Open Middle Fraction Problems

I have often shared the Open Middle website as a source of good thinking problems with the teachers that I work with. This week I created a couple of my own and used them with a grade 6 class who have been working on equivalence in fractions, decimals and percentages. The first problem (given using verbal instructions) I gave was this: 

Use any of the digits 0 to 6 once only to satisfy the statement below.

We used visibly random groups to get the students into threes and gave each group seven tiles to work with. As we began to walk around the room, a few students asked 'What do I have to do?' Taking the lead from Peter Liljedahl's work, I smiled and told them to find out from a friend.

We could see some students getting stuck or making mistakes but resisted the urge to jump in and show them how to get a solution. 

We let them think their way out of it.

Sure enough, we soon heard some 'A-ha's around the room as students realised that the first space had to be a zero, and as they figured out the decimal equivalent of a fraction:

Once they had found one solution, I asked them to find me another. Some used two-fourths as the fraction, others used one-fifth or two-fifths:

One group bent the rules a little bit:

Another group tried something similar but put 2.5 instead of 0.25. I simply pointed to this and asked 'Is this more than one or less than one?' and walked away. When I came back, they had corrected it.

Another group tried this:

When I asked about this, they knew that two-sixths was one-third but they also thought that its decimal equivalent was exactly 0.3. As they knew how to use a calculator to change a fraction into a decimal, I asked them to do this for one-third. This made it easier for me to convince them that 0.3 is different to 0.333333333... 

As we had fifteen minutes left, I gave a variation of a question that I tweeted last week that proved popular:

Any thoughts we might have had about this question being tricky for them were soon dispelled:

As we circulated, we could hear the students justify their solutions and, if we were unsure, simply asked a question like "Can you convince me that two-sixths is less than five-eighths?"

One student asked if it was OK to use a fraction whose numerator was larger than its denominator. This led into a nice discussion about improper fractions and how these are all larger than one:

We could see a couple of groups who found solutions quite quickly so we gave them an added challenge by removing the '1' tile. They relished the added level of difficulty:

All in all, it was a really pleasing lesson that allowed the students to show us their problem solving skills as well as allow us as teachers to assess their understanding of fractional equivalence.