# Alex Reeve and Peter Whyte

These free lessons show how the study of architecture supports the teaching of maths in junior & secondary schools. If you are attracted to the use of geometry of architecture to support your lesson plans but this is your first time, you may experience an unaccustomed enthusiasm in the classroom with a high demand for your attention. This may put you under pressure, leading you to give up. Be patient. Keep going. Have an assistant. Students will soon grasp the concepts.

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These free lessons show how the study of architecture supports the teaching of maths in junior & secondary schools. If you are attracted to the use of geometry of architecture to support your lesson plans but this is your first time, you may experience an unaccustomed enthusiasm in the classroom with a high demand for your attention. This may put you under pressure, leading you to give up. Be patient. Keep going. Have an assistant. Students will soon grasp the concepts.

#### Construction Design Mathematics: a Gothic vesica & a triquetra

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How to draw a vesica piscis School curriculum: Key stage 3 mathematics Mathematically this shape is termed an ellipse and is produced when two circles of the same radius intersect such that the centre point of each circle lies on the circumference of the other, with both circular elements being smaller than a semicircle. The two circular elements would also join along their chords. It extends the introductory exercise constructing a perpendicular bisector. The shape should recall the first steps in the construction of an isosceles or equilateral triangle (see separate lesson on drawing a lancet arch). The equilateral triangle has as its verteces the centre points of the two circles and one of the two sharp corners of the vesica piscis The area of an ellipse is also twice that of a the pointed arch section of a lancet window. Drawing a triquetra (a pointed trefoil) extends the subject. School Curriculum: Key Stage 2 &amp; 3 mathematics: A pointed trefoil may be turned about its centre point into three identical positions, i.e. it has rotational symmetry of order three. A triquetra, from the Latin for three corners. comprises three circular arcs of equal radius. The exercise extends the work on a purpendicular bisector and a vesica piscis.

#### Construction design mathematics: Pythagoras and a Classical temple stylobate

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This lesson is designed for mathematics students at Key Stage 3. It fuses the study of Pythagoras’ Theorem with the study and design of the stylobates - or floor plans - of several Classical temples. The Theorem is approached in an easy to understand step-by-step way . Pythagorean triples are introduced through the medium of a plan of the Classical temple stylobate. The teacher and student are then guided through the process of drawing a floor plan using Ancient Greek units using a pair of compasses, pencil and ruler. The lesson also includes information on the siting and development of the Classical temple.

#### A single lancet with trefoiled apex

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How to draw a lancet window with trefoiled apex The instructions for the drawing on the following pages are based of a window in the nave at St Botolph’s Church in Northfleet Kent. The dimensions produce a drawing of a window that fits on an A4 page. Precision is required, and the task is easier on an A3 page, in which case the dimensions should be doubled. Botolph was a saint venerated in the seventh century in the Kingdom of East Anglia, where he had lived as a monk in an abbey endowed by estates in the Kingdom of Mercia. Botolph mediated a fragile peace between these two warring states. After his death, Botolph’s relics were conveyed to many different places where churches were later dedicated to St Botolph. As these churches were erected next to the cities’ gates or by fords, bridges and shire boundaries, he has become associated with travellers.

#### Construction Design Mathematics: Gothic architecture: a lancet window

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Introduction to Gothic architecture How to draw a pointed arch or a pointed (lancet) window School curriculum: Key Stage 2 mathematics: Equilateral and isosceles triangles. Draw attention to the lancet arch being set out on the vertices of an equilateral triangle . Placing the compass point on wider or narrower loci on the horizontal plane will produce broader or narrower arches respectively. In these cases the lancet arch is set out on the vertices of an isosceles triangle. KS4 Mathematics: The drawing can be a suitable adjunct to a lesson on constructing an equilateral triangle or isosceles on a given straight line, or where the altitude or height of the triangle is given. For proof of centroid of triangle see appropriate lesson plan. The drawing can also support teaching the formula for measuring arc length, usually denoted by s. s = θ° ÷ 360° x 2 π r or sector area, where A = θ° ÷ 360° x π r² An elegant window is usually achieved when the height of the window is eight times the width. Further lessons developing the study of a lancet arch can be seen on lesson plan Gothic architecture: lancet triplets, and Gothic architecture: window with intersecting or Y-shaped tracery.

