Applying Fractions to Polygons

In this programming assignment, we shall draw polygons and divide them into fractions using Pro-Bot. This assignment can be a follow-up to the one where we learned to draw various polygons using Pro-Bot. We shall use sequential programming for drawing irregular polygons and Repeat Loops to draw regular polygons. Next, we shall divide the polygons into specified fractions using Pro-Bot as our drawing tool.

Computer Science concepts involved:   Sequential programming, Repeat loops

Math concepts involved:   Polygons (regular and irregular), Linear & Angular Measurements, Fractions, Area, Congruence

Grade levels:   3, 4, 5

Hours required:   2 or more

Polygons

In geometry, a polygon refers to a closed, two-dimensional figure formed by a set of straight line segments. The straight line segments are called the polygon’s edges or sides, and the points where two edges meet are the polygon's vertices or corners.

If all edges are equal and all angles are equal, then it is a regular polygon. Else, it is an irregular polygon.

Square:

1. Use Pro-Bot to draw a square of sides 8 cm. Each interior angle is 90 degrees.
2. Divide the square horizontally into quarters (1/4) using Pro-Bot as your drawing tool. 
3. Divide the square vertically into quarters (1/4) using Pro-Bot as your drawing tool.
4. Divide the square into 4 equal squares, using Pro-Bot as your drawing tool. What fraction of the large square does each small square occupy?
5. Divide the square into 2 equal sized triangles, using Pro-Bot as your drawing tool. Are the triangles congruent? 

Rectangle:

1. Use Pro-Bot to draw a rectangle of sides 4 cm and 6 cm. Each interior angle is 90 degrees.
2. Divide the rectangle into thirds along the longer side using Pro-Bot as your drawing tool. 
3. Divide the rectangle into halves along the shorter side using Pro-Bot as your drawing tool.
4. Divide the rectangle into quarters using Pro-Bot as your drawing tool.
5. Divide the rectangle into 2 triangles along a diagonal, using Pro-Bot as your drawing tool. Are the triangles congruent? Are they equal in area?

Rhombus:

1. Use Pro-Bot to draw a rhombus of sides 6 cm. One pair of opposite interior angles is 60 degrees each and the other pair is 120 degrees each. 
2. Divide the rhombus into thirds, using Pro-Bot as your drawing tool.
3. Divide the rhombus into sixths (1/6) using Pro-Bot as your drawing tool. Are the fractions congruent shapes? Are they rhombuses? If not, what shapes are they?

Parallelogram:

1. Use Pro-Bot to draw a parallelogram of sides 4 cm and 6 cm. One pair of interior opposite angles is 45 degrees each and the other pair is 135 degrees each (Remember that for a parallelogram, opposite angles are equal).
2. Divide the parallelogram into 2 equal parallelograms, using Pro-Bot as your drawing tool. What fraction of the original parallelogram is occupied by the smaller ones?
3. Considering that the minimum dimension that can be provided by Pro-Bot is 1 cm, what is the maximum number of smaller parallelograms that you can divide up the larger parallelogram into, using Pro-Bot as your drawing tool? What fraction of the larger parallelogram is occupied by each of the smaller units?

Equilateral Triangle:

1. Use Pro-Bot to draw an equilateral triangle of sides 6 cm. Each interior angle is 60 degrees.
2. Divide the triangle into 2 equal triangles, using Pro-Bot as your drawing tool.
3. Divide the triangle into 3 triangles of equal area, using Pro-Bot as your drawing tool. You can do this in a couple of ways: 

• Divide one side of the triangle into 3 equal parts. Then draw lines connecting the opposite vertex to the two points that trisect the side. 
• Find the centroid: If you draw a line from each vertex to the midpoint of the opposite side, the point at which all three lines meet is the centroid. Draw a line from each vertex to the centroid to divide up the triangle into 3 equal triangles.

Drawing Polygons using Pro-Bot given Area or Perimeter

In this programming assignment, we shall draw polygons using Pro-Bot, given the area or perimeter of the polygons. Your first task is to figure out the dimensions of the polygons from the given data and your second task is to write programs to draw the polygons using Pro-Bot. You can use either sequential programming or Repeat loops to draw the polygons, depending on whether they are irregular or regular.

