Geometry

# Angles

## Concept and Language (Types of Angles)

Material Description:

For the oral introduction a length of cord with a mark for the angle.

The Box of Sticks:  This is a box which is divided and contains a collection of:

Wooden sticks

Plastic Arcs

Pins and Tacks of different colours

Nails

Paper Fasteners

Plumb Line

Plane is a board made of cork, for the plane, it should not have edges, measuring tool (right angle (snipped on the hypotenuse to distinguish is from a triangle, it is called a measuring angle).

Method:

### Presentation of the Angles

Say to the children, ‘Look, we’ve cut off a piece of the environment.  Would you like to move your hand up, we’ve cut off a different part of the environment’ Ask one child to move, ‘now we’ve cut off a different part of the environment’.  The part of the environment we have cut off is called an angle.  The word angle is a very old world, we think it comes from a very Old English word ‘angul’ which means fisherman, what do fishermen use? (a rod, and angle) Show the children a fishing hook with a cork.  In Sanskrit the word ‘angle’ means limb, we can use our bodies to make angles, can anyone make an angle with their body?  Now lets make angles on the plane.’

Remove two sticks of different lengths and attach them on the board with a pin, the pink in two places, the longer blue one on top, attached at one side. ‘We are going the make an angle, an angle is really about the amount of turn’ .  Turn the top line once completely. ‘If I make a whole turn I make a whole angle’. Repeat the turn a few times, the direction doesn’t matter, turn the lines back to the initial 12 o’clock position

If I make a half turn I make a straight angle’, turn the top line 180 degrees.

‘Now I am going to make a quarter turn’. Turn the line 90 degrees, ‘We call this a ‘right angle’, it probably comes from the Sanskrit, when they talk about someone lying flat on the ground, and then sitting up, and they describe this as a ‘right angle’’. Here I have a special angle called the ‘measuring angle’, we check that it is a right angle.’  Place the measuring angle under the lines, then reorient the lines.

‘If I make a turn less than one quarter I have made an acute angle, it comes from the Latin acus, which means ‘needle’, let’s check with the measuring angle that this is less than a right angle’.

‘If I make an angle larger than the right angle but not as big as the straight line, this angle is called an obtuse angle, which comes from the Latin ‘obtuses’, which means to make blunt’. Check with the measuring angle and reorient the lines.

‘Let’s recap on all the angles’, make an angle and ask the children to identify them, try all five.  Then do a loose three period lesson, making angles and asking the children to identify them, always reorient the lines. Repeat the idea that, ’the essence of the angle is the amount of turn’.

### Represent the angles on the plane

Represent each of the angles you have shown simultaneously on the plane.  Ask the children to select a type of angle, and show them how to form it with you, using two lines, show them to use the hammer to pin it and check it with the measuring angle.

With a pencil and back strips show the children how to write the labels for the word names of the angles.  The children read one name and ask a volunteer to label the angles on the plane.

‘How many right angles are in a whole angle? (4) show the children moving the lines.

How many right angles are in a straight line? (2)

What is the relationship between the obtuse angle and the right angle?’ (it is larger)

‘How many right angles are in an obtuse angle?’ (1)

‘What is the relationship between the acute angle and the right angle?‘ (it is larger)

Say, ‘Whenever we make an angle, I can make more angles.  A reflect angle is greater than a straight angle, but less than a whole angle.  A right angle creates a reflect angle, an obtuse angle creates a reflect angle and so does an acute’. Indicate the external angles of each angle, apart from the straight angle.

Notes:

If the children have started polygons they might have met the measuring angle

The reflex angle is not given yet because it is the product of two angles

After the lesson:

The children can make the angles on paper.  To introduce this take two sticks, one with a series of holes along it.  Put the point of a pencil and draw a line using the second hole from the top and form an obtuse angle.  Put the pencil in another hole and draw a line parallel with the previous one, demonstrate towards the right and left to show the direction isn’t important.  This shows that, ‘The length of the stick is not what matters it is the amount of turn.’ Give other examples orienting the lines in different directions.

The children can work with the nomenclature cards and the box of sticks independently.

The children can draw and decorate angles

They can explore the angles of the polygons they know

## Measuring Angles

Material Description:

Montessori Protractor, a frame divided into 360 degrees, the white line corresponds to the 0 mark and to 360.  The circler fraction material, 3 compasses with a lead pencil, blue pencil and red pencil.

