chocolate chip, peanut butter, or sugar cookies?
Which sport would you rather play? Is it soccer, baseball, or volleyball?

Connections: Surveys, polls, and observations are a critical concept for social studies and science. The teacher should attempt to connect the math concepts to a real-world context through social studies and science.

Read the book If the World Were a Village: A Book about the World's People, by David J. Smith. The book presents many opportunities for discussing and graphing the world's population by language, nationality, age, and other headings.

D 2-2
Data Collection, Graphing, and Probability/Statistics

Objective: The student will record simple tally/frequency counts of the data.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should sort and classify objects according to their attributes and organize data about the objects.

Suggested Materials:
*   Perkins Brailler 
*   Braille paper 
*   Wheatley Tactile Diagramming Kit 
*   DRAFTSMAN Tactile Drawing Board 
*   Wikki Stix (Cut in 1" lengths) 
*   Counters 
*   Work-Play Tray 

Materials: Listed materials are only a suggestion. Feel free to substitute other appropriate materials including those from other math programs.

Strategies: The student should have determined a survey question and the number of people he plans to survey in lesson D 2-1. The student must now survey his classmates and collect the data. Help the student select the most appropriate tool for collecting the data.

Review with the student some options for collecting data:
*   Perkins Brailler and Braille paper 
*   DRAFTSMAN Tactile Drawing Board 
*   Wheatley Diagramming Kit 
*   Wikki Stix and Braille paper 
*   Counters & Work-Play Tray 

If the student selects the Perkins Brailler, remind him to make tally marks with dots 4, 5, 6 and to leave a space between each group of five tallies. Help the student set up his paper into appropriate sections for data collection.
If the student chooses to use the DRAFTSMAN, Wheatley, or Wikki Stix, help him to determine the best way to mark responses, section his paper, and note headings. If the student chooses to use counters and a Work-Play Tray, help the student determine which sections will represent each response.

Have the student survey the predetermined number of students and collect the responses.

Connections: Take the student for a walk in the neighborhood or have the Orientation and Mobility Specialist help you with this exercise. Talk about the different types of places in the neighborhood. Are there places to live, to shop, or to eat? Are there any unusual buildings in the neighborhood? Help the student count and chart the different types of buildings in the neighborhood or a particular block.

D 2-3
Data Collection, Graphing, and Probability/Statistics

Objective: The student will record data on simple charts or tables.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should represent data using concrete objects, pictures, and graphs.

Suggested Materials:
*   Feel 'n Peel Stickers from the kit 
Worksheet D 2-3

Strategies: The student should have selected a survey question in lesson D 2-1 and collected the data in lesson D 2-2. In this lesson, the student will record the data on a simple chart. Review with the student the data collected. Each of the sample survey questions allow for at least three possible responses. Have the student count the responses for selection A, selection B, and selection C.

Provide the student with worksheet D 2-3. Have the student note the three sections of the chart and the area at the top for labeling the columns. Have the student determine what information she should use for each label and place braille labels with the information into each section or use Feel 'n Peel Alphabet Stickers to note headings. Review with the student the total count for each response. Have the student record the responses on the chart. You may wish to use three types of Feel 'n Peel Stickers to represent the responses to the survey. Help the student make a one-to-one correspondence between each answer to the survey and to each tactile sticker placed on the chart. Have the student count the total number in each group. Help the student determine a title for the chart.

Alternatively, you may wish to have the student place the appropriate number under the corresponding label. Have the student count the total number of responses in each group, label each column, insert the correct number of responses under the appropriate column heading, and determine a title for the chart.

Connections: Provide the student an opportunity to integrate science and math by charting change over time. Help the student plant a fast growing flower such a Four O'clock, Morning Glory, or Sunflowers in a classroom garden. Have the student measure the plant each week and chart the growth.
The student may want to chart her growth. Ask the student to bring her birth length and any other height measurements from home and use these data to chart her growth.

D 2-4
Data Collection, Graphing, and Probability/Statistics

Objective: The student will construct bar graphs with stacks or lines of blocks or cubes and then change these to lines of single, simple characters, letters, or stickers to make bar graphs on paper.

