Unit 4: Proportions with Geometric Applications

Scale drawings visually represent proportionality & can build conceptual understanding, which in turn relates to the constant of proportionality k in y = kx.  So starting with scale drawings may help students grasp proportionality as a concept before going deeper.  Alternatively, percent equations can act a transition into this unit as students learn to write them as proportions or saved for last as another application of their learning of proportionality.  Proportionality instruction should include multiple representations with connections between them such as tables, graphs, equations, diagrams, and verbal descriptions to build conceptual understanding (7.RP.2).  Sixth grade ratios compare whole numbers, and seventh grade ratios compare one fraction to another (creating a complex fraction).  Work with fractions in Units 2 & 3 prepare students for unit rates & complex fractions.  Instruction should connect to sixth grade’s reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations (6.RP.3); the difference being the rate of change now includes fractional changes rather than just whole number changes.  Time permitting, depth can be explored through looking at multiple cross sections, which can be proportional either with a scale factor of 1 or otherwise. Drawing, constructing and describing geometric shapes is a chance to reengage with similar figures as well.  Constructing triangles in particular is an opportunity to explore the conditions leading to one triangle, many triangles or no triangles. 

In seventh grade students should become fluent with fractions, take care to avoid jumping to the cross multiplication algorithm without first engaging with fractional tape diagrams and double numberline diagrams.  Scale diagrams can be used to understand proportionality conceptually, and might be a useful way to begin the unit; these too can be extended to fractional side lengths and fractional scale factors.