Problem... A circle is inscribed in a square that has a side length of one unit, which is also the radius of a quarter-circle. Find the area of as many of the different shapes and regions that you can see. (Which shapes, or regions, are you finding difficult? What is it that is stopping you find their areas?)
Students typically define the square, the inscribed circle, the quarter circle and often a quarter of the inscribed circle as shapes and regions to find the areas of, and proceed to do so without issue. Students also note the dark orange 'crescent' region as an area to find, and, indeed, the lighter orange regions around it, and these delineate the aspects of the problem that have the potential to develop into the most formative discussions.
Students, with the teacher's guidance / encouragement of course, can outline a strategy of how to find these more complex areas, ignoring whether or not they yet have the knowledge, skills or understanding to follow such a strategy through. Depending on where students are in their mathematical careers, this can then lead into more direct teaching, or, moreover, can function as a segue into a body of work that will provide students with the knowledge and skills they need to apply their strategy.
To see a 'thinking through' of a typical solution / approach with students to the problem of the 'crescent' area, click on 'Read more' below.
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