NCTM Standard: Data Analysis for Pre-K - 2 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should represent data using concrete objects, pictures, and graphs.

Curriculum Focal Point

Suggested Materials:
*   Large Tactile Table Graph from the kit 
*   Graphing Squares from the kit 
*   Embossed Graphing Sheets from the kit 
*   Feel 'n Peel Stickers from the kit 

Strategies: This activity is a two-step project composed of making a bar graph using objects then moving to a more abstract representation of bar graphs on paper.

Provide the student with the Large Tactile Table Graph and the Graphing Squares from the kit. Have the student graph the data from the activity that he chose in lesson D 2-1. Help the student determine the number of squares he will need, the number of columns he should use, and the labels he should place on his graph. Have the student graph the data collected for lesson D 2-1.

Introduce the Embossed Graph Sheets. Help the student explore the sheet noting the columns, rows, the areas for labels, and the correlation between the Large Tactile Table Graph and the Embossed Graph Sheets. Have the student graph the same set of data using the Embossed Graph Sheet and the tactile stickers. Remind the student to label and title his graph. Have the student compare the two graphs he has made. How are they different? How are they alike?

Connections: Provide the student additional opportunities to make bar graphs using the Large Tactile Table Graph. Have the student chart the weather for one week using the categories of rainy, sunny, and cloudy days. Have the student graph the total number of students that are absent from his class for one week. Let the student determine an event he would like to graph for a week. Ask the student to determine the labels he would use in each graphing example and a title for his graph.

D 2-5
Data Collection, Graphing, and Probability/Statistics

Objective: The student will identify the mode (the number that occurs most often) and the range (the smallest to largest) of the data.

NCTM Standard: Data Analysis for Pre-K - 2 - Select and use appropriate statistical methods to analyze data. All students should describe parts of the data and the set of data as a whole to determine what the data show.

Suggested Materials:
Worksheet D 2-5

Strategies: Provide the student with a set of numbers and have the student determine the mode and the range. An example of a set of numbers is {1, 3, 4, 4, 5, 6, 7, 7, 7, 8, 9, 14}. The sets of numbers should have multiples (such as "4" and "7" in the example) and the student should put the numbers in order from least to greatest. This will enable her to determine the number that occurs the most often (mode) and the distance from the smallest and largest group (range). 

A good way to explore this exercise is to collect a set of numbers from things such as number of siblings, scores of 10 basketball games, test grades, etc. Have the student determine the mode and the range for each set of numbers.

Provide the student with the worksheet, D 2-5, and have the student determine the following:
What is the title of the graph?
What information does the graph provide?
What kind of graph is this?
What is the number that occurs most often on the graph (mode)? (Answer: 3)
What is the distance from the smallest to the largest group (range)? (Answer: 4)
It is appropriate to use the term range at this point, as in, "What is the range of ages in this school?"

Connections: Bring the scores of the student's favorite baseball or basketball team to class. Have the student determine the number of runs per inning or the number of points per quarter in a particular game. What is the highest scoring inning or quarter? What is the lowest scoring inning or quarter? What is the range, the difference between the lowest scoring inning or quarter to the highest scoring inning or quarter? Are there any quarters or innings that have the same score? Have the student think of other areas of interest that she could use to compute the mode and the range.

D 2-6
Data Collection, Graphing, and Probability/Statistics

Objective: The student will locate information on simple charts or tables and bar graphs.

NCTM Standard: Data Analysis for Pre-K - 2 - Select and use appropriate statistical methods to analyze data. All students should describe parts of the data and the set of data as a whole to determine what the data show.

Suggested Materials:
Worksheets D 2-6 (a) and D 2-6 (b)

Strategies: Provide the student with the worksheets D 2-6 (a) and D 2-6 (b). Have the student explore the worksheets and answer the following questions:
What is the title of the chart/graph? 
What information does the chart/graph provide?
How many children are in each family? D 2-6 (a)
What is the length of each animal? D 2-6 (b)
Is there a mode? If so, what is the mode?
What is the largest number in each group?
What is the smallest number in each group?
What is the range, the distance from the smallest to the largest group of data?