Computer Science concepts involved:  Sequential programming, Repeat loops

Math concepts involved:  Polygons (regular and irregular), Area, Perimeter, Measurement, Angles

Grade levels:  4, 5

Hours required: 3 - 4

Draw Polygons using Pro-Bot given the perimeter:

1. The perimeter of a regular, 4 sided polygon is 32 cm. The interior angles are all 90 degrees each. Can you write a program for Pro-Bot to draw this polygon? What kind of a polygon is it?
2. The perimeter of a regular, 4 sided polygon is 32 cm. A pair of opposite interior angles are 60 degrees each. Can you write a program for Pro-Bot to draw this polygon? What kind of a polygon is it?
3. The perimeter of a regular 3 sided polygon is 18 cm. The interior angles are all 60 degrees each. Can you write a program for Pro-Bot to draw this polygon? What kind of a polygon is it?
4. The perimeter of a 3 sided polygon is 24 cm. The interior angle between the shortest sides is 90 degrees. The interior angle between the shortest and longest sides is 54 degrees. The sum of the lengths of the shortest sides is 14 cm. The difference between the lengths of the shortest sides is 2 cm. What kind of a polygon is it? Can you write a program for Pro-Bot to draw this polygon? 
5. The perimeter of a 4 sided polygon is 24 cm. Two of its sides are parallel to each other and the other two are not. The sides that are not parallel are equal in length. Both the angles coming from a parallel side are equal: the pair of angles coming from the longer parallel side is 54 degrees each and the pair of angles coming from the shorter parallel side is 126 degrees each. The sum of the lengths of the parallel sides is 14 cm. The shorter of the parallel sides is 4 cm. Can you identify the polygon? Can you write a program for Pro-Bot to draw this polygon?
6. The perimeter of a 4 sided polygon is 24 cm. Both sets of opposite sides are parallel. What kind of a polygon do you think it is? The ratio of the shorter side to the longer side of this polygon is 1/2; i.e.; each shorter side is half the length of the longer side. One pair of interior opposite angles is 60 degrees and the other pair of interior opposite angles is 120 degrees. Can you write a program for Pro-Bot to draw this polygon?
7. The perimeter of a 5 sided regular polygon is 25 cm. Can you write a program for Pro-Bot to draw this polygon? What kind of a polygon is it? What is the value of each interior angle?

Drawing Polygons using Pro-Bot given the area:

1. The area of a regular, 4 sided polygon is 24 square cm. The interior angles are all 90 degrees each. Can you write a program for Pro-Bot to draw this polygon? What kind of a polygon is it?
2. The area of a 4 sided polygon is 24 square cm. The opposite sides are equal in length and the interior angles are all 90 degrees each. The longer side measures 2 cm more than the shorter side. The sum of the lengths of the long side and the short side is 10 cm. Can you write a program for ProBot to draw this polygon? What kind of a polygon is it?
3. The area of a 3 sided polygon is 24 square cm. The interior angle between the shortest sides is 90 degrees. The interior angle between the shortest and longest sides is 54 degrees. The sum of the lengths of the shortest sides is 14 cm. The difference between the lengths of the shortest sides is 2 cm. What kind of a polygon is it? Can you write a program for ProBot to draw this polygon? 
4. The area of a 4 sided polygon is 24 square cm. Both sets of opposite sides are parallel to each other. What kind of a polygon is it? The distance between the longer parallel sides is 4 cm. The perimeter of this polygon is 20 cm. One pair of interior opposite angles is 60 degrees and the other pair of interior opposite angles is 120 degrees. Can you write a program for ProBot to draw this polygon?

Solutions: Drawing Polygons using Pro-Bot

Here is a subset of the possible solutions for drawing the polygons in this assignment. Please keep in mind that there are multiple correct ways of programming each of these figures.


The following programs draw the polygons without using Repeat Loops.

1. Square of sides 8 cm:

Fd 8
Rt
Fd 8
Rt
Fd 8
Rt
Fd 8

2. Rectangle 4 cm x 6 cm: 

Fd 4
Rt
Fd 6
Rt
Fd 4
Rt
Fd 6

3. Parallelogram 4 cm x 6 cm:

Fd 4
Rt 45
Fd 6
Rt 135
Fd 4
Rt 65
Fd 6

4. Rhombus 6 cm:

Fd 6
Rt 60
Fd 6
Rt 120
Fd 6
Rt 60
Fd 6

5. Equilateral Triangle of sides 6 cm:

Fd 6
Rt 120
Fd 6
Rt 120
Fd 6 

6. Right Triangle of sides 3 cm, 4 cm, 5 cm:

Fd 3
Rt 126
Fd 5
Rt 144
Fd 4 

Drawing Regular Polygons using Repeat Loops in Pro-Bot

All of the regular polygons below have sides of 6 cm.

