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:

"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.

How Big Is Your Classroom - Part 2

So we spent some time collecting data from the P-4 classrooms. This is a bit of a journey thanks to the way our school is set up. P-4 are over one the western boundary of the school - about 250m from Year 5 + 6.

Data Collecting

When we got back we pooled out data. Here is what we found:

Room              Side 1              Side 2               Area

PK                   9.82m            8.51m             8356.82m2

PK                  not measured           

MH                  8.95m             9.25m             82.7875m2

KM                  not measured

1JH                  9.63m             8.2m               78.963m2

1RB                 10.5m             8.0m               84m2

2PM                10.49m           8.5m              

2AH                 9.85                8.0m               78.88m2

3BR                 32FEET          27FEET         

3EB                 8.0m               10.4m             83.20m2

3TM                10.64m           8.1m               86.9288m2

3HB                 8.14m             9.02                73.4228m2

4RB                 8.0m               9.7m               77.6m2

4CS                 10.58              7.75m             81.995m2

4JG                  8.94m             8.9m               79.566m2

Some interesting points:

• looks like there was trouble with decimal point in one of the PK classrooms - unless their room is actually the size of a football field.

• One of the Year 4 rooms was missed out. Not sure why.

• The group measuring 3BR measured in feet and inches. One of them said, "So is there like 30cm in 1 foot so I could try to figure it out that way?" The teacher prompted them to just turn the tape over and use the metric scale that was on the other side.

• Lots of kids struggled with converting from metres to cms - need to go over that one again.

• And also the need to take about square centimetres when discussing area.

• We had a good talk about why the rooms are not all exactly the same. What things were done differently that might account for diversity of responses. And if we all got different results, what statements can we make about this data? Is it fair to say that all the rooms are about 80 square metres? And what room should Mr Black have?

So what?

In a recent chat on #pstchat, a weekly time on Tuesday evenings at 7.30pm EST, I was discussing listening as an assessment tool in maths. An interesting question came up - do the parents think that this kind of assessment is "soft"? Does sitting around listening to kids talk about their actions constitute rigorous assessment? Is it as valid as a pen and paper test?

Here's my response:

1. If I listen to what the kids are saying as they explain how the went about solving a problem, I will learn more than I would if I just looked at written responses on a page.
2. By listening, and discretely probing with questions, I can get the students to think deeper than they will if they are just writing answers in a box.
3. By giving the students the chance to talk about what they have done and to explain the process, I am letting them develop a deeper understanding of the concept that they are engaged with.
4. There are many students who are disenfranchised by poor literacy skills - their opportunities to progress in Maths are limited because they cannot read or write effectively. These children need to communicate their understanding through talking.
5. By talking with the whole class, children can gain a broader understanding of the concept by hearing what other students have done.
6. From this activity that we have engaged in this week, I have been able to modify my teaching (I won't be using tape measures with imperial measurements again), reinforce skills that students were struggling with and clarify concepts that were not fully understood.
7. I also have a good idea of who can do what - I have notes on individual children and what they said and did. This is qualitative assessment.

So many positives.

And of course we had some fun.

Measure Twice, Cut Once

Any builder, or patchworker, or person who deals with things that are sold by the length, will tell you this old saying is true: "Measure twice, cut once."

I have arrived in Denver, Colorado to attend the National Council for Teachers of Mathematics (NCTM) Annual Conference later in the week. Since is was freezing outside and everything was covered in a thick layer of melted slush, I decided to take a tour of the State Capitol building.

Denver, as you may be aware, is known as the Mile High city, given its height above sea level. This fact is advertised in every second shop you pass and on all of the t-shirts and tourist paraphernalia on sale in the mall.

Anyway, part of the tour of the building took us to the stairs at the front of the building where the official brass plate showing the actual height of one mile above sea level is located - well, two brass plaques and a stone engraving. And they're all on different steps.

In 1909 when the building was completed, it was decided to mark where the actual "one mile above sea level" was so the inscription was made on the 15th step.

The 1909 engraving

In 1969, someone decided to check the accuracy of the first measurement and found that it should have been on the 18th step, so a brass plaque was installed on that step.

The 1969 brass plaque

This was again checked in 2003 using GPS technology, which found that the actual marker should be on the 13th step, so a new brass plaque was put in.

The 2003 plaques - and you can just about see the other

indicators in the background

And the way global warming is going, in 50 years time the marker will need to be somewhere up near the top of the dome...

Da Vinci and the Vitruvian Man

You've seen the picture before, on posters, t-shirts, in books and movies. It's pretty famous. I even used it last week in this blog!

It's known as the "Vitruvian Man." Da Vinci did the drawing and made some notes based on the ideas of Vitruvius, who may sound like a famous volcano near Naples but who was in fact a famous architect in Rome. He had some ideas about the proportions of the ideal man. Da Vinci was able to use these to draw this diagram inscribed inside a square and a circle.

