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Mrs Humphries
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Post Number: 1
 Posted on Tuesday, 25 February, 2014 - 08:17 pm:

Firstly, I am a primary school teacher but my class have started a debate which has led to some incredible discussions. I will outline the points which they were making here and would love to hear from an expect. Ideally, we'd like to Skype an expert so the children can explain their thinking and receive feedback live rather than just a written response.

During a 'start of topic' shape lesson on Monday I set my year 5/6 (9-11 year olds) high ability (levels 5/6) maths class the challenge of naming 20 2D shapes. After they had completed the obvious, and a few that they wracked their brains to remember the proper names for, they began using their imaginations. Here we got things like 'star' or 'arrow head' which lead us to describe shape families and regular/irregular shapes. We concluded a 5 point star would be a decagon, as it has 10 sides, and an arrowhead would be an inverted kite and part of the quadrilaterals family. Then one of the class suggested a heart shape...

First we started with, what properties would a heart have? 2 sides? Ok, what other shapes have 2 sides? Then, the suggestion of a semi-circle - good thinking so far! Next, the question 'Is a heart an irregular semi-circle?' And from here we have had a long and detailed debate, clarifying our understanding of shape vocabulary and challenging what we knew.

If you are interested you can look at our Padlet wall here showing some of our thinking: http://padlet.com/wall/kh2n9kcfyc We know that not all of our information is accurate but it's all learning!

Our main points were:
Is a heart a 2D shape? Or just one line that meets?
If a heart is a 2D shape, what properties does it have to have?
Is a heart like a semi circle as they both have 2 sides?
Is a heart like a circle that's been squashed, even though it has 2 sides?
Does a heart belong to a shape family like a rectangle belongs to quadrilaterals?
Does a typical heart have one vertex? The vertex at the top is a reflex angle so it might be the opposite of a vertex, or an inverted vertex.
Can you measure an angle of the two lines that meet are curved?
If you can't measure an angle on a curved line, can you say that a heart has 2 vertices?
If the circumference of a circle is 'the linear difference around the edge of a closed curve' then that could surely be applied to a heart shape as well? If it can, then are we saying a heart is related to a circle or perhaps an irregular type of circle?

You can see some of our explanations and thinking behind these questions on the Padlet wall. We also discussed what we meant when we talked about 1D, 2D, 3D and defining what a vertex was.

We look forward to hearing back from you and really hope you can get in touch and potentially arrange a time to Skype. We're eager to learn and think very carefully about what we're told.

Mrs H, on behalf of SH Maths
TheConvolution
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Post Number: 212
 Posted on Wednesday, 26 February, 2014 - 01:24 am:

I am not an expert on these things but a heart is defined on $R^2$ but not on $R$ which means it lies in a plane and so is a 2-D shape. Anything which has length and breadth is a 2-D shape. The only required property for a 2-D shape is having 2 countable dimensions (e.g. length and breadth) and so area which is not 0.

Firstly you have to agree which heart you are talking about, there are many general hearts which have different properties but a good way to make a heart is to use the equation: $$(x+y-1)^3-x^{2}y^3 = 0$$.
This gives a standard heart although other variations are possible. This means it belongs to a family of shapes, the particular name for this family is cardioid which itself means heart like. The heart is not very similar to a circle or a semicircle except that they are both defined using circles. A cardioid is basically the resultant shape when you roll a fixed point on a circle around another circle for one complete revolution. A semicircle is half of a conic section ( a shape made from the intersection of a plane and a cone) while a cardioid is not a conic section but defined in terms of conic sections.( It can also be made inverting a parabola over a circle)

How exactly do you define a vertex, as a point at which the shape is non differentiable or something else, if you take the first definition then the heart made by using the definition given above will have 2 vertices.
As far as I know the angle will approach 360 degrees if you zoom into the top vertex but I am not an expert at this and so am not certain about this but that point is certainly a vertex as it is clearly non-differentiable at that point.

As I have said before, a heart is related to a circle by the fact that if you rotate a fixed point on one circle around another circle for one complete revolution you will get a cardioid, although this one will only have 1 vertex and does not look exactly like a heart. That curve will be defined by the polar equation $$r = 1- sin(\theta)$$

Rachel Hudson
Moderator

Post Number: 4
 Posted on Wednesday, 26 February, 2014 - 11:27 am:

Mrs Humphries, please note that this message forum is only via posts and not via Skype. Our posters do not exchange e mail addresses and other personal information.
Mrs Humphries
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Post Number: 2
 Posted on Wednesday, 26 February, 2014 - 11:33 am:

azerbajdzan
Veteran poster

Post Number: 1651
 Posted on Wednesday, 26 February, 2014 - 11:50 am:

heart
It answers no questions I know, but you can see there heart-like shape.

