One way to work through the following exercises is to obtain a paper print-out and use Logo software. You may use any Logo software or that which is given on this website. Only Logo commands from the most common versions of Logo are contained in these exercises. The answers to the odd-numbered exercises are at the end of these exercises, and the answers to all the Logo exercises are in the Instructor's Resource Manual which accompanies Mathematics for Elementary Teachers: A Conceptual Approach, Fifth Edition.

LOGO EXERCISES

Sketch the figure that the turtle will draw by carrying out the commands in exercises 1 and 2.

   
1.
FD 60 BK 120 FD 60 RT 90 FD 50 LT 90 FD 60 BK 120 FD 60
PENUP RT 90 FD 50 LT 90 PENDOWN FD 60 BK 120 FD 60
 

   
2.
FD 50 LT 90 FD 30 RT 90 BK 40 LT 90 FD 20 LT 90 FD 30 HOME

In Example A, the turtle drew six spokes of a wheel when given the command LINE and instructed to make right turns of 60E . Use this approach and the REPEAT command to write the commands for drawing the figures in exercises 3 and 4.

  1. 10 spokes of a wheel
  2. 18 spokes of a wheel
  3. Write the commands for drawing the isosceles triangle in the following figure. The two equal sides have a length of 70 units.

  4. Write the commands for drawing the parallelogram in the following figure. The sides have lengths of 40 and 70 units.

Write the commands for drawing a scalene triangle that satisfies the given conditions in exercises 7
and 8. Sketch the triangle.

  1. Three acute angles
  2. Two acute angles and one obtuse angle

Write the commands for drawing the polygons in exercises 9 and 10:

  1. A nonconvex hexagon
  2. A convex pentagon whose sides are not all congruent

Define a procedure for drawing regular polygons in exercises 11 and 12.

  1. a. Pentagon b. Octagon
  2. a. Dodecagon b. Decagon

  3. Write a procedure called RSTAIRCASE for drawing the right side of the following figure, and have the turtle return to the home position. Then define a procedure called LSTAIRCASE by interchanging the RT and LT commands. The two procedures RSTAIRCASE and LSTAIRCASE should produce the complete figure.

  4. Write a procedure called RTREE for drawing the right side of a tree. Have the turtle return to its start position. Then define a procedure called LTREE by interchanging the RT and LT commands. The two procedures RTREE LTREE should procedure the complete tree.

Use the procedure FLAG from Figure 15 to write the commands for drawing figures with the given numbers of rotation symmetries in exercises 15 and 16.

  1. 10 rotation symmetries
  2. 9 rotation symmetries
  3. Revise the procedure CIRCLE by replacing FD 1 by FD .8 so that it will draw a smaller circle. Then use this procedure to define a procedure called VENN for drawing three circles for a Venn diagram.

  4. Use the procedure CIRCLE to define a procedure called SLINKY for drawing overlapping circles, as shown here.

The procedure ARC from Figure 11 can be used twice to obtain a petal. Define a procedure named PETAL, use it to define the procedures in exercises 19 and 20.

  1. Define a procedure named FLOWER to draw a flower with eight petals, as shown here.

  2. Define a procedure called STARLIGHT to draw a flower with five petals.
  3. Write the commands for drawing three concentric squares, as shown in the following figure.

  4. Write a procedure called GRIDSQUARES that instructs the turtle to draw a 3 x 5 array of nonintersecting squares of the same size.
    1. Understanding the Problem Here is a 2 x 3 array of squares. Sketch a 3 x 5 array. Notice that the size of the squares and the distance between them are not given.

    2. Devising a Plan First we need a procedure to draw a square. Write a procedure to draw a 10 x 10 square. Then since the same size square is needed 15 times, you can use the REPEAT command.
    3. Carrying out the Plan Write the commands to draw the array. (Hint: Do one row or one column at a time.)
    4. Looking Back Revise your program to increase the spaces between the squares.

  5. Mr. Carbrero uses base-five pieces for activities with his elementary school students to provide background for place value. He wants to obtain a procedure so his computer will print out grids of various sizes for other bases. Define a procedure that will print grids of size 12 x 12.

  6. Write a procedure called PENTATRI to draw a pentagon with triangles as shown in the following figure. Then show how this procedure can be revised to obtain procedures called HEXATRI and HEPTATRI to obtain a hexagon and heptagon, respectively, with triangles as shown in the figures.

  7. Write a procedure for printing six rows of six circles each, as shown in the figure, and call it CIRCLEGRID.

  8. A procedure for drawing a figure with many parts, such as a face, can best be created by using subprocedures. The following procedure called FACE has eight subprocedures. In each of these subprocedures, except possibly the one for the nose (see the hint below), the turtle should start from and return to its home. Design a face and define the subprocedures for drawing it. (Hint: Once a subprocedure for the left eye, left ear, or left side of mouth has been defined, the right eye, etc., can be drawn by using the concept of symmetry to obtain a new procedure. The turtle can be used for the nose.)

