Constructive Triangles

The materials of the Constructive Triangles help the child consolidate into her conscious understanding shape, size  (dimension) and colours and help her to realise that the triangle has the roles of a constructor, divisor and consequence.  The child first needs to be aware of shape and size and familiar with the names presented in the ‘Three Period Lesson’ accompanying the Geometric Cabinet, so that she can verbally express her ideas.

Material Description

Five boxes of coloured wooden triangles:

• Two rectangular boxes
• One triangular box
• One large hexagonal box
• One small hexagonal box

The triangles are called ‘constructive’ because they construct other rectilinear shapes.

Extension

After the children have experienced working with each of the Constructive Triangle Boxes, they may combine the boxes to create new shapes using many different types and shape of a triangle.  To put away the triangles the children are guided by colour dots on the back of the triangles and the lids of the boxes.

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First Rectangular Box 

Material Description

• Two yellow isosceles right-angles triangles with a black line painted on one of the sides enclosing the right angle
• Two green isosceles right-angles triangles equal to the yellow ones, with a black line painted on the hypotenuse of each
• Two yellow scalene right-angles triangles with a black line painted on the shorter of the two sides enclosing the right angle
• Two green scalene right-angles, equal to the yellow scalene, with black lines painted on the longer of the two sides enclosing the right angle
• Two grey scalene right-angles triangles, the same size as the scalene triangles with a black line painted on the hypotenuse
• Two yellow equilateral triangles, with black lines on one side
• One red scalene right-angled triangle with a black line painted on one of the sides enclosing the right angle
• One red scalene obtuse-angled triangle with a black line painted on the the side opposite the obtuse angle

Presentation

• Bring the First Rectangular Box to the working mat or table
• Show the child how to handle the triangle, removing three pairs of identical triangles and placing them randomly
• Isolate any one triangle and pair it; placing the with black lines facing, then slowly bring them together
• Repeat with the other two pairs of triangle
• After joining the pairs mix them and invite the child to repeat
• Gradually introduce more triangles to the child as she works till she has handled each pair
• After she has had sufficient practice invite the child to name them

Exercises

• The child independently repeats the presentation, making further explorations

Language

No new language, this activity reinforces language introduced in the Geometric Cabinet

Criteria of Perfection (Control of Error)

• Lies within the child’s visual sense guided by the black lines which function as a control

Direct Aim

• To show that by joining together different triangles quadrilaterals are formed
• To help the child discover the function of a triangle as a constructor

Indirect Aim

• Preparation for Geometry: to show that rectilinear plane figures are composed of triangles
• Preparation for understanding the concept of equivalence and it’s application in finding the area of plane figures
• To help the child prepare herself to identify equilibriums that are; identical, similar and different
• To look at comparative figures
• To prepare for fractions

Age at Presentation

After the child has had sufficient experience with the Geometric Cabinet, at around four to five years

Footnote

We give the Geometric cabinet before the Constructive Triangles because the Cabinet introduces quadrilateral shapes consciousness, which help her intelligence recognise patterns made here and perceive the triangles role as a constructor (rather than mechanically join the lines).

Blue Triangles or the Second Rectangular Box

The Second Box is introduced after the child has discovered that a pair of triangles joined along different sides produces different shapes.

Material Description

• All triangles are blue with no lines, they are the same size as those from the first Box
• There are:
• One pair of isosceles right-angles triangles
• One pair of scalene right-angles triangles
• One pair of equilateral triangles
• One scalene right-angled triangle
• One scalene obtuse-angled triangle

Presentation I

• Invite the child saying, “You have been working with the first box, let us see what we can do with these blue triangles’
• Bring the Second Rectangular Box to the working mat or table
• Remove one pairs of identical triangles and placing them randomly
• Show the child how to join them along any side
• Hold one triangle with the left index finger and slide the other triangle against it’s side, when a new shape forms pause and wait to let the child identify it
• Continue moving along all sides until you reach the original position
• Repeat with the other pairs one at a time
• Leave the non-identical triangles till last.
• Allow the child to participate and to name the figures that she makes

Presentation II

After sufficient practice show her this variation either the same day or later on;

• Remove one pairs of identical triangles and placing them randomly on the mat, adjacent to each other
• Suggest, ‘You turn one triangle upside down and see what shapes are formed’
• The child flips one triangle over and joins it with it’s identical pair to see a new shape
• Slide the triangle to make other figures
• Leave the non-identical triangles till last.

