## Triangle Exploration 1

Students should be encouraged to describe their thinking thoroughly and in detail. The goal is not simply to judge the correctness of students' thinking, but rather to help students develop reflective skills so they can describe, analyze, test, and reconsider their own thinking. Prompts such as "describe more about that idea" or "explain what you mean to another student and then try to write as if you were speaking" can be used to encourage students to write more. Helping students create detailed and lengthy explanations here will communicate high expectations for reflection throughout this unit.

## Triangle Exploration 2

Triangle areas range from greater than zero to approximately 4926 square feet. Side lengths range from greater than zero to less than 160 feet. The graph of side length versus area of the isosceles triangle should be a parabola with maximum at (106, 4926) or ((107, 4926). The actual maximum x-coordinate is between 106 and 107. The maximized triangle is equilateral; the sketch approximates this triangle when its side lengths are between 106 and 107 feet and its area is approximately 4926 square feet. If the triangle perimeter is 240 square feet, the maximized triangle would have side lengths of 80 feet and an area of approximately 2771 square feet. Students could use the Pythagorean Theorem to find the height and then use the area formula for a triangle to calculate this area.

Students' paragraphs should include: (1) descriptions of their thinking about the problem during Exploration 1 and how their thinking changed through the second exploration, (2) explanation of how changes in area and maximization are represented in the graph of side length versus area as well as in the interactive sketch, and (3) to what extent these explorations inform them about the original question. Students should be encouraged to describe their thinking thoroughly and in detail.

## Rectangle Exploration 1

Students should be encouraged to describe their thinking thoroughly and in detail. The goal is not simply to judge the correctness of students' thinking, but rather to help students develop reflective skills so they can describe, analyze, test, and reconsider their own thinking. Prompts such as "describe more about that idea" or "explain what you mean to another student and then try to write as if you were speaking" can be used to encourage students to write more. Helping students create detailed and lengthy explanations here will communicate high expectations for reflection throughout this unit.

## Rectangle Exploration 2

Areas range from greater than zero to 6400 square feet. Side lengths range from greater than zero to less than 160 feet. The graph of length versus area should be a parabola with maximum at (80, 6400). The maximized rectangle is a square with side lengths of 80 feet and area equal to 6400 square feet. If the rectangle perimeter is 240 feet, then the maximized rectangle would have side lengths of 60 feet and area equal to 3600 square feet.

Students' paragraphs should include: (1) reflections about the how the rectangle explorations informed them about the relationship between rectangle side lengths and area, (2) explanations about how their knowledge applies to the original problem including comparisons between possible triangular and rectangular play areas. Students should be encouraged to describe their thinking thoroughly and in detail.

## Pentagon Exploration 1

Many pentagons of varying shape are possible with area less than 2000 square feet. Pentagons with area greater than 7000 square feet should be nearly regular with similar (though not necessarily equal) side lengths. It is unlikely that students will be able to construct a regular pentagon using the interactive sketch. However, they should be able to approximate one with areas approaching 7050 square feet and side lengths of approximately 64 feet. The final questions can be used to determine whether students have generalized the necessity of equal side lengths for achieving maximum area from their work with the triangle, rectangle, and pentagon. It is not necessary or expected that students be able to answer these questions with exactness, but they should be encouraged to make conjectures beyond the limitations of the sketch, and to consider further the relationship between shape, side lengths, and maximum area.

## Hexagon Exploration 1

Many hexagons of varying shape are possible with area less than 2500 square feet. Hexagons with area greater than 7000 square feet should be somewhat regular with similar (though not necessarily equal) side lengths. It is unlikely that students will be able to construct a regular hexagon using the interactive sketch. However, they should be able to approximate one with areas approaching 7390 square feet and side lengths of between 52 and 55 feet.

The last series of questions encourage students to reflect, generalize, and predict. By comparing across shapes, students begin to examine the bigger picture. Students may notice that: (1) the shapes get bigger as sides are added, (2) maximized shapes have approximately equal side lengths, (3) the shapes are equiangular, and (4) the shapes are getting rounder. Expect other responses as well. Encourage students to reexamine the previous sketches, their own writing, and compare their ideas with other students' work. Expect students to reflect and encourage them to expand their lists as much as possible, but do not expect to find more than one or two of the four example responses in each student's list.

Encourage students to be able to explain the reasons for their predictions about the octagon. Ask them what the side lengths of the maximized octagon would be? Ask them to be specific about their side length predictions (since 8 divides easily into 320).

## Octagon Exploration 1

Many octagons of varying shape are possible with area less than 2500 square feet. Octagons with area greater than 7000 square feet may be somewhat regular with similar (though not necessarily equal) side lengths. It is unlikely that students will be able to construct a regular octagon using the interactive sketch. However, they should be able to approximate one with areas approaching 7710 square feet and side lengths of approximately 40 feet.

Responses to the final questions should include: (1) reflections about the how the octagon exploration informed them about the relationship between octagon shape and area, (2) explanations about how their knowledge applies to the original problem including comparisons between possible octagonal and other play areas. Students should be encouraged to describe their thinking thoroughly and in detail.

## Circle Exploration 1

Only one area (approximately 8149 square feet) is possible using exactly 320 feet of fence because the shape of the circle is fixed. The circle provides the greatest area for a given perimeter.

## Summary

Students' writing can be assessed in terms of mathematical correctness, appropriateness of explanation for the intended audience, and completeness of arguments and reasoning. A student who demonstrates a thorough understanding of the mathematical relationships may have difficulty communicating ideas or may skip over steps or ideas. Encourage all students to use examples, develop and organize ideas, and provide reasons for each statement. Students who have not sufficiently reflected and attempted to generalize throughout the explorations, may need to re-examine their work and some sketches in order to develop the understanding necessary to write a complete and reasoned explanation.