#### Construction Design Mathematics: how to draw the front of a Classical temple

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This lesson is suitable for older students at Key Stage 2 and all students at Key Stage 3. It looks at the design of the Doric Temple of Concorde in Agrigento through the eyes of Classical philosophers exploring number theory. While Pythagoras may have thought that the perfect number system was fourness, other mathematicians may have considered it to be fiveness, reinforced by our having five fingers and so enabling us to count in groups of five. Sixness was another idea: six can be made up of components that all agree in their ratios with the number six, i.e. a sixth of six equals one, a third equals two, a half is three. Adding a sixth, a third and a half of six together equals six. Furthermore Greek mathematicians noted that the length of a man’s foot was a sixth of his height. Ancient Greeks applied sixness to the construction of many Doric columns, which were in height six times greater than their diameter. In the case of the Doric Temple of Concorde in Agrigento , builders chose a hexastyle temple, i.e. one with six columns at the front and back and thirteen on the sides. Illustrated easy-to-follow instructions for students on how to draw the front of the Temple are available on the following pages.

#### Construction Design Mathematics: a Gothic window with intersecting or Y-shaped tracery

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This is a lesson suitable when students have mastered the lancet window (see relevant lesson plan). Intersecting tracery Where a dripstone is placed around the outer arcs of the two lancet lights, the area above the lights lends itself to the sculpted opening of a natural rhombus or lozenge type shape, formed with concave sides below, and convex above. This makes for a very elegant window that can be seen on many churches. The overall design is termed intersecting or Y-shaped tracery and emerged towards the end of the thirteenth century and lasted well into the fourteenth. The mullion of a window arcs from the vertical into the window arch, with all neighbouring mullion(s) parallel (or coincident) to it. It is a simple and elegant design, and one that is surprisingly easy to draw. When plain (i.e. uncusped or unfoliated) this style of window is correctly classified as Gothic Early English architecture, but the addition of cusping or foliation classifies it as a Decorated window.

#### Construction Design Mathematics: 2D Shape and an Anglo-Saxon window opening

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How to draw a single Anglo-Saxon church window or door opening The Anglo-Saxon tower at the Church of St Peter, Barton-Upon-Humber, Lincolnshire was built in the late tenth century . The double triangular opening is divided by a bedimmed single shaft. The triangular form is constructed by leaning two stones together at an angle. The blind arcading and vertical pilaster strips of stone emulate beams used in the construction of timber-framed houses. School Curriculum: Key Stage 2 Mathematics: For younger students of the Key Stage, drawing the doors and windows extends understanding of how simple shapes can be combined, manipulated and applied outside the classroom. The drawings require measuring angles with a protractor and the exercise can be extended to calculate the area of a rectangle and a triangle.

#### Construction Design Mathematics: Gothic Geometric tracery: a trefoil within a circle

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How to draw a trefoil within a circle School Curriculum: Key Stage 4 mathematics: The trefoil within a circle extends the work on trefoils. In a trefoil a circle circumscribes the three vertices of a triangle. This enables the placement of the three foils or arcs centred on each of the verteces of the triangle. Here the exercise is repeated but with the addition of a circumcircle, still centred on the circumcentre, i.e. the centre of the triangle, but at a tangent to each of the three foils. Extra decorative arcs embellish the work. The trefoil was brought to England by French masons in the first half of the thirteenth century. Binham Priory in Norfolk, now a romantic ruin, then a thriving Beneditine monastery, vies with Westminster Abbey as the location in which it first appeared.

#### Construction Design Mathematics: an Anglo-Saxon tower with helm roof

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The eleventh century restored Anglo-Saxon tower with a helm roof at St Mary’s Church, Sompting is unique in England. It is a roof having four faces, each of which is steeply pitched so that they form a spire, while the four ridges rise to the point of a spire from a base of four gables. Key Stage 4 Mathematics: The helm is an interesting shape mathematically. Each rectangular side of tower is topped with an equilateral triangular gable. Each of the two upper sides of the triangles adjoins the two lower sides of a rhombus. A rhombus is a parallelogram with equal sides. A net is a 2D representation of a 3D shape which, when folded, forms the 3D shape that it purports to represent. Nets are introduced to children at Key Stage 1. This complex exercise is suitable for able students at Key Stage 4, or IB. It offers a unique insight into the challenges confronting a Norman or Anglo-Saxon builder in the eleventh century. Students will be guided through a precise series of instructions to produce a net, which should then be decorated, folded and glued together.

