**Proportionality**

** **

**Big Idea #11: If two quantities vary proportionally, that relationship can be represented as a linear function.**

**Examples of Mathematical Understandings:**

• A ratio is a multiplicative comparison of quantities.

• Ratios give the relative sizes of the quantities being compared, not necessarily the actual sizes.

• Ratios can be expressed as units by finding an equivalent ratio where the second term is one.

• A proportion is a relationship between relationships.

• If two quantities vary proportionally, the ratio of corresponding terms is constant.

• If two quantities vary proportionally, the constant ratio can be expressed in lowest terms (a composite unit) or as a unit amount; the constant ratio is the slope of the related linear function.

• There are several techniques for solving proportions (e.g., finding the unit amount, cross products).

• When you graph the terms of equal ratios as ordered pairs (first term, second term) and connect the points, the graph is a straight line.

• If two quantities vary proportionally, the quantities are either directly related (as one increases the other increases) or inversely related (as one increases the other decreases).

• Scale drawings involve similar figures, and corresponding parts of similar figures are proportional.

• In any circle, the ratio of the circumference to the diameter is always the same and is represented by the number pi.

• Rates can be related using proportions as can percents and probabilities (Randall, 2005).

**A proportion is a name we give to a statement that two ratios are equal.** It can be written in two ways:

· two equal fractions, **a/b = ^{ }c/d**

or,

· using a colon, ** a:b = c:d**

When two ratios are equal, then the cross products of the ratios are equal.

That is, for the proportion, **a:b = c:d , a ****x d = b x c **

Retrieved from http://www.math.com/school/subject1/lessons/S1U2L2GL.html

Proportional thinking is developed through activities involving comparing and determining the equivalence ratios and solving proportions in a variety of problem-based contexts and situations without recourse to rules or formulas (Van de Walle, 2004).

**Practical Applications:**

Cooking is one way that ratios and proportionality can be applied to a real-life situation. As one ingredient is increased or decreased in a recipe the other ingredients must also be increased or decreased proportionally. Another practical activity is to enlarge or reduce images or photos on the computer. If they are not altered proportionally the image will become distorted. Graph paper or dot paper can be used to draw similar shapes or images in proportion to each other. Maps drawn to scale can be used to show ratios and how the distance on a map correlates to the actual distance. Students can recreate a map to a different scale

(Bahr, D. L., and de Garcia, L. A. 2010).

**Informational Web Sites:**

http://www.learner.org/interactives/dailymath/cooking.html

http://www.mathleague.com/help/ratio/ratio.htm#proportion

http://www.bbc.co.uk/skillswise/numbers/wholenumbers/ratioandproportion/ratio/

http://nrich.maths.org/public/viewer.php?obj_id=4825

**Lesson Plan Web Sites:**

http://www.eduplace.com/math/mathsteps/6/a/index.html

http://www.moneyinstructor.com/lesson/ratio.asp#LESSON_PRINTABLE_MATERIALS_-_WORKSHEETS

http://www.iit.edu/~smile/ma8809.html

http://www.themathpage.com/arith/ratio-and-proportion_1.htm

**More Practice and Games:**

http://www.quia.com/rr/35675.html

http://wps.ablongman.com/ab_vandewalle_math_6/54/13941/3569084.cw/content/index.html

http://math.rice.edu/~lanius/proportions/rate.html

**References:**

Bahr, D. L., and de Garcia, L. A. 2010. *Elementary mathematics is anything but elementary: content and methods from a developmental perspective*. Belmont, CA: Wadsworth.

Randall, Charles I. 2005. Big Ideas and Understandings as the Foundation for Elementary and Middle School Mathematics. *NCSM Journal* (Spring/Summer 2005). Retrieved on April 18, 2009 from http://www4.uwm.edu/Org/mmp/PDFs/Charles-Big%20Ideas_NCSM_Spr05v7(3)p9-24.pdf

Van de Walle, J. A. 2004. *Elementary and middle school mathematics: Teaching developmentall*y, 5^{th} e. Boston: Pearson.