Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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## How Many People Will Fit?

The goals of the NCTM's reasoning process standard are that "in grades K-4, the study of mathematics should emphasize reasoning so that students can-

• draw logical conclusions about mathematics;
• use models, known facts, properties, and relationships to explain their thinking;
• justify their answers and solution processes;
• use patterns and relationships to analyze mathematical situations;
• believe that mathematics makes sense."
(NCTM, Curriculum and Evaluation Standards for School Mathematics, p. 29) Video Overview
This lesson emerged when the school had a tornado drill and students discovered that the hallway was too small to fit them all. Students investigate the concept of area by figuring out how many people will fit in areas in the school building. Specifically, students investigate how many of them will fit in the following areas: (1) their classroom, (2) a hallway, (3) a lobby, and (4) the cafeteria tables.

Students are divided into groups of four and assigned to one of the three areas. The groups choose the materials they want to use and are accompanied to their designated areas by classroom volunteers. The groups estimate the number of students who will fit in an area and then develop a strategy for determining the exact number. During their investigations, students organize and record their solutions and use measuring, counting, and addition to find the total number of people. At the end of the lesson, the class reconvenes and students share their strategies and results.

An Exploration
For Teacher workshops

An Elevator for How Many?

This investigation will familiarize teachers with the mathematical activity that students are working on in this lesson. Each group will need these items:

• paper and pencil
• metersticks (or yardsticks)
• string
• tape
• dot stickers
• calculators

Follow these steps:

1. Before the workshop, prepare a list of spaces in the building, or begin the workshop by creating a list with the teachers. Spaces and situations to consider are sitting in the hallway or other safe location for an emergency drill, standing in the hallway waiting to get into class, standing in an elevator, studying in the lounge, or standing at a reception in the lounge.

2. Divide teachers into groups. Assign each group an area and tell group members to determine how many people can comfortably fit into their area. Have each group record its thinking, reasoning, and results, using the following steps:

• choose the materials they think they will need;
• view the area and make an estimate;
• develop a plan; and
• carry out the plan.

• To close the activity, ask each group to report both its plan and its findings to the whole group.

Topics for Discussion
The following areas provide a focus for discussion after you view the video. You may want to customize these areas or focus on your own discussion ideas.

Using Calculators

1. Some students used calculators in this lesson. Do you think it was appropriate to allow students to use calculators? Why or why not?

2. In the video, two students in a cafeteria told a third student an answer, but this student still insisted on doing the computation on the calculator. What does this situation show?

3. How did calculators help students in their problem solving? What are some appropriate uses of calculators with young learners?

4. Ms. Wheatley has allowed students to use the calculators since the beginning of the school year and always has them available for students. What do you think about this philosophy?

5. A position statement from the National Council of Teachers of Mathematics recommends "the integration of the calculator into the school mathematics program at all grade levels in classwork, homework, and evaluation." What do you think about this statement? What barriers exist to making this vision a reality? What can be done to remove these barriers?

Designing Authentic Tasks

1. Authentic tasks engage students in solving real problems. How were the tasks in the video authentic?

2. Outline the characteristics of authentic tasks. Compare the advantages and disadvantages of providing authentic experiences for learning mathematics.

3. What was the value of asking students to estimate before solving their problems?

4. Describe the mathematical ideas that students use and construct in the video. What instructional support can a teacher provide to assist students with making appropriate connections between the mathematics they know and the mathematical ideas they are constructing?

5. Elaborate on the following statement, drawing on the video to support your view: If students are given the opportunity to think and solve problems on their own, they will do it, enjoy it, and learn from the experience.

6. What was the advantage of letting students decide which methods and tools to use? What materials and strategies did they choose?

7. How would you plan to assess students' knowledge and understanding of mathematics in this classroom?

Extension
Area and Volume Estimation

Think of additional how-many-would-fit investigations for elementary students. Prepare two lists. The first should focus on measurement-of-area activities and the second should focus on measurement-of-volume activities.

For example, how many carpet squares would cover the floor? How many desks would fill up the room when stacked on top of each other and side by side? How many sugar cubes would fill the inside of a teacher's desk?

Authentic Tasks

Create three authentic estimation tasks that relate to your school or classroom. What are some real tasks that students can investigate? For example, can all the children in the school fit into the auditorium to hear a guest speaker, and if so, how many additional guests can be invited? Can two classes of students safely, according to fire codes, watch a video in one classroom?

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