We've continued to work with Cuisenaire rods in upper primary (ages 8-11), introducing the decimal names for numbers alongside multiple fraction names. Last week we did another iteration of the 0-1 ruler task, with the orange rod (10cm) set equal to 1, and I asked the children to include decimal names for each mark on the ruler.
This week we reset the orange rod equal to 1/2, and created another 0-1 ruler (i.e. with one whole = 20 cm).
The children are becoming proficient at finding multiple fraction names for any given mark, but finding decimal names for every mark proved much more difficult today (presumably due to the requirement for hundredths). Here are some early attempts:
This response may have been a hang-over from last week, where every mark was a tenth. The children who did this soon realised that, for example, the 0.5 mark could not also be 1. This prompted a revision of their work. Here is a similar example in which the learner is trying to make sense of what happens after 0.9 (contrast this with her apparent security with fraction names):
Here's another initial attempt at sense-making. I was fascinated by the attempts the children were making to make sense of the shift from last week (i.e. what must be the case if each mark is not 0.1):
The children were realising that it was more difficult to fill in the decimal names this week. I invited them to focus on the fraction names at first, and suggested that the decimal names might then become apparent (by looking at the fraction names). This led to a number of children noticing where the tenths should be:
If a child did not notice the tenths, it provided an opportunity to talk about the equivalence of 1/10 and 0.1.
Filling in the marks between the tenths proved a bit more tricky, suggesting perhaps that most children in the class do not have a secure awareness of hundredths? Here is a wonderful guess as to what might go between the tenths:
This child realised that her guess was not quite right when she got up to 0.09, and tried to fill in the next mark (try it!). I shared this with the class as an example of what mathematicians do when they are stuck. They guess, see if it makes sense, and if not then they modify their conjecture!
After a while, most of the children had found multiple fraction names for each mark, and the decimal names for the tenths. Around five children had found the correct decimal names for the twentieths. This was a hard won realisation in most cases, as you might be able to see from the rubbing out behind Sophie's work:
I made a mental note that we could do 0-1 ruler one more time, but with a metre ruler, perhaps as a class, to draw attention to the relationship between hundredths, tenths and fractions.
I've overheard some mention recently about what is required to be 'secondary ready' - a phrase that concerns me - but if there were certain things I would want children coming out of primary to have a reasonably secure knowledge in, one of them might be fluency in switching between decimal and fraction number names up to hundredths, and be able to identify their position on a number line.
After talking to Sophie about how she worked out the twentieths (she described 1 being "like a hundred"), she made the connection to percentages, and asked if she could include them:
Once everyone had written a problem on the board, they started trying to solve them, and each person was allowed to come up and write the answer to one problem:
We agreed to erase some of the solved problems, until only a few remained which everyone was working on. The end of the lesson came, and a few questions remained unsolved. It was break-time, but a number of children continued to work at the board. These photos were taken during break: