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 he maximize the area?

Lake

length

width

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

Identify the quantities in this problem.

Lake

width

AreaWhich quantity, if any, is varying? Which quantity, if any, is invariant?

How can he maximize the area?

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

Identify a pair of quantities that are related. Lake

width

Area

Identify another pair of quantity that are related.

How can he maximize the area?

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

Lake

width

Area

Describe the relationship between length and width using(a) words, (b) a graph, and (c) an equation.

How can he maximize the area?

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

Lake

width

Area

Describe the relationship between length and area using(a) words, (b) a graph, and (c) an equation.

How can he maximize the area?

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

If the length is 10 meters, what is the width of the rectangle?

Make a prediction:

Lake

width

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

If the length is 20 meters, what is the width of the rectangle?

Make a prediction:

Lake

width

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

If the length is 39 meters, what is the width of the rectangle?

Make a prediction:

Lake

width

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

If the length is x meters, what is the width of the rectangle?

Lake

width

w

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. Write an equation that relates the width, w, of the rectangle to its length, x.

x

Width = 120 – 2 Length w = 120 – 2x

Lake

Mathematical Term:

We say “w is a ________________ of x.”

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

If the length is 10 meters, what is the area of the rectangle?

Make a prediction:

Lake

width

Area

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

If the length is 20 meters, what is the area of the rectangle?

Make a prediction:

Lake

Area

width

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

If the length is 39 meters, what is the area of the rectangle?

Make a prediction:

Lake

Area

width

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

If the length is x meters, what is the area of the rectangle?

Lake

Area

width

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. Write an equation that relates the area of the rectangle, A, to its length, x.

x A

Area = length width A = x w

Lake

w

But I only want to relate A to x. I don’t want w to be in my equation.

What can I do?

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. Write an equation that relates the area of the rectangle, A, to its length, x.

x A

Area = length width A = x (120 – 2)

Lake

w

Mathematical Term:

We say “A is a ________________ of x.”

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

Make a prediction:

Is there a least value of x that he can have?

A

Lake

w

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

Make a prediction:

Is there a greatest value of x that he can have?

A

Lake

w

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

What is the set of all the possible values of x that we can have?

A

Mathematical Term:

The set of all possible input values is called the ______________ of the function?

x 0 60Lake

w

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.

What do you think will happen as we increase the value of x from the least value to the greatest value?

Make a prediction:

A

Lake

Try to sketch a graph to show the relationship between the area, A, and the length, x.

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.

What do you think will happen as we increase the value of x from the least value to the greatest value?

Make a prediction:

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.

x

Make a prediction:

A

Is there a least value of A that we can have?

Lake

w

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

Is there a greatest value of A that we can have?

Make a prediction:

AHow can you find the maximum area?

Lake

w

x A

Lake

Create a table, and use it to plot a graph.

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 (m) w (m) A (m2)

0

10

20

30

40

50

60

w

How can you find the maximum area?

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

What is the set of all the possible values of A that we can have?

A

Mathematical Term:

The set of all possible output values is called the ______________ of the function?

A 0 ?Lake

w

1. Quantitative Analysis of a Situation• Identify co-varying quantities• Establish the invariant relationship

2. Four ways to represent the invariant relationship between two co-varying quantities. i. Verbal Descriptionii. Equationiii. Graphiv. Table

3. Mathematical terms • A function is _________• Domain is _______• Range is _______

Learning Points.

4. How are these equations conceptually different?• 2 50 + 20 = 50 + 50 + 20 • 2x + 50 = 120• 2x + w = 120• A = xw• A = x(120 – 2x)

5. A graph consists of a set of points with each point representing a specific instance relating the values of the two interdependent quantities.

6. Why is creating an equation difficult to represent the relationship between two quantities difficult?

7. What are some strategies that you use to write an equation that relates two quantities?

Learning Points.