This standard is another example of how proportional relationships play an important role in the middle school mathematics curriculum. We say two figures are similar if one can be made to overlap the other exactly through a combination of translation (slide), rotation (turn), reflection (flip), and dilation (magnification). The parts of two similar figures that match up are called corresponding angles, sides, etc. In a pair of similar figures, we know that corresponding angles are congruent and the ratios of corresponding segments are constant - and the value of this ratio is the scale factor. For example, the two quadrilaterals shown below are similar.
Therefore, angles A and E, B and F, C and G and D and H are congruent, respectively. Moreover, the ratios of the lengths of sides, AB: EF, BC:FG, CD:GH, and DA:HE, are constant, and in this case the ratio is 1:2. The scale factor depends on which of the two figure we consider as the base of the comparison. So, if we consider quadrilateral ABCD as the base, the scale factor, in this case, is 2. On the other hand, if we consider quadrilateral EFGH as the base, the scale factor is 1/2.
Suppose AB = 4 cm, BC = 2 cm, CD = 5 cm, and DA = 6 cm. Then, EF = 8 cm, FG = 4cm, GH = 10 cm, and HE = 12 cm. Let's organize these lengths in a table.
Now, you see that as the length in ABCD doubles and triples (from 2 cm to 4 cm or 6 cm), the length of EFGH also doubles and triples. Even when the length becomes 2.5 times as long, from 2 cm to 5 cm, in ABCD, the corresponding length also becomes 2.5 times as long, 4 cm to 10 cm. Thus, the lengths in these two figures are in a proportional relationship. In general, if two figures, X and Y, are similar, the lengths in these two figures are in a proportional relationship. Thus, we can apply all the tools we discussed previously in representing this relationship. So, if we use a double number line, the relationship can be represented something like this:
Thus, if we know a side in Figure X is 15 cm and the corresponding side in Figure Y is 6 cm, we can use that relationship to determine the length of any side can be determined if we know the length of the corresponding sides. Suppose, we know another side in Figure X is 20 cm, the relationship can be represented in a double number line like this:
On the other hand, if you know the length of a side in Figure Y is 4 cm, the relationship will be represented like this:
Another feature of a proportional relationship is that the quotients of corresponding quantities are constant. So, if we divide the lengths in Figure X by the corresponding lengths in Figure Y, the quotients are constant. We can also use this relationship to represent the two situations above like this:
These tables basically show the four values (including the missing value represented by a ?) from the double number line representations above. Note that in the second table, the columns are in the reverse order. A number line has a particular direction, i.e., as you move to the right, the numbers become larger, However, a table does not have such an inherent directionality. So, for students, it might be more natural if we place the relationship as they are presented.
In any event, since 15 x 0.4 = 6, ? = 20 x 0.4 -- 0.4 is the scale factor if we consider Figure X as the base. For the second problem, we can say that since 6 x 2.5 = 15, ? = 4 x 2.5 -- 2.5 is the scale factor if we consider Figure Y as the base.
As is the case with the conversion of measurements from one unit to another, what is important is to help students develop an understanding that mathematics is a web of relationships. The focus of this standard is not just for students to find the missing lengths in similar figures. We also want them to understand that what they have learned previously, namely proportional relationships, can be used to represent, interpret, and investigate new situations.