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.*

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:

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.

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