Describing Analysis of Change
6.EE.9, 7.RP.2, 8EE.5, 8.EE.6, 8.F.4; 8.F.5; 8.SP.3
To understand rate of change, students must understand that rate is a numeric relationship between two changing quantities. See also Understanding speed as rate of change, as speed is an important example of rate of change. This entry will describe common misconceptions and teaching strategies associated with rate of change. See also Understanding Slope, which is a graphical representation of rate of change.
|If I get 8 miles per gallon out of my car during the first leg of a journey but for some reason only get 4 miles per gallon during the second leg, what is my car’s performance over the entire trip?||Many students will add together the values given in the problem, 8 + 4 = 12, and conclude the rate is 12. A more sophisticated response recognizes that two variables are involved. A student finds that the first leg has 8 miles for 1 gallon, the second leg has 4 miles for 1 gallon, so together you have 4 + 8 = 12 miles and 1 + 1 = 2 gallons, so the rate is 12 miles per 2 gallons, or 6 miles per gallon. However, this second method is also incorrect, for it assumes that the trip is exactly 8 miles for the first leg and 4 for the second leg. The number of miles traveled in each leg of the trip is not given in the problem, and so the problem is not solvable. (Thompson 1994)|
Rate as a quality
|I:||Can you give me an example?|
|S:||Ratings on movies and stuff.*|
Rate as a single numeric value
|I:||Can you tell me about the rate that the clown is walking?|
|S:||He’s walking at about 8 seconds.*|
Rate as a relationship between two quantities
|S:||As the area of the sunlight increases, the height of the window shade also increases.*|
Rate as a numerical relationship between two changing quantities
|I:||What is the rate?|
|S:||He’s walking at 4 meters per second. So for every second, he walks 4 meters.*|
* Herbert 2010
Students have trouble coordinating the two quantities required for a rate. The idea of “rate” has stages of development. In the first stage, students do not have a mathematical conception of rate, and instead describe rate based upon everyday usage. For example students may refer to movie “ratings.” At a further stage of development, students see rate as a single numerical quantity. For example, students may say that a man is walking at a rate of 8 seconds, and not 8 feet per second. Next, students may see a relationship between two quantities, but be unable to describe that relationship numerically. At this stage, a student might suggest that as the area of sunlight increases, the height of the window shade also increases. Finally, students might see that rate is the numerical relationship between two changing quantities. For example, a student would be able to identify that the rate that the man is walking is 4 meters per second, because for every second the man walks four meters.
It is important that students recognize that rate is the relationship or ratio between two changing quantities. That involves both an understanding that two different quantities are needed to form a rate, and that a rate should be represented as a numeric value.
Expose students to a number of different types of examples. Students are frequently introduced only to speed (distance per time) as a rate. (See Speed as Rate of Change). Although speed is a familiar context, students frequently have trouble identifying that speed is a ratio of two quantities: distance and time. Multiple contexts help students to see that different types of quantities can be used to calculate rate. For example, finding the number of candies per child is a type of rate that is easily visualized by students and clearly has two different quantities.
In addition, body language can give teachers helpful insights into student thinking when talking about rates. When a student’s words and gestures match, it is likely the student has a clear understanding of the concept. When a student can’t verbalize a concept, but correctly demonstrates a concept through body language the teacher can help the student focus on vocabulary development and symbolic representation. Sometimes a student’s gestures and verbal descriptions do not agree. For example, a student might say that the shadow is getting bigger while their hand motion indicates that the height of the shadow is decreasing . The mismatch between words and gestures indicates that the student does not have a good understanding of the concept.
Herbert, S. (2010). Impact of Context and Representation of Year 10 Students’ Expression of Conceptions of Rate, University of Ballarat.
Herbert, S., & Pierce, R. (2007). Video Evidence: What Gestures Tell us About Students’ Understanding of Rate of Change. Paper presented at the 30th annual conference of the Mathematics Education Research Group of Australasia.
Heller , P., Post, T., & Behr, M. (1985, October). The Effect of Rate Type, Problem Setting and Rational Number Achievement on Seventh Grade Students Performance on Qualitative and Numerical Proportional Reasoning problems. In S. Damarin & M. Shelton (Eds.), Proceedings of the Seventh General Meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 113-122). Columbus, Ohio: PME.
Thompson, P. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 181-234). Albany, NY: SUNY Press.
Thompson, A. G., & Thompson, P. (1996). Talking about rates conceptually, Part II. Journal for Research in Mathematics Education, 27(1), 2-24.
- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?