Category: Geometry

Posted in Geometry

Pentagon to Quadrilateral Puzzle

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Puzzles involving cutting shapes and forming them into different shapes helps reinforce the idea that area do not necessarily change with change in shape. It is also a good activity for developing visualisation skill and spatial ability.

The puzzle below is from one of the leaflets at the booth of Japan Society of Mathematical Education last ICME 12 in Seoul, Korea. The original puzzle is suited for Grade 4. The instruction was to cut the pentagon along the dotted lines and then form them into the shapes shown. The shapes shown in the leaflet is a parallelogram, a rectangle, an isosceles trapezoid, and a general trapezoid. I modified the puzzle for students in the higher level. I have indicated the measure of the two angles just in case you want your students to justify that the pieces really form into quadrilaterals. This is one way to assess your students knowledge of the properties of these parallelograms, trapezoid and trapezoids as they justify each shape formed.

pentagon puzzle

Here are two solutions – rectangle and isosceles trapezoid. Form the other two shapes.

trapezium and rectangle

Posted in Geometry

A problem solving lesson about triangles and circles

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This short lesson was inspired by one of the problems from the blog Five  Triangles Mathematics. The author challenges the reader to construct a circle using only a compass and straight edge, through two points X and Y. The centre of the circle must be a point on the line located between the two given points. If you can’t visualise it, click here to see the diagram and try the problem first and then come back if you are interested to see how you might teach this in your class without losing the essence of the problem solving activity.

Here’s my sequence of tasks. Notice that all three tasks involve geometric constructions in increasing complexity, one building on the previous task.

Problem 1

You can use this as context for reviewing the properties of isosceles triangle after the students have come up with at least two solutions.

Problem 2

Solution

This is one of the solution but I suggest you ask students to come up with other ways of constructing the isosceles triangle. The procedure shown involved constructing the perpendicular bisector of CD. F is any point on the perpendicular bisector.

Problem 3

locating the center of circle

Solution

(Of course I hid some part of the construction to make it a little bit of a challenge. Do you think the location of J is unique?)

In terms of time, this is not actually a short lesson because you need to give students more time to solve the problems. You may also want to read How to scaffold problem solving in geometry. The following book is a good resource for tasks that fosters geometric thinking.

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 Geometry

Regular Polygons Problems

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One  of my favourite lesson design is a sequence of problem solving tasks that requires repetition of same reasoning and analysis by varying the ‘mathematical context’ of the problem in increasing complexity. However the variation in the context of the problem should be such that they still share some properties. In the examples below, the number of sides of the polygons is varying but they are all regular polygons.It is also important that the problems can be solved/ explained in different ways – algebraically, geometrically, arithmetically or a combination of these.

Here is a sample sequence of problems. This lesson is good from Grade 5 up. If you are handling different grade levels and they all reason in the same way as your fifth graders reason, you have a big problem.

Problem 1

The segments in the figure below form equilateral triangles with the dotted line segment. Compare the total lengths of the red segments to the total lengths of the blue segments. You must be able to explain how you arrive at your conclusion or give a justification to it.

equal perimeter

Problem 2

What if the segments form squares instead of equilateral triangles with the dotted line segment? Compare the total lengths of the red segments to the blue segments. Which is longer?

perimeter problem

Problem 3

What if it the line segments form regular pentagons instead of squares? Do you think your conclusion will hold for any regular polygon? Prove.

Problem 4

What if instead of regular polygons, you have a semicircle? Click link to see the problem and solution.

Encourage students to use algebra and geometric constructions to justify their answers. This lesson is not about getting the correct conclusion. That’s the easy part. It is about explaining/ proving it.

You may want to view another similar lesson on quadrilaterals.