Connections: Read Bart's Amazing Charts by Dianne Ochiltree. In this book, a young boy uses different kinds of charts and graphs to present information about his life.

D 2-7
Data Collection, Graphing, and Probability/Statistics

Objective: The student will use data to describe events as likely or unlikely to occur.

NCTM Standard: Data Analysis for Pre-K - 2 - Develop and evaluate inferences and predictions that are based on data. All students should discuss events related to students' experiences as likely or unlikely.

Suggested Materials:
*   Paper bag 
*   Six plastic eggs 
*   Paper 
*   Pipe cleaner 

Strategies: This activity is the beginning of probability, the use of statistics to determine the likelihood of events.
Give the student six plastic eggs. Have the student close four eggs and place the empty eggs in the bag. Have the student place a piece of paper inside one egg and put the egg in the bag. Have the student place a pipe cleaner inside one egg and put it in the bag.

Have the student take one egg out and determine what type of egg it is - empty, paper, or pipe cleaner. Have the student record the selection. Put the egg back in the bag; shake the bag. Ask the student to predict which egg is the most "likely" to be drawn from the bag. Which is the most "unlikely" to be drawn from the bag? Ask the student to predict which type of egg will be picked the most if we repeat this exercise nine times. Write down the prediction then repeat the exercise nine times, having the student record the data. At the end, have the student determine which type of egg was chosen the most often. Was it consistent with the prediction she made? Ask the student why she made her prediction. Ask her why she thinks the empty egg was chosen the most often.

This type of activity can be repeated with various objects as long as the student with visual impairments is unable to determine differences in objects when selecting them from the bag.

Connections: Read the book Pigs at Odds by Amy Axelrod. While trying their luck at various games of chance at the county fair, members of the pig family find out what the odds are that they will go home as winners.

Visit the NCTM Illuminations web site (http://illuminations.nctm.org) for a lesson that ties science and mathematical probability. The lesson, The Beat of Your Heart, asks students to predict the size of their heart based on the measurement of their fist. 

Third Grade
 
Learning Objectives
Third Grade

D 3-1   Collect data from observations, samplings, surveys, polls, etc.
D 3-2   Record simple tally/frequency counts of data.
D 3-3*   Record data on charts or tables.
D 3-4*   Construct bar graphs with a single block, cube, simple character, letter, or sticker corresponding to a single item being studied.
D 3-5*   Construct bar graphs with intervals greater than one.
D 3-6*   Read and label points on a coordinate graph.
D 3-7*   Construct simple line graphs.
D 3-8   Identify the mode, median, mean, and the range of the data.
D 3-9*   Read and interpret information on line graphs and bar graphs.
D 3-10   List some of the possible outcomes based on data and predict whether given outcomes are more, less, or equally likely to occur.
    * Worksheet included

Materials Suggested for Lessons
A limited number of manipulatives are included in the kit that accompanies this unit. Additionally, you may want to use the following items to add variety and to give the child more experiences with objects from his or her environment.
Braille paper D 3-2
DRAFTSMAN Tactile Drawing Board D 3-2
Examples of surveys D 3-1
Perkins Brailler D 3-2
Slate and Stylus D 3-2
Wikki Stix D 3-2, D 3-7

D 3-1
Data Collection, Graphing, and Probability/Statistics

Objective: The student will collect data from observations, samplings, surveys, polls, etc.

CTM Standard: Data Analysis for 3 - 5 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should collect data using observations, surveys, and experiments.
Curriculum Focal Point

Suggested Materials:
*   Examples of surveys 

Materials: Listed materials are only a suggestion. Feel free to substitute other appropriate materials including those from other math programs.

Strategies: Provide the student with examples of observations, surveys, and polls from newspapers and magazines such as the Weekly Reader and Squid. The examples may need to be simplified by providing the student less complex examples such as "a survey found that 3 out of 4 students like juice over milk."