According to Wikipedia, my first port of call when I can't read Leonardo's unintelligible scrawl, the notes by Da Vinci around the picture are a summary of the key ratios of the perfectly proportioned human body:

• a palm is four fingers
• a foot is four palms
• a cubit is six palms
• four cubits make a man
• a pace is four cubits
• a man is 24 palms
• the length of the outspread arms is equal to the height of a man
• from the hairline to the bottom of the chin is one-tenth of the height of a man
• from below the chin to the top of the head is one-eighth of the height of a man
• from above the chest to the top of the head is one-sixth of the height of a man
• from above the chest to the hairline is one-seventh of the height of a man.
• the maximum width of the shoulders is a quarter of the height of a man.
• from the breasts to the top of the head is a quarter of the height of a man.
• the distance from the elbow to the tip of the hand is a quarter of the height of a man.
• the distance from the elbow to the armpit is one-eighth of the height of a man.
• the length of the hand is one-tenth of the height of a man.
• the root of the penis is at half the height of a man.
• the foot is one-seventh of the height of a man.
• from below the foot to below the knee is a quarter of the height of a man.
• from below the knee to the root of the penis is a quarter of the height of a man.
• the distances from the below the chin to the nose and the eyebrows and the hairline are equal to the ears and to one-third of the face.

   These measurements are amazing. I could share most of them with my class at school but there might be a few giggles about using the penis as a point of reference so I might hold off on that one.

   Anyway, a useful insight into the mind of a genius. Maybe if I worked backwards, I could get the kids to discover some of these relationships for themselves....

How Wide is Your Class?

In Year 6 we are currently inquiring into "Who We Are", a unit of inquiry that is giving us lots of room to work with measurement.

Last week we explored "How tall is your class?" Today we decided to think laterally - and see how wide our class is.

What did da Vinci say?

Seen this picture before? It is known as the "Vitruvian Man", drawn by da Vinci based on the work of Vitruvius, a Roman architect. (This picture is fascinating - a quick google gave me enough data to write a separate post - stay tuned!)

Anyway, one of the points of this diagram is to show that the height of a correctly proportioned man is equal to the span of his arms.

Is that true? And would it be true of everyone in our class? And would it be true of the other Year 6 classes?

We already knew how tall we were from our previous investigation.

Time to get inquiring...

So how wide are we?

Using a similar method to last time, we first measured each other individually and added together our measurements.

Then we went outside and lay down on the oval again - I know it looks like the kids are just lying there in the sun but really they are flat out doing some hard maths.

Interestingly we found that the distance from finger tip to finger tip was less than when we lay end to end.

Year 6 working hard again

Total height = 3714cm

Total width = 3686cm

Not a big difference but enough to ask a few questions.

Why are the two numbers different?

Was da Vinci wrong?

Were we wrong/inaccurate?

How would we test our accuracy?

Is there a reasonable margin for error?

Will it change over time?

So many questions!

Where will we go next?

How tall is your class?

Well, a new year, a new class, a new grade. What new adventures will befall us?

This year I have moved up to the big kids - Year 6!

And like a lot of teachers I suppose, we on Year 6 decided that it might be good to start the year with a review of number skills.

And the first one we chose was addition.

So, how can you do an inquiry into a review of addition?

Making Authentic Links

Our year group was starting the year with an inquiry into the transdisciplinary theme of "Who We Are", focusing on the physical, mental and social changes of adolescence.

As part of our exploration of physical change, I thought we could measure the combined heights of all the students in the class, then compare each class to see which was tallest.

This would involve lots of addition.

And some mathematical thinking.

Then if we repeat the process in about 6 or 7 weeks, we can see if there has been any change. Maybe we could even do it again at the end of the year.

Tuning In With Some Basic Addition

We started our brains up, or began the tuning in, with a 10 minute challenge. Students were provided with a series of questions that focused on addition skills:

i) What do I need to add to these numbers to get to 50?

ii) What do I need to add to these numbers to get to 100?

iii) 2-digit vertical addition

iv) Simple word problems involving addition of 2-digit numbers

In days gone by, this would have been the end of my maths lesson - do a page of algorithms - mission accomplished.

I still use this type of material - and the questions we used were from a textbook - but it is as a starting point for further investigation, not as an end in itself.

Thinking Mathematically

Now our brains were engaged and our skills were tuned in, we began the "finding out" part of the inquiry.

I asked the kids to list questions that they would need to answer in order to direct their inquiry.

Here's a few of the questions they came up with:

• How many people are in our class?
• How many classes are we comparing ourselves to?
• How tall is each person?
• What is the average height of each person in our class?
• How can we make it a fair comparison between classes?

Methodology

So, how were we going to measure the height of a class?

Well, the kids came up with three suggestions:

1. We all stand on each others shoulders and then hold a tape measure up to see how tall we are

2. We all lie on the ground in a long line and use a tape measure to see how long the line is

3. We measure each individual and then add them together

We took a vote and decided on Option #3 so we all got to measure each other and came up with a total (well, we came up with several totals and then needed to go back and check that we were measuring accurately and adding correctly).

Our total was 3637cm.

 How we measured the height of our partner

Sorting it all out in a table

And so that we didn't miss out on the fun, and also so that we could try out another methodology, we all went outside and lay down on the grass and used Option #2.

Yes! Looks like 6BF is taller than 6TL

Rolling out the tape measures

Interestingly, when we measured the length of our line of bodies outside, we got 3714cm. This led to much discussion about the accuracy of each measurement, how mathematical we had been and which answer to the question was right.