Mass - Tuning In

We have started a new inquiry - this time we are looking at "mass".

I am keen to avoid the confusion between "weight" and "mass" and as a result have deliberately steered all conversation away from this point.

In our initial discussions, I asked the kids what they thought mass was.

They had lots of good ideas, all worth exploring:

It's a measurement of objects.

All things have it.

It is to do with the space a shape takes up.

Bigger things have bigger mass.

You can measure it and then compare things.

You measure it with scales and things.

All these were good but a bit general. We needed to get more specific and to get our hands dirty. Time for some inquiry...

The Task

We got a range of different balls out of the PE store room as well as a few from home. The job was to arrange them in order according to their mass. Sadly, the teachers had taken away all the weights so the kids had to use the balance scales to compare the balls against each other.

We chose to use a tennis ball, a golf ball, a marble, a ping pong ball, a volleyball, an AFL ball, a bocce ball, a softball and a cricket ball. I was keen to get some examples where the order of size did equate with the order of mass.

Finding Out/Sorting Out

So, we got the gear out and broke up into small groups.

Lots of talk. Lots of conversation. Lots of, "Try this one next, I think it's heavier than that one."

The yellow ball is an AFL ball, for those out there who were puzzled. 

AFL is a sport from Australia, which over the last 100 years has been 

dominated by the mighty St Kilda Football Club.

And pretty quickly we got to the realisation that it was unnecessary to compare each ball against all the others. Once you had organised 4 or 5 of them, you could start near the middle of the line each time you selected a new ball. If it was lighter, you could compare it to the ones at the lighter end. If it was heavier, you went the other way. 

What would be the least number of comparisons needed to add a new ball to the completed line?

The final result - an interesting arrangement

Having got consensus on the line, the kids were then asked to  make an estimate of the mass of each ball. To assist this process, they were told that the golf ball weighed 35g and the bocce ball weighed 4kg - this provided a frame of reference for the estimates.

Going Further

Finally, many of the students started to play with other ideas. One student came up with a suggestion: 

"I want to see how many marbles equal one ping pong ball."

A few minutes later this was modified.

"I want to see how many ping pong balls equal one marble."

Here's the result:

And so began the inquiry into mass.

Taking Action

The next morning. one student came into class and said, "Last night I was talking to my big sister about mass. She told me that mass is different from weight. If you were on the moon..."

Ah! Home Learning at its very best!

(see http://4thgradebigquestions.blogspot.com.au by Capitano Amazing for more information about our school's journey along the road to Home Learning)

Fractions Assessment - Part 2

So, we talked yesterday about how I had done some assessment of our learning about fractions. I posted some student comments on a few diagrams that had inaccuracies and problems. The responses were very illuminating.

Anyway, the assessment continued with a few pretty standard questions sorting out some knowledge about adding and subtracting fractions - nothing ground breaking. 

The last question is what I want to share. It asked students to complete some sentence starters. 

Here are a few things they said.

Complete these sentences...

a) Fractions are....

a way of representing something that is not quite a whole

pieces of things that are divided up

a way of showing non-integers

a part/portion/piece of a whole

dividing between the denominator and the numerator

b) When you add two fractions, you have to....

add the numerators together if they have the same denominator

GUESS

add the numerators together unless the denominator is different then you have to x it by something else to make it the same

I don't know exactly what to do but I suggest you draw a pizza

If they have a common denominator then you  merely add up the top numbers but if they don't you can draw a picture or find a common denominator

c) You use fractions when you....

divide up things evenly or to represent something that is not whole

are cooking. e.g. pour 2/3 cup of cocoa powder in the cake mixture

cut up fruit

are trying to divide things into different parts

can't make a whole

cut a pizza, build a building, pour some milk

have a pizza and your friend says I want 3/4 you have to work out that you don't get much

There were lots of other responses - too numerous to mention here - but by being able to hear what they all said gave me a real insight into their thinking.

I am in a  position now to identify who needs a bit more support, who has really got it and who is ready for the next challenge.

In so many ways, a successful assessment.

Fractions Assessment

Well, we got to the point where we were ready to assess our work on Fractions, to see what we had learnt.

We use a variety of assessment tools, including pencil and paper.

I was interested to see if the kids could explain why some understandings of fractions might be inconsistent, fallacious or even straight-out wrong. So I gave them a few non-examples and asked them to explain what they thought was wrong with the diagram.

Here's some of the responses.

Task - Can you explain what is wrong with each of these examples?

Example 1

The quarters are not equal.

?

The fractions aren't shaded fully.

Not equal = no fractions

All of the fractions are different sizes so they're not fractions.

The four sections are not divided equal. It is important for each section to be equal to make the fractions true.

The bottom and the top of the circle are different areas from the middle.

Bad colouring.

