Reasoning and Proving Grade 3 Style

I had the most enjoyable hour with a group of Grade 3 students the other day. I had them reasoning and proving using their number sense with a very basic Excel spreadsheet that I made.

Some random numbers were put in the bricks on the bottom row and they were then asked how are all the other numbers generated? I stood back and let them argue for a couple of minutes before they figured that the numbers from two bricks add to give the number on the brick above.
"Can you predict what the top brick will be if the bottom bricks are all 1s?" I asked.
A brief but furious debate ended with "The second row will be all 2s, the third row will be all 4s, the next row will be all...8s so the top row will be 16! It will be 16!"
"Prove it," I said which they duly did to hearty cheers. I followed up with asking for their predictions for all 2s on the bottom row then all 10s and again they were spot on with their predictions.
So I decided to ad lib a bit:

"Right, your target is to get 1000 in the top brick but you must have the same number in all the bottom bricks."

After a flurry of trial and error (or trial, feedback and refining to be exact) they got to the point where 63s on the bottom row gave 1008 in the top brick.
"Too big! It's too big! Try 62!"
And then the following happened:

Now decimals are not on the Grade 3 curriculum so this struck me as being not too shabby.
The students loved it and I consolidated by giving them this question from the Problem of the Week section on the University of Waterloo's CEMC Problem of the Week site..

The Great Divide

Here's a question we gave recently to a Grade 6 class:

Three classes are collecting cans for the Thanksgiving food drive. Class A collected 275 cans. Class B collected 225 cans. Class C collected 253 cans. These cans need to be put in boxes of 24 before they can be delivered to the local food bank. How many boxes are needed?

We chose the numbers deliberately. First we wanted to see who would look at the numbers and add them mentally and who who would automatically put them in columns to add. Then we wanted to see what students would do with the remainder. We were also interested in seeing how students would do the division.
As it turned out, all but one pair solved the problem correctly but they did so in varying ways. Yes, some added the numbers in a traditional way which is fair enough. Others though said "I know 25 and 75 is 100 so I added this to the other 400 to get 500 and then added the 253 to get 753," or a variant of that. Most students thought this was neat and it allowed us to give this bit of advice:

Good mathematicians look at the numbers first before they decide what strategy to use.

So, 506+499... do that in my head. 689+7259+164... maybe do that on paper.

With the division, some used a traditional long division approach. Others used an abbreviated version of this (I think this is called short division).

Notice how this student figured out that there were three 24s in 75.
Other students used a chunking method using friendly numbers (this is what I tend to do if I do a division in my head).

 Some chose to start with a chunk of 10, then another and another. Others chose a chunk of 5, then two chunks of 10 followed by another chunk of 5. One group chose a chunk of 9 to begin with. Which got us as teachers thinking 'We tell them to use friendly numbers but do they know what friendly means?'
So in our Consolidation, we asked the students this very question. They were pretty much in agreement that it meant numbers you could work with easily in your head. We also had students comment that a friendly number for you might not be friendly for someone else. With this in mind, I asked the group who started with a chunk of 9 the following question:
If you could go back in time and give yourself a bit of advice before you start this question, what would it be?
Straight away, they replied "Don't use a group of 9. Use a group of 10 instead as it is way easier."
Other groups, having seen the chunk of 20 approach (above) said that they wouldn't just stick to chunks of ten and would look to use 'larger chunks of 10' i.e. multiples of ten.
So there was some really good thinking and understanding going on. With some well chosen practice questions, this understanding will continue to be solidified.

Grade 6 Constructing Shapes

Some interesting things arose out of a geometry lesson in a Grade 6 class. We wanted to get the kids to construct some shapes using different tools. As teachers we tried to do some of these questions ourselves:

All the questions were engaging but especially the last two. The discussion that led to proving that such shapes are impossible actually uncovered some new geometric ideas for us. We decided to ask students some similar questions:
1) Make a shape with three right angles.
2) Make a quadrilateral with three acute angles.
Most of the students used geoboards but a couple of pairs had access to an iPad. Some answered the first question with a square...

