Breaking the Stick

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$By: Melissa Hanel Math B/Grade 11 5 Day Unit Plan Technologies ...$

G 1 6 S: No, because one of the pieces would be 10 cm long. Illustrate the two possible situations on the board. T: Which points in the triangle correspond to having one breaking point at 10 cm? Ask a student to show them in the picture on the board. The points lie on two line segments. Draw them in red. Point to the small triangle at the lower left. T: There are red dots in this region. Could there be a blue dot in this region? S: No; the rightmost break would be at a number less than 10, making the piece on the right longer than 10 cm. S: If we are to get a triangle, both breaks cannot be on the same half of the stick. Illustrate the situation. If some students are having difficulty, choose several points in the lower left triangular region and illustrate where the breaks would be in each case. When the class is convinced that no blue dot belongs in that small triangle, color its interior red. Refer to the small triangle at the upper right. T: There are red dots in this region. Could there be a blue dot in this region? S: No; the leftmost break would be at a number more than 10, making the piece on the left longer than 10 cm. S: Again, if we are to get a triangle, both breaks cannot be on the same half of the stick. G-72 IG-VI
When the class is convinced that no blue dot belongs in the small upper-right triangle, color its interior red. G 1 6 Point to the square inside the large triangle. T: In this region, we have some blue dots and some red dots. Is there any pattern? Perhaps a student will indicate that blue dots seem to fall above the diagonal from (10, 0) to (20, 0) and red dots fall below it. T: Let’s look at the situation more carefully. If the rightmost break is at 15, where could the leftmost break be to yield a success? Illustrate the cases as students discuss them. S: The leftmost breaking point cannot be at 5 because then the middle segment would be 10 cm long. S: The leftmost breaking point cannot be at a number less than 5 because then the middle segment would be longer than 10 cm. S: The leftmost breaking point can be at any number more than 5 but less than 10. For example, if the leftmost break were at 8, then the three pieces would be 8 cm, 7 cm, and 5 cm long. T: Why does the leftmost breaking point have to be at a number less than 10? Why not at 10? Why not at 12? S: The leftmost breaking point cannot be at 10 because then the left piece would be 10 cm long. S: The leftmost break cannot be at 12 because then the left piece would be longer than 10 cm. For a situation in which the rightmost break is at 15, indicate success and failure points with a line segment partially red and partially blue. Mark the point (15, 5) with a red dot. IG-VI G-73
Page 1 and 2: G13 BREAKING A STICK #1 G 1 3 Capsu
Page 3 and 4: G 1 3 Distribute copies of Workshee
Page 7 and 8: Your table should resemble this one
Page 11 and 12: Choose a student to come to the boa
Page 13 and 14: Shade the interior of the (red) tri
Page 15: G16 BREAKING A STICK #4 G 1 6 Capsu
Page 19 and 20: G 1 6 By now the class should suspe

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