Return to Mathematics IndexFractions and Paper Folding

Bill Wagner Hyde Park Career Academy

6220 S. Stony Island

Chicago Il 60637

(312) 535-0880Objectives:

The student will be able to develop the concept and properties of fractions

using paper folding (origami).Materials needed:

Each individual in a class should have the following:

1 ruler (in both english and metric units)

paper squares of varying sizes (at least 1 - 4 inch by 4 inch square)

1 sheet of directionsStrategy:

1. Taking the 4x4 square first, the student shall locate the midpoint of each

side by folding the square in half. Next, fold in each of the corners of the

square so that the vertices of the square meet in the center of the square.

2. Start a discussion about the new shape(s) by asking the following questions:

1. What new shape(s) have been formed?

2. How does the area of the new shape(s) compare to the area of the old

shape?

3. Answering the first question, students may see triangles (the folded sides),

squares (the final shape), and even other shapes depending upon the accuracy of

the folds. The second question will show the relationship between the original

shape and the new shape(s) formed. The students should be able to see that the

four flaps cover the new square and, therefore, each flap is 1/4 of the new

square. Also by either observation or by geometric proof, the students should

be able to see that the new square has an area equal to half of the original

square. Your level of vocabulary and mathematical concepts should be adjusted

according to grade level.

4. Taking the vertex of the folded flap, fold the flap back so that the vertex

now touches the midpoint of the outer edge of the new square.

5. Asking the same two questions you started with, start a new discussion.

Students may see some new shapes, such as trapezoids, have now been added to

the mix. Draw their attention to the new shape inside of the second square.

Hopefully, they will now see a new, even smaller square. Have the students try

to find the area of the smallest square. If they have trouble, show that four

of the new little tabs formed by the last fold will cover the smallest square.

Then demonstrate how many of these little tabs it takes to cover the second

square (16). Therefore, the area of the smallest square must be 4/16 ths of

the second square. Since four of the smaller tabs equal one of the larger

tabs, 4/16 = 1/4 and the area of smallest square equals 1/4 th of the second

square. You may choose to go further and demonstrate how the smallest square

is 1/8 th of the original square.

6. With an advanced group, you can even introduce irrationals by looking at the

lengths of sides of the squares produced and using the Pythagorean theorem to

determine their value.

7. Finally, fold the smallest tabs back under, producing a small picture frame

which the students can now use to frame the picture of their choice. The above

steps can be repeated with different sizes of squares to show that the above

fractions (ratios) are constant and to produce different sizes of picture

frames.Performance Assessment:

K-3Students will be able to fold any square piece of paper, following the

instructions on the direction sheet, into a picture frame for the picture

of their choice.

4-6Students will be able to find the areas of the resulting figures created

by the paper folding and determine what fractional part of the whole

square is represented by the new sections created.

7-10Students will be able to do all of the above plus name the shapes

created and use the Pythagorean formula to find the dimensions of the new

shapes.

References:

Sobel and Maletsky,TEACHING MATHEMATICS:A Sourcebook of Aids, Activities,and Strategies, Prentice Hall, 1988