# Authentic Inquiry Maths

## Thursday, 16 August 2018

### 400 000

Data from the blog - cracked 400 000 views yesterday, thanks to some enthusiastic students at ACU Canberra looking at some of the work my kids had done - thanks people.

Looking forward to the next 400 000.

PS - Something happened about 18th March 2016. That's the big spike. Lots of traffic came from France - not sure why.

A PYP event? Maths conference? Cyber hacker?

### Kids - They Never Cease to Amaze

We were making patterns yesterday. I wanted the kids to make a staircase pattern using Cuisenaire rods. I do this each year with my kids - I just like to see how they will interpret "staircase" and how they will use the materials provided.

For some unknown reason, I seem to expect my present class to be "less" than my previous classes - less creative, less perceptive, less able. Maybe I glamourise my previous students and forget their falibilities, remembering only their moments of glory.

Recycling old tasks gives me the opportunity to be surprised - even though I have expectations of what the new students will do, they never cease to amaze me with their own, individual responses.

So when we sat down to make a staircase pattern, I thought it was not going to be as good as last year.

Here is what we did in 2017:

Click here

And here is what my current class came up with. Just as creative. Just as intuitive. Just as good.

Kids - they never cease to amaze.

## Friday, 27 July 2018

### Knoks, Moks and Snocks

We have been looking at measurement of length this week.

Today when they came into the room, the kids were faced with this challenge:

We were using our interlocking cubes to make patterns so it was natural that these blocks became out units.

I wasn't happy to have this unit be called a "block" so the kids decided to call it a "knok".

Next step was to decide on the other units. Here is what they came up with:

It looked like this:

Today when they came into the room, the kids were faced with this challenge:

*"Overnight, there was an international disaster - all millimetres, centimetres and metres had disappeared. No-one could remember how long the units were. We had to make some new units of measurement. What would we do?"**After a few suggestions, like, "Let's use feet and inches." I encouraged the students to use their creativity to make a new system.*

We were using our interlocking cubes to make patterns so it was natural that these blocks became out units.

I wasn't happy to have this unit be called a "block" so the kids decided to call it a "knok".

Next step was to decide on the other units. Here is what they came up with:

It looked like this:

2 peas = 1 knok

8 knoks = 1 mok

6 moks = 1 snock

And once we had a measurement system, we needed to measure something.

"What shall we measure?" I asked.

"YOU!" was the emphatic reply.

So I became the first of many to be measured with our new measurement system.

Turns out I am 2 snocks, a mok, 3 knoks and 2 peas tall. (Astute students of place value will notice that the 2 peas should be converted to another knok to make 4 knoks.)

I don't think it will ever become an internationally recognised system of measurement, but it gave us an opportunity to explore the idea of standard units and how to convert between units of different sizes.

## Monday, 12 February 2018

### Is a picture a pattern?

So we are back into the swing of things here in Canberra, Australia.

And I wanted to push on with the spatial reasoning and pattern making. Once again, it was going to be pattern making first up each morning.

Day 1 of making patterns and my new class surprised me. I put out a variety of materials - pattern blocks, Cuisenaire rods and tangram shapes - to see what they would produce.

And I could not believe how many of them produced pictures - symmetrical undoubtedly - but pictures just the same. So were they making patterns? Is a symmetrical picture a pattern?

Rocket ships:

So - nice pictures. Were they patterns?

I'll leave that one with you.

And I wanted to push on with the spatial reasoning and pattern making. Once again, it was going to be pattern making first up each morning.

Day 1 of making patterns and my new class surprised me. I put out a variety of materials - pattern blocks, Cuisenaire rods and tangram shapes - to see what they would produce.

And I could not believe how many of them produced pictures - symmetrical undoubtedly - but pictures just the same. So were they making patterns? Is a symmetrical picture a pattern?

Rocket ships:

Monsters and funny faces:

Puppy dog:

So - nice pictures. Were they patterns?

I'll leave that one with you.

## Thursday, 14 December 2017

### Closing down for Christmas and Summer Hols

If you are wondering, I am on schools holidays until the first week in February.

Lots to look forward to next year:

- I'm back on Year 2 - yay!

- I will be doing some part time lecturing at ACU in Canberra

- I will be doing a lightning presentation at the SERC conference in Canberra

- I will be involved with presenting Maths300 workshops

- I hope to be playing bagpipes at the War Memorial on ANZAC Day

Let's see what happens.

Have a great holiday.

BF

## Tuesday, 21 November 2017

### A Proportional Reasoning Sequence

This is a sequence that went over several days. I don't think we are finished (yet) but I wanted to get the initial stages written to clarify my thinking.

I was interested in looking at proportional reasoning. We had done a fair bit with fractions (halves, quarters etc). It was becoming obvious that we might benefit from some time looking at the relationships between the values of numbers and how we could represent this with size/length/area to make the relationship visual.

I have represented this sequence as "phases". They might be lessons. They might be something else.