#### Construction Design Mathematics: the Norman or Romanesque arch

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A rich and elaborate construction of a Norman or Romanesque window. School Curriculum: Mathematics at Key Stage 3: Constructing a Romanesque arch develops the geometric drawing skills of students. The resulting drawings can be a resource for the calculation of the area of a semi-circle using pi (π), and the perimeter of the semi-circle when the base (d or 2r) is known. The construction can also be an application of the trapezoidal rule to calculate approximate area . The exercises extend the work on π, circles, semicircles and arcs following the drawings of a Roman arch and brace &amp; the horseshoe arch, available elsewhere. Maths covered circumference of a circle = πd where d = diameter of the circle, where π = 3.14 = 2πr where r = radius of the circle perimeter of a semicircle = (πd ÷ 2) + d area of a circle = πr² area of a semi circle = (πr²) + 2 Constructing a Romanesque arch can be a practical application of the calculation of the area of a semi-circle using pi (π), and the perimeter of the semi-circle when the base (d or 2r) is known. This lesson would be a development or extension of the lessons on a Roman arch &amp; brace and / or the horseshoe arch.

#### Construction Design Mathematics: the geometry of a volute of an Ionic column

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The volute is the principal distinguishing feature of the Ionic order, the spiral located on either side of the capital of a column, School Curriculum: Mathematics at Key Stages 2,3 &amp; 4. Drawing a volute with semicircles will enable students at Key Stages 2 &amp; 3 to create a pattern with repeating shapes in different sizes and orientations. In doing so, students will become familiar the the properties of a circle (circumference, radius, diameter). The drawing will also provide a visual element for the calculation of the perimeter of a semicircle. At Key Stage 4 drawing a volute with quadrants will facilitate the calculation of arc length subtended by those quadrants. There is also a brief history of the Ionic volute and its symbolism.

#### Construction Design Mathematics: how to draw Saxon blind arcading

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In blind arcading Saxon masons have combined a number of simple shapes, viz. semicircles, rectangles and trapezia to produce a pleasing and interesting architectural feature. School Curriculum: At Key Stages 2 &amp; 3 Manipulating simple shapes to produce complex designs. Content: definition of a trapezium, area of a trapezium, illustrated easy-to-follow instructions on how to draw a trapezium. Examples of blind arcading on Saxon church churches.

#### Construction Design Mathematics: how to draw a large Geometric window in the Gothic style

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How to draw a large Geometric window in the Decorated style School curriculum: Key Stage 4 Mathematics: Drawing the Altrincham window would support the teaching of the 2D stacking of circles. It is concerned with the optimal arrangement of placing four congruent circles within a circle, and related problem solving using Pythagoras. Altrincham Baptist Church, Cheshire was constructed towards the end of the Gothic Revival movement in 1904; its west window is shown on the accompanying pages. The window comprises five lancet lights. From the left, there are two lights under a circle within a sub-arcuation; then a single light under a circle circumscribing four congruent circles; then two lights under a circle within a sub-arcuation; all within a larger arch.

#### Construction Design Mathematics: the history & geometry of Roman arch & brace

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This lesson is suitable for students at Key Stage 3. It comprises an experiment to determine a figure for π, and Archimedes’ calculation of the area of a circle. There is an historical component on the importance of the arch in Roman architecture along with illustrated and easy-to-follow instructions on how to draw a Roman arch and brace using a pair of compasses, a protractor, ruler and pencil. A suitable follow-on lesson is the horseshoe arch and / or the Norman or Romanesques arch.

#### Construction Design Mathematics: the reticulated Curvilinear church window

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This lesson develops the theme of the ogee or s-shaped arch (see relevant lesson plan). In the church window at Finedon, the apex comprises a vesica atop two vesicas that are curved at top and bottom into ogees, all in a net-like arrangement. The apex is set above three lights with ogee arches. This type of tracery is termed reticulated Curvilinear. School Curriculum: This is the first of four studies of different types of Decorative window. Each has a design that is an arrangement of circles or arcs with the same radii, within a defined perimeter, with or without overlaps. An analysis of the windows at Finedon reveals that the architect simply stacked a series of congruent windows and equilateral triangles. The windows studied are within: the Church of St Mary the Virgin in Finedon: the only window with overlaps; St Thomas of Canterbury, Northaw (available on another page); Altrincham Baptist Church (ditto); and Lincoln Cathedral (ditto). This series of lessons is well suited to project work at IB / GCSE level. However it arose, the ogee arch rapidly led to the development of many different tracery designs. Whereas earlier windows had a static non-directional element, now the ogee enable both mullions and tracery to intersect, curve and flow from one shape to another with an arabesque character that became known as Curvilinear tracery.

