Before the wise builder builds – the wise builder thinks. Here we are going to build the foundations of a great skyscraper – Taipei 101.

Start with a 6×6 grid.

Add numbers or letters or colors to the top left part of the grid as shown.

Now is the creative part. Choose a rectangle with one of its corners in the top left. Label all the corners the same as in the top left. Here we have chosen a rectangle and colored its corners red.

Make nine rectangles in total.

There are nine rectangles – but just in case you don’t see them, I’ll show them quickly…

There are nine rectangles – but just in case you don’t see them, I’ll show them quickly…

There are nine rectangles – but just in case you don’t see them, I’ll show them quickly…

As I’m going through – calculate the area of each rectangle.

As I’m going through – calculate the area of each rectangle.

As I’m going through – calculate the area of each rectangle.

As I’m going through – calculate the area of each rectangle.

As I’m going through – calculate the area of each rectangle.

As I’m going through – calculate the area of each rectangle.

There I’m finished. Your students should be creative – choosing their own rectangles, Calculating their areas… and then the final task is to add up their results and compare…

This is an excellent time to hand out a prize for the student who gets the highest result. They will be surprised to find that they all get exactly the same score of 81. Older students should try to prove this.

Taipei 101 is a puzzle by Zeng Cin Jhih. Let’s look at other examples of skyscraper foundations…

In Busan, South Korea, we see the triangular foundations for the Lotte World Tower.

Repeat Zeng Cin Jhih’s experiment here… this time making equilateral triangles along the grid lines. Is there still a consistent result, or does the triangular foundation behave differently from the square foundation?

Students struggling to to find a proof of this phenomena, should always try to simplify what’s going on. One way to simplify is to use smaller squares / rectangles. Another is to explore the phenomena on a line. Here I’ve labeled the first ten points of the line A-J and shuffled these up for the last 10 points on the line.

Then I’ve counted the distance between the identically colored points and am about to add up the result. Is it always the same?

Taipei 101 Foundations

(Zeng Cin Jhih, 2016)

This activity can be used to give students practice with multiplication up to 5x5. It can also be used for older students doing proofs. The last activity on the slides is excellent for students learning to add up to 100.

Go here for a pdf file with blank sheets for students.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

 
MP1 Toughen up!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

 
MP6 Be precise!

This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.

(http://www.corestandards.org/Math/Practice/)

Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)

 

Lora Saarnio

(CEO)