In this activity the aim is for the students to be fully aware of the equality of various fractions, decimals and percents. They are aware to some extent but only for certain specific fractions (rather than as a concept of them really being the same amount). 2 x 45 mins lessons.

A sorting exercise was used. Materials are here.

Students were divided into groups of 2 and each group was given a set of number cards (see here).

First: They were asked to divide the cards into groups. No further instructions from me at this stage on number of groups or criteria.

Students initially found an assortment of groups e.g. 'fractions', 'decimals' and 'whole numbers' and had to defend them. Now I discussed with the students their use of certain vocabulary and what the words mean e.g. 'whole numbers' would include such 'fractions' as "100/100" as it is = 1 (a 'whole number') and so students then had to refine their definitions (because "100/100" belonged to 2 different groups). This step served to make them aware of how tricky certain definitions can be and as a warm-up.

Students were then allowed to choose further sets of groups.

Students were then challenged to place all the cards into only 2 groups. Not so easy since they had been looking at them as 3 groups already based on the 'number format' (fractions decimals and integers). They finally succeeded and didn't require help beyond encouragement and questions. This part helped them shift their viewing of the cards slightly.

Note: Some students did almost become stuck at this point and perhaps in another teaching group they would.

Step 1 (increase room for learning or make the task 'impossible')

Having tried a variety of groups student were asked to divide the cards into 15 equal groups.

At this point they became stuck as it wasn't immediately obvious to them how they could proceed (because they had a total of 75 cards). They tried certain combinations but always got stuck.

Note: One group did come up with a very novel answer at this point - they divided them into 15 'random' groups. Brilliant idea! However I then asked them to come up with another grouping of 15 so that they got stuck too. Another group tried putting them in groups as follows; 3 fractions plus 1 decimal plus 1 integer card; another nice idea I thought. Unfortunately it didn't work as there are 16 decimals not 15...(but it looks like it will work at first glance)

Now all student groups were unable to proceed with the task.

Step 2: (teach or remind students of thinking model ENV)

I went to each group in turn (it happened at different times in each group) and reminded them of the 'variables and values' model. We tried together to look at the variables they had tried so far. The students did need at this point to refocused from looking at 'values' (decimal/fraction, 2/3/4/5, 0-9,  to looking at the 'variables' (number format/number of digits/digits used). First we examined variables in light of the groups they had already done well e.g. one group did 2 groups based on 'fractions' and 'non-fractions' clearly the variable 'number format' and we then looked at home many other '2-groups' possibilities there were using just this one variable (other possible groupings = decimals/non-decimals and integers/non-integers). (yes this is partly a reflection (step 3) but there's a more obvious reflection later).

Then we started to make a list of the variables and their values using just 3 or 4 number cards for reference. This allowed students to focus on the variables-'what's changing from one card to the next' and 'the values' without it being too overwhelming. I left them to continue the listing task after being sure they knew what they were doing and moved onto another group. They could also attempt to complete the task of 15 equal groups if they could.

The variable they need to look at is the 'number value' as there are 5 cards of equal value (and 15 sets of those). e.g. 1/2, 4/8, 50%, 0.5, 50/100

BUT complicating matters is the fact that the 'percents' cards haven't got a % sign on the cards, just the numbers.

After some help in looking for variables students started grouping cards in sets of 5 equal amounts. At this point they included the 50 in the groups although they "didn't know why" or "it just fits, I know it" which when questioned  turned into "well 50 is half of 100, and this group are all the same - 'a half'" . This is good because they are recognising the relationship without needing the sign %.

Note: This part of the exercise also served as a diagnostic assessment because some student were able to do the groups of 5 more quickly as they were more easily familiar with the fractions used. Some students found this very slow going as they were less familiar with many of the fractions or percentages used.

Step 3: We reflected in each group on how the 'variables' of the ENV model had helped us solve the grouping by opening up more possibilities for dividing up the cards. Students recognised they had been looking at a limited number of variables. Students appeared to appreciate that focussing the search using variables was helpful in solving the task. This will be reinforced using second task below.

Next lesson: Further Step 3 activities and new step 1:

We will look at building an algorithm to follow for dividing cards into groups. Then I will ask the students to use the algorithm to divide the cards into 6 groups (this can only be done based on the variable 'number of digits').

Then I will ask them how many possibilities there are for dividing the cards into 2 groups. This should be not easy to answer (step 1 - increasing room again - I haven't worked it out myself yet!) and will involve returning to both the ENV model and the algorithm. It should require that students have to list all the variables and values and use them to calculate the number of possibilities. They will then be able to reflect (step 3) on how the ENV and the algorithm helped them solve this second difficult task.

Part 2 is here.

#### Comments

# Alexander Sokol 2011-03-02 12:16
Deirdre, thanks for an interesting post. I am sure it is very useful for those wishing to see how the teacher can control the process of moving from step to step. What might be useful though is having a few words about the aim of getting the students to do this or that sorting. This would help the colleagues see the process of running a TA lesson in a more systemic way. What do you think?
# Deirdre Jennings 2011-03-20 22:01
The main reason for doing different sortings was as follows: the number of possible groups and size of the groups had been worked out in advance by me, the first sorting tasks were to recheck the students use of the terms 'variable' (or parameter) and 'value' . The second reason is for practice in identifying variables and listing values. The third type of sorting e.g. organise the cards into 6 groups, was used ot illustrate that using the ENV tool they could reach the answer faster than by trial and error method i.e. to establish the superiority of this tool.
The next challenge 'how many ways are there for dividing the cards in 3 groups' will intensify the difference between trial and error and using ENV but only if the student recognises that the number of values for any one variable = the number of groups that will result. So they list variables and values and then simply count the number of variables with 3 values :)
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