Method:

‘Today we are going to see how we measure angles.  When we do work in mathematics, we work in the base 10, it is very important to us.  When we work with angles we use a different form of measurement which is very, very old.  Do you remember the Babylonians?  They were interested in many things, one of the things they were interested in was the study of the stars. They watched the stars every night and they discovered that when they watched a particular star it was at a different position each time.’

Stand and say, ‘it was at a particular position’ many times while moving round.

‘After some time it came back to the first place, the Babylonians noticed it made a particular path, it had a circular orbit, going all the way round from where it started, It is where we get the idea of a circle from.  The Babylonians used their observations of the stars to make the calendar.  Their calendar had three-hundred and sixty days in a year.  Considering it was so long ago it was quite an accurate calendar.  How many days do we have (365).  The total journey of the star was made up of 360 little marks.’

‘Let’s look at this piece of equipment it is called a ‘protractor’, we will use it to measure angles.  The name protractor comes from the Latin ‘pro’ meaning forward’ an ‘trahere’ meaning to trace.  We measure angles in parts we are called degrees. Can you say that word? It comes from the Latin ‘de’ meaning one and ‘gradus’ which means a step, so degree is a step, like the steps the star took in the sky.  When we want to write degree we make a tiny, little circle.’ Draw a degree symbol.’

‘Let’s look at out protractor, how is it made up? it starts at ‘0’  and goes all the way back to ‘0’, we can count it in tens, ’10’, ’20’, ’30’ and so on.  We also have the smallest little while line, they are for the unit steps, we also have some lines which help us count up in fives.  We have a white line which goes from the centre and points to the ‘0’ degree, this is also 360 degrees.’

‘We are going to use this protractor now to measure our angles.  I will begin with the thirds’.

Take a metal inset of 1/3, say, ‘I have to line up one side with the ‘0’ line and the vertex must coincide with the centre, what does it look like I’ve got?  It looks like I have an angle of 120 degrees.’

‘Let’s try with the sixths’. Ask the children to help you remember how you place the metal inset.  Give further examples leave 1/7, whole and half till last, using only one inset.  The 1/7 results in half a degree and the last two leave important impressions.

Redo the placing of 1/3, this time writing a ticket with    , and 1/4 with a ticket writing    .  Continue writing tickets for each of the ten metal pieces, after the others record 1/7 as    and a whole as

When to give the lesson:

After the work with parts of the angle with the stick material.

After the lesson:

The children can continue to explore, taking two insets at a time, writing their own tickets

They can draw them on paper and write their measurements according to the Montessori Protractor.

They can add fractions together systematically

Introduce the regular protractor, have different sizes of protractor available. Children can then draw angles on paper and measure them with the conventional protector (there is no set Montessori presentation for this

## Parts of an Angle

Material Description:

The Box of Sticks:  This is a box which is divided and contains a collection of:

Wooden sticks

Plastic Arcs

Pins and Tacks of different colours

Nails

Paper Fasteners

Plumb Line

Plane is a board made of cork, for the plane, it should not have edges, measuring tool (right angle (snipped on the hypotenuse to distinguish is from a triangle, it is called a measuring angle).

Method:

Attach paper to the plane and ask the child to pin the sticks to make an acute angle.  Say, ‘Now we are going to look at the parts of an angle, this is a side and this is a side, the point where the lines meet is called the vertex, from the Latin ‘vertare’ which means ‘to turn’’.  Ask the child to colour in the part between the sticks. ‘This part that we coloured in is called the amplitude’.  Write the terms, ‘side’, ‘side’, ‘vertex’ and amplitude’, ask the children,‘’What is the point where the sides meet called?’ and ask the child to place the label, ‘What is the name of this part here?’ and ask the child to place the label.

Notes:

The part of an angle which gets bigger, it’s side is called the amplitude.

For the terms ‘sides’ the term ‘arms’ can also be given.

When to give the lesson:

After the concept and language has been given.

After the lesson:

Now it is possible to do the work with the polygons

Independent work with the box of sticks, colouring, labelling and making there own charts.

Using the ‘Parts of Angles’ nomenclature

## Operations (including bisecting an angle and construction of angles)

Material Description:

The circle fraction material, Montessori protractor, paper and pencil, skittles for the division

Method:

Say, ‘Today we are going to add these angles’.

