“A mathematician, like a painter or a poet, is a maker of patterns.
If his patterns are more permanent than theirs,
it is because they are made of ideas.
His patterns, like the painter's or the poet's must be beautiful;
the ideas, like the colors or the words,
must fit together in a harmonious way.”
English mathematician G. H. Hardy (1877-1947)
Collaborate and Reflect
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Extend Your Understanding
Watch the video excerpt below to view an example of a teacher engaging students in looking for and making use of structure.
Cathy Humphreys leads an extended exploration of a proof of the properties of quadrilaterals, helping students learn to investigate, formulate, conjecture, justify, and ultimately prove mathematical theorems. In these clips, students engage in the first of two block-length explorations of their proofs. Humphreys observes, “The square, rectangle, and rhombus appeared to be the most straightforward for the students. Mathematically, if two of the diagonals form right angles, then at least a pair of sides of the quadrilateral will be equal in length. If the diagonals intersect at the midpoint of both diagonals, then the figure formed will be some parallelogram. In order for two diagonals to form a non-isosceles trapezoid, the following relationships must hold true: If AB is one diagonal and DE is the other diagonal, then trapezoid ADBE is formed only if the diagonals intersect at point P, which is not the midpoint, and AP/PB = DP/PE. This relationship was quite difficult for the students to investigate and conclude. The students did not choose to measure the diagonals with rulers, and therefore did not pick up on the proportional aspects of the diagonals in a non-isosceles trapezoid.” This clip is also indicative of Mathematical Practices 1, 3, and 6.