Perimeter, Area, Volume, and All That: A Study of Measurement

Measurement rationale

The aim of this unit is to strengthen the area in my math teaching which I consider the weakest: measurement. Ever since I have been teaching math in the classroom, my units in measurement were not thoroughly organized or strongly assembled. Among the list of core concepts to teach in a given school year, the measurement curriculum ranked towards the bottom of math units. My biggest objective is for this unit on measurement to represent a well-organized, researched and thoughtful curriculum to serve as a model for future units involving math. My desire is for this unit to be the example I look up to when writing math curriculum.

The unit I am writing will be on understanding the array structure of rectangles and the array structures of rectangular boxes and, understanding that the surface of boxes can be cut apart, and unfolded to make an arrangement, or net, of rectangles in the plane. By reversing this procedure, these geometric nets can be folded up to make the surface of geometric rectangular prisms.

I will teach my students to recognize various characteristics of objects as measurable. I am aware that some content might be basic and foundational at first, for example counting square units and visualizing rows and columns of arrays, but this will allow me to address the many concerns surrounding student perplexity and remedying of previous student inaccuracies in dealing with area and volume.

These are some of the key understandings I want my students to absorb:

1. Measurement involves a selected attribute of an object (length, area, mass, volume, capacity) and a comparison of the object being measured against a unit of the same attribute.
2. The larger the unit of measure, the fewer units it takes to measure the object.
3. In understanding the rectangular array, determine the correct number of unit squares required to fill in the space being measured.
4. During the activity students will clarify and incorporate key operations such as compare, combine, and replicate.

If a rectangle is subdivided into square units, and the squares are decomposed and rearranged, the area will not be changed. Students will choose a unit of measurement (a square unit) which is appropriate and conducive to the rectangle being measured.

Students will begin to understand measurement by organizing arrays into units and sets of rows and columns. My students will begin to visualize the area as represented by the total number of rows and columns within a space inside an array, (see figure 6).

Figure 6: Visualizing rows and columns in the same rectangular array

Students will understand that if the complete entirety of the surface area of the shape is covered with the chosen units, then the total area can be determined by counting the number of units used.

I will segue from the sum of the units being used to the mathematical relationship of multiplication to determine the same area. Once these understandings are clear, I will be able to introduce the elements of algebraic representation of values and variables found in formulas.

I will be focusing on the Common Core State Standards with an emphasis on measurement involving area and volume. This curriculum will be aimed at solving real-world and mathematical problems. The primary understandings are the array structure of rectangles, and the analogous structure of boxes (rectangular prisms). The first standard I will be utilizing will be CCSS.MATH.CONTENT.6.G.A.1 with an emphasis on how to “find the area of ….polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.” (CCSS)

The second standard I will be utilizing will be CCSS.MATH.CONTENT.6.G.A.2 with an emphasis on how to “find the volume of a right rectangular prism ….packing it with unit cubes of the appropriate unit edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.” (CCSS)

“Values are often meaningless without units and there are often multiple options for the unit in which something is measured (e.g., meters, yards), creating potential ambiguities if quantities are reported without units. For example, knowing what units the answer must be in can help one determine which quantities to combine to obtain the answer,” (Dorko and Speer).

I will have my students construct a two-dimensional rectangle and a three dimensional rectangular prism while applying the idea and the visual of measurement. Students will choose units when constructing and de-constructing both area and volume of rectangles and prisms. I hope that we can give them the perspective that rectangles are simple and nice shapes in the huge zoo of planar figures. This is the reason that there being such nice formula for the area of a rectangle.

I am planning to have my students make boxes by making nets for them and then folding them up into boxes. This is halfway between activities that I hope will lead my students to understanding.

I will have my students understand that a rectangle with whole number side lengths can be sub-divided into a regular array of unit squares. I expect that this is not an idea which comes naturally to every student. I will follow this by having them understand that the area means the number of unit squares that can fit inside a rectangular array.