Exercises

• The child independently repeats the presentation, making further explorations

Language

No new language, this activity reinforces language introduced in the Geometric Cabinet

Criteria of Perfection (Control of Error)

• No control of Error is necessary

Direct Aim

• To show the Constructive Power (Analysing Power) of the triangle by exploring al possible shapes using only two triangles

Indirect Aim

• Preparation for Geometry: to show that rectilinear plane figures are composed of triangles
• Preparation for understanding the concept of equivalence and it’s application in finding the area of plane figures
• To help the child prepare herself to identify equilibriums that are; identical, similar and different
• To look at comparative figures
• To prepare for fractions

Age at Presentation

After the child has had sufficient experience with the First Box, at around four to five years

Triangular Box

How to replace the Triangles

Place the red triangles, the yellow, green and the grey on top.

Note

The order given follows the psychological principal of starting with the whole

Material Description

A triangular box containing:

• One large grey equilateral triangle
• Two green scalene right-angled triangles; black lines on the longer of the two sides enclosing the obtuse angle.  Each green triangle is equal to one-half of the large grey one.
• Three yellow isosceles right-angled triangles; black lines on the each of the two sides enclosing the right angle. Each yellow triangle is equal to one-third of the large grey one.
• Four red equilateral triangles; three have black lines on one of the sides, one has black lines on the all of the sides  Each red triangle is equal to one-fourth of the large grey one.

Presentation

• Bring the Triangular Box to the working mat or table
• Let the child remove all of the triangles and placing them randomly at one side of the mat
• Isolate the grey triangle
• Isolate the green triangles and match them along the black lines
• Place the green triangles below the grey one (the child will observe pairing and equivalence)
• Show the child how to superimpose the green onto the grey to verify the equivalence
• Follow the same steps for the yellow and red triangles
• Repeat with the other two pairs of triangle
• After joining the pairs mix them and invite the child to repeat
• Gradually introduce more triangles to the child as she works till she has handled each pair
• After she has had sufficient practice invite the child to name them

Exercises

• The child independently repeats the presentation, making further explorations

Language

Later on the terms side, vertex, base can be used.

Note

Parts of the triangle (point out and name) the following;

Angle – a geometric figure made by two intersecting lines

Altitude (or height) – with the green set – the perpendicular distance from the vertex to it’s opposite side, perpendicular to the base

Bisectors – yellow set – a straight line segment that divides an angle or another line segment into two equal parts

Apex – the highest point of altitude

Base – the line or surface on which the figure is assumed to stand

Side – the line between the angles

Centre – yellow set – sometimes known as the centroid of a triangle, which is the point of intersection of the medians of the triangle (a median is a line segment from any vertex of a triangle to the midpoint of it’s opposite side).  The Centroid is two-thirds of the distance from each vertex to the mid-point o the opposite side.

Midpoint – red set – in a line segment or side, the point that bisects the line segment or side

Perimeter – grey triangle – the sum of the sides

Criteria of Perfection (Control of Error)

• The black lines and grey triangle

Direct Aim

• To realise that the Equilateral triangle can be sub-divided into other types of triangle

Indirect Aim

• Preparation for Geometry: to show that rectilinear plane figures are composed of triangles
• Preparation for understanding the concept of equivalence and it’s application in finding the area of plane figures
• If the triangle is analysed each element is also a triangle

Age at Presentation

After the child has had sufficient experience with the Geometric Cabinet, at around four to five years

Large Hexagonal Box 

How to replace the Triangles

Place large yellow, equal yellow, yellow obtuse, grey and red on the top.

Material Description

A box containing;

• Two red isosceles obtuse-angled triangles; black lines on the side opposite to the obtuse angle
• Two grey isosceles obtuse-angled triangles; black lines on one of the two equal sides enclosing the obtuse angle
• Six yellow isosceles obtuse-angled triangles; three with black lines on the side opposite to the obtuse angle and three with black lines on one both equal sides

**All above triangles are the same shape and size**

• One large yellow equilateral triangle; black lines on all three sides

Presentation

• Bring the Large Hexagonal Box to the working mat or table
• Let the child remove all of the triangles and placing them randomly at one side of the mat

Make a red rhombus, a grey parallelogram and a yellow hexagon by

• Isolate the two red triangles, join them along the black lines to form a rhombus
• Isolate the two grey triangles, join them along the black lines to form a parallelogram
• Place the large yellow equilateral triangle in front of the child and look for three yellow obtuse-angled triangles, isolate them and match to forma hexagon

Explore the Hexagon

• Fold the obtuse over to create two large equilateral triangles (one below the other) and unfold them to create a hexagon again
• Match the black lines of the obtuse angles to form a equilateral triangle and superimpose them over the large yellow equilateral triangle within the hexagon

Explore the rhombi

• Show the child that the hexagon is composed of three rhombi by moving the pairs of yellow obtuse-angle triangles apart
• Remove the large yellow equilateral triangle
• Superimpose the red rhombus over each yellow rhombi in turn
• Bring the grey parallelogram and slide one triangle to form a rhombus, then superimpose the red rhombus over it
• Bring the grey rhombus back to a parallelogram
• Gradually introduce more triangles to the child as she works till she has handled each pair
• After she has had sufficient practice invite the child to name them

Exercises

• The child independently repeats the presentation, making further explorations

Language

No new language, this activity reinforces language introduced in the Geometric Cabinet

Criteria of Perfection (Control of Error)

• Lies within the child’s visual sense guided by the black lines which function as a control

Direct Aim

• To show that joining together different types of triangle quadrilaterals are formed.