  9. Regular polygons can be drawn by using Logo commands to specify constant forward moves and constant turns, where the number of degrees in the turn is a factor of 360 E. For example, the following command produces a regular pentagon with sides of length 50.

    REPEAT 5 [FD 50 RT 72]

    Some interesting results occur when the number of degrees in the turn is not a factor of 360E, as shown in the following figure.

    1. Experiment with the following command by selecting different whole numbers for N and different degrees for X. (Note: Select N large enough so the turtle will complete the figure and select the forward move small enough so the figure is printed on the screen.)

      REPEAT N [FD 80 RT X]

    2. What happens when the number of degrees for X is less than or equal to 120 E? Between 120E and 180 E?

  10. Logo commands can be used to instruct the computer to create this five-pointed star. Beginning at H (home), the turtle moves north to I, takes a right turn and moves to C, takes a right turn and moves to D, etc., making five equal forward moves and five equal turns and ending at H, facing north. What is the measure of each right turn, and what is the measure of the angle at I.

     

     

    Answers to Odd-numbered Logo Exercises

    1.

    3. REPEAT 5 [LINE RT 36]

    5. RT 52 FD 70 RT 76 FD 70 HOME

    7. RT 50 FD 40 RT 110 FD 60 HOME

    9. Other answers are possible.

    FD 40 RT 45 FD 40 RT 80 FD 60 RT 120 FD 30 LT 110 FD 50 HOME

    11.

    a. TO PENTAGON
    REPEAT 5 [FD 50 RT 72]
    END

    b. TO OCTAGON
    REPEAT 8 [FD 40 RT 45]
    END

    13. TO RSTAIRCASE

    REPEAT 3 [RT 90 FD 15 LT 90 FD 15]
    RT 90 FD 15 PENUP HOME
    PENDOWN
    END
    TO LSTAIRCASE
    REPEAT 3 [LT 90 FD 15 RT 90 FD 15]
    LT 90 FD 15
    END

    15. Other answers are possible.

    REPEAT 10 [FLAG RT 36]

    17. TO CIRCLE

    REPEAT 360 [FD .8 RT 1]
    END
          TO VENN
    PENUP BK 22 PENDOWN CIRCLE
    PENUP LT 90 FD 40 RT 90
    PENDOWN CIRCLE PENUP RT 90
    FD 40 LT 180 PENDOWN CIRCLE
    END

    19. TO PETAL

    ARC RT 90 ARC RT 90
    END
          TO FLOWER
    REPEAT 8 [PETAL RT 45]
    END

    21.     PENUP BK 10 RT 90 BK 10 LT 90

    PENDOWN REPEAT 4 [FD 20 RT 90]
    PENUP BK 10 RT 90 BK 10 LT 90
    PENDOWN REPEAT 4 [FD 40 RT 90]
    PENUP BK 10 RT 90 BK 10 LT 90
    PENDOWN REPEAT 4 [FD 60 RT 90]

    23.     TO LINE

    FD 60 BK 120 FD 60
    END
    TO VERTLINES
    REPEAT 13 [LINE PENUP RT 90 FD 10 LT 90 PENDOWN]
    END
    TO GRID
    VERTLINES PENUP BK 60 LT 90 FD 70 PENDOWN
    VERTLINES END

    25.     TO SMALL CIRCLE

    REPEAT 45 [FD 1 RT 8]
    END
    TO ROWOFCIRCLES
    REPEAT 6 [SMALLCIRCLE RT 90 PENUP FD 15 LT 90 PENDOWN]
    LT 90 PENUP FD 90 RT 90 BK 15 PENDOWN
    END

    Note: ROWOFCIRCLES prints a row of six circles and places the turtle on the left edge of the next row of circles.

    TO CIRCLESGRID

    LT 90 PENUP FD 60 RT 90 FD 40 PENDOWN REPEAT 6
    [ROWOFCIRCLES]
    END

    27b. If the number of degrees for X is less than or equal to 120 and a factor of 360, a polygon should be produced. However, the polygon may not fit onto the computer screen if its sides are too long, and if its sides are too short it may look like a circle. The number of sides for the polygon is 360 divided by the number of degrees.

    If the number of degrees for X is less than 120 and not a factor of 360, a star-shaped figure (or a figure that appears to be a disc with a hole in it) will be formed. For example, the following three figures were produced by letting N = 50 and using the number of degrees shown under the figure.


    If the number for X is between 120 and 180, the figure obtained is a star. Here are three examples for the number of degrees shown below each figure.