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list, including: a) when the slope is positive, the line goes up from left to right; b) the graph crosses the y - axis at the y - intercept; c) the magnitude of the slope determines the “ slantiness ” of the line; or d) a vertical line has no y - intercept and horizontal lines have zero slope. These relationships are recorded using the learners ’ words and posted for everyone to see. In order to create a variety of arrow diagrams for linear functions of the form f ( x ) = ax + b (for all real numbers), we must create a list of linear func tions (see Table 1). I ask learners to use small coef ficients and constants (e.g. - 3 ≤ x ≤ 3), and to in clude both fractional and negative values. To in crease the likelihood of a successful inquiry, in clude arrow diagrams for two or three of each type of function; patterns will be more obvious. When facilitating this inquiry, I always have a list of equations on hand, just in case more functions are needed for learners to see patterns. Typically, 15 or 20 functions are written and posted for all to see, and then learners are ready to create arrow dia grams for the listed functions. Learners are provid ed large paper, colored markers, meter sticks for straight lines, and materials for posting their crea tions around the room. An effective way to launch an inquiry is by posing an interesting question or two and then letting the learners go. Questions similar to the following have been used to launch this inquiry in my courses: what type of patterns can you imagine finding? How might the arrow diagram patterns compare to those we know about in the Cartesian coordinate plane? How many predict that there will be no pat terns? After posing the launch question(s), learners collect materials and select a function from the posted list, check it off so we do not duplicate ef forts, and then they create arrow diagrams. It is very important that everyone be encouraged to work independently to create arrow diagrams for two or more of the listed functions and then post them around the room as they are completed. We used sticky chart paper and we had lots of wall space.

For learners not ready to begin immediately, do an interactive demonstration of creating an arrow dia gram using a function not on the list, such as y = x + 1. Usually, before the arrow diagram demonstra tion can be completed for about six ordered pairs, the observers abandon their observation in favor of producing their own diagrams. Post the demonstra tion arrow diagram on the wall as part of the col lection. Each student should be encouraged to cre ate at least two arrow diagrams during this part of the inquiry. Options for faster workers include en couraging them to make up additional equations to diagram, or empowering them to begin looking for patterns among the completed diagrams. After most learners have completed two to four arrow di agrams, they are ready to move on. Typically, I task them to begin thinking about how to group the posted diagrams to show similarities and to record their conjectures about observed patterns on their own paper, so that they do not spoil or interrupt the thinking of those who are still creating arrow dia grams. If time is an issue, the arrow diagram creation pro cess may be done outside of class. Prepare printed handouts with functions and labeled axes, one per page. Distribute three to four function handouts to each learner to draw the arrow diagrams for each function. However, take care to provide an inten tionally strategic set of functions to each learner, as well as to consider differentiation concerns for in dividual learners. For example, decide if learners should have a set of functions that will show a spe cific pattern or not. After arrow diagrams are posted, the learners are given time to rearrange and group the arrow dia grams in search of similarities or patterns. After which, the learners take time to make and record conjectures about their observations. The learners in my courses were encouraged to rearrange the di agrams to support their efforts to find patterns. Adult learners used to this style of collaborative re flection, naturally grouped themselves, and dis cussed their observations.

Virginia Mathematics Teacher vol. 47, no. 1

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