Have the student conduct a survey by:
1.   Asking a question (Development of research question) 
2.   Guessing/predicting what the answer will be (Stating a hypothesis) 
3.   Conducting the survey through questioning (Conduct research) 
4.   Documenting the data 
5.   Determining the answer to the question 

Example:
1.   What is your favorite kind of ice cream of these three: vanilla, chocolate, strawberry? 
2.   I think the answer will be vanilla. 
3.   The student conducts the survey by asking her classmates what type of ice cream they would choose. 
4.   The student records the findings on a table or chart. 
5.   The student determines the answer to the question. 

If the student is having difficulty deciding on a question for her survey, refer to lesson D 1-1 and D 2-1 or suggest one of the following: favorite TV show, fruit, season, sports team; the number of pets, or siblings; the hours spent watching TV, hours of sleep or bedtime. Help the student determine a survey question, how many people she will survey, and how she will state the question.

Connections: Read the book Tiger Math by Ann Whitehead Nagda. The story of a Siberian tiger cub is accompanied by graphs on facing pages that chart T.J.'s growth, successfully showing math in "real-world" terms.

D 3-2
Data Collection, Graphing, and Probability/Statistics

Objective: The student will record simple tally/frequency counts of data.

NCTM Standard: Data Analysis for 3 - 5 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should collect data using observations, surveys, and experiments.

Curriculum Focal Point

Suggested Materials:
*   Perkins Brailler 
*   Slate and Stylus 
*   Braille paper 
*   DRAFTSMAN Tactile Drawing Board 
*   Wikki Stix (Cut in 1" lengths) 

Strategies: The student should have determined a survey question and the number of people he plans to survey in lesson D 3-1. The student must now survey his classmates and collect the data. Help the student select the most appropriate tool for collecting the data.

Review with the student some options for collecting data:
*   Perkins Brailler and Braille paper 
*   Slate and Stylus and Braille paper 
*   DRAFTSMAN Tactile Drawing Board 
*   Wikki Stix and Braille paper 
If the student selects the Perkins Brailler or the Slate and Stylus, remind him to make tally marks with dots 4, 5, 6 and to leave a space between each group of five tallies. Help the student set up his paper into appropriate sections for data collection.

If the student chooses to use the DRAFTSMAN or Wikki Stix, help him to determine the best way to mark responses, section his paper, and note headings.
Have the student survey or poll his classmates.

Connections: Read the book Alligator Shoes by Arthur Dorros. While searching for the perfect pair of shoes, Alvin the alligator is mistakenly locked in a shoe store overnight. He spends the whole night trying on different types of footwear. Discuss the different types of footwear in the story. What types of shoes does the student own? Have the student collect data to chart the different types of shoes in his home.

D 3-3
Data Collection, Graphing, and Probability/Statistics

Objective: The student will record data on charts or tables.

NCTM Standard: Data Analysis for 3 - 5 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should represent data using tables and graphs such as line plots, bar graphs, and line graphs.
Curriculum Focal Point

Suggested Materials:
*   Feel 'n Peel Stickers from the kit 
Worksheet D 3-3

Strategies: The student should have selected a survey question in lesson D 3-1 and collected the data in lesson D 3-2. In this lesson, the student will record the data on a simple chart.

Provide the student with worksheet D 3-3. Have the student note the three sections of the chart and the area at the top for labeling the columns.
Have the student determine what information she should use for each label and place a braille label of the information into each section or use Feel 'n Peel Alphabet Stickers to note headings. Review with the student the total count for each response. Have the student place a braille label with the appropriate number in the correct column. Have the student determine a title for the chart.

Connections: Tables are a system to organize data commonly used in social studies, language arts, and science. Have the student review one of her textbooks to locate an example of a table.

D 3-4
Data Collection, Graphing, and Probability/Statistics

Objective: The student will construct bar graphs with a single block, cube, simple character, letter, or sticker corresponding to a single item being studied.

NCTM Standard: Data Analysis for 3 - 5 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should represent data using tables and graphs such as line plots, bar graphs, and line graphs.
Curriculum Focal Point

Suggested Materials:
*   Embossed Graph Sheet from the kit 
*   Feel 'n Peel Stickers from the kit 
*   Large Tactile Table Graph from the kit 
*   Graphing Squares from the kit 
Worksheet D 3-4

Strategies: Provide the student with an example of a bar graph, D 3-4. Explore the graph with the student, reviewing the labels, the title, and the data recorded.