A remark on vertex:
To define vertex not too much technically I would say vertex is something on which you can cut your finger... it is a place on a curve (or surface) that is not smooth.
Michael Grayling
Moderator

Post Number: 28
 Posted on Wednesday, 26 February, 2014 - 02:01 pm:

Mrs Humphries - if they told you asking for assistance via Skype was possible they shouldn't have. Asking for help on this discussion board though is highly encouraged, and hopefully there'll be other posters who can offer further advice on how to explain the problem to your students.
Liz Woodham
Moderator

Post Number: 40
 Posted on Wednesday, 26 February, 2014 - 04:04 pm:

I don't feel able to try to answer any of your group's questions as that is certainly not my area of expertise. However, I know a little about primary maths and a little about Year 5/6 children, and your post warmed the cockles of my heart!

It is wonderful to see learners being curious about mathematics and asking such good questions. They are clearly used to thinking like mathematicians: noticing similarities and differences, making conjectures and generalisations, applying what they already know to new contexts ... How lucky they are to have an environment in which all that is encouraged.

Might you be able to share their thoughts with the wider school community somehow? It would be a real shame if it got lost and, in my opinion, it doesn't matter that you may not be able to answer all the questions posed.
Mrs Humphries
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Post Number: 3
 Posted on Wednesday, 26 February, 2014 - 06:20 pm:

Thank you so much for your comment Liz. You're right our learning has come much more from the discussion than the answers. We don't even really need an answer I just wanted them to be able to connect with a Maths expert that some of them aspire to be! It's a shame that I haven't find anyone yet!

We have spent a lot of time working brought discussion techniques, focussing on enquiry and modelling thought processes and they are definitely an inquisitive lot!

We have a blog which their discussions are shared on with parents and use Edmodo for within class discussions.

The children will be proud to know you acknowledged their hard work, thanks!
Andre Rzym
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Post Number: 1875
 Posted on Friday, 28 February, 2014 - 07:19 am:

I have a few thoughts on your many wonderful questions:

> Is a heart a 2D shape? Or just one line that meets?

A line (e.g. edge of a ruler) is 1-d, a piece of paper is 2-d, and the space we live in is 3-d. So a needle (as in needle and thread) is 1-d, because it can line up with the edge of my ruler. In contrast, a circle (or heart) cannot lie along the edge of a ruler but it can lie on a piece of paper. So it is 2-d. For the same reason, a paperclip (which is not a closed curve) is 2-d.

> Is a heart like a semi circle as they both have 2 sides?

I think that is a very sensible way to think about it. There's something very different about a curved line and a vertex: if I take a microscope to a curved line, it looks more and more straight, but the vertex does not go away.

> Can you measure an angle of the two lines that meet are curved?

Great question. You most definitely can. Think about looking at the vertex with a microscope - the lines that meet now look very straight. It is on this basis that the angle should be defined.

> If you can't measure an angle on a curved line

See the point above. As you enlarge the curve, it looks more and more straight.

This brings us to a point that you do not appear to have discussed: "Is a circle the limit of a polygon with a number of sides getting larger and larger?". Think microscope again - a circle has no vertices.

A similar question would be: "Is a staircase with an arbitrarily large number of stairs the same as a straight line?". From the perspective of the length of carpet you need (i.e. curve length) they are not the same. E.g. if depth and height of the staircase is 10m, then the straight line has length 14.1m, but you need 20m of carpet no matter how many stairs there are.

Andre
Nicola Coles
Veteran poster

Post Number: 625
 Posted on Friday, 28 February, 2014 - 01:48 pm:

Here is a parametric heart spreadsheet. It might be a bit old for primary children, but thought I'd link to it for others that might be reading this.
MrCole
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Post Number: 2
 Posted on Monday, 31 March, 2014 - 02:38 pm:

There's a discussion of what we refer to as a side to be had within this conversation. Strictly speaking, as I understand it, a side must be a straight line. What you call a curved line on the edge of a shape is (it would seem) the topic of several heated internet forums. This means that a circle doesn't have one side, a semi-circle/heart doesn't have two sides and (apparently) a cone isn't a pyramid and a cylinder isn't a prism.
Anyway, I liked the answer about cardioids, and thought that would lead to some cool experiments. Whilst perusing I found a discussion of sides of circles where somebody posted
"Now, on our round dinner table at home my wife sits on the other side of the table to me and when we have guests they sit either side of each other but not on the same side as either my wife or I."
and another asked how many guests were needed before two people sat on the same side.
I liked the ideas!