Congruency, Similarity and Equivalence

Small metal insets

Material Description:

There are 9 square metal insets.  Base is white, inset is red, border is green.

  1. Complete square (one used with both sets)
  2. 8 Divided squares
    1. Squares divided into triangles
      1. Square divided along 1 diagonal
        1. result is 2 right-angled isosceles triangles, each of which is half the whole square
      2. Square divided along 2 diagonals
        1. result is 4 right-angled isosceles triangles, each of which is one quarter/one fourth the whole square
      3. Square divided along 2 diagonals and the mid-points of opposing sides are joined
        1. result is 8 right-angled isosceles triangles, each of which is one eighth the whole square.
      4. Square divided along 2 diagonals, opposing mid-points joined and adjacent mid-points joined.
        1. result is 16 right-angled isosceles triangles, each of which is one sixteenth of the whole square.
    2. Squares divided into quadrilaterals:
      1. Square divided by one pair of opposing mid-points
        1. result is 2 congruent rectangles each of which is one half of the whole square
      2. Square divided by both pairs of opposing mid-points
        1. result is 4 congruent squares each of which is one fourth/one quarter of the whole square
      3. Square divided by mid-points of opposing sides and bi-sectors of the squares (on one pair of opposite sides)
        1. result is 8 congruent rectangles, each of which is one eighth of the whole square.
      4. Square divided by mid-points of opposite sides and bi-sectors of squares using both pairs of opposite sides
        1. result is 16 congruent squares, each of which is one sixteenth of the whole square.

Congruent: (a broad definition) is identical in every respect: same size, same shape.

Similar: this is more difficult to define.  Two geometric figures are said to be similar when they are identical in one or more aspects, such as angles in the same ratio, sides in the same ratio.  They are always different sizes but they must have the same shape.  This is the definition we use with the children. (~)

Equivalent: this is the huge exploration that occurs in geometry.  Equivalent figures are alike in some particular way, they have the same value, cover the same area, but they have different shapes.

Method:

Congruency

Ask the children to describe the first pair of triangles, show that they are as large of the square and rectangles.

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Superimpose the two large triangles and show that each point, line and angle corresponds, ‘They identical in every respect, the same in every way, so lets just leave them out’.

Leave them together, outside the frame, so it is clear they are the same.

Superimpose the two large rectangles and show that each point, line and angle corresponds, ‘They identical in every respect, the same in every way, so lets just leave them out’.

Leave them together, outside the frame, so it is clear they are the same.

Superimpose the two large squares and show that each point, line and angle corresponds, ‘They identical in every respect, the same in every way, so lets just leave them out’.

Leave them together, outside the frame, so it is clear they are the same.

‘We have discovered that the triangles, rectangles and squares are the same in every respect, when you have two shapes which are identical in every way, we say they are congruent.  We can say the triangles are congruent to each other, the rectangles are congruent to each other and the squares are congruent to each other’

Replace all six pieces

Give a loose three period lesson; isolate any piece and say the child, ‘I am going to choose a rectangle, Can you find one which is congruent, Why do you think they are congruent?’  Give other examples.  When all the pieces are in the frame ask the child, ‘Can you take out two congruent rectangles. How do you know there are the same?’ Give other examples, using all the figures and sizes.

Similarity 

Remove the 1/2 and 1/8 rectangle.

Ask the child what they notice

‘Could we say that every point corresponds (no) what about every line (no), Do the angles correspond? Holding both shapes superimpose them and move the line shape around, meeting at each angle.  Say, ‘So here we have two rectangles, they have the same shape, they don’t have the same length of sides but they do have the same angles.  If this happens we can say the shape is similar.’

Do the same for the triangles 1/4 and 1/16, with she same questions, conclude, ‘Because they have the same shape and their angles correspond we can say that they are similar’.

Take the 1/4 and 1/16 square, say, ‘They have the same shape, not the same value, the angles correspond, so they are similar’.

Give a loose three period lesson,

Take one shape and ask the child to find a similar one, again ask the child to justify her choice. Recap the characteristics of similar shapes while you do this.

