## Picture This

• Lesson
3-5
2

In this lesson, students participate in an activity in which they conduct a survey, analyze and summarize the data they collect, and draw conclusions from their findings. This lesson plan was adapted from the article "Picture This" by Marty Hopkins, which appeared in Teaching Children Mathematics, February 1998, vol. 4, no. 6, pp. 354-59.

This investigation involves the students in a research project dealing with pictures such as those commonly found in primary levels of elementary school mathematics textbooks and workbooks. The students conduct a survey of people representing various age groups to determine what number stories such pictures bring to their minds.  Completing this activity will (1) sharpen students' appreciation of the need for standardized methods of data collection; (2) reinforce the concept that a picture without a number story can match with a variety of number sentences, and (3) illustrate that several different interpretations of the operations of addition and subtraction are possible.

To start the lesson, bring five people to the front of the room, and then call three more. Ask the group of five to stand on one side of the room; ask the group of three to stand on the other side of the room.

Encourage the class to tell mathematics stories about the situation at the front of the room, and ask for number sentences to match each story.

Write all number sentences generated by the students on the chalkboard. Possible number sentences include:

5 + 3 = 8
3 + 5 = 8
8 – 3 = 5
8 – 5 = 3

After several stories and number sentences have been identified, arrange students in groups of two. One student should draw a simple story, similar to the one enacted in class. The other student should write a number sentence that corresponds to the story. Both students should discuss other possible number sentences. The students then switch roles and repeat the activity.

Using story books from the classroom library, try to find an example of such problems. These books typically have one number sentence that is supposed to match each picture.  Show the examples to the children, and ask them if they can think of other stories and number sentences that might match the picture

Ask students what they think it means to be a researcher? Discuss the students' responses. Ask students how a survey is used. Once again, discuss the students' responses.

Show students an overhead of What's the Story? Activity Sheet.

Ask students what kind of data could be collected by showing people these pictures. As the class discusses survey questions, guide the students into these two main questions:

1. What types of responses do different people give to the pictures?
2. Do people of different ages give different types of responses?

Distribute copies of the What's the Story? Activity Sheet and data recording sheet. Read the directions at the bottom of the What's the Story? Activity Sheet. After each direction, ask the children to explain why they think each of these directions is important. Suggested responses to each direction follow:

• The more people we survey, the more data we will have. The more data we have, the easier it will be to see patterns to help answer our questions.
• If people who are surveyed hear other people's responses, they may be tempted to respond in the same way.
• It will be interesting to see if the mathematics story matches the number sentence.
• Adults might interpret the pictures differently than children do. Also, all the information from one person needs to be on one piece of paper so that all the data are kept together.

After discussing each point, you may wish to pair the children and ask them to role play the parts of researcher and subject.  Researchers ask questions and record responses without comment; subjects "read" the pictures and give responses.

Assign the survey for homework, asking for all data to be returned after two or three days. Students should find at least two people to survey. One of these people should be an adult, and the other person should be a child. Each student will need at least two copies of the Data Recording Sheet, which is found in the same file as the What's the Story? Activity Sheet.

When all surveys have been completed, place the children in groups of three or four to share their data from the first picture (the rabbits).

Ask the children to make a list or table of addition and subtraction number stories or number sentences that their subjects told them while looking at the rabbits picture, along with the number of people who told each kind. A sample data table follows:

Rabbit Story

3 + 1 = 46 People
1 + 3 = 410 People
Subtraction Story (12 People)
4 - 1 = 36 People
3 - 1 = 25 People
4 - 3 = 11 Person

Ask each group to share its data with the rest of the class. On the chalkboard or on chart paper, make a class list of results.

Discuss all stories and number sentences to make sure that they are mathematically correct. Justifiable number sentences for the first problem include the following:

4 ‑ 1 = 3
3 + 1 = 4
3 ‑ 1 = 2
1 + 3 = 4
4 ‑ 3 = 1
These results can lead to a discussion of many different topics, including:
• Commutative property ("order property"); 3 + 1 = 1 + 3
• Number families
• Different meanings for subtraction
• Take away: 4 ‑ 1 = 3; how many are left?
• Comparison: 4 ‑ 1 = 3; how many more are in the bigger group

If you see little variety in the data collected by your students, you might wish to offer some of these number sentences to encourage discussion among your students.

Challenge the children to analyze their data again—first in their small cooperative groups, and then as a class—to determine whether a relationship exists between a subject's age and his or her choice of number story or sentence. You may wish to have each group record the results of these analyses on the reproducible Picture This Data Sheet.

After the entire class has shared their results, place the compiled class data on a clean version of the data sheet.

Follow the same procedure for the second problem.

Possible solutions for this problem include:

4 + 2 = 6
6 - 2 = 4
6 - 4 = 2
6 - 2 = 4
2 + 4 = 6
6 - 6 = 0

Assessment Option

Discuss the processes involved in research with your class.  Researchers (1) ask the question, (2) collect data, (3) organize the data, (4) analyze the data, and (5) generalize the findings to (6) answer the question.

Extensions

1. Repeat all the previous activities with pictures that are typically found in textbooks to represent multiplication and division situations.

Ask the students to locate multiplication and division pictures in books; design a survey by using pictures similar to those they find; and collect, organize, analyze and make generalizations from the data they gather.

2. Younger students might want to make class books containing the stories that they collected while doing the survey. The cover of each book could be the picture used in the survey, and the students could contribute one or two of the problems they like best from their data, along with the appropriate number sentences.  This book could then be used throughout the remaindered of the year to review the various meaning of addition and subtraction.
3. Older students might want to extend their research to explore pictures that tell fraction stories.

Encourage the children to look through textbooks or story books at all elementary-grade levels, searching for pictures that can be used to tell fraction stories.  In what ways are they alike?  Different?  Can all pictures be adapted to tell fraction stories?

4. Many student  experience difficulty with problems involving more than one step. Encourage them to tell multi-step stories.

Record the number sentences that match the stories they tell.

Challenge the students to draw pictures representing their stories.

5. Create a class book containing only pictures.  Make a separate answer book that includes the original stories written by the artists.

Challenge students to tell or write stories from the pictures in the book.  How are they alike?  How are they different?

Questions for Students

1. Did you encounter any problems while collecting your data from the people you surveyed? If so, what were they?

[Student responses may vary.]

2. As you surveyed people, did any person create a number sentence that you had not thought of before you began? If so, what was it? Was it correct?

[Student responses may vary.]

3. Do you think it was a good idea to analyze the data based on the age of the people being surveyed? What other possible ways could you have grouped the responses?

[Student responses may vary; students may suggest gender or occupation.]

### Learning Objectives

Students will:

• Create three figures and measure the number of perimeter pins, number of interior pins, and the resulting area.
• Use the data to construct a system of equations.
• Use algebraic manipulations to solve the system to find the coefficients of Pick’s Theorem.

### NCTM Standards and Expectations

• Collect data using observations, surveys, and experiments.
• Represent data using tables and graphs such as line plots, bar graphs, and line graphs.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP6
Attend to precision.