Describing Analysis of Change
Students benefit from a contextual understanding of slope so that they come to see slope as a graphical representation of the relationship between two quantities. Instead, many students think of slope as a geometric ratio--I go up this much and I go over this much. Or they may have a limited algebraic understanding of slope using the linear equation y = mx + b to identify m as the slope. This entry will describe common misconceptions and teaching strategies associated with slope.
Problem 1: Ski Ramp
The student was given a picture of a ski ramp that was labeled “Height” “Length” and “Width ”.
Problem 2: Road Sign
Students were asked to interpret slope within the context of a real world situation. Students were given a road sign from a street in Colorado that stated, “7% grade.” Explain the meaning of the sign.
a. 3 (65%)*
The slope of the line y=2-3x is:
a. 3 (12%)
The general equation for a straight line is y=mx+b. If the slope of the line is negative, b may be:
a. positive (8%)
The graph of y = 4x + 1 would look like:
**95 19 year old students responding in Barr's (1980) study.
***252 19 year old students responding in Barr's (1981) study.
PROBLEM #1: Ski Ramp
|I:||How would you determine the relative steepness?|
|S:||I don't know.|
|I:||Which of those measurements would be important?|
|S:||I don't know. I would think it would be like an ideal ratio between the height and the length, but I wouldn't know how exactly to go about it. I can't think of a unit that you could use that would be steepness.|
PROBLEM #2: Road Sign
|I:||Explain the meaning of the sign.|
|S1:||"Like 100% might be vertical. Then 7% would be up from 0, which would be flat, horizontal."|
|S2:||Maybe a comparison to like a normal-maybe it's like 7% upgrade or something, the angle.|
|S3:||Maybe like a circle would be 360, or a circle would be 100. Then it would be 7% of that.|
|S4:||I think it's the percent of the steepness of the mountain from like a flat surface. Probably find an angle of steepness.
Slope can represent real life situations such as the steepness of a hill as in Problems 1 and 2, or slope can measure the rate of change of two varying quantities (such as time and distance--see Understanding Speed as Rate of Change). When dealing with physical situations of steepness, students rarely use proportional reasoning comparing the height of an object to the base of the object. Instead, students tend to focus on angles or a single attribute, such as the length of the ramp. Students are more likely to use the idea of rate or ratio when dealing with symbolic equations than in real world situations. They have difficulty seeing a ratio as a measurement.1 Students are most successful calculating rate as a function of time, which in many ways is the most intuitive form of rate. If a student knows that they have traveled 200 miles in four hours they are able to identify the rate as 50 miles in one hour.
Students may get confused with the idea that the slope is a ratio, especially when the slope is a number such as 3 or when it is referred to as a gradient of 30%. These variations on the idea of slope are based in students' understanding of fractions, decimals, and percents, i.e. 3 =3/1 or 3/4 = 0.75 = 75%. Some students may also believe that the slope of the line is influenced by the intercept (i.e. Problem #5). In Problem #4, some students may get confused by the order of the numbers while others ignore the sign in front of m. An answer of -3/2 may indicate rote procedures being used rather than understanding.
As in problem #6, students are often taught slope as a fraction, with the change in y over the change in x. Students with only this understanding of slope have a difficult time with the way a line is positioned in a plane or the idea of rate of change.2
A student with each of the following perspectives has a rich understanding of the concept of slope.3
- Geometric Ratio: slope as vertical change over horizontal change as a geometric property, such as the ramp example above.
- Algebraic Ratio: slope as a an algebraic formula such as or the change in y over the change in x.
- Physical Property: slope through words such as "slant", "steepness", "incline", "pitch", and "angle".
- Functional Property: slope as the rate of change between two variables.
- Parametric Coefficient: slope referenced by m in the equation y = mx + b.
- Real World Situation: slope in a static or physical situation such as a wheel chair ramp.
- Trigonometric Conception: slope as the tangent of the angle of inclination.
- Calculus Conception: slope as it relates to the derivative.
- Determining Property: slope used to determine parallel and perpendicular lines or a line if given a point and slope.
- Behavior Indicator: the number for slope indicates increasing vs. decreasing as well as magnitude of steepness.
- Linear Constant: slope as a constant property, regardless of representation.
Teachers should promote a fluid understanding of slope as a rate of change using real world situations and have students practice connecting those situations to graphical representations. One way to prompt student thinking is to ask, “What does the slope represent in the context of this situation?”
Students struggle with the connection between the angle of a ramp and the slope of its steepness. One activity to promote an understanding of steepness as a ratio is to have students physically measure real world handrails, sidewalks and ramps. Models of ramps that have slopes that are gentle, moderate, and steep can also be used. Compare the ratio of the rise of the ramp with the run. Students should realize that the ratio with the greatest value relates to the ramp with the steepest slope.
Student difficulties with slope reflect an emphasis on the geometric representation. Research suggests that students have difficulty interpreting slope. There is often a disconnect between being able to calculate slope and actually knowing what slope represents, so teachers need to ask students to explain the meaning of the slope in a given situation.
Some essential questions that can be asked include: What is the relationship between the measurements of the angles and the calculated ratios? What does this tell you about slope?”
Questions a teacher should reflect on when preparing lessons to teach the concept of slope to students are:
a. What representations of slope are in my concept definitions?
b. Do I have a flexible understanding of representations for slope?
c. What representations of slope do I use?
d. What real-world situations do I use to illustrate concept of slope?
e. What is my knowledge of student difficulties with slope?
An app that provides contextualized practice with slope (and thus may help student come to see slope as a graphical representation of the relationship between two quantities) is Submariner Algebra, available for free in the iTunes Store.
1 Swafford and Langrall (2000)
2Walter& Gerson (2007).
3Stanton, M., & Moore-Russo, D. (2012). Conceptualizations of slope: A look at state standards.School Science and Mathematics, 112(5), 270-277.and Moore-Russo, D., Conner, A., & Rugg, K. I. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educational Studies in Mathematics, 76(1). 3-21..
Barr, G. (1980). Graphs, Gradients and Intercepts. Mathematics in School, 9(1), 5-6.
Barr, G. (1981). Some Student Ideas on the Concept of Gradient. Mathematics in School, 10(1), 14,16-17.
Berg, C. A., & Smith, P. (1994). Assessing Students' Abilities to Construct and Interpret Line Graphs: Disparities between Multiple-Choice and Free-Response Instruments. Science Education, 78(6), 527-554. Nemirovsky, R. (1998). Body motion and graphing. Cognition and Instruction, 16(2), 119-172.
Hauger, G. S. (1997). Growth of knowledge of rate in four precalculus students. Presented at the Annual Meeting of the American Educational Research Association, Chicago, IL.
Lobato, J., & Thanheiser, E. (1999). Rethinking slope from quantitative and phenomenological perspectives. In 21. Presented at the North American Chapter of the Psychology of Mathematics Education Conference.
Moore-Russo, D., Conner, A., & Rugg, K. (2011). Can slope be negative in 3-space? Studying concept image of slope through collective definition construction. Educational Studies in Mathematics, 76(1), 3-21.
Stump, S. L. (2001). High school precalculus students' understanding of slope as measure. School Science and Mathematics, 101(2), 81-89.
Stanton, M., & Moore-Russo, D. (2012). Conceptualizations of Slope: A Review of State Standards. School Science and Mathematics, 112(5), 270-277.
Swafford, J. O., & Langrall, C. W. . (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89-112.
Walter, J., & Gerson, H. (2007). Teachers’ Personal Agency: Making Sense of Slope Through Additive Structures. Educational Studies in Mathematics, 65(2), 203-233.
- 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?