Place 1/3 into the Montessori protractor and read the size 120 degrees, remove it and place 1/4, read the size 90 degrees, then place them both, say, ‘an angle of 120 and 90 gives me an angle of ‘210 degrees’.

Say, ‘Now I am going to do it again and this time write tickets’. Measure the 1/3 in the protractor, say, ‘I have an angle of 90 degrees’, Write on a strip an angle symbol and 90 degrees, and an angle of 1/5, ‘Write a ticket with the symbol and 70 degrees

‘What is addition about, it is about putting together and these two are together in the frame, together they have an angle of 160 degree’.

90  =         70        160

After give an example with 1/7 showing the decimal notation.

### Subtraction

Say, ‘Suppose I have an angle of 180 degrees and I want to take away an angle of 120 degrees?

Write the problem

Put the minuend in the Montessori protractor and because we only ever put out the minuend, we never represent the subtrahend,

EITHER

skip count backwards, ten at a time, marking it with your finger, the amount of the subtrahend to calculate the difference and record it

OR

rotate the angle anti-clockwise through the ‘0’ the number of degrees of the subtrahend skip counting in the tens, and read and record the new figure.

### Multiplication

Place the 1/4 angle into the protractor

Say, ‘What is multiplication?’ (We take a number a certain amount of times)  ‘Today I am going to taken it three times’. Write the problem

Say, ‘Here is my angle of 90 degrees taken once’, indicate the angle, I take it two times’, place another 1/4 into the protractor, ‘and a third time’ place another 1/4 angle into the protractor.

### Distributive Division

Place the whole angle in the Montessori Protractor, say, ‘I want to find 1/3 of a whole.  I will need three skittles, what angle do you each skittle will need?’  (the children suggest 1/3)

Place the thirds in the green frame for a whole, say, ‘yes it fits’, divide the pieces among the skittles.  Find how many angles does each skittles has by measuring it in the Montessori protractor.  Record 120 degrees.

### Group Division

Place the 1/2 angle into the protractor, this is what I have 180 degrees

Say, ‘I have an angle of 180 degrees and what I want to find out is how many groups of sixty degrees are contained in 180 degrees ’ Write the problem ‘What fraction has a hundred and eighty degree angle?’ Indicate the straight angle. (the children who have suggest 1/6) Say, ‘Let’s see how many of these I can get into the space in the protractor. How many groups did we succeed in making? (3) record this to introduce the idea of the check.

‘The answer is always the share of one unit’, take 1/3 and measure it in the protector, ‘so my quotient is 120 degrees’ record the answer.

Notes:

This reinforces the idea of measuring angles in degrees and the idea of operations.

The recording can be done on blank paper, squared paper or strips.

When to give the lesson:

After fractions work, giving the opportunity to review fractions and repetition through variety for work with angles

## Using an ordinary protractor

Material Description:

The circle fraction material, Montessori protractor, pencil, ruler, plain paper

Introduction

Find a 180 degree protractor which fits into the Montessori protractor.

The children can measure any other the angles they know.

Measuring an angle

Start with an example the children know. Place an angle of 1/3 into the Montessori protractor, lining up the vertex with the centre, then draw a line on paper, take the ordinary procreator and place the horizontal line over it and count backwards to 120 degrees, mark it.

Constructing an angle

Start with an example the children know. Draw a line, place the ordinary procreator and place the horizontal line over it, make a mark above the number of degrees you want to make an angle and join in.

## Bisecting an angle – of 120 degrees

Material Description:

The circle fraction material, Montessori protractor, two ordinary protractors with a red and a blue pencil in each, pencil, ruler, plain paper

Method:

Take the frame for 1/3. Draw an angle of 120 degrees in lead pencil. Take a blue compass and construct two arcs, placing the point of the compass on the vertex.  Now take the compass with the red pencil and align it with the blue one, make two more arcs with the red compass, placing the point on the blue arcs.  Draw a line between the vertex and where the red arcs meet, this bisects the angle.

Now you can say and write it, ‘I had an angle of 120 degrees and I divided it by two and now I have an angle of 60 degrees’.

After the lesson:

Give Command cards on constructing and bisecting an angle

Geometry

Geometry

Geometry