Intent This activity revisits and pulls together ideas in the Patterns unit. It reinforces In-Out tables and the search for patterns as powerful problem-solving tools. Students use an In-Out table in a geometric context to find a functional relationship and are challenged to prove why their pattern holds.

Mathematics Students explore geometric ideas in this activity, such as the definition of a polygon diagonal and the connection between the number of sides and the number of vertices of a polygon. They use their developing algebra skills to analyze a geometric situation: the number of diagonals in any polygon is a function of the number of sides (or vertices) of the polygon. And finally, returning to the developing notion of proof, students utilize the geometric properties of polygon angles to prove that their numeric pattern must hold for all polygons.

Progression Students gather and organize data, search for patterns and use them to make predictions, and use the problem context to explain why the patterns they found hold. This activity can be used as a small-group activity or assigned for homework and then discussed in class.

Approximate Time 20 minutes for introduction 20 minutes for activity (at home or in class) 20 minutes for discussion

Classroom Organization Individuals or small groups, followed by whole-class discussion

Doing the Activity This activity illustrates a common use of tables in mathematics—to organize information about a complex situation in order to gain insight into the situation itself. The context determines a well-defined, unique function. Part of students’ task is to examine how to use that context to justify and explain any pattern or rule they find. Arouse students’ curiosity by posing the simple question that forms the basis for this activity. Can you predict how many diagonals a polygon has? Encourage everyone to draw a polygon (suggest from four to seven sides) and count its diagonals. Have a few volunteers report their findings, and then turn students to their groups and encourage them to continue their investigation.

Discussing and Debriefing the Activity You might begin by drawing a table on the board and calling on someone to fill in one row of the table based on that group’s work. Pause after each row and ask students if they agree or if they have any concerns about the numbers just recorded. After several rows have been added, the table will look something like this. Number of sides Number of diagonals 3 0 4 2 5 5 6 9 7 14

A convex polygon is defined as one in which all diagonals are inside the figure. A concave polygon is one in which all diagonals are not inside the figure. Some students might notice that for a concave polygon, one might argue that a segment should not be considered a diagonal if it goes outside the figure. If so, assure them that such a segment does not violate the definition of a diagonal: a line segment that connects two vertices of a polygon and is not one of its sides. Ask for volunteers to describe patterns they found. Encourage them to draw on the board to aid their descriptions. During the discussion, draw out the rule that defines the pattern, and press students to develop a justification for why that rule must always hold. Some examples of patterns, rules, and possible justifications follow. If students organize the table with increasing inputs, they will more easily notice that the number of diagonals increases by 2, then by 3, then by 4, and so forth, as the number of sides goes up by 1. This vertical or recursive pattern will allow many students to predict that a polygon with 12 sides has 54 diagonals and given that a 20-sided polygon has 170 diagonals, a 21-sided polygon must have 189 diagonals—results they would have great trouble obtaining by drawing the figures and counting diagonals. The recursive pattern can be summarized in several ways. Two equations that express the relationship are these. Out = previous Out + (previous In – 1) Out = previous Out + (In – 2) Some students, drawing on their work in Consecutive Sums, might see the following pattern in their tables and predict, correctly, that the number of diagonals in a 12-sided polygon is given by the sum 2 + 3 + ... + 10. Number of sides Number of diagonals 3 0 4 2 5 5 = 2 + 3 6 9 = 2 + 3 + 4 7 14 = 2 + 3 + 4 + 5

If so, you might ask, How can you use summation notation to express this pattern of consecutive sums? The key is determining how to use the In. Each Out is the sum of whole numbers from 2 to In – 2. Some students might use their previous experience with In-Out tables to try to find a way to relate each Out to its corresponding In and notice that the following pattern emerges. Number of sides Number of diagonals 3 0 = 3(0) 4 2 = 4(.5) 5 5 = 5(1) 6 9 = 6(1.5) 7 14 = 7(2)

From this, they might see that the multipliers in parentheses are one-half the quantity (In – 3), which leads to this closed-form rule. Out=In(In−3)2 size 12{ ital "Out"= { { ital "In" \( ital "In"-3 \) } over {2} } } {} Why does this rule make sense? The figure itself provides a clue. If students focus on a single vertex, they will see that the number of diagonals from that vertex is 3 fewer than the number of sides, because no diagonal is drawn to that vertex or to those immediately adjacent to it.

Key Questions Can you predict how many diagonals a polygon has? Can you use the table to predict the number of diagonals for an 8-sided polygon, without drawing one? Why must all 5-sided polygons have the same number of diagonals? Why would a 7-sided polygon have five diagonals more than a 6-sided polygon? Why would a 12-sided polygon have ten diagonals more than an 11-sided polygon? Why does your rule make sense?

Supplemental Activity Diagonals Illuminated (extension) is a follow-up activity that draws a distinction between recursive and closed-form rules and asks students to develop a closed-form rule for the number of diagonals of any polygon and to explain why it makes sense.