Teaching with and through Maps

Scaling Up or Down

The Unit 1 RCT assesses whether students can identify scale copies of triangles and irregular polygons when sides are labeled, and when they are unlabeled on a coordinate plane. To assist students in contextually developing this skill, they will be exposed to maps displaying triangulation and asked to identify similar triangles. They will argue a position before they learn to confirm similarity by calculating scale factor. Students will use their reasoning and explain why one triangle is a scale copy in contrast to a triangle that is not. Making a reasonable argument before being offered mathematical proof helps to concretize the proof for students because it is anchored in what is observable to them.  Eventually, when presented with triangles placed on a grid, students will determine side lengths of the triangles (in units), identify corresponding parts, and explain that two triangles are scale copies if all the corresponding parts dilate based upon the same scale factor.

To calculate the scale factor that transforms a preimage to its image, students will discover that they can divide the length of a side of the new image by the length of the corresponding side on the preimage, or in short, ^new⁄_old. Emphasis should be placed on expressing division in fraction form to increase familiarity with fractions as representative of division. This understanding is fundamental to being able to express a scale factor when the multiplicative relationship is not readily seen. For example, when considering the scale factor that takes a side length from 2 cm to 10 cm, students can easily recognize that the new length is 5 times bigger than the old length because of their familiarity with skip counting and recognize that the scale factor is 5. However, when a side changes from 10 cm to 2 cm, students are apt to explain that the side length was divided by 5. While this is mathematically true, it does not provide a scale factor which is a multiplicative term. Since in the first problem, they did not express the scale factor as ¹⁰⁄₂before arriving at its simplified form, 5, they do not automatically express the reverse relationship as ²⁄₁₀ or ¹⁄₅, the reciprocal of 5. When students demonstrate this partial understanding, referring to the divisive relationship, it serves to help them uncover the identity property of multiplication by prompting them to consider that multiplying any number by 1 allows the number to keep its value or identity. Students should investigate multiplying a side length by a number greater than 1 and recognize that it increases. Further, knowing that any number multiplied by zero is zero (zero property of multiplication), students should deduce that one of the factors must fall between 0 and 1 for multiplication to result in a smaller number, and that numbers between zero and 1 are fractions.

Students who struggle with identifying numbers that fall between 0 and 1 will need access to a number line to practice ordering numbers including fractions, a skill introduced in 3^rd grade. To maintain rigor while remediating this skill, teachers should present the number line as part of the coordinate plane. Students will practice ordering fractions on the x-axis and the y-axis, addressing gaps in their learning while expanding their grade level coordinate plane content knowledge. The following activity presents students with an opportunity to transfer their knowledge of identifying scale copies to becoming more precise with determining scale factor when presented with images of maps.