Unit Learning Outcomes
Attending to and Coordinating Two Quantities- Essential Understanding #1
Students will first develop the concept of ratio by solving a set of comparing and missing value problems. Ultimately students need to see ratios as describing the relationship between two quantities. “Before children are able to reason with ratios, they typically reason with a single quantity.” (Lobato et al. 2010) Students must first reach this understanding of attending to and coordinating two quantities before they can reason with ratios. Consider the following problem that will be used to develop this concept.
Let the two rectangles represent different sized cow pens and the ovals represent equally sized cows. Pen A has an area of 8 square yards. Pen B has as area of 4 square yards. Which cow pen is more crowded?
The point that will be drawn from this example is that the relationship between two different quantities, in this case number of cows and area, which can be expressed, for example, in number of square meters, needs to be reasoned with in order to answer the question. A first intuition might be to count the number of cows in order to determine crowdedness. In this line of thinking, a student is only attending to one quantity. This logic can be easily proven wrong by drawing a grid over the rectangles showing that pen A has 1 square yard for every cow whereas pen B only has 4 square yards for 5 cows making it more crowded. Another way to approach this problem is by doubling B making the area the same as pen A but the number of cows greater by 2. This too visually shows that pen B is more crowded than A. After students have grasped the concept of attending to different quantities and their relationship, they will be explicitly taught the different ways of notating ratios.
Although the focus of the cow pen problem is on prompting students to begin using ratios to coordinate and compare two different quantities, it also implicitly calls students to consider proportional relationships. For example students might arrive at the conclusion that pen A has 4 cows per 4 square meters, thus arriving at a proportional ratio. This link should be explicitly shown to students in discussing ratio problems. Even though the unit starts with a focus on ratio, ideas of a proportional relationship will be addressed in conjunction. When a solid foundation of ratio and its utility is established, students will move on to explicitly grapple with proportions.
Proportion - Essential Understandings #2 and #7
Two ratios that are in proportion are also called equivalent. This concept will be initially developed by applying it to familiar contexts such as measurement conversions and price per item. Consider the following problem:
John had 3 yardsticks and he noticed that there were 9 feet in 3 yards. He used the yardsticks to measure the length of the classroom. The classroom measured 12 yards long. What is the length of the classroom in feet?
This problem provides an excellent entry point to understand proportions in two different ways. One approach to this problem is to look across the ratios and see that 12 yards is 4 times as many as 3. So to complete this proportion one can multiply 9 feet by 4 to get 36 feet. In addition, students should ultimately see the relationship within both ratios realizing that 9 is 3 times 3 so 12 can be multiplied by 3 to reach the same answer of 36. This, I will later explain, is the constant rate of proportionality (namely 3 feet per yard), a more advanced concept.
(9 ft × 4)/(3 yd × 4) = (36 ft)/(12 yd)
(9 ft)/(3 yd × 3) = (36 ft)/(12 yd ×3)
The first example demonstrates the understanding that two corresponding quantities in one ratio will relate to any two other corresponding quantities by the same factor. In this case the quantity 9 feet relates to the quantity 36 feet by a factor of 4, as does 3 yards to 12 yards. This is the fundamental property of proportional relationships, and should be focused on first. The second example demonstrates that there is a constant relationship within a ratio and that is the same for all other proportional ratios. In this case the constant is 3. So for any number of yards, the number of feet will be 3 times as many. In this example it is obvious, due to its being a simple whole number, but when the constant rate of proportionality is larger or fractional, this concept becomes more difficult for students to grasp. Both examples are important uses of multiplicative comparison and both will be developed and taught explicitly via these types of problems. With this knowledge students can evaluate ratios for proportionality and generate equivalent ratios. In order to develop flexible proportional reasoning students need these understanding before learning any algorithm.
Taught alongside multiplicative reasoning should be the idea of a unit rate. A unit rate is a version of a ratio that expresses the relationship between a quantity and 1 unit of the other quantity. Unit rates are only defined when the units of each quantity are specified. Often there is an obvious or convenient unit to choose for the denominator, but the fact that a choice has been made is important and should be addressed. Once the units of each quantity have been chosen, the unit rate is a well-defined number. Common examples of unit rates seen in an everyday context are miles per 1 hour or cost per 1 item. Consider the following problem:
If 12 sodas cost $18, how much would 20 sodas cost?
Using the unit rate approach, students can find the cost of 1 soda by dividing 18 by 12 to find the unit rate of $1.50 per 1 soda and then multiply this value by 20 to find the cost of 20 sodas. Ratios within a proportional relationship will always have the same unit rate. This is because the unit rate is the constant rate of proportionality, expressed in the given units. Similar to the multiplicative approach, finding the unit rate can help evaluate proportional relationships and find equivalent ratios. Research shows that students used the unit rate strategy more frequently due to its more procedural and intuitive nature. To make sure that students do not narrowly rely on this strategy it is important to show its relation to multiplicative comparison. (Cruz 2013)
Constant Rate of Proportionality- Essential Understanding 6
When comparing multiplicatively within a ratio and when finding the unit rate, the constant rate of proportionality is defined. However this concept will be taught last; it is not enough to simply understand that the unit rate is the constant rate of proportionality. It needs to be understood as a relationship. Consider the following statement letting the variable k represent the constant: if quantities a and b are in proportional relationship, then for any pair of corresponding values,
In any ratio a/b = k so a = kb
The constant explicitly shows the relationship between the two quantities in the ratio, a:b. This relationship is that for every b there are k times as many a. To reach this understanding it is useful to ask students questions such as: How many times bigger/smaller is a compared to b? For every increase to b what is happening to a? Comprehending this idea is an important landmark in proportional reasoning. Ultimately students will be tasked with creating an equation such as the one above to express a proportional relationship. Because of its importance in their future work in grades seven and eight with linear equations and functions, a conceptual understanding of the meaning of a constant rate of proportionality is critical.
A Note on Algorithms
This is an introduction to ratios and proportional reasoning. Students will be developing this sense over the course of the next two years. Teaching of standard algorithms could impede the future development of proportional reasoning. Although my standard curriculum introduces the cross multiply and divide algorithm half way through the unit, I will hold off on teaching this strategy until I am confident in my students’ understanding of what a proportional relationship is.
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