Review with the student how to construct a bar graph using the Large Tactile Table Graph and the Graphing Squares from the kit. Have the student construct bar graphs with the data from the exercise D 3-3 or allow the student to select another example to use for the bar graph. If the student is having difficulty with this activity, refer to lesson D 2-4 for a review.
Next, using the same data, have the student create a 2-D bar graph using Feel 'n Peel Stickers and Embossed Graph Sheets. Have the student label the graph and provide a title for the graph.

Have the student compare the two graphs. How are they alike? How are they different?

Connections: Visit the NCTM Illuminations web site (http://illuminations.nctm.org) for an eight lesson unit entitled Eat Your Veggies. In this unit, students collect and display data in a variety of ways, beginning with tallies and pictographs. Later lessons focus on representing data using bar graphs, line plots, circle graphs, box-and-whisker plots, and glyphs. The students also compare graphs from two sets of data and find the range, median, mean, and mode of each set. The unit highlights the NCTM Process Standards of Representation, Problem Solving, Connections, and Communication.

D 3-5
Data Collection, Graphing, and Probability/Statistics

Objective: The student will construct bar graphs with intervals greater than one.

NCTM Standard: Data Analysis for 3 - 5 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should represent data using tables and graphs such as line plots, bar graphs, and line graphs.

Curriculum Focal Point

Suggested Materials:
*   Large Tactile Table Graph from the kit 
*   Graphing Squares from the kit 
*   Feel 'n Peel Stickers from the kit 
Worksheet D 3-5 (a), D 3-5 (b), and D 3-5 (c)

Strategies: Up to this point, all graphs have been based on an interval of one. Discuss with the student the need to represent more than one item in a group (e.g., number of students in his school, populations of states and countries, pounds of produce imported into the country). Provide the student with examples of bar graphs where the intervals are greater than one, Worksheet D 3-5 (a) and D 3-5 (b). Review the worksheets with the student and determine the interval used on both sheets as well as the information that the graphs provide.
Have the student use the following information to create a bar graph using the Large Tactile Table Graph and the Graphing Squares from the kit:

Number of students going on a field trip to the zoo
Grade 1   20 students
Grade 2   30 students
Grade 3   40 students

Help the student determine the intervals she would use for her graph. Have her construct the graph and label the data on her graph.
Explain to the student that information is not always provided in even numbers such as 20 students. How should we graph 25 students if we are using intervals of 10? Review Worksheet D 3-5 (c) with the student noting that some answers are halfway between the numbers on the graph.
These intervals represent half of a unit or interval. Help the student read the graph.
Provide the student with the graphing square that represents half of a unit. Explain that on the Large Tactile Table Graph these units will represent half of the chosen interval.

Ask the student to graph the following information using intervals of 10:
Number of students going on a field trip to the zoo

Grade 1   15 students
Grade 2   35 students
Grade 3   30 students

Have the student label the graph and provide a title for the graph.
Provide the student additional opportunities to graph in units greater than one and to graph partial intervals.

Connections: Use issue two of Squid magazine to investigate a puzzle that uses bar graphs ("Ms Placed"). In issue five of Squid magazine students build a graph that uses intervals of 2 ("Favorite Season").

D 3-6
Data Collection, Graphing, and Probability/Statistics

Objective: The student will read and label points on a coordinate graph.

NCTM Standard: Data Analysis for 3 - 5 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

Suggested Materials:
*   Embossed Graph Sheets from the kit 
*   Feel 'n Peel Stickers from the kit 

Worksheets D 3-6 (a) and D 3-6 (b)

Strategies: Being able to read and label coordinate graphs is a foundational skill needed for advanced mathematics. The coordinate graph is a grid with two labeled sides.

First, provide the student with a coordinate graph (basic labeled grid). Help the student start at the bottom left-hand corner (the origin). The student should then move along the horizontal axis (moving left to right). Next, have the student explore the vertical axis (moving from the bottom up). Make sure the student can find the beginning point (origin) and can read the labels on axes.
Next, provide the student with Worksheet D 3-6 (a) which includes coordinate points. Have the student find a point on the coordinate graph. It is important that the student understands that the horizontal movement always comes before the vertical movement. To determine the location of the point on the grid, have the student start at the point of origin, move to the right, and then up to the point. Repeat having the student count the number of moves to the right; then count the number of moves to the point.