This is wrong because the are of each shape is different. If it is going to be even then it should be cut like a pizza.

✔ Perfect

The quarters are not coloured in properly.

Example 2

 One is bigger than the rest.

Just like the first example this fraction is uneven. Each of the thirds is its own size and they are not equal.

The middle part is bigger than both the other parts making it unequal.

All the squares aren't measured to the correct measurement so they're everywhere.

The lines are slightly off so it's not equal.

The area is not the same as the other parts.

Using grids to shade fractions makes it look like there are more than 3 sections. The thirds are not equal!

Example 3

The blacks are scattered.

It's wrong because 1/4 isn't shaded. 6 squares are supposed to be shaded.

The quarters are not accurate.

The fractions are split up all over the place.

Not enough fractions are shaded to get the right fraction. The fraction would be 1/6.

It's not showing 1/4 of the rectangle.

A quarter is not shaded in and it would make more sense to put the shaded parts in line.

It is not a quarter of 24.

Not enough squares are coloured in and it is not very easy to understand.

Well, unlike the other example, this one is even as in shape and the dividing up of the box. But it is false. It says it is 1/4 of this box but it is wrong. It would be 1/6 of this box.

Reflection

I have not included every response here - only a select few of those that were interesting for one reason or another. There were many children who got the idea for each picture but it was not important to include all the similar answers.

So, it is interesting to see the things that act as distractors for the kids and get them off topic.

Colouring and shading seem to be pretty significant for some kids. It is the thing they focus on first of all. 

Also, many seemed to want the shaded sections to be grouped together - but instead maybe they should have been exposed to more examples where they were not.

The diagrams themselves need to be unambiguous. The picture is meant to help the kids see the idea - not confuse them even further.

Lots for this teacher to think about. Some good responses but also some areas that still need clarification.

We will revisit this topic again later in the year...

The 21st Century Teacher

I have been thinking about the role of the teacher in the 21st Century, about the challenges we face and how we are going to redefine how and what we do.

This will be a long term project - I don't pretend to have all the answers but here are a few things I've come to realise.

The Challenge to be Creative

It is no longer enough just to know your subject content. There are thousands of textbooks and websites out there crammed full of content. 

The challenge is to find ways to communicate this content in a creative way. 

Kids spend hours each day on interactive computer games, i-pods and x-boxes. we should not be surprised when they are not inspired by a textbook or photocopied worksheet.

So, how to bring creativity into the lesson?

Well, try a few of these...

• use modern images - find new ways to present old ideas 

• go transdisciplinary - bring music, drama, art, literature into your maths lessons

• get outside - take your lesson out the door of your classroom and find out how the learning can happen out there in the real world

• use technology - make movies, record music, create games - and use maths to do it

• go global - we live in a small world - get connected to it and connect your classroom at the same time

Well, it's a start. 

Let me know if you have some suggestions to add to the list.

Fractions - Tuning In

We were tuning in to our inquiry about Fractions. It was interesting to ask them to write down everything they already knew. Here is one example:

This is a pretty comprehensive list.  I liked the way it included text and pictures - a really descriptive account. 

Student questions

I like to ask the students about what they want to learn - it is part of the PYP (Primary Years Programme - IB) framework of inquiry. So we include it in maths.

Here's a few they came up with:

• Does BISMAS apply to fractions?
• How do you add fractions together?
• Could "1" be a fraction?
• Will I become more understanding of how to add and subtract fractions during this unit of inquiry?
• What is the origin of fractions?
• How many different ways of adding fractions together can there be?
• If you have two unproper (sic) fractions, how can you add them together?
• How come you cannot use decimals in a fraction?

These student questions gave us some areas to focus our inquiry on:

1. The nature of fractions - what they are, what they represent
2. Addition and subtraction of fractions
3. Improper fractions
4. Decimals and fractions

Thanks kids - it's great to know what you want to learn.

Equivalence - what's that?

The next part of the conversation focused on a really important aspect of fraction - equivalence. 

Not surprisingly, there was a diversity of understanding of what this meant and how it looked. 

We did a bit of exploring using circle and rectangles, dividing them up to show how we could show 1/2, 1/4, 1/6, 1/8 etc.  Lots of conversation and discussion about what to do and how to get it equal.

Then we tasked the kids with choosing different shapes, not circles or rectangles, to see if they could be used to represent fractions and demonstrate equivalence.

Here are a few examples:

A bit hard to see but it shows 3/4 and 6/8 of a square being equal.

Using a triangle to show that 1/2 = 2/4 = 4/8

Yes, this is true - but are the pictures really showing equal parts?

The hexagon and the octagon were produced by the same Awesome and Adventurous inquirer. He was able to put forward a theory. We called this theory:

Jack's Theory of Making Fractions Out of Polygons

According to the theorist, when you have a polygon, you will definitely be able to divide it into equal pieces with a denominator that is a factor of the number of sides.

