During a ‘start of topic’ shape lesson on Monday we had the challenge of naming 20 2D shapes. After we had completed the obvious, and a few that we wracked our brains to remember the proper names for, we 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 someone 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. We know that not all of our information is accurate but it’s all learning!

Our main questions 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 (irregular), 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?

If you have any answers, or opinions, about our discussions and our questions we would love to hear what you think! Please leave us a comment.

Hi there 5/6. My name is Mrs. Sinnott and I am a grade 5/6 teacher in Australia. I really like the activities you have used to start off your unit on shape!! Mmm, so is a heart an irregular semi-circle…or a line that meets! To be completely honest I have never really thought about it!! Can it just be a heart??

Kind regards

Leah Sinnott

http://www.thelearninglair.weebly.com

Dear

Mrs Sinnott

5SH maths group haven’t come to a conclusion yet but we are still debating if a heart is a shape or not. We ( Vrinda and Rashi) think that a heart is a shape because if you can call a rhombus for example a irregular square and have its own name then a love heart is an irregular semi- circle with its own name. If you half a square it becomes a rectangle but if you half a heart it becomes irregular semi- circle.

From

Vrinda and Rashi

And the rest of 5SH maths group.

I’m not a mathematician, though I sometimes pretend to be one…

1. and 2. It’s a 2D shape, because you can draw it on a flat (2D) sheet of paper. A closed curve is any line that eventually meets itself. A polygon is a 2D closed curve that consists of straight lines joined together. A heart shape, semicircle, and circle are 2D closed curves but aren’t polygons. Your route to school, drawn on a map, is a 2D open curve. (Your route in the real world is a 3D open curve because of hills.)

3. and 4. There’s a whole branch of mathematics called “topology” that’s all about shapes and how you can turn one shape into another. It’s often called “rubber sheet mathematics” because topologists think two shapes are the “same” if you can draw one on a rubber sheet and then, by only stretching the sheet, turn it into the other one. Shapes are different if you have to make or seal holes in the sheet, or tie knots, or similar. Yes, you can transform a heart into a semicircle or a circle by sheet stretching, but interesting things happen at those corners.

6., 7., and 8. A good word here is “tangent”. A tangent is a straight line that just touches a curve, once only, without crossing it (nearby). Curves have many tangents and you talk about the tangent at a particular point. There’s only one tangent at each point. You can find the difference in angle of a curve at two points by finding the difference in angle of their tangents.

For most of the heart, the tangents of two nearby points are pretty similar, and the closer the points, the more similar the tangents. But points either side of the “vertices” have quite different tangents and the tangents will stay different no matter how close the points get, so long as they’re on either side. To me, that makes those “vertex” points quite interesting and it’s quite reasonable that curves with different numbers of vertices are in different groups: circles and ellipses are different from teardrops are different from semicircles and hearts.

Mathematicians talk about smooth curves. Most of the heart is smooth, but it’s not smooth at the vertices.

Going back to topology, turning a smooth curve into a “vertex” would mean infinite stretching of the curve at that point.

(Another way of thinking about smoothness is radius of curvature and the size of circle needed to approximate the curve at a point. How does the size of the circle change as you move around the heart? What’s this got to do with the sharpness of knives? How sharp can a knife possibly be? Sai and Shrey started down this line of thought.)

9. Yes, though I probably shouldn’t mention “convex hulls” here. (Wrap a rubber band around a heart. Are there gaps? Are the lengths the same?)

5. (finally!) I don’t know about families, but I can think of some properties of the cardioid that are different from other 2D shapes. The heart is a closed curve. It has two “vertices” where it’s not smooth. The heart curve doesn’t intersect itself, though a bowtie does. A semicircle is a convex shape, but a heart isn’t. (Convex means nothing “points in”. A kite is convex, and arrowhead isn’t. A test for convexity involves imagining putting a light inside the shape and seeing if any of the perimeter is in shadow. If there are no shadows, no matter where you put the light, the shape is convex. Where would you put the light to show the heart isn’t convex? How does this relate to convex hulls?)

These are deep questions! Sir Isaac Newton is regarded as one of the best mathematicians to have lived because he defined smoothness. You’ll come across these ideas again when you do calculus at A-level. A thorough treatment of topology will probably have to wait until you do a maths degree. Non-convex and self-intersecting shapes cause a lot of problems in computer graphics (what’s “inside” a bow tie? How do you tell a bow tie from a rectangle?).

Hope this helps rather than confuses!

We are Shrey and Jack B from the maths group that are debating this subject .

Our thought on our post are slightly different to yours. We think for number:

1 and 2 a heart is a shape because it has many properties of other shapes for example lines of symmetry and side and vertices. It can’t be a line that meets because there are two lines.

3 and 4 if a heart is a semicircle but irregular if you split it in half it’s going to become two semicircles but is not the same as the heart.

I’m glad you’re having your own thoughts!

I think the confusion is about what a “line” is. I’d call a semicircle as being made of one line, as you can draw it in one go without taking your pencil off the paper. I’d also say a square is one line. But a semicricle has one straight line segment and one curved segment, and a square has four straight line segments. But despite how you define “line”, you can still talk about the other properties.

I don’t understand what you mean with your “3 and 4” paragraph. Could you please think of another way to say it?

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.

Hello 5SH Maths group! It is great to read all your thoughts about shape – it is really important to be ‘curious’ about maths, rather than just learning rules. Have you looked at string art? I did a search on string art cardioid and found some amazing pictures, showing how straight lines can result in curves, or what appear to be curves. One other thought – if you draw shapes on the surface of a ball, are they 2-D? I’m a maths teacher, but I certainly don’t know all there is to know about maths and you have made me think! Keep enjoying your maths!