Next, provide the student with an Embossed Graph Sheet. Help the student to locate the origin (or starting point) and then give the student coordinates such as 2-4, 3-5, 1-2, etc. The student should place a Feel 'n Peel Sticker on the graph paper to demonstrate understanding. Remind the student that the first number tells how many moves to the right. The second number tells the number of moves up the vertical axis.

Worksheet D 3-6 (b) can be used for either reinforcement or for assessment of understanding.

Connections: As an activity, the teacher can develop a "Math BINGO" game that will reinforce coordinate graph movement.
Provide the student practice in locating coordinate points using games such as Battleship and in word searches found in each Squid issue.

D 3-7
Data Collection, Graphing, and Probability/Statistics

Objective: The student will construct simple line graphs.

NCTM Standard: Data Analysis for 3 - 5 - Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. All students should represent data using tables and graphs such as line plots, bar graphs, and line graphs.

Curriculum Focal Point

Suggested Materials:
*   Embossed Graph Sheets from the kit 
*   Feel 'n Peel Stickers from the kit 
*   Wikki Stix 
Worksheet D 3-7

Strategies: A line graph is a graph that shows trends of data by connecting data over time.
 
Provide the student with an example of a line graph, D 3-7. Explore the graph with the student noting the labeling on the graph. Demonstrate to the student how to read the graph. Ask the student to tell you how many hours of TV were watched on each day. Ask her when the number of hours "spiked" or went up.
Provide the student with an Embossed Graph Sheet and have the student plot points on the graph using Feel 'n Peel Stickers and connect the points with Wikki Stix. Have the student label the data on the graph and select a title for the graph.

The graph should show trends. Examples of things the student could graph include:
*   The attendance of the class for one week 
*   The number of books read by the class per month 
*   The number of class birthdays per month 
*   The inches of rainfall by month for the city or state 

Connections: Read Pepper's Journey by Stuart J. Murphy or The Mysterious Tadpole, available in braille, by Steven Kellogg. In both books, students can collect data and chart an animal's growth. Pepper's Journey is about the first year in a kitten's life. The Mysterious Tadpole is about a young boy's birthday present from Scotland.

D 3-8
Data Collection, Graphing, and Probability/Statistics

Objective: The student will identify the mode, median, mean, and the range of the data.

NCTM Standard: Data Analysis for 3 - 5 - Select and use appropriate statistical methods to analyze data. All students should use measures of center, focusing on the median, and understand what each does and does not indicate on the data set.

Suggested Materials:

Strategies: This activity introduces the student to new terminology.
Mode is the number that occurs most often.
Median is the middle number in a list of numbers.
Mean is the average of numbers and calculated by adding all the data and dividing by the number of terms.
Range is the distance from the least to greatest numbers.
At this grade level, provide the student sets of numbers to compute the mode, median, mean, and range of the numbers.

Example:
{2, 4, 5, 7, 3, 11, 3}
1.   Have the student put the numbers in order from least to greatest. {2, 3, 3, 4, 5, 7, 11} 
2.   Then determine the mode. Answer: "3" since it repeats the most number of times. 
3.   Then determine the median. Have the student look at the list of numbers and determine which number is in the middle of the list. Answer: "4" 
4.   Then determine the mean. Answer: Add up all the numbers (35), and then divide by the number of terms (35 / 7), which will give you the mean or average (5). 
5.   Then determine the range. Answer: 11-2 = 9. 

Provide the student additional opportunities to compute the mode, median, mean, and the range. The student may generate the numbers for the exercise.

Note: If provided an odd number of numerals, the median is easy to determine as the middle number in the list. However, if an even number of numerals is provided, then the median is computed differently. The median is determined by computing the mean (average) of the two middle numbers. Therefore, in this situation you would add the two middle numbers and divide the answer by 2.

Brainstorm with the student when he would need to know the mean, the median, the mode, or the range in everyday life.

