 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Solutions for Session 6, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6| C7    Problem C1

 a. All sketches produce a similar triangle: b. No, it is four times larger. In all cases, the area of a doubled polygon is four times the area of the original. Tripling the side lengths results in a polygon nine times larger in area. Quadrupling the side lengths results in a polygon 16 times larger in area -- and so on.   Problem C2

If Ao is the area of the original polygon, then we can write the following:  Polygon Scale Factor of 2 -- Area of the Enlargement in Terms of the Original Shape Scale Factor of 3 -- Area of the Enlargement in Terms of the Original Shape Scale Factor of 4 -- Area of the Enlargement in Terms of the Original Shape         Rectangle C 4 • Ao 9 • Ao 16 • Ao    Parallelo-gram M 4 • Ao 9 • Ao 16 • Ao    Trapezoid K 4 • Ao 9 • Ao 16 • Ao    Problem C3 The number of copies needed is the square of the scale factor. For example, making a copy that is three times larger in each direction will take nine copies of the original shape.   Problem C4 The area of the enlarged figure is the original area multiplied by the square of the scale factor.   Problem C5 Because the scale factor is 3, the area is nine times larger. Therefore, the area of the enlarged figure is 72 cm2. For example, suppose the original figure were a 4-by-2 rectangle (with an area of 8 cm2). The new shape would then be 12 by 6, with an area of 72 cm2 -- nine times the original area. Here's how it breaks down: A = 12 • 6 A = (4 • 3) • (2 • 3) A = 4 • (3 • 2) • 3 associative property A = 4 • (2 • 3) • 3 commutative property A = (4 • 2) • (3 • 3) associative property   Problem C6 One way to think about it is that enlarging an object will require k copies of that object in each direction: k copies in one direction, multiplied by k copies in the other direction, for a total of k2.   Problem C7 All of these polygons are rep-tiles. Most rep-tiles have side lengths that have a common factor, but this is not a requirement.     