Algebra for Secondary Teachers: Engaging Students to Develop Symbolic Meaning

As we examine the concepts of ratio and proportion from a rates perspective, let’s recall our earlier, Activity 1.2.1, from Section 1.2. You can refresh your memory of this activity by watching the video in Figure 1.2.5.

In this activity, students frequently use rates as a means for solving the problem and determining how many seconds the Rover must run to reach the desired distance where the deer is placed. While some students will stick to more conventional rate units like cm/sec, others interpret the rate from an inverse point of view, sec/cm since they wish to have a model where they input a desired distance to obtain the time (in seconds) that they can use to tell the Rover how long it should move forward. The use of sec/cm as a rate of change is also an important deviation from the rate of change that is typically used, cm/sec, where the unit in the denominator is time. Lapp & Cyrus (2000) suggest it is also important for students to experience rates of change that are not time-based in order to abstract beyond physical motion involving velocity or acceleration. For example, in our daily lives we experience rates like that of fuel efficiency miles/gallon. When we measure energy density for the food we eat, we often use Calories/gram. In the next subsection, we will explore other non-time-based rates as a means to illustrate the concept of ratio as a relationship between quantities, but for now, let’s explore the units from Activity 1.2.1 and their connection to our discussion in Section 3.1.

Suppose we consider a scenario where the deer is placed at the location 78 cm along the measuring tape. A common strategy students use in Roadkill is to start with a time of 1 second and see how far the Rover goes. A typical distance found is roughly 19 cm. If we want the Rover to stop just short of the deer, we could view the units of 19 cm/sec as saying we can scale up the a single rectangle/bracket image (see Figure 3.2.1) to 4 such images since \(19 \cdot 4=76\text{.}\)

The use of either cm/sec or sec/cm in the Roadkill activity can be visualized from the scale-up or scale-down perspective in Figure 3.2.1. In this representation, the units cm/sec or sec/cm depend on which part of the image you focus on first. If our eyes are drawn to the 19 cm labeled in the boxes first, we might interpret this as saying every 19 cm chunk takes 1 second to attain. If our eyes are first drawn to the 1 second brackets below, we might interpret this diagram as saying that for each second of movement, we obtain 19 cm of diatance.

Further extending the scale-up/scale-down interpretation to rates, suppose students, while exploring the Roadkill task, started by entering 2 seconds with the Rover going 38 cm. The same visual representation might contain 2 chunks rather than 4 and be broken into 2 second intervals as seen in Figure 3.2.2. Now we have \(38 \cdot 2=76\) and thus we could interpret the rate as \(38 cm/2sec\) or, as a fraction, \(\frac{38}{2}\text{.}\) There is nothing wrong with expressing the rate in this form; however, it is uncommon for a student in a mathematics class to be allowed to leave it unsimplified. As teachers, it is important to realize that if the student experienced the rate by entering 2 seconds into the Rover, expressing the rate as \(\frac{38}{2}\) might actually carry more meaning \(-\) especially during the initial development of the concept. So adhering to convention during the learning process could be counterproductive to concept development.

If we want to draw attention to the rate given in sec/cm, we could consider a diagram such as Figure 3.2.3. Here the total distance is chopped into distinct 1 cm blocks (much like that of each centimeter on a measuring tape) and the time is seen as \(\frac{1}{19}sec\) chunks corresponding to each cm labeled above. Here it would take \(\frac{1}{19} \cdot 76=4\) seconds. Again, this can be viewed as a scaled-up process of the single rectangle/bracket image taking a total of 76 such chunks so that a 4 second time setting on the Rover would correspond to a 76 cm distance stopping just 2 cm short of the 78 cm location of the deer.

In either of these interpretations, the units attached to the numerical values help to provide insight into how the scaling process works. When we think about most of our experience with ratios and proportions in daily life, they almost always involve units and rates of change. It stands to reason that when teaching these concepts to students, we should also include these types of contexts so that our students have experiences to which they can relate the mathematical ideas. If we view mathematics as an abstraction of context-specific scenarios, we must first begin with the specifics so that abstraction can occur. Too often, mathematics classrooms start with definitions that assume the abstraction (i.e. no units are even considered). For example, in teaching rates of change, many texts start with the abstract definition of slope as \(m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}\text{.}\) This sterile presentation is missing the units that make the concept of rate meaningful.

As winter approaches, you may have experienced the need to stop by your neighborhood gas station to add air to your tires. Or maybe you have noticed that on a beautiful Fall day while driving to observe the wonderful colors on the trees that your readings on the dashboard for tire pressure tick up ever so slightly. Or maybe, when camping, you noticed that when using your camping stove connected to a propane cylinder, there is significant condensation on the outside of the propane tank. You touch the tank only to realize that it is very cold. What is the cause of these observations?

In the next activity, we will explore the nature of these observations and see the connections to the concepts of ratio and proportional reasoning. Bassok & Holyoak (1989) suggest, as teachers of mathematics, the culture of the mathematics classroom gives us an advantage since students seem to be more open to investigating, applying, and transferring isomorphic concepts across a variety of subject-matter domains. Therefore, we need to be versed in a myriad contexts where a mathematical abstraction can be seen within a real-world experience. The next activity will give use such an experience so that when we teach, we can make the mathematical relevent to students’ lives.

While Guillaume Amontons lacked the ability to accurately measure temperature, his research did lead him to achieve an estimate of absolute zero of \(-240^{\circ}C\text{,}\) which wasn’t too shabby for the time. Later, Joseph-Louis Gay-Lussac was able to improve on Amontons’ work to state what is often referred to as the Gay-Lussac Law, \(\frac{P_1}{T_1}=\frac{P_2}{T_2}\text{.}\) This is a special case of the more general ideal gas relationship, \(\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}\) where the volumes cancel out if they are indeed the same.

While we can numerically examine the ratio of pressure to temperature to see that we roughly maintain the same value, we can also help students to see this relationship in more than one way. If we see the numerical value of the ratio stay the same, this implies that if one quantity changes, the other quantity must change in a similar manner. In looking at the graphs of both pressure and temperature versus time in Figure 3.2.6, we can notice that the overall "shape" of both graphs are the same. This visual representation conveys the relationship that as one changes, the other changes in a "similar" manner. Therefore their relative ratio should remain fairly constant. Having more than one representation to explain the consistent behavior of the quantities relative to each other gives the learner a means of confirming and deepening the understanding.