Draw Polygons with a Given Perimeter using Pro-Bot

Can you program the Pro-Bot to draw polygons & compound figures with a given perimeter?

This is an assignment that I designed for our Grade 3 students. It relates to the Common Core Math Standards: Geometric measurement: recognize perimeter. For this exercise, I highly recommend using graph paper, as it provides a helpful medium for the kids to work out the math problems. Provide at least one sheet per child to work out the problem and then additional sheets as required for the groups to draw the figures using Pro-Bot. Here is a link to a graph paper with 1 cm grid in PDF format; you can make copies for the students to draw on using the Pro-Bot.

Perimeter of a Figure

A perimeter is a path that surrounds a two-dimensional shape. The term may be used for either the path or its length. It can be thought of as the length of the outline of a shape. (Wiki)

How do you calculate the perimeter of a given figure? You add the length of all the sides of that figure that form its outline. Say you are given a square with sides 3 cm each. The perimeter of the square is 3 + 3 + 3 + 3 = 12 cm. Similarly, a 5 cm x 6 cm rectangle. has a perimeter of 5 + 6 + 5 + 6 = 22 cm. 

Can we do the reverse too? Given the perimeter, can we come up with the design for a figure with that perimeter? 

Let's look at an example. Given a perimeter of 12 cm, how many polygons can we draw? We can draw multiple polygons, all with the exact same perimeter of 12 cm. In the figure below, you can see:

• a square 3 cm x 3 cm,
• a rectangle 5 cm x 1 cm,
• a rectangle 4 cm x 2 cm,
• a hexagon with sides 3 cm, 1 cm, 1 cm, 1 cm, 4 cm, 2 cm

Can you think of more polygons with a perimeter of 12 cm?

Let's now look at a scenario that shows the practical application of the concept of perimeters. And then program the Pro-Bot to draw a few polygons with a given perimeter.

Programming Assignment

Old McDonald lives on a farm and has lots of animals. He would like to build a new set of fences to keep his cows safe.

1. If Old McDonald has 36 meters of fencing available, how many different ways can he build an enclosed area for his cows? Make a rough drawing of the figures that you come up with and then program the Pro-Bot to draw them on graph paper. Use 1 cm to represent 1 m in your figures.
2. Classify the figures that you came up with into the various classes of polygons based on the number of sides.
3. Write programs for Pro-Bot to draw at least 3 of the figures that you came up with.
4. Suppose Old McDonald has only 35 meters of fencing available, but wants to build a square or rectangular fence using all of that fencing material. Would it be possible for him to build it? Why or why not? 
5. You can program the Pro-Bot to draw other figures with the 36 cm perimeter. Or explore other figures with different perimeters.

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?

Programming Tangrams using Pro-Bot

This is a comprehensive programming project spanning over 4 hours, that aims to teach the key concepts that can be introduced via Pro-Bot (namely sequential programming, Repeat Loops and Procedures) while applying them to geometrical concepts that are covered in the classrooms. 

We shall make use of the famous tangram puzzles for this assignment. Let us take a quick look at what tangrams are and then talk about how to write programs for Pro-Bot to create them.   

Tangrams:

The tangram is a dissection puzzle consisting of seven flat shapes, called tans, which are put together to form shapes. The objective of the puzzle is to form a specific shape (given only an outline or silhouette) using all seven pieces, which may not overlap. It is reputed to have been invented in China and is one of the most popular dissection puzzles in the world.(Wikipedia)

Research shows that tangrams are helpful for developing an intuitive sense of geometry and developing visual-spatial skills. Tangrams have both geometric and artistic features. They help with classifying shapes, understanding the spatial rotation of shapes, understanding fractions, understanding the properties of various shapes including angles, area, perimeter, etc.

To create a tangram, a square is divided up into 16 equal squares. Diagonal lines are drawn to produce 7 shapes. The seven pieces (numbered in the above picture) are:

• 2 large right triangles  (A, D)

• 1 medium right triangle (E)

• 2 small right triangles (C, G)

• 1 square (F)

• 1 parallelogram (B)

Some of the properties of the tans (puzzle pieces) are: 

1. The large triangle is twice the area of the medium triangle. 
2. The medium triangle, the square, and the parallelogram are each twice the area of a small triangle. 
3. Each angle of the square measures 90 degrees. 
4. Each triangle contains a 90 degree and two 45 degree angles (isosceles right triangles). 
5. The parallelogram contains two 45 degree and two 135 degree angles. 

The relationships among the pieces enable them to fit together to form many figures and arrangements.

Now that we have taken a look at tangrams, let us write programs for Pro-Bot to draw the tangram puzzles.

We shall start by writing simple sequential programs to generate the seven individual puzzle pieces of the tangram. Then, we shall modify some of those programs to use Repeat Loops. We may also take a look at developing a generalized algorithm for drawing regular polygons (polygons with all sides equal and all angles equal) using Pro-Bot. Next, we shall learn to store programs for the individual puzzle pieces as Procedures in Pro-Bot’s memory, and then “call” them within programs that draw more complex tangram puzzles. 

Note that the programming questions are given in the order of increasing complexity, starting with simple polygons, then combining two or three of those to create compound figures and finally, creating complex tangram puzzles that use all seven pieces. 

Programming Assignment: 

Given a large square of sides 14 cm each, divide it into sections as shown in the figure below and create the 7 pieces of a tangram. Find the dimensions for each of the seven pieces. Make suitable modifications to the dimensions so that they can be used by Pro-Bot. Write programs to draw each piece separately using Pro-Bot. Finally, draw simple/complex tangram puzzles using Pro-Bot, making use of the programs that you wrote for the 7 tans/puzzle pieces.

