Picture Puzzles in Sweden
Year 7
Falkbergsskolan, Botkyrka, Sweden
(It would be useful to have Square Numbers open as you read this article  Ed.)
As part of a professional development day based around Discussion Lessons, the maths staff chose to explore Square Numbers which is one of the Free Tour Picture Puzzles. All the students bring a computer of some sort to class, so their teacher simply downloaded the file and emailed it to the students. That took about 3 minutes. It took longer to go to the storeroom to find the concrete material.
Picture Puzzles use one screen, two learners, concrete material and a challenge (in fact, an unfolding sequence of challenges) to invite students to participate in the process of learning to work like a mathematician.


In this lesson all the students were working at their own pace through the same puzzle. However, the resource is organised in content area menus, so in some classes it may be that different pairs are exploring different Picture Puzzles with related content. The photos show some of the activity from a 50 minute lesson, which the teachers (Lena, Pontus, Joakim & Mikael) later observed had kept almost all the students involved almost all of the time.
The lesson began in the context of learning to work like a mathematician. Lära sig arbeta som en matematiker was written on the board to generate a brief discussion around what students thought a mathematician's work would be. It was quickly agreed that a mathematician's work is to investigate problems that interest them.
Today I hope you will be interested in this problem.
The students opened the file previously sent (concrete material was on the table when they entered) and the teacher projected the same file. A brief introduction to how the slide show worked, then partners were invited to continue to investigate like mathematicians. There was 40 minutes left in the lesson and 30 of that was spent working in pairs.
Using one screen encourages mathematical conversation. Also, the slides seems to encourage recording; a mathematician needs to keep notes as they investigate. However at this point in the slide sequence recording hasn't been suggested. The students on the right chose to make a table to record the data so far. 

These two thought they had finished and were fiddling. But they had missed the request in the slides to check their predictions another way. They had easily seen the multiplication approach 
to finding the total number of blocks, but a discussion was needed to encourage them to investigate the relevance of the shaded section of each diagram. 
When the other students are fascinated, captivated and... 
absorbed by their investigation... 
time is created for the teacher to visit groups... 
and join their mathematical conversation. 
When students are exploring in an open environment  rather than doing sets of closed exercises  it often happens that their work generates 'aha' moments for teachers.
The confidently circled equation in the middle of the page reads
All block + Length + Width + 1 = The Answer
Correct and a perfect preparation for explaining why the Lshape shading must be an odd number of blocks and why
(a + 1)^{2} = a^{2} + 2a + 1
which, by asking What happens if the L is 2, 3, 4, ... tiles wide? is itself preparation for
(a + b)^{2} = a^{2} + 2ab + b^{2}
This student's work could be used to begin the next lesson.

This student is not making what is asked for in the puzzle, but it is the puzzle  and the availability of the materials  that has got him thinking this way. The photo itself is a perfect introduction to a future lesson in this algebra unit.
A swimming pool is surrounded by Size 1 tiles.
If I tell you the size of the swimming pool,
can you tell me the number of tiles needed to surround it?
Can you check your answer another way?
What happens if the surround is 2, 3, 4, ... tiles wide?
What might it do for the selfimage of these students to have the teacher begin a future lesson from their work? Perhaps the students could introduce their lesson?


Back to the original lesson, which after 30 minutes had different pairs at different points in the puzzle. Time to draw some threads together.
Everyone had reached the point where they could calculate a square number by multiplication or by the sum of odd numbers. The challenge now was how to efficiently add the sequence of odd numbers. 'Rainbow Facts'  adding the two ends, then the two second ends, then the two third ends...  was suggested and used to finish the lesson. There is (at least) one other way.
And the next lesson?
That could begin with these three slides... 