## Phase 1

I had a stack of coloured strips cut up.

The red strip is noticeably shorter (24cm) than the other colours (yellow, orange and green are all 29.5cm - the length of a piece of A4 paper). This was deliberate. I wanted the students to refer to the value they were certain of (red = 8) and use this as the basis of their reasoning, rather than simply fold each strip to the fraction of the whole.

eg if yellow = 4 (which is half of 8) then I might be tempted to just fold it in half without having to refer to the red unit at all.

After a time, two strategies emerged;

__Strategy 1:__

Some students decided that all the coloured strips needed to start off as the same length. Then they could fold yellow in half so it equalled 4; fold orange into quarters and then cut off one section to leave 3/4 because 6 = 3/4 of 8; then fold green into 8 parts and cut off 3 parts (5 = 5/8 of 8)

__Strategy 2:__

Other students folded the red strip into 8 equal parts and used this as a "ruler" to measure where to cut each of the other strips.

This was an interesting starting point. 8 was a good number because finding 4 and 6 was relatively easy but finding 5 required a bit more thinking.

## Phase 2

And here are the strips I had prepared:

Look - he's done it again. Can't this guy cut the strips to even lengths?

No - I wanted to play with the kids' heads again. This time the purple strip (=24cm) was given a value of 6. How would you work out how long the other strips should be to show 1-5?

The kids pretty quickly realised they would have to find a way to divide the purple strip into 6 equal parts. Here is what happened:

__Strategy 1:__

Some students decided that a good way to find 6 equal parts was to fold the strip in half, then half again, then half again, making 8 equal parts. They then cut off two of the sections leaving 6 qual parts. This then became the measure showing where to cut the other strips.

__Strategy 2:__

Other students folded the purple strip into 3 parts using the z-fold technique. Then they folded this in half to make 6 equal parts.

__Strategy 3__

Not the most efficient strategy but totally effective, this student worked out how long a unit was, and then proceeded to cut out individual units in each colour and used these to establish the numbers 1-6.

__What we found out__

When we put these staircases into our 1cm grid books, we quickly found out that there was a number pattern. By counting how many squares high each step of the staircase was we found a counting-on pattern.

For students who used Strategy 1 or 3, the purple strip was 18cm long. They had started with 24cm then folded it into 8ths and then cut of 2/8 to make 6 equal parts. So their number pattern was the counting by 3's pattern.

For students who used Strategy 2, the pattern was the counting by 4's pattern, since 6 = 24cm, therefore 1 = 4cm.

Interestingly, no-one had considered using a ruler. They were busy exploring the relationships.

We also found out it was important to be accurate. In this picture (below) pink = 1 and blue = 5. But when they are put together, they are longer than the purple, which is meant to equal 6. This student has not fully understood the task. Their purple is folded into 8 sections. Their other units for 1-5 are a random series of increasing lengths.

At this point we paused and reflected on what we needed to do.

## Phase 3

And so we moved to the next proposition. If you are given a unit (such as pink = 4) can you find the numbers less than and more than?

The students had made the Cuisenaire staircase hundreds of times. They quickly noticed that I had conveniently provided strips of paper using all 10 of the Cuisenaire colours.

*This is the scene that greeted the kids when they came into the room.*

They also noticed that the coloured strips were of varying sizes. Once again, I didn't want them to rely on any value other then pink = 4.

We were now showing that we understood the need to use pink = 4 as our basis for our reasoning about where to cut the other coloured strips.

There was much folding and cutting - and some even began to use additive strategies to find new numbers. For example, once I had calculated 2 and 3, I could work out 5 by putting them together. This showed me where I should cut the yellow for 5.

Some students used the white (=1) to add onto the end of their latest length to make the next length, showing that the number after

*n*is*n*+1.
Some other students developed their own "ruler" once they had found the value of white = 1. They used this standard unit ruler to measure where to cut each strip.

And our friend from yesterday persisted with the strategy of cutting each colour into individual units and the glueing them into the book.

As I observed each of these events unfolding, I stopped the class and we had a moment of reflection to see what we could learn from each strategy.

## Once we were done

After we had produced some nice looking staircases, I brought the class back together for another chat.

We had been thinking that pink = 4 all through this task.

But what if pink = 40?

Or 400?

Or 0.4?

Or 40%

Suddenly it all became clear - the model we had played with for half an hour was just a representation of a relationship based on proportion.

We could use reasoning to work out that...

- if pink = 40 then brown must be 80

- if pink = 400 then red must be 200

- if pink = 0.4 then blue must be 0.9 and orange must be 1.

"I think it should be 1.0," said one student. "We are talking decimals here you know."

## Thursday, 16 November 2017

### Show me a quarter

I asked the kids to show me a quarter.

Here is what they did.

I'm not going to say anything.

Just I love those kids.

*And yes, the yellow line has 3 squares,*

*which is a quarter of the total 12 squares used...*

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