Rectangle Riddle

In this task, students use their knowledge of area and perimeter of rectangles to find rectangles that match the description given (a perimeter that is 4 more than the area). Third and fourth grade students are developing an understanding of area as the number of square units that make up a two-dimensional shape and connect the concept of area to multiplication of two factors (side lengths). Third grade students understand perimeter as the length around a figure, and fourth grade students can use a general formula to calculate the perimeter of a rectangle.

Students will likely experiment with several areas with different lengths and widths that get them closer to the rectangle described. They may use what they know about factors and products to choose a number that could easily be arranged into one or more rectangles and then compare and contrast the perimeters and areas of the different arrangements.

Any rectangle arranged into 2 rows or columns will result in a perimeter with a numerical value that is 4 more than the numerical value of the area, so students may make and test conjectures as they notice this pattern developing.

Various apps may be used to represent the context of this problem.

• In the Pattern Shapes app, students may build a rectangle and calculate the perimeter and area by counting (as shown here) or by using formulas (as shown here). Once they have found one rectangle that fits the requirements, they should explore further to find others.
• In the Geoboard app, students might build a rectangle and calculate the perimeter and area by counting (as shown here) or by using formulas (as shown here).
• In the Number Frames app, students might experiment, using the “Choose a Frame” tool. This tool shows the area as the rectangle changes. Students might work inside this tool, comparing area with perimeter.

This activity gives students an opportunity to look for and express regularity in repeated reasoning. You can ask such questions as: How can you find the perimeter without counting each exterior side? How can you find the area without counting by 1s? Students who have not recognized or used formulas for finding area or perimeter will likely lean into the structures that they have observed when counting the total number of square units (area) or linear units around the rectangle (perimeter), such as skip-counting or noticing and operating with equal groups.

Additionally, students may begin to notice how the relationship between perimeter and area changes as the dimensions of the rectangle approach those of a square. You could ask: Did the perimeter increase or decrease when you made a rectangle with the same area, but with more rows? Fewer rows? As students make multiple rectangles that fit the requirements, ask: What similarities do you notice between the rectangles that fit the requirements? Encourage students to make and explore conjectures: Will that always be true? How do you know?