Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Solutions for Session 5, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6

 Problem B2 The triangles are similar because they both have two equal angles. One equal angle is the right angle formed by the lamppost/friend and the flat ground. The other equal angle is the angle formed by the shadow of each object and the Sun's rays (the Sun's rays are parallel lines that strike the ground at the same angle for either shadow).

 Problem B3 We can take a known measurement for each object (the length of the shadow) to establish the scale factor by setting up a ratio (AB/DE). Then we multiply the height of the person (EF) by the scale factor to get the height of the lamppost (BC). This is equivalent to setting up a proportion: AB/DE = BC/EF Or, using cross multiplication: BC = (AB • EF)/DE

 Problem B4 Answers will vary. You could use the proportion from Problem B3, or alternatively, you could set up a different proportion, which would yield the same result: AB/BC = DE/EF Or, using cross multiplication to solve for BC: BC = (AB • EF)/DE Both proportions will yield the same result.

Problem B5

 a. It is a derived measure since it is determined by calculations on other measures. b. Answers will vary depending upon the actual height of the person and the lengths of the shadows. The upper limit will use the maximum lengths for the person and for the lamppost's shadow, and the minimum length for the person's shadow. The lower limit will use the opposites. Assuming that you could know the accurate measure for each of these lengths, the upper limit would be BC = ((AB + 0.25) • (EF + 0.25))/(DE - 0.25). (Each of these amounts (AB, EF, DE is the absolute height of the object.) The lower limit would be BC = ((AB - 0.25) • (EF - 0.25))/(DE + 0.25). For more information on accuracy, go to Session 2, Part C. c. The best value for the height of the lamppost might be the average of these two limits, since it gives us a reasonable estimate that is close to either limit.

 Problem B6 You could use the methods presented in this part to take a derived measure of the height of the tree.