Ratios in Geometry

Geometry offers a challenging arena in which to wrestle with ideas of ratio. Except for the first resource, the work below is more appropriate for the upper end of middle school than for the younger students. All of the resources include activities that will involve your students in working with visual, geometric figures that they can draw or manipulate online. You will notice the absence of a favorite and most significant ratio: p. You will find several interesting resources on the circumference to diameter ratio in Going in Circles!

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Constant Dimensions http://illuminations.nctm.org/LessonDetail.aspx?id=L572

In this carefully developed lesson, students measure the length and width of a rectangle using standard units of measure as well as nonstandard units such as pennies, beads, and paper clips. When students mark their results on a length-versus-width graph, they find that the ratio of length to width of a rectangle is constant, in spite of the units. For many middle school students, not only is the discovery surprising but also opens up the whole meaning of ratio.

Parallel Lines and Ratio http://www.ies.co.jp/math/java/geo/hiresen/hiresen.html

Three parallel lines are intersected by two straight lines. The classic problem is: If we know the ratio of the segments created by one of the straight lines, what can we know about the ratio of the segments along the other line? An applet allows students to clearly see the geometric reasoning involved. The activity is part of the Manipula Math site. MSP full record

Figure and Ratio of Area http://www.ies.co.jp/math/java/geo/ratioAB/ratioAB.html

A page shows two side-by-side grids, each with a blue rectangle inside. Students can change the height and width of these blue rectangles and then see how their ratios compare--not only of height and width but also, most important, of area. The exercise becomes most impressive visually when a tulip is placed inside the rectangles. As the rectangles' dimensions are changed, the tulips grow tall and widen or shrink and flatten. An excellent visual! The activity is part of the Manipula Math site. MSP full record

Cylinders and Scale Activity http://www.exploratorium.edu/structures/cylinders.html

Using a film canister as a pattern, students create a paper cylinder. They measure its height, circumference, and surface area, then scale up by doubling and even tripling the linear dimensions. They can track the effect on these measurements, on the area, and finally on the amount of sand that fits into each module (volume). The lesson is carefully described and includes handouts. MSP full record

The Fibonacci Numbers and the Golden Section http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Here students can explore the properties of the Fibonacci numbers, find out where they occur in nature, and learn about the golden ratio. Illustrations, diagrams, and graphs are included. MSP full record

The Golden Ratio http://cuip.uchicago.edu/~dlnarain/golden/activities.htm

Another site that introduces the golden ratio, this resource offers seven activities that guide students in constructing a golden rectangle and spiral. Although designed for ninth and tenth graders, the explorations are appropriate for middle school students as well. MSP full record)

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Copyright June 2006 — The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

This work is licensed under a Creative Commons License.