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Shape and Space in Geometry
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Across the Grades
Standards
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Proportion is one of the most important principles in mathematics. It appears in geometry when you do scale drawings or look at similar figures. The figures in this activity—the squares—are all similar, so their parts are proportional. This means that the length of the diagonal divided by the length of the side (not added, not multiplied, not subtracted) will be the same for all squares, no matter what the size.

But what is that constant of proportionality? About 1.4. If you take a shortcut across square field, you go 1.4 sides instead of 2.0; if your head is 21 inches around, you can tie your 18-inch bandana around your forehead, because 18 times 1.4 is 25.2 inches. And what is that "about 1.4"? The square root of 2.

Unit squares and their diagonals, and the half-square (45-45-90 triangles), appear all the time in geometry and in real life. If 1/2 and 3/4 are benchmark fractions that every fourth grader should know, the square root of 2—the length of that diagonal, that is, the hypotenuse of the isosceles right triangle—is a benchmark irrational number (along with pi and a few others).

across the grades

Students can do this activity as soon as they can measure and understand what division is, usually about grade 4. Students who have a more sophisticated grasp of proportion (about grade 6), or have studied the Pythagorean theorem, will get more out of it.

You don't need to know anything about the square root of 2 to do this activity—it's really all about proportion. But, if you are about to introduce students to the Pythagorean theorem, you can point out that the proportion they find is, in fact, that important square root.

standards

From Corner to Corner is the kind of activity designed to give students the background they need for middle-grades mathematics. In the current version of the Principles and Standards document, the NCTM includes this content in the Geometry and Spatial Sense standard. For example, at grades 6–8,

The Pythagorean theorem introduces students to irrational numbers, helps determine the relationships between the legs of 30-60-90 degree triangles and isosceles right triangles, and can be used to indirectly determine distances that would be difficult to measure directly either on the coordinate plane or in the context of other problems.

resources

Even before they measure the diagonal, students can get experience with the shape. For example, in the Teaching Math K–4 library, several of the activities in the Geometry and Spatial Sense section use this shape:

  • At grade 1, Thanksgiving Quilt, page D-3.
  • At grades 2–3, Shapes from Squares, page D-11, and Rocket Shape, page D-15.

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