- fractions
- problem solving
- addition
- trial and error
- manipulating fractions
- writing equations
- solving equations
- visualising
- changing whole
- reasoning and proving
where the pink writing represents non-content related mathematical skills.
In order for this to be an effective task, Gabriel circulated around the room, looking at different solutions, to allow him to lead the lesson by choosing certain students to give their methods of solution. Slides from this session are here.
Dealing with Misconceptions
There are two ways to do this:
- choose questions to avoid these misconceptions coming into play
- choose situations to bring these to the surface and then quash them forever more!
Shortly after lunch came the baffling multiplication and addition...
Anne gave us a bottle of lemonade to drink between two of us?
Have they shared the drink?
Have they divided the drink equally?Following an activity involving equally segregating various items between our tables, we came up with the following list of division models:
- simultaneous counting
- angle division
- division including congruency
- measurement
- division by folding and cutting
It was then noted that there are two main types of division, division resulting in stuff and division resulting in pieces. Additionally, division can often be made easier by looking for common factors, but this is rarely taught until Year 7!
Next came a lot of playing with blocks - bedlam.
"Don't slap them about for playing"a + b = c c = a + b b + a = c c = b + a
These relationships showed that the equals sign doesn't mean work something out, but rather that equality works both ways. In particular, they show 8 ways to represent the same relation, without any reference to numbers. This could also be used an introduction to algebra.
Multiplication is not repeated addition!This is the way that almost all students are taught at primary school, but is stretching a rubber band not also multiplication?
Remember: learning your times tables and long division algorithm by heart is not the answer!
And finally, we briefly discussed exponentials (which is most definitely not repeated multiplication!) but we struggled to find a definition other than scaling by a changing scalar.
