Posted in Geometry

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

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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, Mathematics education

Develop your ‘Variable Sense’

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Number sense refers to a person’s general understanding of number and operations along with the ability to use this understanding in flexible ways to make mathematical judgments and to develop useful strategies for solving complex problems (Burton, 1993; Reys, 1991). Researchers note that number sense develops gradually, and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms (Howden, 1989).” – NCTM

If there’s number sense, there must also be also ‘variable sense’. Number sense is associated with arithmetic and basic to numeracy while variable sense (also called function sense) is associated with algebra. I collected the following set of tasks I believe will develop variable sense. Having a sense or feel of variables helps develop algebraic thinking and functional thinking.

Task 1

What must be true about the numbers in the blanks so that the following equation is always true?

____ + -2 = ____ + -4

Task 2

The following integers are arranged from lowest to highest:

n+1, 2n, n^2.

Do you agree? Explain why.

Task 3

What is the effect of increasing a on the value of x in each of the following equations?

1) x ? a = 0

2) ax = 1

3) ax = a

4) x/a = 1

Reason without solving.

Task 4

Drag the red point. Describe the relationship among the lengths of the line segments in each of the figure below. It would be nice if you can come up with an equation for each.

[iframe https://math4teaching.com/wp-content/uploads/2012/10/meaning_of_variable.html 600 360]

Tasks 1 and 2 are common problems. Task 3 is from a research paper I read more than five years ago. I could not anymore trace the paper and its author. Task 4 is  from Working Mathematically on Teaching Mathematics: Preparing Graduates to Teach Secondary Mathematics by Ann Watson and Liz Bills from the book Constructing Knowledge for Teaching Secondary Mathematics: Tasks to enhance prospective and practicing teacher learning (Mathematics Teacher Education). I just made it dynamic using GeoGebra.

Posted in Algebra

How to derive the quadratic formula

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As I wrote in my  earlier post about solving quadratic equations, introducing the quadratic formula in solving for the roots of a quadratic equation is not advisable because it does not promote conceptual understanding. All the students learn in using the formula is to substitute the values and evaluate the resulting numerical expression. I have seen test questions like “In 2x^2-4x+4=0, what is the value of a, b and c?”  Where is the mathematics in this item?

Not teaching the quadratic formula in solving for the roots of a quadratic equation does not mean that the quadratic formula will not be part of the algebra lesson. It would be a good exercise at the end of the unit to ask students to derive a formula for finding the roots of ax^2+bx+c=0 because you will be talking about vertex and discriminant (if you think they need to know what a discriminant is as this will just add to the terms they need to memorize) in later lessons. However I suggest that you ask the students ‘to solve for x’than ‘derive the formula’.

Problem

Solve the equation ax^2+bx+c=0.

Solution

 

 

 

 

 

 

Express the left hand side as product: x(x+ \frac {b}{a}) = \frac {-c}{a}.

Complete the square:

The rest as they say is pure algebra:

As you can see, deriving the quadratic formula is a beauty. Using it is not. Completing the square and factoring will do for students solving quadratic equations for the first time (ninth grade, for most countries). What is needed at this point is exposure to different problem solving context requiring representations of and solving quadratic equations.

Coming up in the next post is the meaning of this in graphs.

 

Posted in Algebra

Solving quadratic equations by completing the square

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I’m not a fan of  teaching the quadratic formula for solving the roots of quadratic equations because the sight of the outrageous formula itself is enough to make students wish they are invisible in their algebra class. Indeed who wants to have to do withOf course not all quadratic equations can be solved by factoring. Here’s how I try to resolve the situation. Before quadratics, students have been solving linear equations. So if you ask them to solve x^2+4x-3 = 0, chances are, they will use the same technique they learned earlier and this is to put all the x‘s on one side of the equation and the constants on the other side. They will not think of factoring the expression on the left even if they have done hundreds of factoring exercises earlier. For them factoring is another way of representing an algebraic expression and indeed it is. Solving equation means to find the value of x and based on their earlier experience, the technique is to put the x on one side. So this is what they will do:

x^2+4x+3=0

   => x^2 +4x=-3

=> x(x+4)=-3

Students will try to guess and check until they find the values of x that will make the equation true. They will continue to use this technique until you give them something like x^2+4x-3=0 which will make the procedure very tedious. This will be the time to prompt them to think of how easy it would be if the one of the side where the x’s are is a perfect square like in x^2=10 where x = + \sqrt{10} or in (x+2)^2 = 10 so that they will have x+2= + \sqrt{10}. So the problem now is to make the side x^2+4x a perfect square. A visual representation of the equation will be handy. Students should have no problem thinking of a rectangle as visual representation of a product.

Clearly the left hand side is not a square. The way to make one is to cut-off half of the 4x area. But it makes an incomplete square!

Let’s complete it by adding a 2 by 2 square. To keep the balance we add the same amount on the right hand side.

It should be now easy solving for x by extracting the root and using the properties of equality.

I believe that this process will make sense more than using the quadratic formula. Students just memorize the formula without understanding. They also will not remember a piece of it the next day anyway. I’m not saying the quadratic fomula is not completely useful. One application of it is on using the Cosine Rule for ambiguous case.

Should the method of factoring be taught first? I believe it’s best to introduce the students to the method of completing the square first (with the visuals, of course). Once the students get the hang of this procedure, the first thing that they will drop is drawing the rectangle and square and just do it mentally.You can later ask them to investigate the structure of quadratic equations where it is  no longer necessary to transfer the constant on the other side. Solving quadratic equation by factoring therefore is a shortcut students should deduce from the procedure of completing the square.

Any new procedure should be linked to previously learned procedure or it should be an improvement of the first. This is my reason why I think the process I described above is a natural sequence to the process of solving linear equation that students already learned. Another reason is that most of the problems students encounter involving quadratic equation is of the form x^2 +bx=c rather than x^2+bx+c=0. For example, “Two numbers differ by 4 and their product is 3. What are the two numbers?” The major reason of course is that it will always work for all quadratic equations. Check out the visuals for solving ax^2+bx+c=0.

I also developed a geogebra applets Completing the Square Solver and Quadratic Equation Solver that I posted in AgIMat. You can use them to solve quadratic equations and to investigate their roots.