Tag: volume

Posted in Geometry

If One World Trade Center is a prism and not an antiprism, would it be less in volume?

Posted on by

One World Trade Center, more simply known as 1 WTC and previously known as the Freedom Tower, is the lead building of the new World Trade Center complex in Lower Manhattan, New York City. The supertall skyscraper is 104 storey  and is being constructed in the northwest corner of the 16-acre World Trade Center site. The image at the right shows the design as of May 2012.

One World Trade Center is an example of an antiprism. The square edges of the world trade centre tower’s cubic base are chamfered back, transforming the building’s shape into an elongated square antiprism with eight tall isosceles triangles—four in upright position and another 4 in upside down direction. Near its middle, the tower forms a perfect octagon, and then culminates in a glass parapet whose shape is a square oriented 45 degrees from the base. My question is Is this bigger than if it were a square prism? How about in terms of surface area?

Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals. Here’s a model of a square antiprism.

square antiprism

The surface area of the antiprism may be bigger than the prism because of the additional faces but I’m not quite sure. How about the volumes? Cavalieri’s principle does not apply because the cross sections do not have the same area. Anyone wants to resolve and explain this? There is a formula for volumes and surface areas of antiprisms but I’d appreciate a more intuitive solution.

Note: All information and images about 1 WTC is from Wikipedia. Square antiprism model from eusebeia.dyndns.org.

Posted in Algebra, High school mathematics, Math investigations

Teaching combining algebraic expressions with conceptual understanding

Posted on by

In Math investigation about polygons and algebraic expressions I presented possible problems that students can explore. In this post, I will share some ideas on how the simple investigation of drawing polygons with the same area can be used as an introductory lesson to teach operations with algebraic expressions with meaning and understanding. Like the rest of the lessons in this blog, this lesson is not so just about learning the math but also making sense of the math and engaging students in problem solving.

The lesson consists of four problem solving tasks to scaffold  learning of adding, subtracting, multiplying and dividing algebraic expression with conceptual understanding.

Problem 1 – What are the different ways can you find the area of each polygons? Write an algebraic expression that would represent each of your method.

The diagram below are just some of the ways students can find the area of the polygons.

1. by counting each square
2. by dissecting the polygons into parts of a rectangle
3. by completing the polygon into a square or rectangle and take away parts included in the counting
4. by use of formula

The solutions can be represented by the algebraic expressions written below each polygon. Draw the students’ attention to the fact that each of these polygons have the same area of 5x^2 and that all the seven expressions are equal to5x^2 also.

Multiple representations of the same algebraic expressions

Problem 2 – (Ask students to draw polygons with a given area using algebraic expressions with two terms like in the above figure. For example a polygon with area 6x^2-x^2.

Problem 3 – (Ask students to do operations. For example 4.5x^2-x^2.)

Note: Whatever happens, do not give the rule.

Problem 4 – Extension: Draw polygons with area 6xy on an x by y unit grid.

These problem solving tasks not only links geometry and algebra but also concepts and procedures. The lesson also engages students in problem solving and in visualizing solutions and shapes. Visualization is basic to abstraction.

There’s nothing that should prevent you from extending the problem to 3-D. You may want to ask students to show the algebraic expression for calculating the surface area of  solids made of five cubes each with volume x^3. I used Google SketchUp to draw the 3-D models.

some possible shapes made of 5 cubes

Point for reflection

In what way does the lesson show that mathematics is a language?