I asked them to find three rules for the pattern: the number of squares, the perimeter and the area. All groups were very quick to find the rate, most using a table and looking at the first differences even though we haven't talked about them yet. Most groups determined that they needed to multiply the rate by the step number but the answer was off so they adjusted, adding the appropriate amount. I love how much of this they're figuring out on their own.

For groups that finished early I asked them how things would change if we said that each square was 2 unit by 2 units. Their first reaction was that they just needed to double their previous answer. I asked them to prove it to me and they soon discovered that the area of the new pattern was actually four times the area of the original. Such great thinking by a great group of students.

The other day one of my students asked why the rule for dividing fractions worked. I was so happy to hear this. We didn't have time to get into that day so we had a look today.

I started with this visual, then moved into looking at dividing with a common denominator.

From here it was easy to see that the green would fit into the red once (the green rectangles would fit over 8 of the red) leaving one rectangle. So we would need 1 out the 8 green ones. The solution then was that the green fit into the red once plus an eighth. I think many students appreciated the visual nature of this approach. However, when I asked them to try it some of them just wanted to use 'the rule'. We talked a little about how you could do this without drawing a picture. You could find a common denominator then divide the numerators by each other and do the same for the denominators. I love this approach.

Next we talked about adding and subtracting fractions, again starting visually, then becoming more abstract. We were running out of time so I had to forgo doing a problem today. I gave them the last 8 minutes to practice operations with fractions.

One of the students asked today if we were going to be doing any textbook work this year. Since we don't have any books for this course my reply was no. He seemed happy, which seemed odd given that the homework I give is the kind of work you find in a textbook. The boy sitting beside him was the boy that approached me last week saying he wasn't sure what was going on. He said that he really liked all the group work, problem solving and working at the board.

It seems that my daily routine in this class, generally, consists of a warm up, a problem and some skill work. I really like the balance but it's tight to fit it all in everyday. It would be perfect if our classes were 20 minutes longer!

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