## Variety of Perimeters with Fixed Area

1) Overview
After learning the concepts of perimeters and areas, it is easy for students to think that figures with larger perimeters would also have larger areas, and vice versa. This applet helps teachers to explore with students the variety of the perimeters of a figure formed by several congruent squares touching side by side. Together with the complementary applet "28888 Variety of Areas with Fixed Perimeter", teachers can clarify with students that a figure with a larger area may have a smaller perimeter, and areas and perimeters are two different concepts.

2) Learning Objective
1. Recognize figures with same areas could have different perimeters;
2. Recognize the strategy of minimizing the perimeters of figures with the same areas.

3) Teaching Approach
An inquiry teaching approach is expected. Students are asked to arrange 3 to 9 squares to form different figures and find their possible perimeters. Teacher then guide students to express their strategies of getting the largest and smallest perimeter with a certain number of squares.

4) Teacher’s Note
1. For each number of squares, ask students to record the possible perimeters in the table of the applet;
2. Guide students to focus on the change of the perimeter when a square is dragged to a new position;
3. Discuss with students the strategy of minimizing the perimeter, especially for 4 and 9 squares.

## Variety of Areas with Fixed Perimeter

1) Overview
After learning the concepts of perimeters and areas, it is easy for students to think that figures with larger perimeters would also have larger areas, and vice versa. This applet helps teachers to explore with students the variety of the areas of a rectangle with a fixed perimeter. Together with the complementary applet "22723 Variety of Perimeters with Fixed Area", teachers can clarify with students that a figure with a larger perimeter may have a smaller area, and areas and perimeters are two different concepts.

2) Learning Objective
1. Recognize rectangles with same perimeters could have different areas;
2. Recognize that among all the rectangles with the same perimeters, the square has the largest area.

3) Teaching Approach
An inquiry teaching approach is expected. Students are asked to shorten the longer side and lengthen the shorter side by 1 cm each time and visualize the change of the area, and investigate when the area would become largest. Teacher then help students to express arguments which justify their conclusions.

4) Teacher’s Note
1. Ask students to record the dimensions, perimeters and areas in the table of the applet;
2. Remind students that when the longer side and the shorter side are shortened and lengthened by the same amount, the perimeter is unchanged;
3. Guide students to visualize that the area would always be increased if the longer and shorter sides are changed by the same amount until they become equal;
4. Another way of arguing is to begin with a square and then change the longer and shorter sides by the same amount to see that the area must be decreased.