This unit is part of the special topic “Mechanical Linkages and Deductive Geometry”. Mechanical linkages – sets of hinged rods – form the basis of many everyday objects such as folding umbrellas, ironing tables and car jacks and are built using the geometry of triangles and quadrilaterals. They offer rich potential for investigating geometry, starting with real-life objects, then making working models and then using pre-prepared dynamic geometry software. This unit uses the properties of quadrilaterals, especially rhombuses and parallelograms, to explain how these tools and objects move. The unit is especially suited for Year 8. The lessons are independent of each other and can be taught in any combination.

Read the Teachers' Guide, also included in the unit download, for teaching advice and practical hints for the constructions.

### Lesson 1: Car Jack

Students investigate the design and operation of a rhombus car jack (as supplied with many cars) to see how the geometric properties of a rhombus contribute to how it works. Students first examine a real jack or images, and make a physical model to observe how properties of sides and angles enable the operation of the jack. They then examine a dynamic geometry simulation, which reveals the importance of diagonal properties to safety. Students can then use congruent triangles to reason deductively about rhombus properties.

### Lesson 2: Scissor Lift

Students investigate the design and operation of a scissor lift to see how the geometric properties of a rhombus contribute to how the scissor lift works the way it does. They look at images, and make and operate a physical model and a computer simulation to explore the geometric properties on which a scissor lift depends. They then use congruent triangles to prove the key property: that the diagonals of the rhombus are perpendicular.

### Lesson 3: Folding Umbrella

Students investigate the design and operation of a folding umbrella to see how the geometric properties of a parallelogram contribute to neat folding. Students observe an actual umbrella and its ribs, make a physical model and operate a computer simulation, at each stage seeing the geometric features more fully. This linkage is then used as a stimulus for students to engage in deductive reasoning about the properties of quadrilaterals.

### Lesson 4: Toolbox

Students investigate the design and operation of a cantilever toolbox to see how the geometric properties of a parallelogram contribute to how the trays of the toolbox open and close. Students observe an actual toolbox, make a physical model and use a computer simulation to understand how the design based on parallelograms constructed from bars of equal length allows the toolbox to open and close with the trays remaining horizontal. There is an opportunity for deductive reasoning, supported by a slideshow.

### Lesson 5: Cherry Picker

Students investigate the design and operation of an aerial work platform commonly called a cherry picker to see how the geometric properties of parallelograms contribute to how the work platform is maintained in a horizontal position and how changing shapes of the parallelograms allow the work platform to be manouvered into different positions. They construct a physical model and use a computer simulation of the cherry picker. They also see how articulated parallelograms are used in robotic arms in certain machinery and design a robotic arm for cake icing.

### Lesson 6: Angle Bisectors

Students investigate the design and operation of two different angle bisector tools and use the geometry of rhombuses and kites to explain why they work. The tools are useful for carpenters when constructing mitred corners. The lesson begins by observing mitred joints. Students make physical models and observe computer simulations that highlight the geometry and give accurate measurements. They investigate the reasons why the tools work. This stimulates investigation of other angle properties supported diagrammatically. An extension examines a third type of angle bisector tool.

Last updated December 11 2018.