This is a math series follows our unit for introduction to fractions in grade 3. This is one of several. Look for the others by searching “fractions.”
Though we are not completely finished with the number line work, it was Friday and the end of a busy week, so we were both looking for something a little different, a little more fun, and a little less work. With this in mind we skipped to our paper-folding activity. To begin the activity we both started out with 2 – 8.5 inch squares of paper – two for her, and two for me. Because of the two folding activities we were doing, one relied on the other, we started with the vertical fold project first.
Vertical Fold Project
We began with folding the paper in half on a vertical fold. We colored one side with a color of our choice, and opened it to see that ½ of the page was colored now, and then labeled it accordingly. We then folded the paper back, and began the next fold of the paper to make a square by bringing the top to the bottom, leaving the colored part of the paper on the inside. We then colored this part of the paper a different color, opened the sheet to see how many pieces that we had now, and labeled that square 1/4. From here, we flipped the paper back over, folding our previous folds to show the colored 1/4 square. Repeating the process, we folded the top of the 1/4 square down to meet the bottom creating a rectangle. We choose another color for this rectangle, opened the sheet, found the appropriate fraction, and a labeled. We did this once more folding the 1/8 rectangle from the left to right creating the 1/16 square. See the process in the pictures below.
Finding Equivalent Fractions and Expressions
Here we paused to study our creations and come up with equivalent expressions and fractions. You can see below the picture that she placed in her notebook which ones she found. I began by asking her how many of each of the expressions would fit into the 1/2. Then I would move onto 1/4. It might seem counter-intuitive to think to ask how much of a larger fraction would fit into a smaller one, but going through this process with manipulative that make it more noticeable, it is possible for a child to notice things like ½ of ½ fits into ¼. It is helpful in telling the child that when we use “of” in an expression, it is a “x,” the multiplication sign. We also noted the series of halves that added to the whole (great segue into calculus ideas for later.)
Diagonal Fold Project
Though this project is similar in the folds, it is different in that the folds are diagonal and each corner folds into the center. To begin, we folded each corner to its opposite corner creating an “X” in the center of our square. From this point, we folded each corner of the square into its center created by our “X” folds. This creates a square whose area is half of the original. We choose a color and colored the exposed side in that color, which creates 4 triangles of that color when unfolded. In the first project we found the fraction of the paper by opening it up and counting the spaces, but in this project the opening and discovering of the fractions does not happen until the end of its process. We continued to fold the corners to the center and colored the exposed space a different color until we had done this 4 times (though my daughter did this 5 times.)
Cutting and Comparing
At this point we opened our paper to find a series of nesting squares. This is where it became obvious that doing this project with her was good idea. She had no intention of cutting her beautiful project, so we cut mine. We cut the triangles from each of the corners and sorted our triangles by color. When we had all the corners cut, we laid the colors over the vertical-fold project to see what fraction each color was of the whole as seen in the picture below.
After we finished each of the projects, she placed each into her notebook. She found the following equivalent fractions which I transcribed for her.
1/2 = 1/4+1/8+2/16
1/4= 1/2 X 1/2
1/8 = 2/16
1/8 = 1/2 X 1/4
1/8 = 1/4 X 1/2