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- Exploring Standards
- Analyzing a Case Study

The National Council of Teachers of Mathematics (NCTM, 2000) has identified measurement as a strand in its *Principles and Standards for School Mathematics.* In grades pre-K-12, instructional programs should enable all students to do the following:

- Understand measurable attributes of objects and the units, systems, and processes of measurement
- Apply appropriate techniques, tools, and formulas to determine measurements

Understand such attributes as length, area, weight, volume, and size of angle, and select the appropriate type of unit for measuring each attributeIn grades 3-5 classrooms, students are expected to do the following:

- Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems
- Carry out simple unit conversions, such as from centimeters to meters, within a system of measurement
- Understand that measurements are approximations and understand how differences in units affect precision
- Explore what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape is changed in some way
- Develop strategies for estimating the perimeters, areas, and volumes of irregular shapes
- Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles
- Select and use benchmarks to estimate measurements
- Develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms
- Develop strategies to determine the surface areas and volumes of rectangular solids

The NCTM (2000) Measurement Standards suggest that “students in grades 3-5 should explore how measurements are affected when one attribute to be measured is held constant and the other is changed. For example, consider the area of four tiles joined together along adjacent sides. The area of each tile is a square unit. When joined, the area of the resulting polygon is always four square units, but the perimeter varies from eight to ten units, depending on how the tiles are arranged. … This activity provides an opportunity to discuss the relationship of area to perimeter. It also highlights the importance of organizing solutions systematically” (NCTM, 2000, p. 173).

*Principles and Standards for School Mathematics* © 2000 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or redistributed electronically or in other formats without written permission from NCTM. standards.nctm.org

Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.

To begin the exploration of what measurement topics look like in a classroom at your grade level, you will watch a video segment of a teacher who took the Measurement course and then adapted the mathematics to his own teaching situation. We will begin by looking at some of the content addressed in the videotaped lesson. **Note 3**

Mr. Belber uses pentominoes to explore area and perimeter relationships with his class. A pentomino is made from five squares arranged so that each square shares at least one adjacent side with at least one other square. There are 12 unique pentominoes.

On graph paper, draw the 12 pentominoes, or print out the prepared set (PDF). Cut them out so that you can use the 12 pentominoes in the next problem.

**Problem B1**

Put any two pentominoes together and determine both the perimeter and the area of the new shape. What is the largest perimeter possible, and what is the smallest perimeter possible?

**Problem B2**

As students discover different perimeters, Mr. Belber has them record their findings on grid paper. What are the advantages and disadvantages of this recording scheme? What problem-solving strategies are students using to find the possible perimeters?

**Problem B3**

Mr. Belber wanted his students not only to use a guess-and-check strategy to find the possible perimeters, but also to analyze how the placement of the pentominoes next to each other affected the perimeter. How can the perimeter of a new shape made from two pentominoes be determined without counting? How did Mr. Belber help students understand this analytic method? What additional questions could you ask to confirm that the students understand the method?

**Problem B4**

This lesson focuses on the fact that if area remains constant (in this case, 10 in^{2}), the perimeter of shapes constructed with that area can vary. Do you think the mathematical purpose of the lesson is clear? What other factors make a lesson successful?

**Problem B5**

Sometimes we want students to generalize what they have learned. How did Mr. Belber extend the learning from this lesson? What generalizations might he expect students to mention?

**Note 3**

The purpose of the video segments is not to reflect on the teaching style of the teacher portrayed. Instead, look closely at the methods the teacher uses to bring out the ideas of measurement while engaging his students in activities.

**Problem B1**

The largest possible perimeter is 22 units; the smallest is 14 units. This is because all but one pentomino have a perimeter of 12 units. One pentomino has a perimeter of 10 units. A method for determining perimeter when combining two pentominoes is to add the perimeters of the two pentominoes and then subtract the number of sides touching when the pentominoes are combined. To get the largest possible perimeter, combine two 12-unit pentominoes so that the least number of sides touch, which is two. So 12 + 12 – 2 = 22. To get the smallest perimeter, combine one 12-unit pentomino with the 10-unit pentomino so that as many sides as possible are touching. The maximum number of sides touching is eight, making the smallest possible combined perimeter 12 + 10 – 8, or 14 units.

**Problem B2**

One advantage of recording on grid paper is that it allows students to trace the perimeters of the shapes they’ve formed. Recording on grid paper also enables students to keep track of the different shapes. Recording all of their findings also helps them notice patterns. When tracing their combined shapes, however, if students do not also include the detail of all the squares in the pentomino, they will not be able to see the number of sides that are touching to make their new combined shape. This will lessen the power of recording their findings.

Some of the problem-solving strategies students used included the guess-and-check strategy, as well as analyzing shapes and drawings for patterns (e.g., all perimeters must be even, and the combined perimeter will be the sum of the perimeters of both pentominoes less the total number of sides touching).

**Problem B3**

To determine the perimeter without counting, the students first added the perimeters of the two pentominoes being combined for the new shape and then subtracted the number of sides touching between the two pentominoes. To help students see this analytic method, Mr. Belber asked them questions that focused on the number of sides that touched in the new shape and how that number decreased the sum of the two pentomino perimeters.

Additionally, you could ask students the following questions to confirm that they understand the concept: What patterns do you notice in the perimeters of the combined figure when the two pentominoes have two sides touching? Four sides touching? etc. Does your answer depend on which two pentominoes you choose? Can you explain why this pattern makes sense, or why increasing the number of sides touching makes the perimeter smaller?

**Problem B4**

This lesson has a clear mathematical purpose, which is the understanding that if area remains constant, the perimeter of shapes constructed having that area can vary. Using manipulatives helps students see that area stays constant. The tiles also allow students to see and feel the number of sides touching, and how that number affects the perimeter.

**Problem B5**

Mr. Belber extended learning by asking students to explain why they knew they had found the smallest or largest perimeter, how they knew they had found all possible perimeters, and why they got only even perimeters. Some generalizations he might expect students to make include the fact that only even perimeters can be made and that a rule for determining perimeter is that the perimeters of two pentominoes, minus the number of sides touching in the combined new shape, equals the perimeter of the new shape. Mr. Belber’s homework assignment of repeating the task with three pentominoes provides additional generalizing opportunities and could ultimately lead to a rule for determining the maximum and minimum perimeters for any number of pentominoes.