Equivalence

Recap earlier work, ask the child what congruency is, to give examples and to justify her choice.  Do this with similar shapes too.

Say, ‘Today we are going to talk about something new’.  Take out my whole square and move it’s frame forward. ‘I am going to put the large triangles into it’s frame.  They now take up the same space, they are the same, except each triangle is half of the whole (swap the triangles for rectangles) ‘They are the same except the frame, except it is divided into two and they are each half of a square’.

Take out one triangle and the rectangle.  ‘What can we say about these, they are both the half of the square so they have the same value’

Superimpose and ask, ‘Does each point correspond (no) does each line correspond (no) does each angle (no).  But we know they have the same value.

‘If I were to superimpose them and transform them could they have the same shape? Is there any possibility that I could take this bit off my triangle and reattach it here’ Lets try’ . Place the triangle behind the rectangle to reveal the missing small triangle, show that a small triangle is the same as the bit missing from both the large rectangle and triangle.

Continue to arrange many figures to show equivalence, for example take the small triangle and with three other small triangles build up a rectangle the same size as the original one, or use two small triangles and a square to form the rectangle and the large triangle, continue experimenting with the child.

‘So what can we say about these figures? They are the same value, they are not the same shape, we have seen that each is half of the whole, we can say that these are ‘equivalent’, we call figures with the same value but different shapes ‘equivalent figures’.

Give a loose three period lesson,

Take one shape and ask the child to find one which equivalent

Notes:

  • Begin with the large figures as these are easier to transform and later can be transformed on paper

After the lessons:

The child can cut out the rectangle and triangle on paper to reform it.

Key:

To further work with congruence, similarity and equivalence

Notes for all concepts:

  • The whole square belongs to both sets so it is place along side both.
  • The concepts are given in short presentations, one after the other, but without much of a time lag.
  • While the work is simple, the ideas are not, the work appeals to the child’s intellect, these are fundamental concepts which deal with figure relationships, these kinds of concept appeal to the child of the second plane (social relationships and the search for the reasons behind the facts.
  • Design work is given to reinforce basic, fundamental concepts.
  • While we use the concrete material, the ideas being used here are only accessible to the intellect. Firstly, the concrete experience, then the language and finally opportunities to acquire the concept through the child’s own independent work; having given the key we let the children explore independently.
  • Congruency is given first, because it is the most concrete and it serves as a reference point for the other concepts and relationships.  Equivalency is the least obvious concept but is the most important in further work, it brings all the shapes together

When to give the lesson:

Ideally this is the first work at Elementary in Geometry, the children will require the knowledge of the names of the figures and the names of parts of the figures.  Knowledge of the concept and notation of fractions, though not to have abstracted the operations involving fractions.

The child new to Elementary could first do art work filling surface area with points and lines and printing to show that points and lines can meet.

After the lessons:

Children use the frames themselves, they can do drawing, cutting and pasting activities and can be helped to write descriptions.

The children can paste on to charts or into booklets.

They can draw and colour figures.

Children continue to do design work, e.g. use the triangles to build a ‘fir-tree’ shape and rectangles to build a ‘city-scape’.

Children can place figures under written labels from the Nomenclature, or write the labels themselves.

Show the child the handout to show they can explore equivalence with none-geometric shapes.

Follow up work:

An introduction and exploration of the signs

  • The concept is further explored with the constructive triangles.
  • Introduce the signs as later work – exploring the 1/2 rectangle and four 1/8 arranged as a rectangle, they are congruent, bring both halves and put the sign for congruent ≌

Rearrange the four 1/8 triangles and compare them to the 1/2 rectangles, these shapes are no longer congruent but are equivalent, give the sign for equivalent≣

Then show two similar shapes and give their sign.∾

  • This work prepares for later work in area and volume.

Divided Triangles

Material Description:

4 metal inset plates:

  1. Equilateral triangle (undivided)
  2. Equilateral triangle divided along the altitude to form two right-angled scalene triangles.
  3. Equilateral triangle divided along the bisectors of the angles to form three obtuse-angled isosceles triangles.
  4. Equilateral triangle divided by joining the midpoints of adjacent sides to form four equilateral triangles.

