The Laws of Maths - The Associative Law with Multiplication

We're getting close to the end of our time looking at the Laws of Maths (and the kids will be glad to move on to something new).

So, we brought the multiplication process into the Associative spotlight.

Now we were getting into some interesting territory.



Having spent the last two weeks:
1. learning about the Commutative and Associative Laws
and..
2. experimenting with using arrays

...we were now in a place where we could start thinking for ourselves and the teacher could take a bit more of a backseat.

So once again we pulled out the counters and started thinking about the provocation.

 A nice looking display - at the top we have 5 groups of 4 x 2

and underneath we can see 2 groups of 4 x 5

 So when we put them together we can see that the two groups

are the same. Nice proof.

This one was a bit confusing to me. We've got 4 x 5 at the top

and 2 lots of 2 x 5 at the bottom. Not quite the same as the provocation at the top.

So we reorganised things a bit and had a chat about what we were looking for.

I find these little chats that follow moments of confusion really significant -

helps me to assess and diagnose.

And just when I thought everything was going so smoothly, I came across this

example of someone trying to model the symbolic notation using blocks

rather than making an array. 

Time to reflect on my tuning in process for this inquiry.

Having come this far, we are going to finish with the Distributive Law later in the week.

Can't do it tomorrow because we have "Gruffalo Day"

...but that is another story!

  

  

Year 2 Addition with Numberlines

This morning I had the opportunity for some quality time with Year 2. I wanted to see what they could do with numberlines. And how they could use them to explain addition and subtraction.

So we started off with a strip of paper that we wrapped around our heads - seemed like the best place to start.

Using the paper strips, we measured the circumference of our heads and then compared to see who had the biggest. Contrary to expectations that the teacher would have the biggest head because (i) he was older and (ii) he had more brains, we found that while my head has a circumference of 57cm, one of the students' heads had a circumference of 61cm.

We compared strips of paper and found that, yes, one was indeed longer than the other. I asked them if they knew a word that would describe that bit of overlap - but couldn't draw out the word "difference" so I had to tell them.

"Oh yeah, we know that," they said.

Next, we had a general chat about how to draw a numberline and then had a go at representing our head data:

57cm + ? = 61cm

This they did (sorry, no photos of this bit) as it was quite an easy calculation to make. The answer (4cm) was never going to be a surprise but I wanted to see them using a known fact before we explored any further.

Then I asked them to use a numberline to show me:

57 + 14 = ?

And so the fun began!

Here is what we came up with...

So, in one jump we go straight from 57 to 71. This is mathematically correct but shows me nothing about using a numberline or the process behind the operation. I asked for more information - and got a blank stare.

Another student started with this idea - what if I jump by 10 straight to 67? But, seeing that space was running out, she wiped out the line and started again...

...to produce this one. Add one jump of 10 and then another jump of 4. Good work!

 But then we had this solution. According to this representation, 57 + 14 = 70. I put it to the group that this one was the correct answer and that 71 was incorrect, which threw them a bit.

"Look!" I said. "There are 14 jumps under the number line. So 57 + 14 must equal 70."

They knew it couldn't be but couldn't see why.

Here's where it gets messy. As the numbers get closer together, there student runs out of room and ends up skipping one of the jumps that he is numbering, so the jump between 69 and 70 is not labelled. Yes - we do need to be careful when construct a numberline. 

So what?

This was just an introductory activity to get an idea of what the Year 2 students understood about numberlines. 

As a result:

1. we talked about some of the important features, such as labels, direction, accuracy and infinity

2. we talked about how to make "provable" jumps, not just blind leaps to apparently random answers

3. we modelled addition on a numberline

4. we laid the groundwork for future explorations using numberlines for other applications

5. we had some fun and I got to meet a few students who I didn't know before. Looking forward to getting back in to see them on a regular basis.

The Laws of Maths - The Commutative Law with Multiplication

Our conversation about the Laws of Maths has moved on this week to consider multiplication.

How do the Laws that we have discussed so far apply here?

Let's have a look at the Commutative Law.

Here's what our provocation looked like:

So, after a bit of tuning in that looked like this:

We did the "1 Minute Challenge" and then had a look at the grid.

I find this really interesting and helpful for the kids to get them focused

on what they already know and what they need to learn.

As we go on, they can colour in the tables facts they "master".

And then it was on to the fun and games.

How do you prove the provocation is correct?

So, I was really keen to get the kids to use arrays for this one, partly because I don't think they have used them very effectively in the past and also because they are going to need them when we start talking about the Distributive Law - stay tuned for that one!

When we got started, we decided it might be a good idea to make two arrays: 

one that was 4 x 9 

and the other that was 9 x 4

But were they going to look any different? 

And how could we demonstrate that they are equivalent?  

After a mad scramble for the blocks...

...we found that 9 groups of 4...

...and 4 groups of 9 take up the same space.

Seems obvious but it was really good to see it.

Some groups lined all their blocks up to compare the length and see that it was equal.

Not everything went to plan. Here's a few problems we encountered:

So here we have 4 groups of nine, which is obviously longer than 9 groups of 4... 

A simple problem, but an important one. Yes - you actually need to count accurately when using blocks. These guys got a bit carried away and the 4 groups of 9 included one group of 10 and one of 11.

This one is fascinating. It shows the alert teacher that here are some students who don't get the idea of arrays, don't understand how to use blocks to represent number facts and who missed the point of the activity.

Back to the drawing board...

So we made 9 groups of 4 and also 4 groups of 9...

...and with a bit of reorganising we could see they looked the same!

So what?

It's not rocket science, I know, but having done this activity...

1. we can now construct an array

2. we can demonstrate the Commutative Law as it applies to multiplication

3. we can see what we are talking about (abstract idea made concrete through modelling)

4. we all now know that 4 x 9 = 36

5. ...and we got the blocks out and had some fun.