Showing posts with label maths investigation. Show all posts
Showing posts with label maths investigation. Show all posts

Sunday, 2 December 2018

Making Geometric Stars: An Investigative Christmas Maths Teaching Sequence

So Christmas is upon us and, as a conscientious teacher, you don't want the learning to stop (or the craziness to begin), but you also want to make the most of the Christmas context/the kids are pestering you for a 'fun' lesson. You've got something for writing (please don't let it be letters to Santa - so much wrong with that - go with writing a list of presents they'd like to give others, preferably not bought ones either) and RE is sorted - so's reading as there are so many Christmas-themed texts out there.

But what've you got for maths? Some Christmas-themed word problems? If I eat 24 mince pies over the Christmas season and mince pies come in boxes of 6, how many boxes of mince pies have I eaten? Bit tenuous. Let not mighty dread seize your troubled mind. Try this investigative teaching sequence which culminates in making some lovely geometric stars:

Step 1: Investigate the size of internal angles in regular polygons

Begin with triangles, discussing what they already know about the sum of the internal angles in any triangle. A tangent here could be to check a number of different triangles, not just equilateral ones, to see if indeed all the angles add to 180 degrees - doing this will also provide important protractor use practice for later on. It might be worth pointing out that the half-circle protractor also measures 180 degrees.

Move onto squares, rectangles and other quadrilaterals. Again, call on prior knowledge: all of a square's internal angles are right angles, right angles are all 90 degrees and that multiplying 90 by 4 is 360 so a square, and a rectangle, has interior angles which add up to 360 degrees. You could mention at this point that circles also contain 360 degrees. You could also get them to check a range of different quadrilaterals, reinforcing their names and other important shape
vocabulary.

Although probably too early, you could ask children to make conjectures about what the internal angles of a 5-sided shape - a pentagon - might total. Some may point out that from the triangle to the square the number doubled so may predict that a pentagon's internal angles might add to 720 degrees. Others may point out that from the 3-sided shape to the 4-sided shape the number of degrees increased by 180 and therefore predict that the pentagon might have internal angles totalling 540 degrees. If both of these conjectures are brought up you can discuss how it is too early to be sure of any pattern and that it is important to continue testing the hypotheses.

Provide a printed sheet containing at least a pentagon - it's also worth including a hexagon, a heptagon and an octogon (you could even include a nonagon and a decagon). Allow the children to further investigate the sum of the internal angles in these shapes. At the same time ask them to create a table to record their findings.

Shape
Number of sides
Each angle (in a regular shape)
Sum of internal angles




Triangle
3
60 o
180o
Quadrilateral
4
90 o
360o
Pentagon
5
108 o
540o
Hexagon
6
120 o
720o
Heptagon
7
128.57… o
900o
Octagon
8
135 o
1020o

After measuring and totalling (this could be by multiplication or repeated addition - discuss efficient methods) the internal angles of each shape they should make further conjectures about what the next one will total - before moving onto the hexagon they should be almost certain that each time a side is added to a shape another 180 degrees are added to the sum of its internal angles.

It should be noted that some of their measuring almost certainly won't be accurate and that some mathematical reasoning will have to be applied, for example:

"If I measured one angle as 107, another at 109 and another at 108, which is most likely to be?"
"I've predicted the sum of the internal angles to be 720 degrees but it is coming out as 723 degrees - which is wrong, my prediction or my angle measuring?"

Once findings are recorded in a table it becomes easier for children to begin to find a way of expressing a rule for finding the sum of the internal angles of a shape with any number of sides. I have worked with year 6 children who have managed to generate a formula for this. Even if they cannot yet write it down, some will be able to verbalise the rule:

"Number of sides subtract 2, then multiply that by 180"

To get to this point it helps to talk about the triangle being the first shape, the quadrilateral being the second shape, the pentagon being the third shape, and so on. With this as a starting point they can generate something like this:

Shape
Shape Number
Number of sides
Sum of internal angles




Triangle
1
3
180o
Quadrilateral
2
4
360o
Pentagon
3
5
540o
Hexagon
4
6
720o
Heptagon
5
7
900o
Octagon
6
8
1020o

Now that they have the shape number next to the number of sides in the shape they will much more easily be able to see that the difference is two therefore subtracting two from the number of sides results in the number that 180 must be divided by to find the sum of the internal angles.

