activity leap frog

							LEAP FROG
How many slides, jumps,
  and moves must be
  made if you have 30
pairs of frogs in the leap
       frog game?
          How to Play

Set up your two colors of “frogs”
on a playing board like the one
below:
3 2 1 space 1 2 3
             Free
               How to Play
• Begin with 1 pair of frogs of different
  colors.
• Place one frog to the left and one frog to
  the right of the free space in the 1 position.
• Your goal is to get all the frogs of one
  color to the other side of the free space.
                  Rules
• Only one frog can occupy a space at one
  time.
• Frogs can only move forward (toward the
  opposite side), never backward.
• You can slide a frog to the empty space
  beside that frog. This includes the free
  space.
• You cannot jump the same-color frog.
                  Rules
• There must be a space on the other side
  of a frog before you can jump that frog.
  You cannot jump 2 frogs next to each
  other without a space in between the
  frogs.
  As you perform this investigation,
 organize your data in a table like the
             one below:
# of    Sequence of moves    # of    # of # of
frog                        slides jumps moves
pairs
         Recording Data
• As you move your frogs, record your
  slides and jumps using a method you
  develop. Write this data in the
  sequence column of your table.
• Count the number of slides, the
  number of jumps, and the total
  number of moves needed to solve the
  game and record these numbers in
  your table.
After you have solved the
game with 1 pair of frogs,
repeat the game with 2 pairs of
frogs, 3 pairs of frogs, etc.,
until you see a pattern.
  Answer numbered questions in
      complete sentences.
1. What is the relationship
   between the number of frog
   pairs and the number of slides?
2. What is the relationship
   between the number of frog
   pairs and the number of jumps?
3. What is the relationship
   between the number of slides
   and jumps and the number of
   moves?
4. How did you determine the
   number of slides, jumps, and
   moves for 30 pairs of frogs?
5. What are the variable
   expressions for slides,
   jumps, and moves?
6. What labels would you give
   to the x-axis and the y-axis
   in a graph of this data?
             Drawings of frog      Table with variable
Place the           pairs              expressions
following    (at least 4 steps)   (include expressions
                                    at bottom of table)
items on
your chart       Variable                Graph
paper:       expressions and       (include at least 4
                  word                data points)
             representations
          ANSWERS


Answers begin on the next slide.
                    ANSWERS
# of frog      Sequence of moves          # of    # of # of
  pairs                                  slides jumps moves
   1                  S-J-S               2     1      3
   2             S-J-S-J-J-S-J-S          4     4       8
   3        S-J-S-J-J-S-J-J-J-S-J-J-S-    6     9      15
                       J-S
   4        S-J-S-J-J-S-J-J-J-S-J-J-J-    8     16     24
              J-S-J-J-J-S-J-J-S-J-S
   n                                      2n    n2   2n + n2
               ANSWERS
1.   What is the relationship between the
     number of frog pairs and the number of
     slides?
     • Answer: The number of slides is double
       the number of frog pairs.
2.   What is the relationship between the
     number of frog pairs and the number of
     jumps?
     • Answer: The number of jumps is the
       square of the number of frog pairs.
                 ANSWERS
3.   What is the relationship between the number
     of slides and jumps and the number of
     moves?
     •   Answer: If you add the number of slides and
         jumps, you will get the number of moves.
4.   How did you determine the number of slides,
     jumps, and moves for 30 pairs of frogs?
     •   Answer: For 30 frog pairs, you double the
         number of frog pairs for the slides, square the
         number of frog pairs for the jumps, and add the
         slides and jumps together to get the number of
         moves (2 ∙ 30 + 302 = 60 + 900 = 960).
                              slides   jumps moves
                  ANSWERS
5.   What are the variable expressions for slides,
     jumps, and moves?
     •   Answer: The variable expression for slides is 2n,
         for jumps is n2, and for moves is 2n + n2.
6.   What labels would you give to the x-axis and
     the y-axis in a graph of this data?
     •   Answer: Since you choose the number of frog
         pairs, you would label the x-axis # of frog pairs.
         Since you will have 3 sets of dots on your graph
         for slides, jumps, and moves, you would label the
         y-axis # of. In this case, you will need a legend
         or key to explain the different dots.
           ANSWERS

Drawings
       2 slides, 1 jumps, 3 moves


                  4 slides, 4 jumps, 8 moves


                              6 slides, 9 jumps, 15 moves



                                           8 slides, 16 jumps,
                                                24 moves
           ANSWERS
Variable expressions and word
representations:
The number of slides is twice the
number of frog pairs (2n). The
number of jumps is the number of
frog pairs squared (n2). The number
of moves is the number of slides and
jumps added together (2n + n2).
                      20
ANSWERS               18

                      16


Graph                 14

                      12

                      10

        Legend         8
        # of slides    6
        # of jumps
                       4
        # of moves
                       2

                      00   2         4      6    8   10
                               # of frog pairs
                     TEACHER NOTES
•   Materials:
    Snap cubes, cm cubes, or any type of manipulative to represent the two colors
    of frogs; you could have students be the frogs by laying colored paper on the
    floor and giving students the same colored paper to hold
    Chart paper
    Markers/crayons/colored pencils
    Pencil/paper
•   Pair students for this activity.
•   Have students copy the table onto their paper.
•   As students work, circulate to ask questions to guide students during their
    investigation.
•   If chart paper is not available, you can substitute newsprint or butcher paper.
•   For the drawings of the frog pairs, students should include at least 4 steps in
    the sequence.
•   For the graph, students may or may not include (0,0). Students should
    include at least 4 data points. Discuss with students the fact that you do not
    connect the dots because you can’t have a partial pair of frogs; you can only
    have whole number pairs of frogs.
•   Have students present their chart papers to the class. Post chart papers on
    the wall after each presentation, provided that all groups and classes are
    finished with the construction of their chart papers.