 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum           A B C Homework Solutions for Session 6 Homework

See solutions for Problems: H1 | H2 | H3 | H4    Problem H1 Using the fact that each of the tangram pieces can be divided into some number of small triangles, we get the following areas: The small square, the parallelogram, and the medium triangle each have areas of 2 square units, and the large triangles each have areas of 4 square units. The area of the entire tangram is 16 square units.    Problem H2 The trapezoid is divided into two triangles, each with height h, the height of the trapezoid. One triangle has base b1 while the other has base b2: Since the area of a triangle is , the total area is , or , which is also the area of the trapezoid.   Problem H3 Using the Pythagorean theorem, a2 + b2 = c2, we can calculate the lengths of the bases and the height. The bases have lengths of 2 (the hypotenuse of the triangle with legs 2 and 2) and 6 (the hypotenuse of the triangle with legs 6 and 6), respectively. Similarly, the height is 2 (it is the length perpendicular to the bases and the hypotenuse of the triangle with legs 2 and 2). The area is , which equals 16 square units. We can also find the area by drawing a rectangle around the trapezoid and then subtracting the smaller areas, or by dividing the trapezoid into two triangles, or by any of several other methods. But any way you slice it, the area is 16 square units. To learn more about the Pythagorean theorem, go to Learning Math: Geometry, Session 6.   Problem H4

a.

Both shapes have areas of 4.5 square units.

b.  Number of Boundary Dots Area
(in Square Units)  3 0.5

4

1

5

1.5

6

2

7

2.5

b - 1 In each case, the area is half the number of boundary dots minus 1. If there are b boundary dots and one interior dot, the area is - 1.

c.  Number of Boundary Dots Area
(in Square Units)  3 1.5 4 2 5 2.5 6 3 7 3.5 b  In each case, the area is half the number of boundary dots. If there are b boundary dots and one interior dot, the area is .

d.  Number of Boundary Dots Area
(in Square Units)  3 2.5

4

3

5

3.5

6

4

7

4.5

b + 1 In each case, the area is half the number of boundary dots plus 1. If there are b boundary dots and two interior dots, the area is + 1.

e.

Every time you add a boundary dot, the area goes up by half a unit, regardless of the orientation of the shape.

f.

 The formula is A = I + - 1, and it can be used on many of the figures in Part B of this session.     