I will explain to my students that a rectangle can be regarded as an array of unit squares and the number of squares in a row of this array is the length of the base (in linear units), while the number of rows (or equivalently, the number of squares in a column) is the length of the side (in linear units). Therefore, the total number of squares are the length (= number in a row) times the width (= number of rows). To wit, the area of a rectangle is the length times the width times the area of the square, which we define to be 1. One condition for these activities in this unit is that we will be working with only whole numbers.

By the end of this unit, I will have my students feeling confident and clear in how to use and choose appropriate units of measurement when applied to length, area, and volume to determine quantities that are important in our everyday life. Examples of these can be how students will determine the area of floor and classroom space; the volume of various size containers.

The first goal is to understand that a rectangle can be regarded as an array of squares. Not every rectangle can be sub-divided into squares: there must be some relationship between the length and the width. They should be whole numbers. Students will understand the differences in objects by comparing them and asking direct questions of longer or shorter; greater capacity/less capacity/ weighs more/weighs less.

The next goal is to understand that a box (a rectangular prism) can be partitioned into a three dimensional array of unit cubes. (Again, we need the side lengths to be whole numbers). Students can practice by being presented with a variety of carefully selected boxes to fill with unit squares. The key for some is the activity being hands on and concrete.

Students will familiarize themselves with and master measuring tools such as rulers; measurement systems; and formulas to understand the area, surface area and volume.

The underlying factors faced by students today may lie with an absence of attention to detail and meaningful attempts of real life application to what measurement represents. Of the challenges teachers face in today’s classroom, measurement can often times be found atop the list. But as measurement is now being taught across curriculums, teachers look to solutions as varied as exposing students to as many different practical experiences; understanding how to choose units; and how to collaborate and work with peers in teaching measurement across grade level and content areas. (Thompson and Preston, 2004).

In my classroom, I use various types of teaching strategies throughout my math block which will include the following:

• Visualization: a main focus to this unit is to move away from the textbook mentality of teaching and into a visual teaching method of hands on practices, where students are encouraged to leave their seats and move about the classroom. Visualization provides my students the opportunity to make practical connections to the real world. My students will use foam squares to manipulate square units into arrays. They will organize several rectangular arrays, representing the sides of a box, into a net, which they will draw on paper; and will fold paper nets into solid forms of a geometric rectangular prism.
• Cooperative Learning: One of my favorite teaching strategies is cooperative learning. This provides learners of all levels to work collectively on building and decomposing rectangular arrays and rectangular prisms while working collaboratively with each other on the understandings of measurements with the goal of their small group succeeding together.
• Differentiation: The goal of differentiation is to meet the needs of learners while challenging students who have stronger math understandings with rigorous tasks. While at the same time supporting students with lower capacities of comprehension with an increase in individual help and support.

By utilizing technology in the classroom the students will see that there is a stark contrast between the classrooms of today, and the classrooms we as teachers attended as children. The main difference is accessible technology to all students. From laptops to smart boards, technology has moved the classroom big time. The use of technology is key in supporting differentiated instruction, allowing freedom and innovation to bloom and challenging our students with more rigorous tasks. The following are but a couple of examples of websites that students can use to strengthen comprehension in area and volume.

“It is essential that teachers and students have regular access to technologies that support and advance mathematical sense making, reasoning, problem solving, and communication. Effective teachers optimize the potential of technology to develop students’ understanding, stimulate their interest, and increase their proficiency in mathematics. When teachers use technology strategically, they can provide greater access to mathematics for all students,” (NCTM).

https://www.geogebra.org/m/pCv2EvwD

http://www.shodor.org/interactivate/activities/SurfaceAreaAndVolume/

The structure of the unit will include the following key points I will be teaching on measurement:

1. Understanding the concept of an array of any sort of figure arranged in a rectangle
2. Understanding that area of rectangles can be found by decomposing the rectangle into an array of unit squares, and counting the squares, or, more quickly, by multiplying the length and the width.
3. Visualizing groups of rectangles into nets
4. Geometric form of rectangular prisms into the shapes of buildings