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.


 

I ♥ Arrays

Arrays are brilliant.
I'll say that again.
Arrays are brilliant. I love the fact that I can talk about them in a kindergarten class as well as in a Grade 12 class. Arrays allow me as a teacher to help students connect ideas and concepts especially when it comes to multiplicative thinking. Yet I had no real experience of arrays at school but I wish I did; it would have given me a much deeper understanding of some important algebraic concepts.
 
So I was in a Grade 5 class this week. The teacher has been working very hard to address any gaps in the students' additive thinking and was now pretty sure that they were ready for some work on multiplication We wanted to see how they would solve a fairly routine equal groups problem so gave them the following:

A baking tray holds 15 donuts. How many donuts would there be on 13 trays?
 
Now one or two attempted to use an algorithm but were making lots of procedural errors:

Others relied on sketching out the problem:

In the case above the thirteen trays drawn have 10 donuts giving 130. The student then reasoned that the remaining five donuts per tray would give another 65 thus giving 195 in total.
Others used some really neat student-generated procedures:

What was important was that the students sensed that they had to do 15×13. They had also done some work using arrays to represent 6×5, 4×3 etc. So I said that suppose we had to do 16×11 then we would get an array like:

 

By splitting the 16 and 11 into a friendlier 10 and 6, and 10 and 1 the students were able to tell me quickly how much was in each of the four quadrants:

... and from there tell me that the answer was 176 (some added in their head, others wrote their work on paper). We then returned to the donuts question to see how we could use the array to help us here but I pointed out that I didn't want to draw out all the donuts as it would be too time-consuming!

"Wow, that's so much quicker," said one student and many agreed. We consolidated by trying another problem (essentially 17×15) and here is a sample of work:



I love the way that there are no place value lies in this method and that it encourages good number sense. I have seen some students develop this array method into a partial products method:



...which is neat but to me what is even neater is when I can use the array method to explain what happens when you multiply polynomials. When I first came to Canada I learned a new acronym: FOIL. I found a lot of students misunderstood this (it stands for First, Outside, Inside, Last) but they had much greater success when they drew an array:

And when there are more terms in the polynomials it becomes easier to collect the like terms as they lie on the diagonals:

I have shared this method with parents at Math Evenings and it is always neat to see their faces when I give these examples: they finally understand why it works!
I could go on about how arrays make it easier to understand what it means to 'complete the square' or how they can be used to do polynomial division (see James Tanton's excellent video on this).
But for today, it was rewarding to see how quickly the students took to this method and how it fitted in so nicely with their existing knowledge.

Now That's What I Call Feedback

I am slowly but surely working my way through James Joyce's Ulysses (my goal is to finish reading it before the summer!). I find the whole stream-of-consciousness technique fascinating (if at times confusing). Today I was in a Grade 6 class and I was suddenly reminded of Ulysses when we used TodaysMeet and saw the stream-of-consciousness of the students.

The students had been working on probability so we asked them this question:

In this can, there are between 10 and 30 cubes. A third of them are blue. What could be in the can?

This is what was in the can:

What followed blew us away. The class had a set of iPads and the students began posting their solutions immediately.

Or their questions.

Or their revised solutions.

Or advice for other students.

Or requests for help.

As a teacher I could see who needed help just be looking at the feed on TodaysMeet. This stream-of-consciousness eventually ran to over 27 pages! Below is just a sample of what was going on. As you read through it, see if you can link all the conversations.

 Isn't all that feedback just wonderful? It was fantastic to see students not being afraid to say if they are stuck and ask questions and other students helping them and sharing ideas. At this point we gave them some additional info. We told them that there were just two colours, blue and green, and that there were between 10 and 20 cubes (not 10 and 30). This really caused problems for some students:

Yes, some students did get stuck but there was so much feedback available that they overcame these difficulties:

Some used blocks to help their thinking:

At this point we asked the class for all the possible solutions. They told us that you could have A) 4 blue and 8 green, B) 5 blue and 10 green, or C) 6 blue and 12 green. What really pleased us is how they were able to reason why these were the only possible solutions (e.g. because these are the only totals between 10 and 20 that are multiples of 3). 

We then gave them one final piece of info: the total number of  cubes was odd. They then had to vote on what was in the can based on A, B, or C as detailed above:

We were pretty chuffed to see someone reason so clearly here!

At this point we wanted to know what students thought of the whole experience. Dave, the classroom teacher, reminded them that we were looking for descriptive feedback. This is what we got. Then picture the smiles on our faces.



So if this the richness in thinking that is potentially there, can you imagine what a disservice is done when students are asked to simply copy a note? In silence?

The final say goes to this student who summed up how we all felt today:

Hooray indeed.
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TodaysMeet can be found here: http://todaysmeet.com/

Representing Patterns

I reckon that the ability to represent ideas in maths is something that most people underestimate or are even unaware of. Yet it is a crucial tool in a mathematician's backpack: it will help him or her gain a deeper understanding by thinking of a concept in a variety of ways. But it is a skill that I saw in action in a Grade 1/2 split class this week.

I showed students the pattern I created below and asked them to create a similar pattern using the shapes at their disposal:

All students came up with something like this. Some where initially concerned that the colours didn't match but convinced themselves that it was OK as it was still square, triangle, square, triangle. A lot of students also extended the pattern without being prompted to. 

They were then asked to represent this pattern without using squares or rectangles and we got something like this from all the groups:

Then we narrowed the attributes and said that the had to represent the pattern again but this time only use one shape. I wondered if they would find this tricky but over half the students managed to get things like:

We allowed students to go on a scouting mission to see other students' solutions. The students who got stuck really benefited from getting the immediate peer feedback.

I then asked them to tell me how they could represent this pattern not with shapes but letters and they quickly gave me some examples.

We then agreed to call these patterns AB patterns.

But I wanted to push them further so I gave them some red counters and asked them to make an AB pattern with the red counters. Sure enough, many groups came up with something like:

We then asked the students to make an AB pattern using themselves. Firstly the organised themselves into boy, girl, boy, girl. Then after a little more thought, came up with stand, crouch, stand, crouch.

I then asked them to represent the AB pattern using sounds (Maths and music are so connected!) and this is something they really enjoyed:

By the way, I like to tell students that I call an AAB pattern 'We Will Rock You'!

 

And here's the thing which really toasts my crumpet. At the end of the lesson the teacher told me that the students who were the most successful today were her 'weakest' students!
Did I tell you how much I love my job?