 ...as it has three right angles. More than three was the justification given. A simple tweak to the question "... exactly three right angles..." clarified things quickly enough. Our question was deliberately ambiguous to bring out such conversation.
The second question brought out a lot more thinking. Some groups looked to a create a kite and checked for acuteness by using a corner of a piece of paper:

 

Others came up with a chevron type figure:

(Later, the teachers debated as to whether or not a symmetrical chevron is also considered a kite, something I'd never thought about. Now I think it is, unless someone else can convince me otherwise?)
Overall, the students engaged in some really good conversations and were using some very precise geometric language.
There were a couple of considerations though. One student I was watching found it very difficult to use the geoboard. When asked to make a pentagon, he didn't know where to begin and moved the elastic from peg to peg in the hope that some familiar shape would come up. Eventually he made a pentagon but didn't realise that he had done so. I then asked him how many sides his shape had. He proceeded to count all the pegs that the elastic was in contact with! I'd not seen this before so i handed him a pencil and paper and asked him to draw a pentagon. He did this quickly, and then drew another four all of which were irregular. For this student, the geoboard was not a good tool. Yet. Maybe it will be later on I don't know. But at least he knew that pentagons have five sides and could draw them.
Two other groups of students used the geoboard to make a 'pentagon' like this:

They argued it has 5 sides and traced these with their fingers. We shared this with the rest of the class who offered advice along the lines that 'It can't be a pentagon if the lines cross', or 'It's not a pentagon, it's three triangles.' After some debate the class agreed that polygons can only be made up of straight lines that don not cross and that all polygons must be closed.
Now I could have written this down on the board at the start of class and told the kids to learn this by heart. But this would have prevented any of the rich thinking and debate that actually occurred.

How Misconceptions Begin

This is how a group of students began their work in response to the question 'Do you know anyone who has lived 1000 days?' As they are eight years old, they decided to see if they are 1000 days old by checking 8 times 365. Notice how they just write down the 5 and 6 digits in the solution as "there is nothing above them". They then do eight times 'three' to get 32.

Now these are grade 3 students so they are only just beginning to learn about multiplication and should not be coming across any formal algorithm until Grade 4 at the earliest. But when it does come to the time for them to learn how to compute something like this, I am certain that they will be more succesful by being introduced to the array method first.

The neat thing was that even though they got the calculation wrong, they knew that 8 years was way above 1000 days. So they then tried 6 years, then 5 years, then 4 years and finally 3 years. Each time, the calculation was incorrect but their reasoning based on this calculation was perfect. Not bad for Grade 3!

Geometry Centres

I took part in a really engaging geometry lesson in a Grade 1/2 split this week. Five centres have been set up and the students work pretty much independently on these; you can see them building their own geometry knowledge as they do the activities. The talk that these activities provoked was impressive and exciting as the students focussed on the geometric properties and relationships.

Compare this approach to the worksheet approach where kids just write the word square next to a picture of a square. Or even worse, cube next to a 2D picture of a cube: how can anyone truly learn about three-dimensions from a two-dimensional worksheet?

The students' use of Mathematical Processes were in abundance including a surprising amount of reasoning and proving ("I think I'll need four sticks to make a cube and I'll show you") and reflecting ("Oh, I needed 12 sticks because a cube is not a square.")

Even a simple question such as 'Use the two larger triangles (from a set of tangrams) to make different shapes' brought out a wealth of geometric ideas e.g. composing and decomposing shapes, names of different shapes, congruence. The tactile component here was a vital part of the learning: kids need to flip, rotate and move shapes to learn about them.

After all, if the world's best geometer, Donald Coxeter, used manipulatives and concrete materials to help him better understand geometry, then why shouldn't all our students do so?

You don't learn geometry from a blackline master.