Computer Science concepts involved:  Sequential programming, Repeat loops, Procedures/Sub-routines

Math concepts involved:  Polygons (Quadrilaterals, Triangles), Measurement, Fractions, Decimals, Rounding of decimal numbers, Angles, Trigonometry, Area, Perimeter, Congruence, Symmetry

Grade levels:  4, 5, 6

Hours required:  4 or more

Lesson Plan:

Hour 1:   Introduce Pro-Bot and tangrams. 

Allow the students to get familiarized with the Pro-Bot and its control panel. Introduce tangrams. Find out the dimensions of the 7 pieces in the tangram puzzle. Use sequential programming to create each of the 7 pieces in the tangram.

14 cm side square divided into 16 equal parts, 

with the diagonals drawn to create the seven puzzle pieces

• On a piece of paper, draw a square of sides 14 cm each.
• Divide this 14 cm side square into 16 equal squares and trace all 7 shapes, using the above diagram as your guide.     
• For the right triangles, either calculate the hypotenuse using the mathematical formula or measure it using a ruler.
• Find the dimensions of the square and the parallelogram.
• Can you figure out what fraction of the large square is taken up by each of the seven puzzle pieces?
• Check to see if the dimensions obtained for the sides of the seven puzzle pieces are whole numbers or decimals. If they are decimal numbers, think about whether decimals can be used with Pro-Bot. If not, how can you adjust these numbers to work with Pro-Bot?
• Once you have adjusted the dimensions so that they work on Pro-Bot, write programs to draw the puzzle pieces using Pro-Bot.
• You can see that some of the puzzle pieces are congruent. Hence, you need to write the program only once for each set of congruent shapes.
• Test your programs on Pro-Bot, one at a time to see if you get the desired results for each shape.

Hour 2:    Introduce Repeat Loops on Pro-Bot.

• Now that all the puzzle pieces have been drawn using sequential programming, show how the programs for the square and the parallelogram can be rewritten using Repeat Loops in Pro-Bot. 
• Write programs for drawing a few regular polygons using Pro-Bot (polygons with all sides equal and all angles equal) such as pentagon, hexagon, etc. Show how Repeat Loops can be used to write programs for these figures. Develop a generalized algorithm (a set of steps for solving a problem) for drawing regular polygons using Pro-Bot. 

Hour 3:    Introduce Procedures on Pro-Bot. Draw compound figures that use a subset of the seven puzzle pieces.

• Store the program for each puzzle piece as a Procedure on Pro-Bot. Since the puzzle pieces would be used multiple times in the project, it makes it easier for you as the programmer, to write the program for each puzzle piece just once, store it as a Procedure and re-use it multiple times.
• Work on a few compound figures that use a combination of just two or three of the puzzle pieces. Call the procedures for the individual puzzle pieces while writing programs for the compound figures.

A.   Write programs for Pro-Bot to draw the following compound figures that use the square and a small triangle from the tangram. 

B.   Write programs for Pro-Bot to draw the following compound figures that use the parallelogram and two small triangles from the tangram. 

C.   Your class is planning to do a gardening project. You have decided to grow spinach in an area measuring 12.5 square feet and carrots in 25 square feet. The two designs that you are considering for the garden are the ones in Question A above. Would you choose the square or the triangle part of the garden for the carrots? If the cost of fencing the garden is $5 per foot, which design would you choose so that the cost of fencing can be minimized.

D.   Your school has decided to pursue more vegetable gardening projects. This time, the plan is to grow tomatoes and peppers in spaces of equal area and carrots in a space that is double the area of tomatoes. The garden designs being considered are the ones in Question B above. Determine which vegetable would go in the parallelogram part of the garden and which ones would go in the triangles. How many square feet each of tomatoes, peppers and carrots can be grown? If the cost of fencing the garden is $5 per foot, which design would you choose so that the cost of fencing can be minimized.

E.   Can you make a trapezoid using just two pieces from the tangram? Write a program to draw your trapezoid using Pro-Bot.

F.   Can you make a parallelogram using just two pieces from the tangram? Write a program to draw your parallelogram using Pro-Bot.

G.   In each of the compound figures that you worked with, can you see the relationship between the angles of the puzzle pieces that allows them to fit together seamlessly?

Hour 4:    Write programs to draw complex tangram figures using Pro-Bot, using all seven puzzle pieces. You can choose either option A or option B.

Option A:

Create geometric shapes using all seven puzzle pieces of the tangram. Write programs for Pro-Bot that uses some of the Procedures for the puzzle pieces to draw the following figures. Compare the areas and perimeters among the complex figures that you create.

A.   Can you make a square that uses all 7 of the tangram pieces? Write a program to draw your square using Pro-Bot.

B.   Can you make a triangle that uses all 7 of the tangram pieces? Write a program to draw your triangle using Pro-Bot.

C.   Can you make a trapezoid that uses all 7 of the tangram pieces? Write a program to draw your trapezoid using Pro-Bot.

D.   Can you make a parallelogram that uses all 7 of the tangram pieces? Write a program to draw your parallelogram using Pro-Bot.

E.   How many squares can you make using some or all of the tangram puzzle pieces?

Option B:

You can find hundreds of artistic tangram puzzles on Google. Choose the designs that are the most interesting to you and work with them. Write programs for Pro-Bot that uses some of the procedures for the puzzle pieces to draw the figures that you choose.

Here is an example. Can you write a program for Pro-Bot to draw this figure, using some of the procedures that you previously created for the tangram pieces?

Here is a sample set of solutions.