This is part 2 (find part 1 here)

Following day:Students were challenged to explain how they knew that the larger whole number fit with their groups. None at this stage pointed out that it was a percentage! They didn't recognise it or were not thinking in that direction? Anyway I asked them to justify it and they said "well, 50 is half of 100" but they couldn't explain why they thought this was an explanation.

I then asked them to write out 100 in as many different ways as they could in each group. They could rearrange the numbers, the sizes, orientation etc (i.e. they could play with all the variables). They started playing with them and with encouragement became bolder in their designs until after just a couple of minutes they shouted "it's percents!!!" because they had rearranged 100 to look like a %  sign :) we also looked at what the word means again (i.e. per cent = 'out of 100'). Now they understand it's in %s.

Then I asked them to find a graphic representation of the value of the 5 cards in one group. The graphic/diagram/pic had to show how these 5 numbers, which don't actually look alike, represent the same number value.

The objective at this stage is simply to reinforce that the number value is the same.

Step 1 again.

I asked them to create 6 groups. They began to try this activity but I could see they were not using the ENV at all but rather 'just having a go' using the 'trial and error' method. So I introduced a time limit, which effectively made the task much harder, as they didn't have time to move the cards around until the answer 'popped out'. No group could solve it in the time allowed (1 minute).

Then I went directly to step 2and reminded them of the 'variables and values' from the previous lesson. Together as a whole group we identified 4 variables and their values. Now they could do the 6  groups instantly. Did the model help them ? Yes it did! SO why didn't they use it then? Not sure except they were tired and it was the last lesson of the day so they preferred trial and error to really thinking through the challenge. Note: last lesson of the day is not a good choice for such activities! when the reflected (step 3) they really did see that using ENV helped but they were not in the right frame of mind to find that stimulating.

Reflection step 3 again I asked them to divide the cards into 5 groups and pointed at the list of 4 variables already made.

Step 1 again.

I indicated that the next challenge would be to work out the number of ways one can divide the cards into 2 groups. This can be done really laboriously by trial and error but a) you'd never be sure you didn't miss one and b) it will take forever at this grade level. They will need the ENV to make the task a lot simpler. List all the variables and all their values. Then they can divide into 2 groups for each and every value listed. e.g. grouping = fraction/non-fraction, grouping = contains zeros/doesn't contain zeros, grouping = has 6 digits/doesn't have 6 digits, etc.

The variables are (the values are)

number values (in %s, 2, 5, 10, 12.5, 20, 25, 30, 33, 40, 50, 60, 70, 80, 90, 100)

number format (fractions, decimals, percents)

number of digits used (1, 2, 3, 4, 5, 6)

number of different digits (1, 2, 3, 4)

number of 'zeros' (0, 1, 2, 3, 4)

number of 'ones'  (0, 1, 2)

number of 'twos' (0, 1, 2)

number of 'threes' (0,  1, 2, 3, 4)

number of 'fours'  (0, 1)

number of 'fives' (0, 1, 2)

number of 'sixes' (0, 1)

number of 'sevens'  (0, 1)

number of 'eights' (0, 1)

number of 'nines'  (0, 1)

With the other challenges I think they saw that the ENV was useful, BUT given enough time they would have reached the conclusion themselves eventually (and without losing interest or confusing themselves). This task should prove to be more complex as there are more than 50 solutions!

part 3 here