5 Activities For Understanding The Conic Sections

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Right from ice cream, to a party hat, most of the real-life entities are conical. As a math pupil, it may be common to infer these shapes. What’s equally crucial is to also infer that sectioning this shape can give out multiple insights. While we can categorize these as circles, ellipses, hyperbola, and parabola, each of them have real-life applications. 

Accordingly, it may be a fairer idea to learn about these with activities to ensure real-life settings and applications after classroom learning. Here we listed out five distinct activities that may assist the pupils to comprehend conic sections better to implement them later in academics. 

Various conic sections- Comprehending them in detail!

By Definition, conic sections are the two-dimensional shapes that are formed when a plane intersects at a certain angle. As the shape of the cone is not uniform throughout its length, the shapes formed often differ when the angle of plane intersection is altered. Here, let us read through four distinct shapes that may be crafted:

  • Ellipses: These are random shapes that are formed when a plane touches a cone at a particular angle. The angle of intersection is often less than 90 degrees. An example would be: The axis of revolution of various planets is elliptical.
  • Circles: These are special sorts of ellipses that are formed when the cutting plane is parallel to the base of the cone. In other words, we can define it as the shape formed when the plane is perpendicular to the axis of revolution of the cone. In real life, we see various circular entities like steering, pizzas, rings, and many more.
  • Parabola: When the plane strikes the surface of the cone rather than covering its whole breadth, an arc is formed. Rather than a close ellipse, this shape is referred to as Parabola. For instance, a water fountain often opts parabolic paths.
  • Hyperbola: Say we have considered a couple of cones facing away from each other upside down. Now, if a plane intersects them being parallel to the axis of rotation, a hyperbola is formed. It is basically a couple of arcs facing away from each other. Apart from guitar, you may notice cautious hyperbolic shapes in real life. 

Conic sections activity – To discern them better

Being aware of the explorative mindset of young learners, here we listed out a few eminent activities crafted to discern conic sections better:

1. Introduction to the Cones

Introduction to the cones

The formation of various conic sections may not be clear after classroom teaching.  This activity ensures the students learn about the same. 

This activity needs the teacher to arrange five paper cones, paint, play scissors, and a piece of paper. The student takes one cone and cuts at an angle, but they dip the paper into the paint and stamp it on the paper. They can observe an ellipse is formed. Now, they take a second cone and cut it parallel to the base, dip it in paint, and stamp it beside the ellipse. This would form a circle.  

Similarly, the third cone is cut on a surface, dipped in paint, and stamped on paper, which forms a parabola. And finally, a hyperbola is stamped by cutting two cones upside down. 

This activity helps the learner to discern how these shapes are formed from cones.

2. Clay It Up

Aim: To show that a parabola as a conical section lies between that of the ellipse and a hyperbola

Material required: clay, metal wire, two pencils 

Clay it up

How To Do It : 

  • Straighten the wire and wrap it around a pencil on each end with two palm spaces in between. 
  • Take the clay and make a cone out of it. 
  • Imagine that there is an inverted cone that’s set upon the clay cone, with its vertex joint end to end. 
  • Now hold both the pencils and straighten the wire, and begin the dissection. 

1st Conical Section 

  • Move the wire in such a way that the dissection begins just below its conical vertex, and goes straight towards its spherical face. 
  • Detach the dissected part of the clay and set it aside. 
  • If you carefully analyse the shape of the remaining part, you would find that a parabola is formed out of it. 
  • Memorise the shape and share three examples of things that are of the same shape. 

2nd Conical Section

  • This time, take the wire, hold it horizontally but instead slice through at a little angle. 
  • Detach the dissected part of the clay and set it aside. 
  • Again, upon careful review, you would find that an ellipse is formed out of the remaining part. 
  • Memorise the shape and share three examples of things that are of the same shape. 

3rd Conical Section 

  • For this section, hold the wire parallel to the other edge of the cone and slice the contrasting end, through and through to the spherical face of it. 
  • Again, detach the dissected part of the clay and set it aside. 
  • If you would review the remaining part here, you would find that a hyperbola is formed. 
  • Memorise the shape and share three examples of things that are of the same shape.

