In this collection of activities we use ruler and compass constructions to draw a variety of shapes and to construct angles such as 90°, 60°, 45° and 15°.
Angle can be seen as and thought of in at least three ways. These are as:
The final one of these underpins the others and leads on naturally to the definition of degree and the ability to measure angles with a standard unit. This then enables students to be able to apply their knowledge of angle in a variety of situations.
We see angle as developing over the following progression:
Level 1: quarter and half turns as angles
Level 2: quarter and half turns in either a clockwise or anti-clockwise direction
angle as an amount of turning
Level 3: sharp (acute) angles and blunt (obtuse) angles
right angles
degrees applied to simple angles – 90°, 180°, 360°, 45°, 30°, 60°
Level 4: degrees applied to all acute angles
degrees applied to all angles
angles applied in simple practical situations
Level 5: angles applied in more complex practical situations
The concept of angle is something that we see students developing gradually over several years. As their concept matures, they will be able to apply it in a range of situations including giving instructions for directions and finding heights. In the secondary school angle is used extensively in trigonometry (sine, cosine, tangent, etc.) to measure unknown or inaccessible distances. This deals with situations where only right-angled triangles are present in 2-dimensional situations through to more complicated triangles in 3-dimensional applications.
Surprisingly these trigonometric functions are used in abstract settings too. At Level 8 and above they are used extensively in the calculus as a means to integrate certain functions.
Outside school and university, angle is something that is used regularly by surveyors and engineers both as an immediate practical tool and as a means to solve mathematics that arises from practical situations. So angle is important in many applications in the ‘real’ world as well as an ‘abstract’ tool. This all means that angles have a fundamental role to play in mathematics and its application.
In the rest of this unit, students will be using the kinds of constructions that they used in the tasks of Copymaster 1 in order to make other shapes. This will require them to use ruler and compass. Students will also use their knowledge of length and area to calculate various attributes of those shapes. This is to be done in two ways (i) theoretically using their knowledge of the geometry of the shapes involved; and (ii) using measuring instruments (ruler and protractor).
Here students will make two shapes and determine their heights using both theoretical calculations and direct measurement.
This session is very similar to the last one but this time we require the students to calculate area. To construct these drawings using ruler and compass they will need to know (i) how to find the sum of the interior angles of a pentagon; and (ii) how to construct angles of 120°, 135°, and 15°. (Now 120° is two lots of 60°; 135° is 60° plus 75° as well as 90° plus 45°; and 15° can be constructed using a 60° angle and subtracting a 45° angle.)
In this session the students are to use ruler and compasses to construct a regular hexagon (in two ways) and a regular octagon.
The regular hexagon can be completed using equilateral triangles. It can also be made by first drawing a circle and then marking off equal radius lengths around the circle. The six points so formed will make a regular hexagon.
In the final part of this unit the class is asked to undertake two projects that will require the use of ruler and compass.
Project 1: Design pentagonal tiles, where one of the angles is 112.5°, to tile a floor that is 3m by 5 m. Some tiles will need to be cut in order to make them fit.
(It is easiest to make tiles with right angles at the base as in the side of the house shape from earlier Copymasters.)
Project 2: Design a running track whose inside lane is 400 m long. The interior of the track should have space for a hockey field. A scale model of the track should be drawn using ruler and compass.
Allow them to find the dimensions of a hockey field by using the web. These are length 91.4 m and width 55 m. The track should be a rectangle with two semi-circles added, one at each end.
You should realise that there is no unique answer to this project. The shape of the track can be varied quite a lot without changing the fact that it allows a lane of 400 m and that it can contain a hockey pitch.
You could ask students what other playing fields the track could hold.
Printed from https://nzmaths.co.nz/resource/how-high-and-other-problems at 6:49pm on the 25th April 2024