Connections: Have the student use the Internet to locate the website for the local paper. Help the student to determine the predicted temperatures in the city for the next seven days. Have the student determine the mode, median, and the range for the daytime and nighttime temperatures in the city.
D 3-9
Data Collection, Graphing, and Probability/Statistics

Objective: The student will read and interpret information on line graphs and bar graphs.

NCTM Standard: Data Analysis for 3 - 5 - Select and use appropriate statistical methods to analyze data. All students should describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed.
Curriculum Focal Point

Suggested Materials:
Worksheets D 3-9 (a) and D 3-9 (b)

Strategies: Provide the student with examples of line graphs and bar graphs. Have the student read and interpret information.
What is the title of the graph?
What information does the graph provide?
What kind of graph is this?
What is the number that occurs most often on the graph (mode)?
What is the distance from the smallest to the largest group (range)?
What is the median?
What is the mean?
What is the interval used to represent the data (e.g., by 5's, by 10's, by 100's)?

Connections: Read the book Lemonade for Sale by Stuart J. Murphy. In this book, available in braille, the Elm Street Kids decide to sell lemonade to earn money to fix up their clubhouse. They use graphs to track their sales.

D 3-10
Data Collection, Graphing, and Probability/Statistics

Objective: The student will list some of the possible outcomes based on data and predict whether given outcomes are more, less, or equally likely to occur.

NCTM Standard: Data Analysis for 3 - 5 - Understand and apply basic concepts of probability. All students should describe events as likely or unlikely and discuss the likelihood using words as certain, equally likely, and impossible.
Curriculum Focal Point

Suggested Materials:
*   Spinner from the kit 
*   Overlays from the kit 
*   Feel 'n Peel Stickers from the kit 

Strategies: Provide the student with the spinner. Place a tactile sticker on the base of the spinner as a pointer for the spinner. Provide the student with the first overlay and help the student note that the overlay has four equal divisions. Demonstrate how the spinner works, helping the student note that the tactile sticker on the base points to the answer for the spin. Allow the student to spin the spinner several times for practice.
Place two stickers of one type and two stickers of another type on the overlay, one per division. (The smiley and frowny faces work well for this purpose.) Have the student predict the probability of the spinner landing on one type of sticker. Use the terms "likely," "unlikely," or "equally likely" to predict the outcome of the spins. Ask the student why he made his prediction. Have the student spin the spinner 10 times recording the responses each time. Was the prediction correct?

Change the tactile stickers to a configuration of three of one type and one of another type. Now have the student predict the probability of the spinner landing on a certain type of sticker. Ask the student why he made his prediction. Have the student spin the spinner ten times recording the outcome each time. Was the prediction correct?

Repeat this procedure reducing one type of sticker while increasing the other type of sticker until the configuration is four of one type and none of the second type. On the last set of spins introduce the words "certain" and "impossible" to help the student predict the outcome of the spins.
Repeat the exercise using the overlay with six divisions. Have the student make predictions of "likely," "unlikely," "equally likely," "certain," and "impossible". Have the student compare the two overlays. Was it easier to predict with one of the overlays? If so, which one was easiest? Why?

Connections: Read the book Probably Pistachio by Stuart J. Murphy. Readers are introduced to the concept of probability in a story about a boy who has a day in which nothing goes right.

Read the book Who's Got Spots? by Linda Aber. In this book, Kip takes a survey and organizes his data using tallies and graphs to forecast whether chicken pox will keep him and his friends out of the autumn fest. 
Assessment Check List
 
Objective   Date/Rating Notes

Kindergarten
D K-1 Count class members and real objects to determine answers to real life questions.    
D K-2 Arrange objects on a floor or table graph according to attributes.    
D K-3 Construct pictographs using people or real objects in lines.    
D K-4 Look for largest group, the smallest group, and groups with the same amount.    

