The Paper Folding Problem Gift

1

Problem by Karen Campe

Edited by Tom Reardon

This gift is to be used in conjunction with the TI-Nspire file: Campe_PaperFolding_Edited_2008.tns

This activity requires a standard sheet of computer paper.

You also need a ruler to measure to the nearest sixteenth of an inch.

Part 1

The paper should be in landscape mode, that is, the longer sides are horizontal and the vertical sides are vertical. We will refer to the longer side (horizontal) as the length and the shorter side (vertical) as the width. Record the length and width below to the nearest quarter of an inch:

Length = ______Width = ______

How many one-inch squares could fit on this sheet of paper? ______

Now fold the paper to model the problem. Use the Nspire document to understand the problem. Record the values that you obtained when you folded your paper, again measuring to the nearest sixteenth of an inch.

NOTE: If the value to enter is , please enter it into Nspire as

If the value to enter is , please enter it into Nspire as

Notice the decimal point in the whole number part of the value. It is very necessary.

It makes the calculator give the decimal approximation for the area, which is what we want.

Base DE = ______Height DF = ______Area = ______

Part 2 If the value to enter is , please enter it into Nspire as 2

Base / Height / Area

Based on your observations from page 2.8 of the Nspire document, what are the dimensions of the triangle that yield the maximum area and what is that maximum area (to the nearest thousandth)?

Maximum Area = ______

Its Base = ______

Its Height = ______

Part 3

You need to develop an algebraic formula for the area of the triangle.

Let the length of base DE = x.

Let the length of height DF = y.

On this gift below, represent the lengths of FA and FE in terms of x and/or y.

Then below, represent all lengths, DE, DF, FA, FE,in terms of just the variable x.

For what values of x does the triangle DEF not exist? Why?3

What is the domain of the formula we will write for Area(x)?

Create the formula for Area(x) in simplest form below. Show your work below.

Part 4

The next step is to capture data values from the geometric model.

Follow the directions in the Nspire document.

What is the equation that you typed into f1(x) that actually modeled the data points?

f1(x) = ______

When you traced on the correct graph, list the coordinates that approximated the maximum area:

Maximum area is approximately = ______when the base is approximately = ______.

Part 5 4

In this section, you are asked to use the CAS tools to finish solving the problem, that is, find the maximum area of the triangle, and find the base and height that yield that maximum area.

Round all final answers to the nearest thousandth on this paper below.

But show all work clearly on the pages of Nspire. This will be graded.

Maximum Area = ______when the base = ______and the height = ______.

Compare these answers to the ones obtained in Part 4.

On page 5.5, using CAS, answer this question: Using non-rounded values, what is the ratio of the height of the maximum area triangle to the original width of the paper, 8.5 inches?

Ratio = ______.

This might be a bit surprising to you.

Continue to page 5.7 and figure out what is special about triangle FDE. Show any work on this page below.

List all information that you know about triangle FDE and include justification for how you know it is true.

Enjoy… 

C 2008 Campe-Reardon Gifts, Inc.