by Dr Sharon Whitton
In her article, Sharon Whitton focuses on the middle grades. Her intriguing suggestions for work with proportional reasoning all stand on their own and could be used by teachers whose students had little or no prior experience with manipulative materials in school. But how much richer would this work be if students had played with tangrams in preceding years, had worked with unit blocks and other materials that lent themselves to thinking and talking about proportion and scale?
In todays world, opportunities for proportional reasoning are everywhere. Proportional reasoning is essential in the sciences, in economics, and in our daily lives. It is used in the construction of such common items as maps, blueprints, and copy machines. In the applications of mathematics and science, it arises in numerous situations such as unit conversions, trigonometry, and percentage relationships.
The development of middle-grade students competence in proportional reasoning is so paramount that educators and psychologists have identified it as a primary benchmark into higher-order thinking (Bar, 1987; Tourniaire & Pulos, 1985). Thinking at this level requires an understanding of complex ideas and their underlying principles, the ability to categorize and symbolize ideas, a sense of when and how to unite ideas in new and novel ways, and competence in representing ideas symbolically and applying them in solving new problems. Since proportional reasoning entails an understanding of the concepts of ratios and proportions, and the ability to use these concepts appropriately for solving and evaluating new problem situations, it is indeed a form of higher-order thinking.
Too often, school experiences with proportions are only presented procedurally, using textbook examples disassociated from real-world contexts. Without engaging students in the analysis of proportional situations in real-world contexts, proportional reasoning may not develop fully as a thinking and problem-solving tool for middle-grade students. An objective of this article is to illustrate several situations that engage students physically in the applications and informal analyses of proportional relationships. These applications are specifically designed to lead to the development and advancement of middle-grade students proportional reasoning skills.
Students enjoy using rubber band stretchers to draw similar enlarged figures. The following activity provides an informal introduction to the concepts of scale factor and similarity.
First, each student creates a two-band stretcher. This is accomplished by connecting two rubber bands of equal size, looped together, as shown below.
The rubber bands should be two and one half inches to three inches in diameter.
When the right- and left-hand ends of the two-band stretcher are pulled away from each other, a knot is formed in the center of the stretcher. Sometimes adjustments need to be made to center and tighten the knot.
Initially, students follow along with the teacher as he or she demonstrates the following instructions for drawing an enlargement of the given figure (in this case, a bird) with the two-band stretcher. Later, students can supply pictures of their own and draw enlargements using the same procedure.
After the enlarged figure is drawn, students compare the two figures with respect to their lengths, widths, distances from the base point, and their angles. These measurements are made using rulers and protractors. Since this enlargement was created with a two-band stretcher, students will discover that the dimensions of the enlarged figure are approximately two times the dimensions of the original figure and that the corresponding angles are approximately equal in measure.
When the two figures are enclosed in triangles, as indicated in the diagram below, students may also be guided to discover the properties of similar triangles and scale factors.
In this case, students should observe that the corresponding dimensions of the enlarged triangle (PRS) are two times the dimensions of the original triangle (PQT) and that all three angles of the original triangle are equal in degree measure to their corresponding angles in the enlarged triangle. This implies that the triangles are similar and the scale factor of the enlarged triangle is two.
To expand on this activity, students can create a three-band stretcher to enlarge the same figures. Typically, they will find it necessary to work in groups and to use very large paper (or they can tape several pieces of paper together) because their new pictures will be very large. In the creation of these new figures, they will discover that the scale factor of these enlargements will be three and that all of the angles remain approximately the same.
This activity is especially exciting to middle-grade students. They are simultaneously making mathematical discoveries while creating visual representations of proportional relationships. It requires them to compare the characteristics of two figures and to determine their common and related features. More importantly, students participate in measurement activities that lead to their own discovery of the properties of similar triangles and scale factors.
Earth Ball Catch
Another activity for applying the concepts of ratios and proportions is Earth Ball Catch. This activity can be conducted within the classroom or outside of the building. A beach ball designed as a globe is randomly tossed to students. After catching the ball, each student will locate the position of his or her right-hand thumb on the globe and call-out "land" or "water."
