Published December 2011,February 2011.
As teachers, we appreciate the need to have clear objectives at the start of lessons but have been aware of the limitations this sometimes seems to place on our ability to get the most out of using rich tasks. Through our case study we tried to address two issues:
In our schools we have been working towards creating a balance of content and process-led objectives, moving away from an ethos of just having formal, explicit, content-based objectives at the start of each lesson. Raising the status of process objectives has allowed teachers to be more flexible in their view of what students might achieve in each lesson.
In our planning, we have:
Try a few more times with different sets of three numbers until you understand what is happening.
Link to interactive applet
Using the same starter numbers $2$, $1$, $4$ and $6$, rearrange them to find the largest and smallest possible totals at the top of the pyramid.
Can you get these totals in different ways?
Keep a record of the arrangements of the starting numbers which generate each total.
What do you notice about these arrangements?
Test out your observations and insights with other sets of four numbers.
You could include big numbers, small numbers, negatives and decimals.
Can you explain what is happening? Can you explain why it is happening? Can you explain it algebraically?
In lessons where new mathematics was being developed, or mathematical connections were being made, teachers felt that giving content objectives spoilt the discovery. Teachers focussed initially on process objectives, sharing and agreeing content objectives at the end of the sequence of lessons.
There are some sets of coloured balls with numbers on them:
A set is mixed up and two balls taken at random. For example:
There are five positive whole numbers with the following properties:
How many different sets of five numbers can you find that satisfy these conditions? Can you convince us you have found them all?
If I also tell you that the range is 10, can you identify my numbers?
Another five positive whole numbers have the following properties:
How many different sets of five numbers can you find that fit these criteria?
For similar articles about teachers using NRICH go here.