3. Conic Zone

Aim: To understand the existence of conic sections in our immediate environment and how we are completely surrounded by them. 

Material Required: Printed captured pictures

Conic zone

How To Do It :

  1. In this activity, students will be given a basic understanding of how many conic sections there are and how they look, either through simplified notes or by an educator, without the introduction of any equations yet. 
  2. Now, the students will be asked to take a day and identify these sections in their immediate environment and click pictures of them. For example, It could involve the handles of a gate.
  1. Students will have to print these pictures, without mentioning which shape it is.
  2. Each student will have to bring at least a picture per conic section mandatorily. (One student will be uploading a minimum of 4 pictures, one for circle, parabola, hyperbola, and ellipse, respectively)
  3. After a day, once there are several pictures to the rescue, a presentation will be arranged in the classroom where all the pictures will be viewed. 
  4. The students will be asked to identify each picture in the presentation. 
  5. The picture that would have the maximum correct answers for the identification of the conic section they represent without any confusion, will be rewarded. 

4. Graphication

Aim: To understand and practice the equations in regards to various conic sections through art.

Material Required: Graph papers, pre-drawn sketches, pens, and pencils

Graphication

Tak for the teacher: The educator will have to create various sceneries or objects with multiple conic sections. Example: If you are drawing a flower, make sure the petals are made of two parabolas joined together, similarly for the leaves, for the stem, one could make sure to give spiral texture to it with colours, and the centre of the petals will, of course, be a circle.

How to do it: 

  1. This activity requires the students to have a basic knowledge of the shapes and definitions of the various conic sections. 
  2. The teacher will be asked to divide the students into groups of 4. 
  3. Each group will be given a pre-drawn scenery. 
  4. The groups will then be provided with graph papers for each time, and they will be required to copy that scenery on the graph paper, either by redrawing its boundaries or by simply putting the drawing over the graph paper and tracing it. 
  5. After recreating it on the graph, each group will be required to mark the coordinate points of each vertex of the outline of the drawing. 
  6. These groups will now be asked to write the formula-based equations of each conic section and figure out the various conic sections in the picture. 
  7. They will then have to derive those equations through the formulas, by using the coordinates they have identified for the respective shapes. 

5. Ride The Rope

Aim: To identify the various conic sections, namely the circle and the ellipse through their focal point and associated properties. 

Material required: Chalk, Rope, and Volunteers 

Ride the rope

How to do it : 

  1. The teacher will have to divide the student volunteers into two groups of two and three each. 
  2. Now, the group with two students will be given chalk and a rope. One student will be asked to stand at a point and hold the rope while the other student moves in a circular motion around him/her, leaving an imprint with chalk as he/she moves. 
  3. This is just like drawing a circle with a compass where the tip represents the student at the center and the pencil/pen attached is the student who will move in a circle. To make this more easily understandable, one can think of a stone tied with a thread and students start rotating the stone in the air by holding the other loose end of the thread.
  4. The students will then be asked to observe the shape which is a circle and list the properties of the focal point associated with that conic section. By this, students will observe that every point on the circle is equidistant from the center of the circle.
  5. For the next section, the group will comprise 3 students, A, B, and C. Students A and B will be asked to stand at parallel points to each other attached with a rope to their waist, so they remain equidistant through the activity. 
  6. Now, student C will be connected with students A and B respectively with long ropes. 
  7. Student C will then be asked to move around in a circular motion, but circumfer points A and B from one side (sort of a semicircle), then move to the other side and circumfer them forming another semicircle from the changed side. 
  8. When the two chalked lines will be joined, they will form an elliptical conic section with two focal points. 
  9. The students will then be asked to enlist the properties of an ellipse based on the two focal points. 

Wrapping up,

Sometimes, despite efficient assignments and worksheets, pupils may need additional practice sessions to clear out minor doubts and ensure gripping mastery of the subject. Activities like the above-mentioned choices may succeed not only in better learning but also in understanding how these notes are practically applicable. Accordingly, read through these activities and check out which of these may be befitting for you, even with minor changes. 


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