The ball should be tossed 40 or more times in order to collect a representative sample of data. The teacher, an aide, or a student can tally the number of land catches and water catches as the students play the game. When complete, students can use their data to approximate answers to these questions:
Middle-grade students respond well to this activity because it gets them out of their desks and keeps them physically active and alert. They also get to participate in creating ratios from their own data and apply these ratios in determining land measures without actually measuring anything. Additionally, students have opportunities to compute with fractions, percents, and scientific notation. Whats more, students may be encouraged to report their answers in a variety of formats. Activities such as these take mathematics out of the realm of textbooks and into the world of the student.
Tangram puzzle activities and probability
By the middle-grade level, most students have gained some familiarity with the tangram puzzle. This is a seven-piece puzzle invented by the Chinese more than two thousand years ago. Typically, elementary and middle-grade students use the puzzle pieces to examine the properties of squares, triangles, parallelograms, trapezoids and symmetry relationships. In elementary education the primary use of the tangram puzzle is to foster students spatial reasoning. One way this is accomplished is by challenging students to arrange all seven pieces into a square. (Try it yourself. Its not so easy! The answer is found at the end of this article.) After succeeding in this task, some teachers invite students to create their own original designs with the tangram pieces. These designs are subsequently displayed on a bulletin board. Beyond these activities, the tangram puzzle is seldom used in elementary and middle-grade education. This is unfortunate, since all seven pieces are related proportionally. The following activities employ the concepts of area, ratio, and probability. Students will also need to compute with whole numbers, fractions and decimals.
Tangram puzzle pieces
Initially, the teacher guides the students in comparing the various pieces of the tangram puzzle. For example, teachers might ask questions such as, "How many little triangles does it take to make a little square? How many ways can you cover the large triangle with other puzzle pieces? How is the parallelogram related to the little square?" At this point, the teacher might remind students of the concept of area, which provides a measure of the interior of a region. Without introducing area formulas, the teacher can ask students to consider the following five situations given in columns IV in the table below. For each column in the table, the student is given the area of a single puzzle piece and asked to find the corresponding areas of the other puzzle pieces in that column, Table 1.
While completing this activity, students will naturally begin to reason proportionally and realize that once the first column is complete, columns II-V can be determined by using the proportional relationships found in column I, as shown in Table 2.
Specifically, they will discover that the areas of the small square, medium triangle and parallelogram are always the same; the area of the small triangle is always half that of the small square; and the area of the large triangle is always two times the area of the small square.
Additionally, this table was specifically designed to provide opportunities for students to gain facility in computing with whole numbers, fractions and decimals. Answers for these five situations are given in Table 3.
A final, but related, activity involves the concept of probability. Students are given the following scenario:
Suppose a dart board is in the shape of a completed tangram puzzle (See the diagram at end of this article for the completed puzzle). Suppose further that you are always able to hit the dart board whenever you throw a dart toward the board. If you randomly throw 40 darts at the board, approximately how many darts would you expect to land in the small square region?
Without formerly considering the principles of probability, there are several ways that middle-grade students might solve this problem.
These activities were created to move mathematics beyond the realm of textbooks into real-world contexts. Students involvement in these activities will serve to enhance and nurture the development of their higher-order thinking skills, particularly their proportional reasoning skills.
Completed tangram puzzle
Bar, Varda. "Comparison of the Development of Ratio Concepts in Two Domains," Science Education, 71 (4), pp.599613. 1987.
Curcio, Frances, et al. Understanding rational numbers and proportion: Addenda Series, Grades 58 National Council of Teachers of Mathematics, Inc. 1994.
Erickson, Sheldon. AIMS Activities grades 69: Proportional Reasoning. AIMS Educational Foundation. 2000.
Lappan, Glenda, et al. Stretching and Shrinking. Prentice Hall. 2002.
Tourniaire, Francoise & Pulos, Steven. (1985). "Proportional reasoning: A review of the literature," Educational Studies in Mathematics, 16 (2), pp.181204. 1985.
Dr Sharon Whitton
- Dr Sharon Whitton is a professor of mathematics education at Hofstra University in Hempstead, New York. She has conducted numerous curriculum projects which focus on methods for engaging students (K-12) in discovering the meaning and beauty of mathematics.