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.
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