Indirect Aim

• Preparation for Geometry: to show that rectilinear plane figures are composed of triangles
• Preparation for understanding the concept of equivalence and it’s application in finding the area of plane figures

Age at Presentation

At around four to five years

Small Hexagonal Box 

How to replace the Triangles

Place grey, red and green on the top.

Material Description

• A box containing;
• Six grey equilateral triangles; black lines on two sides
• Two red equilateral triangles; the same size as above with black lines on two sides
• Three green equilateral triangles; the same size as above; two with black lines on one side, one with black lines on two sides

** note: Additional triangles come with the box, these are not to be used at the primary level, but are to illustrate mathematical theorems at the Elementary Level.  Remove from the box in the Children’s House one large yellow equilateral triangle without any black lines and six red obtuse-angled triangles with black lines on the hypotenuse.

Presentation

• Bring the Small Hexagonal Box to the working mat or table
• Sort them according to their colour and size
• Join them along their respective black lines to form various shapes, including a red rhombus, a grey hexagon and a green trapezium
• Show the child how to superimpose the red rhombus and, in turn, green trapezium onto the onto the grey hexagon and remove

The child may discover that the hexagon contains six equilateral triangles,three rhombi and two trapezium.

Exercises

• The child independently repeats the presentation, making further explorations

Variation

• The child can work independently with multiple boxes to compare figures

Language

No new language, this activity reinforces language introduced in the Geometric Cabinet

Criteria of Perfection (Control of Error)

• Lies within the child’s visual sense guided by the black lines which function as a control

Direct Aim

• To further develop the child’s exploration of the Equilateral angle.

Indirect Aim

• Preparation for Geometry: to show that rectilinear plane figures are composed of triangles
• Preparation for understanding the concept of equivalence and it’s application in finding the area of plane figures

Age at Presentation

At around four to five years

Blue Triangles

Material Description

• Twelve blue identical right-angled scalene triangles, which allow the child to build experimentally with right-angled scalene triangles

DO NOT PRESENT EACH AND EVERY POSSIBILITY, LET THE CHILD EXPLORE

Presentation

• Bring the Box to the working mat or table
• Let the child remove all of the triangles and placing them randomly at one side of the mat
• Isolate any four triangles and make any shape
• Invite the child to take another four and do the same
• Leave them for the child to explore

Exercises

• The child independently repeats the presentation, using more than four triangles to build shapes, eventually she will use all of them at the same time

Language

No new language, this activity reinforces language introduced in the Geometric Cabinet

Criteria of Perfection (Control of Error)

• Lies within the child’s visual sense guided by the black lines which function as a control

Direct Aim

• To further develop the child’s experience of the right-angled scalene triangles

Indirect Aim

• Preparation for Geometry: to show that rectilinear plane figures are composed of triangles
• Preparation for understanding the concept of equivalence and it’s application in finding the area of plane figures

Age at Presentation

After all of the Constructive Triangles have been shown

Footnote

The work with this material extends the child’s sensorial experience of equivalence

Constructive Triangles: Note

Working with the Constructive Triangles the child applies his knowledge of shape and colours.

• These are not sensorial activities but intelligent ones in which the child makes discoveries to reach realisations

• While working with the blue triangles the child can be shown the compass to hep her further her discoveries

• The child makes the following discoveries with the large hexagonal box
• A pair of red triangles make a rhombus
• A pair of grey triangles make a parallelogram
• The large yellow equilateral triangle and three yellow obtuse ones with single black lines make a hexagon
• These obtuse triangles can be folded in to show that the area of the hexagon can be split into two equivalent equilateral triangles
• The hexagon is made up of three rhombi
• The rhombi are made up of pairs of equilateral triangles
• By superimposing the rhombi onto all the three pairs of equilateral triangles (rhombi formed from the hexagon and since split) the child sees their equivalence
• The parallelogram can be made to form a rhombus by sliding one triangle
• The grey, red and yellow rhombi all have equivalence

• The child makes the following discoveries with the small hexagonal box
• The six grey equilateral triangles form a hexagon,
• The three green equilateral triangles form a trapezium
• The pair of red equilateral triangles form a rhombus
• Two grey trapeziums and three pairs of red rhombi form hexagons
• The red rhombus is equivalent to each of the three grey rhombi
• The green trapezium is equivalent to each of the grey trapeziums