Cindy started the meeting by asking participants to draw rectangles in Jamboard using the graph paper background.
Tip: To make a straight line in Jamboard, hold down the shift key!
Cindy then posed the MoMATH puzzle above.
Initial responses:
- We appreciated the puzzle’s wording of area and perimeter “happening to be the same number” instead of being the same. Area and perimeter can never be the same, since they are measured with different units.
- Is a square a rectangle? Yes.
Noticings from our early work:
- A 3×8 rectangle has an area of 24 square units and a perimeter of 22 linear units, but 24 and 22 are not the same number.
- A 4×7 rectangle has an area of 28 square units and a perimeter of 22 linear units, but 28 and 22 are not the same number.
- A 2.5 x 10 rectangle has an area of 25 square units and a perimeter of 25 linear units. However, 2.5 isn’t an integer.
Working on the problem:
One participant approached the problem by looking for patterns when testing individual cases. In her early work, she noticed that area grew much more quickly than perimeter, so wondered if the edge lengths would need to be less than 10. She noticed than when an edge length is 1, the perimeter is always larger than the area and, as the other edge length increases, the difference increases. She also noticed that when one edge length is 2, the area and perimeter numbers always differ by 4. Therefore, there are no solutions with edge lengths of 1 or 2.
More testing yielded the following solutions: 4 x 4, 6 x 3 and by symmetry, 3 x 6. The rest of the discussion focused on proving that there are no other solutions.
If the area and perimeter are the same number, then ab = 2a+2b. Solving for a, yields the following:
We considered several different graphs using the Desmos Graphing Calculator, including the graph of y=(2x)/(x-2) (see below).
We noticed asymptotes at x=2 and y=2. We added green dots on the curve where the x value is a whole number. Here we see the solutions (3,6), (4,4) and (6,3). Because of the vertical asymptote, we won’t have a solution with x<3. (Technically (1, -2) has integer solutions, but a rectangle with a negative side length doesn’t make sense.) When x=5, the y value isn’t an integer. Since there is an asymptote at y=2, we can see that when x>6, the y value is a non-integer between 2 and 3. Therefore we have found all 3 positive integer solutions.