Method:

Part 1

Ask the child, to describe each one, adding to the description the following ideas;

  1. Proof for equilateral is rotation
  2. This looks like an equilateral but it is divided along the altitude into two right angled scalene triangles
  3. This looks like an equilateral triangle but it is bisected along the diagonal, giving three obtuse angled isosceles triangles
  4. This looks like an equilateral triangle but it  is divided by joining the mid points of the sides to form four equilateral triangles.

Part 2

Review the triangles adding the following;

2. I want to have a look at something, what can I say about my to right angles scalene triangles, they are each a 1/2 of the whole,

3. What about my obtuse angled isosceles, they are each 1/3 of the whole

  1. Now lets look at the triangle divided into four, each are 1/4

Part 3

Ask the child the following questions, ask her to make a few examples and review what makes the examples accurate as necessary

‘Do you think you can find any congruent shapes?

What about similar figures, can you find any?

Bow I am looking for some equivalent figures? form a kite with the right angled scalene

Part 4

The children continue to make examples and explore of the table and then they use paper.

Display the charts to help the child with her own work

Aim:

To give the key concepts of congruency, similarity and equivalence

Preparation for later work with area and volume

Notes:

The charts were used by Montessori before the metal insets were invented

When to give the lesson:

This work continues throughout Elementary, ages 6-7 to introduce concepts and above 7 for free exploration and expression of the concepts.

Constructive Triangles

Triangular Box

Material Description:

A whole grey equilateral triangle

A green equilateral triangle divided into halves by a line from the apex to the midpoint of the base, we get two right-angled scalene triangles.

A yellow equilateral triangle divided into thirds when all the angles are bisected and the bisectors meet in the centre, we get three obtuse-angled isosceles triangles.

A red triangle divided into fourths by joining the mid-points of the sides, we get four equilateral triangles.

Ten green extra copies of the green right-angled scalene triangle.

These boxes lend them selves to further exploration of congruency, similarity and equivalence.  The five boxes will have been used by the children in Casa, making figures by joining them along the black lines and using the names of the figures.

The first series of the constructive triangles is all five, at Elementary the child uses the second set, the Triangular Box and the Large and Small Hexagonal Box, she used these to develop her consciousness of the constructive role of the triangle.

Method:

Introduction, recapping the work at Casa

Begin by asking the children to remove the triangles from the boxes, grouping them according to colour, joining them along the black lines and identifying them.

Superimpose the grey to verify that they are three identical equilateral triangles.

Recap fractions, terminology and give precise descriptions

Discuss the fractional valve saying, ‘We have a whole grey triangle, we have a whole triangle divided into thirds and a whole triangle divided into fourths.’ Indicating them one at a time.  ‘We have a whole green triangle divided into halves with a line from the apex to the mid-point of the base, we call this the altitude of the triangle.  What does that make? (two scalene right angled triangles) We have a whole yellow triangle divided into thirds, all the angles are bi-sected What does that make? (three obtuse angled, isosceles triangles) We have a whole red triangle divided into fourths by lines on the mid point of the sides.  What does that make?’ ( four equilateral triangles).

Grey and Green triangle – equivalence

We are interested in the grey and green triangle which are both equilateral triangles, the green have been divided into two tight angled scalene triangles.

Looking at the right angled triangles, ask, ‘What could we say about these? (congruent)  Remember that we were talking about equivalency that if things have the same value but different shapes.  Let’s see what I can make if I ignore the black lines and turn one upside down’ Form a rectangle, two parallelograms, isosceles triangles and kite, asking the child if it is equivalent to they grey triangle after the first shape. ‘I have now made five figures which are all equivalent to the grey equilateral triangle’

‘Now we are going to explore the properties of equivalency’

Aim:

Further exploration of the concepts, especially equivalency

Notes:

Each of the four triangles relates to the Divided Squares

This is given as a group presentation

When to give the lesson:

After the concepts of congruency, similarity and equivalence have been given with the divided figures, a knowledge of fractions.  The children can classify triangles and parts of triangles from their work in Casa and introduction to Elementary (including, the terms apex, mid-point, base)

Properties of Equivalence

These are explorations of the concepts, not specific presentations giving rise to further independent work.

