Concepts of Area; or: how to link parallelograms and rectangles

The year 7 teachers have this week been doing some work on the links between the area of a parallelogram and the area of a rectangle with the same dimensions.  This has caused issues for a number of learners, most notably when trying to calculate the height of the shape.  Spending time drawing and cutting up rectangles and parallelograms has proved fruitful but has taken longer than a lot of the teachers expected. We also found that the phrasing of the question also made a big difference to learners' attempts: "Does the rectangle fit inside the parallelogram?" led to a much different exercise than "Who can do it in the fewest cuts?".

Whilst observing a class who had been given a sheet of one 8cm x 4cm rectangle and three 8cm x 4cm parallelograms, I was struck by the necessity of there being some forcing of awareness to make sure that connections between concrete and abstract forms were made.  Learners set about measuring the sides of both the rectangles and the parallelograms getting 32 square cms for the rectangle and other answers for the parallelograms. Learners then set about cutting up the rectangle to fit it over one of the parallelograms; some very proficiently. However, at this point I noticed that conversation had changed to who had done it in the fewest cuts and the implications of the rectangle fitting perfectly were not being appreciated.

It occured to me that some stages need to be moved through in order for there to be a deep understanding:

-Establish clearly the area of a rectangle (what it means and how to find it)
-Make predictions of what you think the areas of the parallelograms will be
-Use concrete forms to rearrange one or the other area
-establish what the implications from the concrete forms are
-think about whether the original predictions need to be altered in the light of the implications.

Discussion and Follow up in the Mathematics Lesson

Some of the difficulties that teachers have reported at the beginning of using Big Ideas has been what to do after a class has had a discussion, what to do to ensure that a discussion has been a fruitful activity. Whilst believing that the act of taking part and engaging in a discussion in a mathematics lesson is positive and important in itself, I shared their concerns.


For those of you who have been similarly unsure I thought it would be helpful to share some of the ideas that I suggested with you on the blog:


1)Ask the learners to do some writing based upon their experience of the discussion: what have they learnt that they did not know before? What did they think before? What did other people think?

2)  Ask learners to summarise the debate for someone who wasn't there (and then pick and share some of them). Reading which points were important to different learners can highlight a lot for the teacher.

3) Be sure to leave enough time at the end of the lesson for there to be a shift in mode of working, i.e. you might be expecting the learners to go from sitting in a circle discussing as a whole group to sitting and writing reflectively on their own. This require both a physical and mental shift and requires some effort.


4) Be prepared to answer questions about what form the responses should take but give other learners the opportunity to make suggestions as well.

How do you frame discussion in your lessons? What do learners do after discussing? What do you think of these suggestions? Leave a message...

Big Ideas Released

 My first publication has just been published with the ATM and is now shipping. Apologies about the delay of the CD but if you've already received the book you should receive the CD this week.

I am really excited to hear from those of you who have picked up the book and have started using some of the ideas. Please leave a comment below to let myself and others here about any successful lessons or possible pitfalls that you have had.

Looking forward to working with you all!