1. Equilateral Triangle

Rpt 3 [
Fd 6
Rt 120
]

2.  Square

Rpt 4 [
Fd 6
Rt 90
] 

// You can also use the instruction "Rt", instead of "Rt 90" in the code above,                                           // as the default turn is 90 degrees for Pro-Bot.

3. Pentagon

Rpt 5 [
Fd 6
Rt 72
]

4. Hexagon

Rpt 6 [
Fd 6
Rt 60
]

5. Octagon

Rpt 8 [
Fd 6
Rt 45
]

6. Nonagon

Rpt 9 [
Fd 6
Rt 40
] 

7. Decagon

Rpt 10 [
Fd 6
Rt 36
]

A Circle using a Repeat Loop

Rpt 360 [
Fd 1                 // Move 1 cm
Rt 1                 // Turn 1 degree
]

An Algorithm for Drawing Regular Polygons using Repeat Loops 

From the pieces of code above for various regular polygons, the relation between the number of sides of a regular polygon and the angle of turn can be deduced to 360/N where N is the number of sides of the polygon.

The generalized algorithm for drawing regular polygons is:

Rpt N [
Fd X
Rt 360/N
]

where N is the number of sides of the regular polygon, X is the length of each side of the polygon.  

Drawing Polygons using Pro-Bot

In this programming assignment, we shall draw polygons using Pro-Bot. We shall start by using sequential programming to draw the polygons. Next, we shall look at some regular polygons and learn how to use Pro-Bot’s Repeat Loops to draw them. We shall also develop a generalized algorithm to draw regular polygons using Pro-Bot. We can store programs for the polygons as Procedures on Pro-Bot.

Computer Science concepts involved:  Sequential programming, Repeat loops, Algorithm development

Math concepts involved:  Polygons (regular and irregular), Linear & Angular Measurements, Interior angles, Supplementary angles

Grade levels:  3, 4, 5

Hours required:  3 or more

Polygons

In geometry, a polygon refers to a closed, two-dimensional figure formed by a set of straight line segments. The straight line segments are called the polygon’s edges or sides, and the points where two edges meet are the polygon's vertices or corners.

We see polygons of different types all around us in our daily lives. Here is an interesting article that I found on mathforum.org, as to where we see different types of polygons in our daily lives.

If all edges are equal and all angles are equal, then it is a regular polygon. Else, it is an irregular polygon.

Square:

Use Pro-Bot to draw a square of sides 8 cm. Each interior angle is 90 degrees.

Rectangle:

Use Pro-Bot to draw a rectangle of sides 4 cm and 6 cm. Each interior angle is 90 degrees.

Parallelogram:

Use Pro-Bot to draw a parallelogram of sides 4 cm and 6 cm. One pair of interior opposite angles is 45 degrees each and the other pair is 135 degrees each.

Note: Remember that for a parallelogram, opposite angles are equal. 

In the figure of the parallelogram above, angles marked ‘a’ are equal; they are opposite angles. Similarly, angles marked ‘b’ are equal; they too are opposite angles. The angles ‘a’ and ‘b’ add up to 180 degrees; ‘a’ and ‘b’ are supplementary angles.

Note: Remember that when Pro-Bot has to draw a 45 degree interior vertex, it would have to turn 135 degrees and not 45 degrees, assuming it was moving in the forward direction before making the turn. Similarly for the 135 degree vertex, Pro-Bot would have to turn 45 degrees. Can you see why?

Rhombus:

Use Pro-Bot to draw a rhombus of sides 6 cm. One pair of opposite interior angles is 60 degrees each and the other pair is 120 degrees each. 

Next, use Pro-Bot to draw a rhombus of sides 6 cm. One pair of opposite interior angles is 90 degrees each. What shape do you get?

Equilateral Triangle:

Use Pro-Bot to draw an equilateral triangle of sides 6 cm. Each interior angle is 60 degrees.

