PANTOGRAPH ON SKETCHPAD AND MATHEMATICS

i. Objectives:

M.6.1.7.2. Bir bütünün iki parçaya ayrıldığı durumlarda iki parçanın birbirine veya her bir parçanın bütüne oranını belirler, problem durumlarında oranlardan biri verildiğinde diğerini bulur.

M.6.1.7.3. Aynı veya farklı birimlerdeki iki çokluğun birbirine oranını belirler.

M.8.3.3.2. Benzer çokgenlerin benzerlik oranını belirler, bir çokgene eş ve benzer çokgenler oluşturur.

ii. Pedagogical Explanation:

My tool is a pantograph. Its working principle depends on the ratios. For example,

 You draw a segment and you put a point on it that is B. When you move the point B, point A will move too.  (In Sketchpad, when you put a point to a segment; even you make bigger the segment, the ratio of AB to BC does not change.) However, there is difference in their movement. If you move B, A will move more. When it is investigated, it is seen that the ratio of CB to CA is the same with the ratio of drawings with B to A (the ratio of one side of the little house to the same side of the big house). The pantograph works like this, too. The construction I made above is not possible in real life because when you fixed C and when you put a pencil to B, you cannot move freely as in this app because your movement capability is limited. To benefit from this ratio property, people improved the pantograph in real life.

I constructed a pantograph on Sketchpad that is similar to real life version.

 This tool can be helpful to teach the objectives of M.6.1.7.2 and M.6.1.7.3 because it works according to ratios. Students can find the ratios of FA to FB, AC to BE and FC to FE with the Measure-length and Number-calculations parts of the Sketchpad. Also, the teacher can relate the topic with pantograph by saying ‘There is interesting ratios in pantograph, can this be related with the working principle of the pantographs?’. Thus, it will be seen that the topic of ratio has relations with the real world and the topic can help making a tool that makes drawings bigger or smaller. On the other hand, this tool is very helpful for teaching the objective of M.8.3.3.2 because this tool works according to similarity principle in mathematics. While we are constructing this tool, we draw AC as parallel to BE and DC parallel to BF. At the end we see that AFC and BFE are similar triangles. Also, we see that EDC and EBF are similar triangles, too. This property of pantograph will provide students’ better understanding of similarity and similarity ratio.

iii. User Manual:

 Firstly, you can watch the video that I put on to the construction steps below and you can construct a tool that is same with my tool. You can download this construction to all computers in PC lab. For 6^th grades, all students can open on Sketchpad this construction. The teacher can start to lesson by introducing what a pantograph is. For example it can be showed a video about how it works. This can be an example of this video: https://www.youtube.com/watch?v=yZEg7cnDejM

 Then, the teacher can say ‘We will try to discover more this tool.’ Students can try in their PC whether the tool makes the thing bigger or smaller and then the teacher can say ‘How this tool can work? Let’s make some measurements.’ Students can measure the lengths of FA, FB, AC, BE, FC, FE from the measure-length part of the Sketchpad. The teacher can ask them to find the division of FA to FB, AC to BE and FC to FE with Number-calculations parts of the Sketchpad. During this the teacher can say ‘The division of one segment to other segment or the division of one part of a segment to whole segment named as ratio. We can find the ratio of various lengths.’ Also, the teacher can ask ‘Did you aware of these three ratios that we measured are equal to each other?’ and can say ‘These ratios form the base of the working principle of a pantograph.’ If it is possible students can try to improve different tools with ratios.

 On the other hand, for 8^th grades, the teacher can start to lesson with the same thing like by introducing the pantograph and watching a video. Then, they can try this tool on their computers whether it works or not. In that case, students already know what is ratio, so the teacher can say ‘Can you find the ratios of FA to FB, AC to BE and FC to FE?’ They will calculate these from Sketchpad. Also, the teacher will say in this tool it is drawn AC as parallel to BE and DC parallel to BF. Because they learnt what similarity is in the previous objective the teacher will ask ‘Which triangles can be similar here?’ After they find similar triangles, the teacher will ask ‘Do you see any relationship between the ratios of the sides of the triangles?’ We are expecting that they will say all are equal. At the end, it will be concluded as we name this common ratio as similarity ratio. Students can try to change these ratios by moving the points of B and C. They can make estimations about the magnitude of drawing. It can be asked ‘How the magnitude of drawings changes when we changed the ratio?’ The lesson can be concluded as pantographs work with this principle and in the real life it was beneficial to use pantographs in the past.

iv. Construction Steps:

 Here there is the video of construction steps of the pantograph on Sketchpad:

https://drive.google.com/file/d/1UQS5mFYps4e8aXupeJsdGNhJwSpTyjMi/view

Also, here is the file of Sketchpad : 

https://drive.google.com/open?id=1yZ246RmJO1cgU6aVEUItwZNF30wcokP2


If you want, you can draw segments of AD, BC, CE, too. And then you can hide the rays that you constructed at the beginning by right clicking on them and choosing ‘hide the ray’. Also, you can add a picture by dragging the picture to the Sketchpad screen. Then, you can put a J close to point C on CD and from the edit menu and merge option, merge the picture and J.

Second way to construct a pantograph:

Link: https://drive.google.com/open?id=1G4_8GkiHwZZC50cFbea-RrfmorO1DcVV

The steps can be listed as,

- construct segment AB.

- Construct ray CD

- Construct circles with centers C and D and radius AB.

- Construct ray CE, where E is one of the intersection points of these circles.

- Construct segment CE and segment DE.

- Construct segment EF, where F is any point past E on ray CE.

- Construct a line through point F parallel to segment DE and also construct point G where the new line and ray CD intersect.

- Construct a line through point D parallel to segment EF.

- Construct segment DH, where point H is the intersection of the lines constructed in steps 7 and 8.

- Hide the circles, lines, and rays so that your pantograph consists only of segments.

-Trace points D and G and drag point D to write something.