#### Construction Design Mathematics: the Angel Choir Window in Lincoln Cathedral

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The Angel Choir window in Lincoln Cathedral Once a single or pair of lights surmounted by a quatrefoil within a circle had been constructed (see relevant lesson plans), masons were quick to realise its potential for larger areas, by increasing the number of lights and circles. Large C13th windows of the Geometric period had an even number of lights arranged in pairs, each pair with its own circle. The largest example is the eight-light Angel Choir window in Lincoln Cathedral where the window is made up of eight lights, arranged singly, yet in pairs, and also in fours, i.e. two times two pairs, or four + four lights, with a total of thirteen circles of different sizes in the window apex. School Curriculum: Key Stage 4 Mathematics: The Sine Rule. In a similar way to the apex window at Altrincham Baptist Church, and the mouchette wheel at St Thomas of Canterbury, Northaw, (see relevant lesson plans) the circular window in the apex of the Angel Choir window at Lincoln Cathedral required builders to find appropriate centres of the inscribed congruent, circular windows, or circles. In the Angel Choir window, there are six circles, each tangental to the circumscribing circle, and to two neighbouring inscribed circles. There is also one inner circle; with the inner circle being inscribed by the six inscribed outer circles. Applying the sine rule provides a formula for finding the radius (rs) of any number of inscribed circles where (rl) = radius of the circumscribing circle, and angle x are known. Photo credit: Jules &amp; Jenny

#### Construction Design Mathematics: the development of a horseshoe arch

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In Asia Minor the horseshoe arch has long been a structural and decorative motif adorning tombs, sarcophagi and stele. After the Eastern Christian Church adopted the shape it quickly spread to North Africa, Spain, Gaul, Italy and Rome itself. How to draw a horseshoe arch: School curriculum: Key Stage 4: measurement of arc length Maths covered: Arc length is the distance between two points on a curve. and is usually denoted by l or s, the latter from the Latin spatium meaning length or size. s = (θ° ÷ 360°) x 2 π r where theta θ is a measure of the angle subtended by either ci-e-di &amp; co-e-do (fig. on lesson plan in degrees, π = 3.14 and r = the distance ae (fig.) in centimetres If the arc is a semicircle then s = π r This lesson would serve as an extension of the lesson on a Roman arch and brace, itself extended by the lesson on a Romanesque or Norman arch

#### Construction Design Mathematics: two trefoiled lights with a quatrefoil apex

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This exercise builds on earlier lessons on Gothic Early English architecture. It is a complex drawing suitable for top set project work for GCSE students Content: the first exercise is a simple construction of a quatrefoil. It is followed by detailed and illustrated instructions on how to draw two adjoining trefoiled lights under a arched dripstone. The instructions that follow fills the apex above the two lights with a quatrefoil, with an option to do so with a pointed quatrefoil.

#### Construction Design Mathematics: Gothic architecture: lancet triplets

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How to draw lancet triplets Once the students are at ease with drawing a lancet window (see relevant lesson plan) then try lancet triplets using the illustrated ieasy-to-follow instructions. A very elegant window arrangement is to draw the triplets together with the centre light being taller and slightly wider than the outer lights. A good example can be seen at Temple Church in London, completed in 1240. Each light has its own dripstone. Within the dripstone are mouldings that descend onto the capitals. The capitals sever the multiplicity of mouldings from the single shaft below thereby producing a pleasing contrast.

#### Construction Design Mathematics: a circle, elliptical & ogee curve or mouchette

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School Curriculum: Key Stage 3 &amp; 4 Mathematics: drawing angles using a pair of compasses and a straight edge; centroid of a triangle; congruent circles within a circle: manipulating shapes, Pythagoras Theorem and the sine rule; Properties of 30°-60° 90° triangle: Drawing a mouchette is based on circle stacking, i.e. arranging three congruent circles within a circle. The window at St Thomas of Canterbury, Northaw required builders to find appropriate centres of the smaller congruent cirlces when the larger circle had been designated. To effect this the design could often be manipulated mathematically to produce a triangle that in turn enabled calculation of sides or angles by the 30° -60° -90° rule if a right angle was present. It was a short step from the reticulated Curvilinear tracery (see relevant lesson plan) to the use of a mouchette as a motif. A mouchette is a daggerlike motif found most often in 14th century Decorated church tracery. It is formed by elliptical and ogee curves that produce a point at one end and an incomplete circle at the other.