Having realised last week that a deeper understanding of equivalent fractions would be desirable, I planned this task (inspired by this lovely post):
Give out a nearly-blank ruler with marks at only 1 and 2 (12cm and 24cm, but the children didn't know that). Ask children to mark on as many fractions as possible using one set of red to orange (no white) Cuisenaire rods.
I resolved to follow where the children went with it, resisting my urge to over-plan. That said, I had envisaged possible next steps, including measuring, or perhaps a back and forth 'chanting' session inspired by this Gattegno video (e.g. 7:24), and also this paper by @AlfColes (open access). [Further inspiration from the Gattegno video led to a request for clear desks, apart from the rulers, rods, and a sheet of paper on which the children could write questions as they had them].
I presented the task with the minimum of information. There was initial confusion, but it didn't last very long; the children soon worked out what to do by talking to each other.
As I went round the room, the following things happened:
- Some children were working through the rods systematically, starting with reds, which was nice to see given earlier lessons.
- Most children realised that you could make marks along the ruler using the same rod (as in one sixth, two sixths, ...), although a few had only written unit fractions at first. Having only one of each rod led to this counting along, which led to writing more than one fraction name for some of the marks (see images below), a desired outcome of the task. One child said "You can't have 9/6!", in response to which I modelled counting along in reds, and we said in unison: "One sixth, two sixths, ... nine sixths!"
- Most children left gaps for yellow (5/12) and black (7/12) marks, as well as other blanks at 1/12, 11/12 and 13/12. They wrote questions such as: "What is the length of yellow/black?" and "Are the black sticks useless in the ruler?" One child asked: "Can I have 1/2.4 for yellow?", to which I said: "Yes, but can you write this fraction using only whole numbers?" A few children wrote 1/5 and 1/7 for yellow and black marks respectively.
- I think the withholding of whites was useful. It directed attention towards the other rods and the relationships between them. It became clear that some children would benefit from having some whites (one child wrote: "Please can I have some white ones?"), so I offered them some. This helped them 'complete' the ruler. We then talked about this as a class, and identified the yellow rod as being the same length as five twelfths, and so on.
Here are some of the rulers:
As they finished creating these, I invited the children to start making their own ruler with their own choice of unit. Here's a particularly interesting one, the unit being called a 'blob', but the creators were a bit disappointed with it!
There's plenty to explore here. In hindsight we might have dwelled here much longer, but the curious (lusty?) part of me wanted to try the chanting idea. I held up various rods, a la Gattegno, and the children called out the fraction. Holding up a pink, they called out "One third" (with a few alternatives), and I said: "OK, one third... and another name... and another?" We did a few of these. When I held up yellow, all of the children called out "five twelfths" in unison; when I asked for another, one child offered 2.5/6, and another child (who earlier in the lesson had said he was "no good at fractions") offered "ten twenty-fourths".
Then I held up two rods in a train. A large majority of the children seemed to be able to add any pair of rods together, always giving their answers in twelfths. I asked them to explain an easy way for finding the length of two rods, and one child replied: "You imagine the number of whites in each rod, and then add them up." This was the intuitive approach to adding fractions that I was hoping for a couple of lessons ago. I suspect this was made possible by the design of the task, which focussed attention on comparing the key attribute of the rods, their length, coupled with the act of counting along.
Buoyed by success, I again moved forward, but too quickly. Teacher lust is difficult to resist. I presented the example 1/2 + 1/3, we talked about visualising it as "dark green and pink", and then "imagine counting the whites", leading to the answer 6/12 + 4/12 = 10/12. I then wrote a few questions on the board for them to solve:
There was a sense in which I presented these questions out of habit. This pedagogical 'choice' came from wanting to find out if they could use what they had been learning to add fractions, which was my original aim. But why do this, now? I was aware in this moment that there were other possible choices, such as allowing (more) time for reflection. I also considered allowing some time to compare rulers, but I couldn't work out how to do this, practically.
We then went through the answers, and the reasons for those answers. There was lots of 'success', and some beautiful explanations, but I feel a few of the children might have benefitted from more time spent on the previous tasks:
Finally, I asked the children to reflect on what they had learned. There was a range of comments:
"It didn't make sense to me"
"It's quite hard!"
"Sometimes it was confusing but I think we finished the ruler in the end"
"I was confused but I got it eventually"
"It was challenging at times but mostly fun"
"We learned a lot about fractions and converting and simplifying fractions"
"I learned something new, how to add fractions easily"
"I enjoyed working out all of the questions on the board, we also enjoyed doing the thingy on the ruler"
"It was good working out hard sums using the sticks and my brain"
"I found it very fun transforming the sticks into fractions"
Next lesson I'd like first of all to come back to the rulers. What are they all about? Are they all correct? I'd like to give children more time to create their own ones, and explore which ones lead to marks with many fraction names and which ones don't (I'm personally interested in a 1 unit = 30cm ruler). I feel as though my desire to move onwards too quickly, whether that was to satisfy my own curiosity, or to reach an aim I had set, resulted in missed opportunities.
After this, I'd also like to explore this image (via @ProfSmudge):