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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.

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.
A single lancet with trefoiled apex
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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: 2D Shape and an Anglo-Saxon window opening
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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: the Norman or Romanesque arch
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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 & 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 & brace and / or the horseshoe arch.
Construction Design Mathematics: how to draw Saxon blind arcading
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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 & 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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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 reticulated Curvilinear church window
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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 development of a horseshoe arch
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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 & 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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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.