Measurement
In my unit, I want my students to understand how architects and engineers use measurement in finding the area, surface area and volume of structures they will compose and construct. When determining any given space, we use measurement to describe its size. After learning this unit, students will know architects and engineers and the use of volume of a box. They will know the amount of space, and that the product of the side lengths in the 3 directions (length, width, height) tells what any box or rectangular prism can hold. Students will understand to use different ideas like form, space, and quantity when conceptualizing a new structure.
Students will work with two dimensional arrays and three dimensional rectangular prisms or boxes. Area is introduced with manipulative hands on activities such as working with foam square units in composing and decomposing rectangular arrays. Surface area of rectangles is understood by forming arrays of columns and rows of unit squares. The focus on volume is centered on how unit cubes occupy solid prisms as layers and will conclude by investigating how formulas are used.
As most elementary math teachers can attest, teaching measurement and hoping for student comprehension, particular with area and volume, can be challenging and frustrating. The mathematical strand of measurement has tended to become the “black sheep” of the math family. Often neglected and overlooked in its teaching, its profound impact on the overall understanding of elementary math can leave students with gaping holes in grasping the entirety of Common Core Mathematics curriculum, with some students unable to overcome the deficits. From "my" experience, measurement can be overlooked as a unit but as we continue, we can determine how measurement impacts all mathematics curriculum.
A simple rectangular array is basic in appearance, a polygon we have drawn as students since early childhood (see figure 1). Well, rectangles maybe, but the array structure is something that many students need to be shown explicitly. A student may determine the area of the rectangle in multiple ways: by counting each individual square unit to get a total of six square units; by decomposing the array into three groups of two and adding each of the three groups (2 + 2 + 2 = 6); others will count the length of square units (3) and count the width of square units (2) then solve by using the formula to find the area (A=L ∙ W).
Figure 1. Rectangular Array
In my experience as a math teacher of over ten years measurement curriculum is taught to students quickly without the opportunity to explore. The scenario unfolds as such: state testing has been completed in late spring towards the end of the school year, and you as a teacher have completed teaching the units found in the front of the curriculum such as algebraic thinking, base ten number systems, fractions, and geometry, etc. You’re feeling pretty good about the school year coming to an end, when you realize you still have 3-4 weeks left of school to teach. As a teacher, I am committed to a strong finish to the school year but I anticipate half-completed assignments and my student’s attention wavering away towards their summer vacation.
A talk I often hold with the grade band teachers I work with will focus on how my middle school math students come ill-prepared in basic foundational skills at the beginning of a school year. Often these conversations lead to full out rants on how some of my students still have not mastered the multiplication table, how the base ten number system is too complex, and their struggle in understanding the difference between a numerator and a denominator.
A student's misconceptions of new mathematical knowledge can often be overwhelming for them. This lack of a general, foundational understanding of math skills becomes increasingly challenging to a student as he/she moves from primary, intermediate, middle, and high school math. As students move through one grade to the next, the curriculum becomes more complex and rigorous. Through informal and formal assessments, I can identify a student’s lack of understanding of measurement and how this contributes to mathematical misconceptions.
Education research attests to the difficulties of the measurement concepts that students face in the classroom. “Units are difficult for elementary school students and these difficulties seem to stem from trouble understanding area and volume. Specifically, elementary school students have trouble with arrays. Researchers have found that students have trouble visualizing and using the unit structure of an array” (Battista & Clements, 1998).
Students need to be provided extra support to re-learn basic measurement concepts. “Rather, they add the lengths together to get the area, misappropriating length units for other spatial measures seems indicative of trouble with dimensionality, e.g., length is one-dimensional but area is two-dimensional.” (Dorko and Speer).
The intention of this unit is to have my students understand the structure of rectangles as rectangular arrays: an arrangement of unit squares into rows and columns. My students will know the characteristics of a rectangle and how it can be formed by defining rows and columns of squares. I want to clarify to my students that area is the two-dimensional space contained within a region.
When the figures are rectangles whose sides are whole numbers, and in related cases, this area can be measured by the unit squares that cover the space. My focus will be on students with misunderstandings about rectangular arrays and prisms. Rectangles do not need to have whole number side lengths and unit squares only. I will focus on both whole number side lengths and unit squares to introduce learning goals of measurement of rectangular arrays. Students will learn that area applies to all types of general polygons, and other figures, such as circles as well.
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