Construction of 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.7.1.4.2. Birbirine oranı verilen iki çokluktan biri verildiğinde diğerini bulur.

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 which is a simple device that uses two pens to enlarge or reduce drawings and maps. It can be constructed by computer software programs (see sketchpad below) or constructed from four strips of wood, metal, or poster board.

Its working principle is related to the ratio in Mathematics.

The arms of the pantograph are hinged at points D, E, F, and H so that they move freely. Point C is the projection point and should be held fixed. As point D traces the original figure, a pencil at point G(its image) traces the enlargement. To reduce or enlarge the figure, the pencil is positioned at D, and G is moved around the boundary of the original figure. The pantograph can be changed to obtain different scale factors by adjusting the locations of points E and H. 

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The scale factor is the ratio distance from the fixed point to the drawing point distance from the fixed point to the tracing point. The mechanism of the pantograph serve to keep that ratio constant as the mechanism is rotated, expanded, and/or contracted.

** while responding to this, they can use the measure and calculate tab of the sketchpad.

This tool can be helpful to teach the objectives of M.6.1.7.2, and M.7.1.4.2 because it works according to the ratio. To give an example, for an enlargement with a scale factor of 2, point E is halfway between C and F, and point H is halfway between F and G. Students can find the ratios of FE to FC and FC to FG with the Measure-length and Number-calculations parts of the Sketchpad and see the relationship. You can give distance and ask for ratio or vice-versa to investigate the working principle of pantographs and their relation to ratio. While doing this, the teacher can ask “Do you see any ratio while enlarging or reducing your image on pantograph? Is it changing or stable?” Therefore students can make relations with pantograph and ratio. The questions may engage students in the topic and see the relationship between the images and their ratios.

Moreover, this tool is helpful for teaching the objective M.8.3.3.2 in the curriculum, since this tool also works with the similarity principle. While we are constructing this tool, we draw ED as parallel to FG and HD parallel to FE. In the end, we may not hide the segment between CG and see that ACD and HDG are similar triangles. Also, we see that HDG and FGC are similar triangles, too. This property of the pantograph will provide students’ better understanding of similarity and similarity ratio.

iii. User Manual: 

Before showing how you will use this tool, I am going to introduce you to the parts of the pantograph I created above.

PARTS:

(C) Pivot Base - To mount Pantograph to the working surface

(D) Tracing Point - To trace original

(G) Lead Holder - To draw a picture lead must be inserted in the lead holder and must be held in contact with the blank drawing paper.

(F) Balancing Pin - Must be on a working surface

(E, H) Ratio Screws - Used to set the desired ratio of enlargement or reduction.

First of all, you can watch my construction video which explains how to construct a pantograph on a sketchpad then create your own pantograph. You can download this construction to all computers in your lab for your students. https://drive.google.com/file/d/1UQS5mFYps4e8aXupeJsdGNhJwSpTyjMi/view

For 6th and 7th graders you can start by showing the working mechanism of the pantograph and ask them to measure EF, DH, FH, and ED. Then ask them to measure CE, CF, CD, CG, ED, and FG to find the constant ratio. After measuring, they will find  EF=DH and FH=ED. Also, when point D moves G moves accordingly, and CE/CF=CD/CG= ED/FG. At this point, you can say "The division of one segment to another segment or the division of one part of a segment to the whole segment called as a ratio. We can find the ratio of various lengths." Also, you can ask "Did you notice that these three ratios that we measured are equal to each other?" and  you can explain "These ratios form the base of the working principle of a pantograph, If you want to change this ratio, you can change the distance AB or move free points." For more investigations you can ask your students;

1. What happens to the scale factor for enlargement as point E is moved toward D and point H is moved toward F?

2. Where should points E and H be placed for an enlargement with a scale factor of 8?

3. What happens to the scale factor for enlargement as point E is moved toward F and point H is moved toward G?

For your 8th graders, you can create different similar triangles on pantograph with changing distance AB or moving free points..For example, triangle CED and  CFG are similar in my construction. You can start to lesson by introducing the pantograph and watching the video in this link https://www.youtube.com/watch?v=raQs62xOmXk. Then, you can want them to try the working mechanism of the pantograph on their own computers. In this step, students have already know what is ratio thus, you may ask them to find ratios. Also, you can mention a little about the construction and say EF// DH and ED//FH because it is important for the similarity. Then, you can ask them "How many different triangles do you see? " Which ones are similar to each other?" After finding similar triangles, you can ask them their strategy. It is expected that they will relate the similarity with ratio. For example,  they may say "triangle CED and  CFG are similar because the ratio of side CE to CF, ED to FG and CD to CG are the same." After that, you can call this ratio a similarity ratio.

iv. Construction Steps: 

We found 2 different ways of constructing a pantograph on a sketchpad. You can choose one of them and construct your own pantograph.

1. You can click the link and see our construction video. In this video, we vocalize the construction.

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

Here is the file of Sketchpad

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

2. You can also watch this video that we record on sketchpad as an alternative method.

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 the previous 2 steps.

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

-Tracepoints D and G and drag point D to write something.