How can he maximize the area?

1. How can he maximize the area? Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length width

2. How can he maximize the area? Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. Identify the quantities in this problem. length Area Which quantity, if any, is varying? Which quantity, if any, is invariant? width

3. How can he maximize the area? Identify a pair of quantities that are related. Lake Identify another pair of quantity that are related. A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length Area width

4. How can he maximize the area? Describe the relationship between length and width using (a) words, (b) a graph, and (c) an equation. Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length Area width

5. How can he maximize the area? Describe the relationship between length and area using (a) words, (b) a graph, and (c) an equation. Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length Area width

6. Make a prediction: Lake If the length is 10 meters, what is the width of the rectangle? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length width

7. Make a prediction: Lake If the length is 20 meters, what is the width of the rectangle? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length width

8. Make a prediction: Lake If the length is 39 meters, what is the width of the rectangle? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length width

9. If the length is x meters, what is the width of the rectangle? Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length width

10. Write an equation that relates the width, w, of the rectangle to its length, x. Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. Width = 120 – 2  Length w = 120 – 2x Mathematical Term: We say “w is a ________________ of x.” x w

11. Make a prediction: Lake If the length is 10 meters, what is the area of the rectangle? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length Area width

12. Make a prediction: Lake If the length is 20 meters, what is the area of the rectangle? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length Area width

13. Make a prediction: Lake If the length is 39 meters, what is the area of the rectangle? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length Area width

14. If the length is x meters, what is the area of the rectangle? Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. length Area width

15. Write an equation that relates the area of the rectangle, A, to its length, x. Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. Area = length  width A =  x w But I only want to relate A to x.I don’t want w to be in my equation. What can I do? x A w

16. Write an equation that relates the area of the rectangle, A, to its length, x. Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. Area = length  width A =  x (120 – 2) Mathematical Term: We say “A is a ________________ of x.” x A w

17. Make a prediction: Lake Is there a least value of x that he can have? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. x A w

18. Make a prediction: Lake Is there a greatest value of x that he can have? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. x A w

19. What is the set of all the possible values of x that we can have? Lake  x  0 60 A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. Mathematical Term: The set of all possible input values is called the ______________ of the function? x A w

20. Make a prediction: Lake What do you think will happen as we increase the value of x from the least value to the greatest value? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. A Try to sketch a graph to show the relationship between the area, A, and the length, x.

21. Make a prediction: Lake What do you think will happen as we increase the value of x from the least value to the greatest value? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing.

22. Make a prediction: Lake Is there a least value of A that we can have? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. x A w

23. Make a prediction: Lake Is there a greatest value of A that we can have? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. How can you find the maximum area? x A w

24. Lake A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. How can you find the maximum area? x A Create a table, and use it to plot a graph. w

25. What is the set of all the possible values of A that we can have? Lake  A  0 ? A man is planning to fence in three sides of a rectangular region using the straight part of a lake shoreline as the fourth side of the region (i.e., no fencing is required for the fourth side). The man has 120 meters of fencing. Mathematical Term: The set of all possible output values is called the ______________ of the function? x A w

26. Learning Points. Quantitative Analysis of a Situation Identify co-varying quantities Establish the invariant relationship Four ways to represent the invariant relationship between two co-varying quantities. Verbal Description Equation Graph Table Mathematical terms A function is _________ Domain is _______ Range is _______

27. Learning Points. How are these equations conceptually different? 2  50 + 20 = 50 + 50 + 20 2x + 50 = 120 2x + w = 120 A = xw A = x(120 – 2x) A graph consists of a set of points with each point representing a specific instance relating the values of the two interdependent quantities. Why is creating an equation difficult to represent the relationship between two quantities difficult? What are some strategies that you use to write an equation that relates two quantities?