Intent

In this activity, students are asked to add reasonable numeric scales to the axes of graph sketches from previous activities. The key idea is that each axis is to be treated as a number line, or part of one.

Mathematics

This activity continues the development of graphing strategies by introducing the process of quantifying graphs using scales. Students begin to understand that each axis is a number line. They read information from graphs and use this information to add scales to graph sketches. Students must justify the reasonableness of their scales. In the process, they employ basic ideas and terminology about coordinate graphing, in particular the term coordinate. Scaling the axes of a graph and reading quantitative information from graphs are conceptual sticking points for many algebra students, so the groundwork laid in this activity is particularly important for the mathematical development of the unit.

Progression

After students have shared ideas from Graph Sketches, the teacher introduces the process of scaling the axes on a graph. Students then do the activity in groupsand begin to wrestle with the difficulties associated with scaling axes. The Issues Involved is an immediate follow-up activity in which issues about scaling are raised and more conventions for graphing are defined.

Approximate Time

25 minutes

Classroom Organization

Whole-class introduction, followed by small groups

Materials

Transparencies of the graphs[link to BLMs]

Graph paper

Doing the Activity

You might use the graph from Question 1 of Wagon Train Sketches and Situations to demonstrate the level of thought and detail you expect in this activity. The initial, unscaled graph looks like this.

Figure 1

To choose appropriate scales, students will need to make estimates. They might start by estimating how much coffee the family might have begun with and how long it might have taken for them to use it up. For example, if the family started with 25 pounds of coffee and consumed half a pound of coffee per day (perhaps for a large Overland Trail family), the family will have run out of coffee after 50 days. For those values, the graph with scaled axes might look like this.

Figure 2

You can ask whether it’s permissible for “10 days” and “10 pounds” to be represented by different lengths on the two axes. In general, if the units on the two scales are not measuring the same type of item, there is no reason for the lengths to match.

Once the class has decided how to scale the axes, ask some questions that can be answered from the graph, such as, How much coffee is left after 10 days? After 20 days?

Have students both articulate and demonstrate how they are finding these answers. For instance, to determine how much coffee is left after 20 days, they would locate 20 on the horizontal axis, go up (vertically) from there to the graphed line, and then go over (horizontally) to the vertical axis and read off the value (in this case, 15).

Identify these two numbers, 20 and 15, as the coordinates of the associated point on the graph. Also bring out the units associated with these numbers: this point on the graph represents that after 20 days since leaving Fort Laramie, the family has 15 pounds of coffee remaining. You can introduce ordered pair notation (20, 15) now or delay this until the activity The Issues Involved.

Discussing and Debriefing the Activity

Circulate as groups work, using the opportunity to encourage the use of successful group strategies, such as collaboration and checking in with one another.

This activity will not need a whole-class debriefing. The next activity, The Issues Involved, raises general issues about creating scales that may lead the class to look back at the graph sketches from this activity.

Key Questions

What would be an appropriate scale for this axis?