He later produced an addendum to his theory to include multiples of the number of sides.

A very interesting exercise.

Fractions and Pizza

We were starting some inquiring into fractions in Year 6. We began our tuning in by looking at a Youtube clip:

This is the image to select when you search for "Fractions in real life"

and here is the link....

http://www.youtube.com/watch?v=8hmb0FyGe4Y

As with any discussion of fractions, the conversation turned to pizza. We all agreed that pizza was a poor example because of that pizza guy who cuts the pieces so carelessly that you always end up with some pieces that are obviously smaller than the others. 

How can this be a good example of fractions, we asked ourselves.

The conversation then moved on to the things that we already knew about fractions.

Here's what we knew:

• fractions are used to measure
• fractions represent equal parts of a whole (unlike the pizza above)
• the top number can be bigger than the bottom number
• fractions are related to decimals and percentages
• they are a way of dividing things up
• the bigger the denominator, the smaller the fraction

I wanted to explore this last idea a bit further, so I put the following list of fractions up on the smartboard:

1/2   1/3   1/4    1/5    1/6    1/7

We talked about what was happening and how, if the numerator was the same, the smallest fraction was the one with the biggest denominator.

"Yes, so it is," I agreed. "But why?"

"Well," replied one Wise and Wonderful student....

"You can't make the pizza bigger, so the pieces must be getting smaller."

Inquiry into Graphs and Data

Well, first week back from "The World Tour of Maths" and Tina the Awesome from next door had our classes sorted for the week. We were going to launch into an inquiry into graphs and data. Here's how it looked - you may see some references to Australian Curriculum here:

Inquiry - How I am going to organise this data?

An inquiry into - how we can collect, organise, represent and draw conclusions from data.

Skills - Addition; Subtraction; Collection of Data; Graphs

Learning Intention - Collect and organise data and draw conclusions. To understand data can be represented in different ways and some ways are more appropriate than others.

Success Criteria - 

• Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)
• Interpret and compare a range of data displays, insulting side-by-side graphs for two categorical variables (ACMSP147)

Teacher Questions - 

• What is the percentage of people in Australia are aged 0-14 years?
• What is the percentage of people aged 0-14 years in 9 other countries?
• How do these countries compare with Australia?

Student Questions - 

Children generated 3 questions of their own.

My plan - 

Children devised a plan. How were they going to get answers to these questions? What strategies would they use? How would they represent their information?

Basic equipment I will need - 

Students made a short list of equipment they were going to use.

Running the Inquiry

So, we launched into the inquiry. 

Our chosen data source was the World Fact Book on the CIA website - lots of data on lots of countries, and population data had age categories including 0-14 years. Convenient huh? You would almost think Tina had organised this...

Anyway, lots of discussion, lots of planning, lots of collaboration and lots of fun.

Here are a few work samples from the kids:

We started by collecting data and putting it into a table

A bar graph  - courtesy of Microsoft Excel

A simple column graph - the simple things in life are often the best

A line graph - this generated lots of discussion. Is it the right type of graph for this data?

A pie graph - took ages to work it out but looks very busy.

Can too much information be a bad thing?

Reflection - What did the kids say?

All good inquiries allow space for reflection. We had a few questions as prompts to get the kids to write about some of their experiences - the choices and decisions they made. Here are a few comments from them: 


Which graph was the best type to represent our data?

"I think maybe a bar graph would have been the best choice because you can accurately see the results of the data."

"I found that a column graph was the best to represent my data as it was easy to read and simple to make and information was clear to represent. Here's why: the height of the columns are identifiable and variable, while the vital points on the side containing numbers and/or percentages as a part of information extending knowledge of the topic."



Why is one type of graph better than another?

"All types are good in their own way and it depends on what data you have. Different graphs are useful for different things."



What would you do differently next time?

"I drew a bar graph. This was a good decision because people will understand and interpret my data better. Next time I would do nothing different. I am proud of my decision and I will stick with it."

"I drew a pie chart. This was probably not a good choice because there was too much data to be shown and it would not give the person reading it a fast, visual impact. Next time I would draw a bar or column graph because it would be easier to compare the data and it would be clear and quick to read."

"The column graph was much clearer to read than a pie or line graph in the situation we used it but it depends on what sort of information you need to show."

"The graph I represented my data with was a line graph. This was not a very good choice because line graphs are supposed to show results over time. Next time I would use a bar graph because it can clearly show the data."

So what?

Well, you heard it from the kids. They know that different graphs have different purposes. They know that their decisions will determine how effectively they communicate their data. And it all links back to the learning intention.

Great stuff.

Can't wait for next week...