Site Planning Phase I, Cut and Fill, is the first part of this lesson and explores mathematical concepts such as area and volume using a realistic building site development project. Through site excavation, students will discover and analyze the necessary volume of land to be moved for the cut and fill process on a given site.
Materials needed for this activity include: a calculator, and a ruler or straight edge. To introduce this activity the teacher should discuss the introduction with the class. The students can work on this activity in small groups and complete the remaining problems for homework. A follow up class discussion of the activity will be necessary to identify key mathematical ideas within the lesson and address any student difficulties.
Before beginning this lesson, students will already be familiar with several necessary mathematical concepts. Students should understand the concept of estimation as an approximation for an actual value. They should also be familiar with graphing in a coordinate plane and the concept of numbers along the x and y axes representing a set amount of distance from the origin.
Students should understand conceptually that area is a two dimensional expression for the quantity of space, or surface, contained by a set of two dimensional boundaries; height and width. The area of a space can be found by multiplying these two values together. The value of height should be understood as a distance in the coordinate plane that is overlaid on the section cuts (i.e. the y value). Similarly, the value of width should also be understood as a distance in the coordinate plane (i.e. the x value). Students should also know area is expressed in square units.
In general, they should also understand conceptually that volume is a three dimensional expression for the quantity of space contained by a set of three dimensional boundaries; height, width, and depth. The volume of a space can be found by multiplying these three values together. Students should already know that volume is expressed in cubic units.
They should be familiar with extracting information from applied problems. Given a mathematical situation in text format, students should be comfortable with translating from text to numeric and formulation value for problem solving.
There are several mathematical concepts in this workplace that students may have difficulty understanding. Students may have difficulty understanding that estimating the area under a curve requires an over estimation on one side of the curve and an under estimation on the other side of the curve. These over and under estimations should almost balance each other out. For the closest approximation to area under the curve, it is very important to maintain consistency in the method in which the area is partitioned. If the students' method is inconsistent, their estimation will not be accurate because the over estimation and the under estimation will not balance.
Another concept that students may struggle with is how to determine the third value (depth) when calculating the site volume. Since this activity is taking multiple cut sections (or slices) of land to calculate the area of cut and fill, the depth portion of the formulation represents the distance between cut sections. Students will most likely struggle with the idea that the depth measurement for the first and last cut section comes from measuring half the distance to the next closest cut section (this is a one sided measurement); while the depth measurement for all other cut sections is the sum of half the distance to the cut sections on either side of them (this is a two sided measurement).
The goal of this lesson is to utilize and develop students' prerequisite knowledge. From this lesson, students should realize that although an estimate is not an exact measure, by using a proper and consistent method, they can produce results that are very close to accurate. They should also realize that estimating using smaller intervals for area and volume can produce a more accurate estimation. Students will visually and conceptually understand how the mathematical ideas of area and volume play a role in the cut and fill process. Students will discover that the new variable introduced for volume relates to the distance between section cuts, i.e. the depth of the site.
In a practical sense, estimates are valuable when some margin of error is allowed in a calculation. When producing an estimating method that will remain within this margin of error, estimates can save time and money in many real world applications.
Through this activity, students will be able to apply their knowledge of area and volume to a real world application. They will gain practice in visualizing their mathematical ideas of area and volume through the visual images that are required in understanding the cut and fill process. Students will learn that methods can be developed to produce relatively accurate estimations and that these estimation methods can be valuable assets in the design industry. Most importantly, students will understand that mathematical concepts learned in the classroom can be applied practically in the world they live in.
This activity could be extended by asking students what would happen if the site was cut into sections parallel to the tiered levels of the site. How would students account for the drop offs between tiers? Would the estimates be as accurate? Why or why not?
Students could also explore the issue of estimating the cut and fill for a different site that tiered from the middle of the site down to all four sides. In other words, the center of the site is the tallest point. How would students have to adjust the cut sections to produce an accurate estimation of cut and fill? Students should realize that there is practical reasoning required in the method in which section cuts are taken.
An additional extension of this activity could
involve Geometer's Sketchpad as a tool for exploring area under
the curve, and investigating site volume. Students can use this
program to visually represent area and volume, given certain site
conditions as a classroom activity.
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