New Fibonacci-problem, product sum

James Tanton published a new problem about Fibonacci sequence  on Thursday. I started to play with it with GeoGebra. After I published my article in  Finnish some people asked about having it in English, so they could use it easier with their students. So here it is. Here I am working only with the traditional Fibonacci sequence from Liber Abaci.

In Tanton’s example  a = 4, F(3) = 3, b = 9, F(9) = 34, F(5) = 5 ja F(10) = 55 ja 377 = F(14).

flista = {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …} 

When I was playing with this with GeoGebra I noticed a simple connection how far a and b gets T(a,b) in the Fibonacci sequence. Kind of happy about that. Still no idea how to prove it.

At this point I will leave you some problems.

  1. Tanton’s Theorem. Let F(n) be the n’th number in the traditional Fibonacci sequence and T(a, b) = F(a)*F(b) + F(a+1)*F(b+1), a>0, b>0 and of course they are natural numbers. Then T(a, b) is a Fibonacci number. So there exist a number n so that T(a, b) = F(n).
  2. Find a relation between n, a and b, if the theorem is OK. In Tanton’s example a = 4 ja b = 9, T(4 , 9) = 377 = F(14).
  3. Is thera a simple equation for T(a, b), without recursion.
  4. Are there any other initial values F(1), F(2) for the sequence to have this property.

For me it is a little bit hard to find T(a,b) (problem 3). Tinkering (haha learnt a new word from Thomas’s book) with this idea I found that if a + b is constant, then T(a, b) is constant. So T(1,9) = T(2, 8) = T(3,7) = T (4,6) =T(5,5) = 89

I will guess that proving those theorems/problems are kind of hard.

I will be back. 

Dr. Bea – the public examination

Me and my wife tried to travel to Iceland last summer. Because of the covid we had to postpone it to my Autumn holiday. It just happened that friend Bea’s doctor public examination was on that week at the University of Iceland, so of course we had to go to see it.

Her dissertation title was “Silent video tasks – their definition, development, and implementation in upper secondary school mathematics classrooms”. I have not yet read the thesis/articles, so I will not comment on that.

The dissertation can be found at

After the formalities Bea had her lecture, she explained what silent videos are, how they are used in classroom and about the research and the methods.

The opponents, prof. Dr Merrilyn Goos (Australia) and Prof. Dr. Morten Misfelt (Denmark) had their statements remotely and there was very interesting discussion about the subject. 

And finally, the dissertation was accepted.

There are many ideas that started to wake in my brain while listening the discussions. Like what are we teaching when we teach math, physics, or programming? Facts, methods, processes, problem solving, algorithms, how to use programs, how to use pen, pencil, protractor, and ruler, …? Is Bea’s silent videos the same thing like when I use simulations or videos in physics to learn about new concepts? Why some teachers want to change their ways of teaching and why they return to old ways so often? Is it a good or a bad thing? 

I hope I will write about those ideas later. 

more photos from the examination

some tourist photos

Layers in GeoGebra

When you create objects that overlap in Graphics window, then the newest will be on top. In this article I will show how to change the order of the objects. Revathy, my Fulbright sister asked me to translate this article from Finnish. So I had to do it.  

Create two triangles and change the colors by selecting them and using the Style bar at the top of the Graphics window. The red triangle t1 is created first and the green t2 triangle second.

Here we see that the sides of the red triangle are on top of the green triangle, so there is something wrong in the logic. Is it a bug or feature, I don’t know?

I will use the Layers setting to change the order. Click the red triangle with the right button of the mouse and select Object properties…. In Object properties window click the right arrow to the right and select Advanced. All objects are on layer 0 by default. There are 10 possible layers numbered 0, 1, …, 9.

Change the Layer setting to 1 for the triangle t1. Now the red triangle is on top of the green triangle. Actually the sides are still in 0 Layer, so change the layer value for the red sides also.

When working with layers there is a handy shortcut to select all the objects on a layer Ctrl-L. Create a pentagon and set it to layer 1. 

Select the pentagram by clicking it and then use shortcut Ctrl – L (Cmd – L on a Mac). Now the triangle and the pentagram are selected, so you can move and change the colors if you like.

The Layer help page on GeoGebra wiki is at.