Yesterday, in my planning, I was bouncing around between a couple of ideas for the lessons I was teaching today. I decided to go with this Illustrative task that was the basis for the lesson study project I did last year with The Teaching Channel and Illustrative. I though it would be a great formative assessment as students move from thinking about fraction of a fraction on a fraction bar to an area model using a square unit.
I opened with a choral count in which the students counted by 3/4. I played around with ending each row on a whole number (12/4, 24/4, 36/4…etc) or ending on something not as “nice.” I opted for the second, but in hindsight, probably should have gone with the first. I really wanted more conversation around when the whole number occurred and why and possibly the distributive property (4 x 3/4) + (1 x 3/4) = 15/4, but not as obvious as I felt the first option made it, but it didn’t happen. The count looked like this (I didn’t get a chance to take a pic of the board):
I asked students what they noticed and I got a variety of responses. I got many variations of equations such as 3/4 + 6/4 = 9/4 and 12 x 3/4 = 36/4, and with each one, I asked how we could show that within the count. One student said she did 20 x 3/4 and just when I thought she was going to explain by saying because we added 3/4 20 times, she surprised me in her wording. She said, “I knew it was a 4 by 5 to get 20 and then each one was 3/4, so I got 60/4.” This struck me different than the “jumps” or repeated addition of 3/4. It reminded me of this diagram from the learning progressions:
I know it wasn’t fraction of a fraction but just they was she described the size of each one as 3/4, it reminded me of this. Then, I got noticings around the patterns in the ones place of the numerators as you go down the columns, adding 15/4 down every column, the sum of any two fractions also showing up on the board and many others about multiples of 3, but nothing about where the whole numbers came up or any mixed of operations in their equations.
Next I gave them the task to engage in alone for 5 minutes before I let them move into group work. I was really impressed by the way they jumped right in and easily could find 1/3 of 1/4. I was not surprised that many ignored the “square pan” piece and went with fraction bars. Much of the work looked like this, cut vertically:
While there were also quite a few students who quartered it by cutting vertically and horizontally in half and then splitting that quarter into thirds:
This is something for me to think (learn) more about because if we are thinking area model, the dimensions of the piece, to me, looks like 1/2 x 1/6. That seems like it could be problematic to me when the square has dimensions in units of length.
Another interesting thing that always comes up in this work is the difference between dividing by a fraction and dividing it up into fractional parts. I saw those equations sneaking there way in like this…
That will be something to keep in mind in future lessons. What it means to divide by a number versus divide into parts? Is it different when we are thinking about whole numbers versus fractions? Cool convos to be had around that!
There was one student who was not working with the original whole in her work. She was working with the 1/4 in the first part and then the 1/2 in the second part. When I asked her about how she was determining 12ths, she said it was just like her phone, she took the whole thing and just zoomed in (she did the fingers swipes as you would do on your phone) on the part she needed.
There was some really great proportional reasoning going on with the cost of the cornbread pieces. When the pan price changed from $12 to $24, $6, and $18, student used great reasoning in relation to $12. In the example above you can see that work as well as here:
I left them with a question about the denominator, why is not ending up in any of the denominators we are using the problem? I only had a chance to snap one pic before I headed to another classroom.
Lots of great stuff to keep in mind and I think comparing the ways in which the students divided the models would be a really interesting conversation. Then after they move to an area work, I wonder if it would be great to bring that back for a comparison.
Also, I forgot to add that the substitute who was in for Leigh today was so incredible about taking notes and talking to the students because he was going to be in charge of teaching this same lesson to the next math class that came in. He said he learned so much and I thought this could be such a powerful way to have substitutes involved in learning more about the way in which math is taught.