STAIRCASES
Once again, this investigation is not about the answers, it’s about the seeing, thinking, talking, and discovery of patterns. It takes weeks to break students’ misconceptions that math is all about the answers at the back of the book. Children need daily excursions into the territory of “I wonder…”, “What if…”, and “I don’t know; what shall I do?” (best answer – an intrigued shrug!) . As they discover that the patterns are there, on the table (or floor, if that’s where they’re working!) – that they’re there, to be seen, their confidence grows.
We have to be careful not to push, not to squash the delicate tendrils of discovery. We mustn’t judge one student’s work as better than another’s because it is more abstract. Everyone sees what they see. Trust that insights are happening.
Investigation pages: CW staircases
Possible answers: key cw patterns
There will be a large spectrum of speeds and depth in this activity. Some students transfer to the abstract quickly, and see patterns quickly. Others will need a lot of time, and will continue to build with blocks long after we adults feel they ‘should’ have seen the pattern. We must allow this – they are building meaning, and we can’t force neurons to connect.
On the bottom of page one, only a few students can see that the pattern in column 1 is “add 2 to the figure number, then square it”. More students see the second pattern, and pretty much everyone sees the 3rd column (the figure # squared). This is all developmentally normal at this age.
The development of ability to describe pattern relationships is fundamental to success in algebra later, so we spend time building that skill.
The following videos are a tribute to the power of manipulatives:
1) This child realized the “Square” pattern after building 7 rows:
This child proved a 5-high staircase = 5squared by rotating some of the blocks:
These 2 proved a 10-high staircase = 10squared by rotating some of the blocks:
Challenge: Does anyone know what you’re talking about if you write up the question “I wonder what the sum of 2 consecutive triangular numbers is?”
Just shrug and let it simmer. Someone will see the triangles in the rotating blocks above. (If not, that’s fine too – maybe tomorrow…)