Areas and Factors Puzzle

The object of this puzzle is to fit rectangles on the grid that are unique, have different areas and all of the areas are multiples of 3.

I wanted a task where students consider multiples and factors. They need to know which length have a product that is a multiple of 3 and also consider that all the areas have to fit on a 9 by 12 grid so the areas all need to add to 108rect prob

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Factors

I saw a task where students had to play a game against a partner. They rolled two dice and multiplied the numbers together, they could then draw a rectangle on the grid with a perimeter equal to the product of these two numbers.

I decided to create a task loosely based on this idea but I decided I wanted to look at areas instead. I then realised my task wasn’t really about area but about factors. Students need to also consider which numbers have only one or two solutions. If students look at the first two areas in the list they should note that 14 has factors; 1, 2, 7, 14 but a 1 by 14 rectangle isn’t possible in this grid so it needs to 2 by 7. Likewise 3 by 9 is the only possibility here. Once they have drawn in 3 by 9 on the grid it leaves a thin rectangle with a width of 1 so 1 by 6 is the only rectangle that fits here (or a 1 by 1)

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rect3

rectangles