Material Description:

Triangular Box, supply tray, ten additional green triangles

Method:

Reflexive Property A=A

Ask the children to make the equivalent figures to the grey equilateral using the green triangles.

Form the five equivalent figures with the additional green triangles.

Symmetric Property A=B, B =A

With the grey triangle and one equivalent green shape, use the paper to write labels, say, ‘I am going to call my grey triangle A and my green triangles B. We know that A is equivalent to B’ Place the letters and symbols.  ‘I am going to move these figures around.  Exchange one green shape for another Is it still true, does A still equal B?’ (yes).  Use all the figures and ask, ‘Why is A  equivalent to B?’, (because the figures have the same value).

Give the term Symmetric Property

Transitive Property A=B, B=C, C=A

With the rectangle and parallelogram, made from green triangles as before and the grey equilateral triangle, give the labels ‘A’, ‘B’ and ‘C’, (the grey triangle remains B because we have established that it is equivalent to the green figures).  Ask the children, ‘What can we say about our rectangle and our equilateral triangle?’ (They are equivalent) Put the sign for equivalent. ‘What can we say about our parallelogram and our equilateral’ triangle (they are equivalent). Put the sign for this. So, ‘What can we say about our rectangle and our parallelogram?’ They are equivalent too).

Arrange the tickets,  ∴  A=B, A=C, B=C

Give the term Transitive Property

Using the grey equilateral triangle as the key figure the children explore Symmetric and Transitive Properties, using the other figures made with the green triangles.  Firstly, only change the parallelogram, later change both figures.

Key:

This work gives a key to; Reflexive Property A=A, Symmetric Property A=B, B =A

and the Transitive Property A=B, A=C, B=A.

Aim:

Further exploration of the concepts, especially equivalency

When to give the lesson:

Later than the Constructive Triangles, with older children who have explored these concepts and can use the symbol for equivalency. The children are now familiar with more the terminology.

After the lesson:

The children continue to explore,

Follow up work:

Work with mediators

Relationship of the Lines

Material Description:

Grey equilateral triangle and ten green scalene right angled triangles

Method:

Recall with the children the equivalent figures made with the additional green triangles

Say, ‘Lets have a look at the rectangle, let’s look at the lines, what I want to know is there a relationship between the lines on the grey equilateral triangle and the lines we have identified on our rectangle’.

Superimpose one green triangle on the grey triangle to show that the base of the rectangle is half of the base of the triangle and their altitudes are the same.  Say, ‘We can declare that the rectangle is equivalent to the triangle when the base of the rectangle, is half of that of the triangle and the altitudes are the same.’

The children can go onto explore the relationship between all the other figures based on the relationship of line.

Charts T9,11, 12 and 13 are both a reminder and a source.

Aim:

To explore the relationship between the lines of the figures

Indirect preparation for insets of equivalence

When to give the lesson:

When the children work with properties and the work with lines.

Notes:

Here exploration with equivalency is more important than the other two concepts, as it is used for area and volume.

Follow up work:

Cutting and pasting, constructing figures using geometrical instruments (especially the ruler), drawing and writing the statements declaring equivalences.

Large Hexagonal Box

Material Description:

A yellow equilateral triangle (same as for the Triangular Box)

Ten obtuse-angled isosceles triangles (same as the thirds in the Triangular Box)

Six are yellow, two are red and two are grey.

Method:

Introduction

Ask the children to remove the triangles and group them according to colour, and then ask the children to make the figures they know using the black lines.  The children form the hexagon, equilateral triangle, rhombus and parallelogram, ask the children to name them.

Lines

Tell the children, ‘Let’s look at the lines, in the hexagon I see diagonals, in the triangle I see angle bisectors, in the rhombus I see a major diagonal and in the parallelogram I see a minor diagonal’.

‘I am going to take my yellow isosceles triangles and see what I can make with them, using two at a time and ignoring the black lines’, form a deltoid, rhombus and parallelogram. 