Use the following rating scale to indicate the student's current level of performance of each objective:
1 = Beginning level of performance
2 = Developing level of performance
3 = Accomplished/Mastery level of performance

Objective   Date/Rating Notes

First Grade
D 1-1 Collect data from class members, observations, etc. to answer real life questions.    
D 1-2 Record data on simple charts or tables.    
D 1-3 Construct pictographs or bar graphs with blocks, cubes, or symbols in stacks or lines to represent individual people, objects, etc.    
D 1-4 Look for the largest grouping and the extremes - smallest to largest amount of the data (range).    
D 1-5 Answer questions related to data displayed in graphs (e.g., how many more in a group compared to another group, how many in two groups combined, etc.).    

Use the following rating scale to indicate the student's current level of performance of each objective:
1 = Beginning level of performance
2 = Developing level of performance
3 = Accomplished/Mastery level of performance

Objective   Date/Rating Notes

Second Grade
D 2-1 Collect and organize data from observations, samplings, surveys, polls, etc.    
D 2-2 Record simple tally/frequency counts of the data.    
D 2-3 Record data on simple charts or tables.    
D 2-4 Construct bar graphs with stacks or lines of blocks or cubes and then change these to lines of single, simple characters, letters, or stickers to make bar graphs on paper.    
D 2-5 Identify the mode (the number that occurs most often) and the range (the smallest to largest) of the data.    
D 2-6 Locate information on simple charts or tables and bar graphs.    
D 2-7 Use data to describe events as likely or unlikely to occur.    

Use the following rating scale to indicate the student's current level of performance of each objective:
1 = Beginning level of performance
2 = Developing level of performance
3 = Accomplished/Mastery level of performance

Objective   Date/Rating Notes

Third Grade
D 3-1 Collect data from observations, samplings, surveys, polls, etc.    
D 3-2 Record simple tally/frequency counts of data.    
D 3-3 Record data on charts or tables.    
D 3-4 Construct bar graphs with a single block, cube, simple character, letter, or sticker corresponding to a single item being studied.    
D 3-5 Construct bar graphs with intervals greater than one.    
D 3-6 Read and label points on a coordinate graph.    
D 3-7 Construct simple line graphs.    
D 3-8 Identify the mode, median, mean, and the range of the data.    
D 3-9 Read and interpret information on line graphs and bar graphs.    
D 3-10 List some of the possible outcomes based on data and predict whether given outcomes are more, less, or equally likely to occur.    

Use the following rating scale to indicate the student's current level of performance of each objective:
1 = Beginning level of performance
2 = Developing level of performance
3 = Accomplished/Mastery level of performance

Appendix A
 
Related Math Materials available from American Printing House for the Blind
Data Collection, Graphing, and Probability/Statistics   Catalog Number
Chang Tactual Diagram Kit   1-03130-00
DRAFTSMAN Tactile Drawing Board   1-08857-00
Geometry Tactile Graphics Kit   1-08841-00
Graph Sheets, embossed   1-04058-00
Graphic Aid for Mathematics   1-00460-00
Hundreds Boards and Manipulatives   1-03105-00
Tangible Graphs   1-08860-00
Wheatley Tactile Diagramming Kit: Picture Maker   1-08838-00
Copies of Student Worksheets
 
D 1-2 Data
 
D 1-3 (a) Data
Pets in My Neighborhood
 
D 1-3 (b) Data
Pets in My Neighborhood
 
D 1-3 (c) Data
Pets in My Neighborhood

Legend:
Fish    
Cats    
Dogs    
 
D 1-5 (a) Data
My Favorite Season
 
D 1-5 (b) Data
Dollars Earned
 
D 1-5 (c) Data
Dollars Earned
 
D 2-3 Data
 
D 2-5 Data
Number of Fish Caught
 
D 2-6 (a) Data
Children in the Family
 
D 2-6 (b) Data
Animal Length
 
D 3-3 Data
 
D 3-4 Data
Dollars in Candy Sales
 
D 3-5 (a) Data
Students Per Grade
 
D 3-5 (b) Data
Abby's Math Scores
 
D 3-5 (c) Data
Trees on Campus
 
D 3-6 (a) Data
 
D 3-6 (b) Data
 
D 3-7 Data
Watching TV
 
D 3-9 (a) Data
Average Temperatures
 
D 3-9 (b) Data
Favorite Pizza in Our Class
 

