Which Cow Gets Most Grass?

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!

 

How a Question Evolved

Some of the math problems I give kids (and adults) have been begged, borrowed and stolen from many unsuspecting folk. Other questions however, I have created and developed by myself or with colleagues. It strikes me that this is an important skill yet it is one which I don't recall ever learning about in teachers' college. And part of the skill in developing a question is being able to reflect after the fact if the said question actually got the kids to learn what you hope they would.

So I was in a Grade 4 class before March Break and the teacher was just beginning to start a unit on division. In Ontario, this involves solving 2-digit divided by 1-digit problems. We wanted to create a question with a context so that students would be forced to consider any remainders and what they might do with them. Our first suggestion was:
A large pizza has eight slices. If 40 pieces of pepperoni are used, how many pepperoni pieces would be on each slice?
We quickly dismissed this as a) There are no remainders to think about and b) is pepperoni ever distributed evenly anyway?
The next suggestion was:
I pay $47 for five hats. How much is it per hat?
Whilst this does have a remainder to deal with we wondered if this would be a question that students would engage in. And, of course, why would you buy five hats anyway?!
The classroom teacher then mentioned that there were 19 students in her class and that they liked going to Canada's Wonderland. That got us thinking:
If a roller coaster car holds 4 people, how many cars would be needed for the whole class?
What if the whole school went? How many many roller coaster cars would be needed?
Then, because we realised that there are height restrictions for these rides:
If a roller coaster car holds 4 people, how many cars would be needed for 93 kids?
Someone mused "I wonder how long they'd have to wait to all get on?" and the question then evolved again:
There are 93 people waiting in line for a roller coaster. Each roller coaster car hold four people and there are 6 cars to a 'train'. There are five minutes between each roller coaster train. How long will the person at the end of the line have to wait before they go on the ride?
We were very pleased with our brilliant efforts and, after we had opened the champagne, even found a video of people lining up for a ride just as a 'train' leaves which we used as our Minds On.
However, as the kids began working on the question we noticed something that was quite glaring: they weren't using division as a strategy. Most student realised that there were 24 people on each train so they either counted up by 24s till they got to 93 or counted back by 24s from 93 till they got to zero. There was also a five-minute discrepancy in the times they worked out but this was because some kids thought that at time t=0, a train takes the first 24 away while others thought that at time t=0, a train has just left without the first 24. The students were able to justify this though either way so we were OK with this ambiguity.
Our group went back to the library to talk about what we saw. We had thought that we had developed a brilliant division question. We were wrong. It was a great problem solving question for sure and the kids were engaged in solving it. But, no division was evident.
So we followed the advice we often give our students:
We tried, we made a mistake, we learned from it, we moved on.
So we came up with two other questions that would allow us to be more intentional about division:
a) There are 74 students in Grade 4 and they will be split into 4 tchoukball teams. How many will be on each team?
b) There are 74 students in Grade 4 and they will be split into curling teams with 4 to a team. How many teams will there be?
The first question is a sharing problem whilst the second is a grouping problem and student need to experience both of these types of division.
A couple of students solved it like this:

 

... but they admitted it was difficult to keep track of the numbers.
One student kept a running total like this:

...but again felt that it was a pain drawing a tally for each of the 74 children.
One student used her multiplication table to help figure the answer out:

Neat eh?
Others set their work out like so:

What was powerful was that when students who used one of the first two methods above got to see other ways of solving the question they really liked the last way as it was much more efficient. Yet no matter what method was used we were able to show the students that what they had all done was in effect 74 ÷ 4. In other words, this question actually got the students to think about division, something our first question failed to do.