Now I will see what I can make when I use three isosceles triangles, (take the grey,red and yellow ones) to makes a I have made an isosceles trapezium, an equilateral triangle and an obtuse angled trapezium.  I am going to stack all of my isosceles triangles, what can I say about them?’ (they are congruent)

Explore the relationship between equivalent figures 

Part A

‘Let’s make figures using two yellow isosceles triangles’ the child forms figures and names them.

‘What can we say about these three figures’, each are made of the same two congruent triangles so they are definitely equivalent. Now we are going to explore the lines, the relationship between the equilateral triangle and each of these figures in turn.

Systematically take the rhombus and place it adjacent to the triangle, ask ‘What can I say about the relationship between the lines, the major diagonal of the rhombus is the same as the base of the triangle’, move the triangle over to prove it.

Now let’s try with the parallelogram, we could say that the major diagonal of the parallelogram is the same as the base of the triangle’.  Move the triangle to check, each time.

Now let’s compare with the deltoid, the longest side of the deltoid is the same length as the base of the triangle.

Now I have compared all three of these with the triangle, lets compare the rhombus with the parallelogram, the short sides are the same, the major diagonals are also the same.

Now let’s try the rhombus and the deltoid, the axis of symmetry of the deltoid is the same as the sides of the rhombus, the major diagonal of the rhombus is the same as two of the long sides of the deltoid.

Now, let’s compare the parallelogram and the deltoid, the major diagonal of the parallelogram are the same as the long side of the deltoid and the sides of the parallelogram is the same as two of the shorter sides of the deltoid.’

Part B

‘This time I want to make the three figures with three obtuse angles in each, equilateral triangle, isosceles trapezium and obtuse angled trapezium’.  Ask the child to form them and use the formed equilateral triangle this time, not the solid one.

‘Let’s compare the triangle and ‘The isosceles trapezium, the major base of the trapezium is double the base of the triangle, the minor base of the trapezium is the same as the base of the triangle

Compare the triangle and the obtuse angled trapezium, this is a bit more tricky, the base of the triangle is the same as only one of the sides of this trapezium and double that of the bisectors of the triangle, the bisector of the triangle is the same as one of the sides of the trapezium’.

Continue with a total exploration of the two trapeziums

Final stage Fractional Relationship

Part 1

  • Ask the children to form a hexagon with the equilateral triangle and three isosceles ones and an equilateral triangle made with the three remaining yellow obtuse triangles.
  • The equilateral triangle formed in the hexagon has diagonals, I am going to remove the equilateral triangle and place it on the divided triangle.  Place it over the three triangles and ask the child what can be said about them (they are congruent)
  • Place the solid equilateral back inside to form a hexagon and fold the isosceles triangles over it, ask the child what is formed (another equilateral triangle) and they are also congruent.
  • Unfold the isosceles triangle, remove the equilateral triangle and put it to onside, bring the isosceles triangles into the space to form a new hexagon.

Part 2

Ask the child to describe it, add, ‘The equilateral triangle is inscribed in the hexagon by diagonals which touch none consecutive vertices’.

Separate out the hexagon into three rhombi, and say, ‘The major diagonal of the rhombus is the same as the side of the equilateral triangle. Say, ‘Now let’s look at the fractional relationship between the inscribed triangle and the hexagon.  We can say that the inscribed triangle is half of the hexagon, the hexagon is double that of the triangle. How many triangles are contained in the equilateral triangle? (3) so each isosceles triangles is one-third of the equilateral triangle. The hexagon is the same as six isosceles triangles, so the isosceles triangle is one-sixth of the hexagon. How many rhombi are in the hexagon (3) so we can say each rhombus is one-third of the hexagon’

Aim:

Further exploration of congruency, similarity and particularly equivalence.

