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Representing Half of a Rectangle


Alignments to Content Standards: 2.G.A.3

Task

Ms. Nim gave her students a picture of a rectangle. Then she asked them to shade in one half of the rectangle. Here are three pictures:

Halves_f5c8a39f3f1fd005fdbcedce453ad20e

Which ones show one half? Explain.

IM Commentary

This task is for assessment purposes, providing a context for indentifying different ways of representing half of an object, a rectangle in this case. The task may also be used for instructional purposes but if so the teacher may wish to introduce some other ways of showing one half of the rectangle, such as dividing along a diagonal (and shading in one piece) or dividing it into four equal pieces, shading in two pieces that only touch at a corner. Teachers are referred to ''Which pictures show half of a circle?'' for more variants on this theme.

Solution

  1. In the picture on the top left, the rectangle has been divided in half vertically while in the picture on the top right it has been divided in half horizontally. In each case, the big rectangle has been divided into two equal pieces, one shaded and one unshaded. So the shaded area represents one half of the big rectangle in both cases.

    In the third picture, each side of the rectangle has been divided in half but the small rectangle does not represent one half of the area of the large rectangle. As can be seen in the picture below, the large rectangle can be divided into four rectangles, each equal to the small one so the shaded area only represents one fourth of the area of the large rectangle:

    4squares_f0b2deece166b9f753e04d532d92b6b1

    The student who drew the lower incorrect picture does show some understanding of the fraction one half. Each side of the rectangle has been cut in half, demonstrating an understanding of the fraction one half in the context of measuring a length. The mistake is that when both linear measurements are cut in half, the area is only one fourth the area of the original shape.

Anthony says:

over 1 year

It is important to use the same shape when introducing halves. I see too often confusing students with multiple shapes at one time. Some students think there is only one way to cut a shape in half. Hopefully students come up with the diagonal on their own. Also stressing the word/idea of dividing the shape.

Catherine Parker says:

about 4 years

Would this task also be appropriate for 1.G.3?

Kristin says:

about 4 years

It could be. The big difference between the first and second grade expectations is that in first grade it is enough for students to be able to show one way of partitioning a whole into two equal shares, whereas in second grade they should understand that different ways of showing one-half of the same whole may not have the same shape. This task pushes in that direction, which is why it is aligned to second grade; I think in first grade it would only be appropriate as an instructional task that is heavily supported by classroom discussion.

Kentucky Department of Education Math Specialist says:

about 4 years

Replace the symbol 1/2 in the student question to the words "one half." In grade 2, the CCSSM do NOT expect students to understand the symbol for one half (1/2). The symbol, itself, the way I understand the standards, is not introduced until 3rd grade. Students are merely developing the language of halves, fourths, quarters...half of, fourth of, and quarter of.

Kristin says:

about 4 years

Thanks for pointing that out--I've fixed it now.