Note: Remember that when you draw an equilateral triangle using Pro-Bot, at each vertex Pro-Bot has to turn 120 degrees (supplementary angle), and not 60 degrees, assuming it was moving in the forward direction before making the turn. Can you see why?

Right Triangle:

Use Pro-Bot to draw a right triangle of sides 3 cm, 4 cm and 5 cm. The interior angle between the 4 cm and 5 cm sides is 36 degrees. The interior angle between the 3 cm and 5 cm sides is 54 degrees. The interior angle between the 3 cm and 4 cm sides is 90 degrees.

Note: Remember that when you draw a right triangle using Pro-Bot, at each vertex Pro-Bot has to turn at its supplementary angle, and not the interior angle, assuming it was moving in the forward direction before making the turn. 

Drawing Regular Polygons using Repeat Loops in Pro-Bot

For the various shapes above that you drew using ProBot, you could see that in the case of programs for for the regular polygons (polygons with equal sides and equal interior angles), there is a pattern of instructions that keeps repeating multiple times. 

You can rewrite your programs for the square and the equilateral triangle using Repeat Loops in Pro-Bot. 

Use Pro-Bot to draw each of the shapes below. Use Repeat Loops to write your programs.

1. Equilateral Triangle:   Use Pro-Bot to draw an equilateral triangle of sides 6 cm. Each interior angle is 60 degrees.
2. Square:    Use Pro-Bot to draw a square of sides 6 cm. Each interior angle is 90 degrees.
3. Pentagon:   Use Pro-Bot to draw a pentagon of sides 6 cm. Each interior angle is 108 degrees.
4. Hexagon:    Use Pro-Bot to draw a hexagon of sides 6 cm. Each interior angle is 120 degrees.
5. Octagon:    Use Pro-Bot to draw a octagon of sides 6 cm. Each interior angle is 135 degrees.
6. Nonagon:   Use Pro-Bot to draw a nonagon of sides 6 cm. Each interior angle is 140 degrees.
7. Decagon:   Use Pro-Bot to draw a decagon of sides 6 cm. Each interior angle is 144 degrees.

After drawing the various polygons, ask the students to identify a few examples of where they might see these shapes in their daily lives. For example: a sandwich cut in half is a triangle, roof trusses are triangles, a STOP sign is an octagon, a YIELD sign is a triangle, cells in beehives/ honeycombs are hexagons, etc.

Here is a set of solutions for this assignment.

Develop an Algorithm for drawing Regular Polygons using Pro-Bot

Now that you have written programs for Pro-Bot to draw various regular polygons, let us develop a generalized algorithm for the same.

1. Is it more efficient to use Repeat Loops or Sequential Programming when you write programs for regular polygons on Pro-Bot?
2. From your programs above, can you identify a technique to find the angle at which Pro-Bot should turn at each vertex for a regular polygon?
3. Given the length of a side of a regular polygon and the angle you found using your above technique, can you write a program for drawing a regular polygon of N sides (N being any whole number) using Pro-Bot?
4. Can you now generalize your method and write down the various steps involved, as an algorithm.

Experiment more with Regular Polygons using Pro-Bot:

1. While developing your algorithm, did you notice that as the number of sides of the polygon increases, the shape of the polygon tends to a circle? What happens to the interior angle of the polygon in this case; does it increase or decrease?
2. What happens to the angle at which Pro-Bot has to turn as the number of sides of the polygon increases? Does it increase or decrease?
3. What is the smallest measurement of angle that you can use on Pro-Bot? What is the smallest length that you can draw using Pro-Bot?
4. Using the algorithm that you developed above for Pro-Bot, draw a regular polygon using that uses the smallest angle and the smallest length that Pro-Bot can provide as its dimensions. What shape do you get? 
5. Draw a regular polygon using Pro-Bot, just like the one in the previous question, that uses the smallest degree of angle, but with the length of the side increased by 1 unit. What shape do you get? 
6. What happens to the shape if you keep increasing the length of the side by 1 unit at a time, but keep the angle constant at the smallest value?
7. What happens if you increase both the length and the angle by 1 unit each time? What difference do you see?

Challenge Question: 

1. Can you write a program for Pro-Bot to draw a circular maze (a spiral shape) using all of the ideas that you learned from the above questions? It is easier to start the spiral from inside and start moving outwards. Start the spiral with the smallest length and angle that Pro-Bot can provide and then build outwards.