If children don't have prior experience of writing this as a formula they can be shown how to record this:

Sum of interior angles = (n-2) × 180° (where n = number of sides)

And that each angle (of a regular polygon) = (n-2) × 180° / n

Step 2: Investigate drawing stars within circles

Nrich has a couple of great activities for this:

Path to the Stars: https://nrich.maths.org/1097 (their printable resources page has circles with pre-printed dots on it for this ativity: https://nrich.maths.org/8506)

Stars: https://nrich.maths.org/2669 (this is an interactive resource)

Round and Round the Circle: https://nrich.maths.org/86

In these activities it is worth drawing out rules such as:
  • if you draw a line straight to the next dot you get a regular polygon
  • with an odd number of dots, if you a draw a line which skips a dot you get a star shape
  • with an odd number of dots which isn't a mulitple of 3, if you draw a line which skips two dots you get a star shape with longer points than when you just skip one dot (doesn't work for 5 dots as there aren't enough dots - skipping two is the same as skipping one in the opposite direction)
  • if you skip just one dot when there are an even number of dots you get a regular polygon with have the number of sides and vertices as the the original number of dots
There are many more rules such as these to notice and discuss.

You could also experiment with Nrich's Mystic Rose activity (another great interactive resource: https://nrich.maths.org/6703) which does more than just create star shapes.

Step 3: Practise drawing regular shapes using a protractor


Model to children how to draw the shapes from step 1 using the angles they discovered and by deciding on a particular side length. Impress upon them the importance of accuracy in measurements - perhaps demonstrate how even being a few degrees/millimetres off once or twice will result in an irregular shape.

Children should mark a starting point and draw a line of the side length they have decided. Then they should measure the internal angle according to their findings in step 1 and draw a second side of the same length to the shape. This should be repeated until they reach their starting point again.

It will be best to do this exercise on scrap (and/or large) paper as often the children will find that their chosen side lengths lead to their shape becoming too big for the paper! In this case they will have to readjust and start again.

Step 4: Make stars!

Once children have mastered the drawing of regular shapes in step 3 they can move onto making their stars. Give children coloured/decorated card in festive hues/patterns to draw their regular shapes out onto.

From protractor to tree!

Once they have drawn out their shapes they can use what they experimented with in step 2 to join the vertices of their shape in different patterns to form stars.

If you carried out the Mystic Rose activity in step 2 you will need to ensure the children can identify a regular pattern where they will cut (there will be many options). It is a good idea to give them a pen to go over the lines that they want to cut before they take their scissors to their carefully drawn out shape. The mystic rose patterns that they have drawn will provide interesting decoration to their finished stars.

Alternatives:

If you don't have time to follow the whole sequence, or teach children too young to be able to do all aspects of it, there are alternatives to the above sequence which avoid the lengthier steps 1 and 3:
  • Just do one of the activities from step 2 then make large print outs of the Nrich templates on coloured card so that children can make their stars.
  • Teach children to use a pair of compasses to draw their own circles then teach them to use a protractor to divide the circle into equal sections by dividing 360 by the number of points they want ( 5 points = 72°; 6 points = 60°; 7 points = 51.42...°; 8 points = 45°; 9 points = 40°; 10 points = 36°). They can then use these to create star or Mystic Rose patterns on coloured card to cut out.
A colourful hexagonal mystic rose pattern - the green part will have been cut off to form a star.

A well-drawn nonagon with mystic rose pattern and a heptagon with mystic rose pattern that has already been transformed into a star.

If completed on plain card children can decorate their stars - perhaps in more festive colours than this one!

If children can draw a perfect hexagon then they can also make and fold shapes which can be cut into snowflake shapes which will have a realistic six lines of symmetry.