Intent

This activity reverses the process of moving from graph to rule followed in Out Numbered, helping students gain fluency interpreting representations and transferring between representations.

Mathematics

The multiple representations of a function—through situations, tables, graphs, and equations—is an important theme that will be revisited often in this unit and throughout the IMP curriculum. Later in this unit, the activity All Four, One provides an opportunity for students to synthesize their understanding of the connections among these representations.

In this activity, students create graphs for equations that are given symbolically. In the process, they are reminded that graphs can continue into quadrants other than the first, that a graph is like a picture of an In-Out table, that linear graphs have a constant rate of change (increase, decrease, or no change), and that not every graph is linear. The vocabulary and conventions of graphing are emphasized during discussion of the activity, encouraging students to use these terms to communicate their ideas.

Progression

Students were introduced to the phrase “graph of an equation” at the conclusion of Out Numbered. In that activity, students identified a rule from a graph; now they reverse the process by graphing given equations. The class discussion focuses on clarifying how students did their work, while reinforcing several mathematical ideas.

Approximate Time

5 minutes for introduction

25 minutes for activity (at home or in class)

20 minutes for discussion

Classroom Organization

Individuals, then small groups, followed by whole-class discussion

Materials

Grid transparencies or similar resource for presentations

Doing the Activity

In Out Numbered, students started from graphs, made In-Out tables, and then found rules for those tables. Tell students that this activity reverses that process, and help them to clarify the instructions.

Discussing and Debriefing the Activity

Have students compare their graphs within their groups. Observe to see how students managed the instruction “sketch the graph.” Ask, How did you interpret the instruction “sketch the graph”? Do you think this graph will be discrete or continuous?

Ask students from various groups to present their results. During the presentations, as students clarify and discuss their results, maintain the emphasis on students making sense of each other’s work as you highlight key ideas, monitor the use of terminology and conventions, and review fundamental graphing concepts and terms introduced during the past few activities. Encourage use of vocabulary as appropriate, including x- and y-coordinate, ordered pair, x- and y-axis, coordinate system, plotting the point, quadrant, andrate.

One key idea to bring out is that the graphs involve more than just the first quadrant. In particular, for Question 1b, students need to use negative as well as positive inputs to see that the graph “goes up at both ends.” You could introduce the word parabola for this shape; some students may recognize the term from previous math classes.

Help students see the connections between tables and graphs. How is looking at a graph like looking at an In-Out table?

Finally, focus attention on the graph for Question 1b to avert the potential misconception that all situations and all graphs are linear. Point out that the graph is not a straight line and connect this aspect of the graph with the behavior of the outputs in the table for this equation. For instance, the table might look like this (use integer inputs for simplicity).

Ask how the fact that the graph is not a straight line is related to the values in the table. Students should see that, although the inputs are changing by the same amount at each step, the outputs are increasing by different amounts. They should gradually be coming to associate a constant change in outputs for a regular change in inputs with a linear graph.

Key Questions

How did you interpret the instruction “sketch the graph”?

Do you think this graph will be discrete or continuous?

Why do some rules give linear graphs that go up to the right while others give graphs that go down to the right?

How is the fact that the graph is not a straight line related to the table?