Children learn algebra by doing Maths and trying to generalise their observations. These tasks give them opportunities to do that.
They can be used in different ways, and it may be best to vary them. You can either give the sheet out as it is, and just go round working individually and prompting, or you can just use the sheet as a guide. In the latter case, just set up the problem and say “Investigate” then feed the problems in one by one as they need them.
All tasks involve the setting up of relations and the inversion of these to solve problems. Note that algebra is not explicitly mentioned, except in Pyramids. It is possible to do the whole thing in words. After all Cardano solved the cubic without algebra, but it is so much easier with it.
The problems gradually get harder as you work through the sheet. In each case the introduction of algebra greatly helps the tasks, but they can be done by trial and error for the persistent. Sometimes trial and error becomes a problem when the numbers get too big. Sometimes it becomes a problem when there is a conceptual leap.
At these times teacher led introduction of algebra would show the notation is powerful for its neatness, precision and ability to be manipulated to solve problems backwards. Obviously if the kids can do this without the teacher, then more learning is achieved.
The following order is how I would use them, but it does not have to be this way. Each task has harder stuff at the end that is much harder than the beginning of the next task.
If they use the sheet they can fill the answers in on it. However they should be prompted to write down the patterns in words, and this cannot and should not be done on the sheet. As Fermat said “This margin is not large enough to contain the marvellous demonstration”.
Note that the last one has a line of solutions. Some kids will be able to find the relation, some a set of whole numbers and others will find some negative ones as well. “Are there any more” and “how can we describe all the solutions” are good questions here.
Getting students to make some up is a recurrent and productive activity in these tasks. You can ask them to make some up with whole number solutions, or without – positive or negative.
Definitely the most productive way forward with this is for students to solve for one row first, then two rows etc. They get a series of linear functions that they can generalise with a second variable. Clever ones can multiply out the bracket and spot the symmetry.
Setting puzzles for each other is very important here.
It is up to you when you push kids into algebra on this one. It is similar to Number Cells, but the linear functions have bigger numbers, and they can add three functions together to get the total for the whole staircase.
This is the only one that has algebra in the questions. By now, kids should be able to formulate simple linear expressions and manipulate them.
Maths Thoughts 