Here is a little activity that I've used in a number of classes that has always given us great information about what the students do and don't know about number sense and area. It starts with the question below. Note that I go to great lengths not to use the word area at any point.
Most students work on somehow counting the squares inside each 'pen'. Occasionally. some students will make the mistake of finding the perimeter of each pen. Usually I get them to reflect on this error by asking "Can you shade the grass that the cow is going to eat?"
Sometimes they interpret the question as "Which cow has eaten the most grass (in the past)?" and will respond like this:
Other times, students will try a 'count-all' approach and sometimes will not include the area udderneath the cow.
Here, I remind the students that the cow can move around which usually is enough to get them to realise to include the missing area.
But I really want them to move away from this 'count-all' approach. I want them to see that there are more efficient ways of finding the area and have thus chosen the dimensions of the pens quite deliberately. When I chat to the students I often find that they know that counting all is time-consuming and prone to error. Now the students I was working recently with were grade 3s and there was certainly now way that I was going to chuck a 'just do length times width' at them. However, we consolidated a few of their strategies and this is what we got:
Here, the student split the pen into a 10 by 6 pen and 1 by 6 pen. The area of the former is 60 and the extra 6 of the latter gives a total of 66. Neat, eh? Now look at what this student did and wrote and try to figure out what they 'saw'.
I don't know about you, but I'm quite impressed that a grade 3 student is comfortable writing
12×5+6 and this gives a clue to what they saw: there are 12 squares in the top two rows of the pen and there are five such rows (hence 12×5) with the extra 6 on the bottom row being added at the end. This student actually counted by 12s too ("12, 24, 36, 48, 60!")
This student more clearly split the pen into equal sections of 8 to get the total area. In fact looking at the three examples above it is clear to me now that the ability to decompose the pen into smaller pens is a really important strategy (the same way that we sometimes decompose numbers into smaller numbers in order to make calculations easier).
But it is also so powerful that students (and teachers) see these different approaches as it does help expose inconsistencies and misconceptions. All these strategies are solidifying their multiplicative understanding and preparing the groundwork that will allow them to develop the formula for the area of a rectangle.
Finally, listen to this student's reasoning:
Again, I love my job!