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Answering the Question, “When Is Halving Not Halving?”

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It’s late in the school year, but I hope you had a chance to try out the perimeter and area comparison problem with your students. If not, you might try it early next year.

Recall that students were going to start with a rectangle and then make another one with half the area. The task was to see what fraction of the old perimeter the new perimeter could be.

I tried this problem in a sixth-grade class and had a really interesting experience. The regular classroom teacher had four students who normally did not stay with the others for math; they usually went to a special education classroom. We asked if, for this problem, the students could stay and, in fact, each of them was successful in creating the second rectangle and in realizing that the new perimeter was not only not half the old perimeter but also that it was greater than half.

Some students figured out that if the cutting in half was on a line of symmetry, the new perimeter had to be more than half of, but less than all of the old perimeter. That’s because even if you cut the length (or width) in half, you retain the full width (or length) of the original.

For example, in the two rectangles below, note that the length of the top edge and bottom edge do not change when the shape is cut on a line of symmetry; however, the lengths of the left and right sides (shown in blue on the original figure) are half the length of the new figure (shown in green on the halved figure).

      Figure 1 - comparison of edge lengths 

fig. 1                    Perimeter = 2 full red + 2 full blue       Perimeter = 2 full red + 2 half-blues

 

Some students realized that if you have a long rectangle or a tall and skinny oneand cut it in half by making it even skinnier, you end up keeping most of the perimeter, so the fraction of the old perimeter and that of the new perimeter turn out to be really close to 1.

Compare the change in perimeter on the left versus on the right. It is much less of a change on the left.

Figure 2: Graphic comparing change of perimeter on left vs.right
 

fig. 2 

Keeping the same perimeter or even increasing the perimeter actually is possible, but only if the original shape is not cut on a line of symmetry. For example, a 1 ´ 6 rectangle has half the area of a 3 ´ 4 rectangle but the same perimeter. And a rectangle that is 6 ´ 5 with a perimeter of 22 can be halved in area by using a rectangle that is 1 ´ 15. But the new perimeter is 32, even bigger than the original perimeter.

How did your students respond to the task? 


Marian SmallMarian Small is the former dean of education at the University of New Brunswick, where she taught mathematics and math education courses to elementary and secondary school teachers. She has been involved as an NCTM writer on the Navigations series, has served on the editorial panel of a recent NCTM yearbook, and has served as the NCTM representative on the MathCounts writing team. She has written many professional resources including Good Questions: Great Ways to Differentiate Mathematics Instruction (2012), Eyes on Math (2012), and Uncomplicating Fractions to Meet Common Core Standards in Math, K–7 (2013), all co-published by NCTM.


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