The use of nomenclature of the figures, including lines and fractions

Notes:

The significant triangle here is the obtuse triangle

After the lesson:

The children continue to work with T 6, 7, 8, 15 and 16

Small Hexagonal Box

Material Description:

Six grey equilateral triangles

Three green equilateral triangles

2 red equilateral triangles

6 obtuse-angled isosceles triangles

1 large yellow equilateral triangle.(not the same as that for the other two boxes)

Method:

Introduction

Ask the children to take out the equilateral triangles and stack them according to colour, ask the child to name them as ‘congruent, equilateral triangles’ and arrange them according to the black lines in the way they are familiar with.  They form a grey hexagon, green trapezium and red rhombus.

Presentation 1

Fractional Relationship

‘Let’s see if there is a fractional relationship between these figures in a systematic way, we used six triangles to make the hexagon, three to make the trapezium and two to make the rhombus.  Isolating each shape by pulling it in from of you, say.,Let’s look at the hexagon, each triangle is 1/6 of the hexagon, so the hexagon is made up of six-sixths.  Now let’s look at the trapezium, each of the triangles is 1/3 of the trapezium, so the trapezium is equal to three-thirds.  Each triangle in the Rhombus is 1/2, the rhombus is made up of two halves.

Isolate the trapezium and the Hexagon, ask the child, ‘What can we say about these shapes? trapezium is half of the hexagon, there are two trapezium in the hexagons, the hexagon is double the trapezium’. Separate the hexagon to show this’

Isolate the rhombus and the Hexagon, ask the child, ‘What can we say about these shapes?The rhombus is one-third of the hexagon, I have three rhombi in my hexagon, the hexagon is three times the rhombus’. Separate the hexagon to show this.

Relationship of Line

Pointing to the lines of the hexagon say,’In the hexagon we have six sides, and each side is the same as the side of an equilateral triangle.  We have three diameters.’

Pointing to the lines of the trapezium say,‘This is the major base and this is the minor base, the major base has twice the side of an equilateral triangle, whereas the minor base is equal to one side of the triangle.’

Pointing to the lines of the rhombus say,‘This is the minor diagonal of the rhombus is the same as half of the diameter of the hexagon, the side of the rhombus is the same as the side of the triangle.’

Presentation 2

Material Description:

Small Hexagonal Box; six grey equilateral triangles, three green and two red and six red obtuse angles isosceles triangles and a large yellow equilateral triangle (here we being using only the extra material).

Method:

  • With the large equilateral triangle and the grey hexagon, take the red obtuse triangles, ask the child to stack them and notice that they are congruent notice that they are congruent.
  • Join the obtuse angled triangles to form three rhombi, then use them to form a red equilateral triangle inscribed in a hexagon.  Superimpose the yellow equilateral triangle on the red inscribed one, ask the child what they notice (they are congruent).
  • Exchange the red equilateral triangle for the yellow one.  In terms of the hexagon, what can I say about the yellow equilateral triangle (it is half) and the hexagon (it is double the triangle).
  • Replace the red triangles and say,’ I want to see how these two hexagons are related’ (they are congruent). Put the six grey equilateral triangles over the red hexagon.
  • Put out the green equilateral triangles to make a trapezium and red rhombus to use when making the statements.

I can now make some statements;

 ‘The yellow equilateral triangle is half red hexagon, so it must be half of the grey hexagon.  

The yellow equilateral triangle must be equivalent to the green hexagon and to the grey hexagon.  

If the yellow equilateral triangle is equivalent to the green trapezium it must be equivalent to three equilateral triangles, so each of the three equilateral triangles must be 1/3 of the trapezium.

The rhombus is 2/3 of the equilateral triangle’

Notes:

Say some of the relationships you see to begin the exploration

The minor diameter of the rhombus is the black line the shortest internal line, the major diameter is the longest possible diameter.

When making the mathematical statements we do not superimpose to prove this, we say it and let the child continue to explore.

Charts T10 and T14

The difference between the large and small hexagonal box, is that the key triangle in the small hexagonal box is the equilateral triangle, in the large hexagonal box it is the obtuse angled isosceles, but both of these can be used to make hexagons.

We are now using transative properties, you can check the knowledge of this term with the children.

When to give the lesson:

After the Large Hexagonal Box and Triangular Box

After the lesson:

Further exploration with the box

Draw and colour figures, cutting them and pasting them into booklets or charts.

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