standard_description: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
problem_activity: Warm-up
Students reflect on equivalent expressions that represent the area of a shaded rectangle which is part of a larger rectangle of unknown width.
Launch
Allow students 2 minutes of quiet work time, followed by a whole-class discussion.
Student Facing
A rectangle with dimensions 6 cm and
\(w\)
cm is partitioned into two smaller rectangles.
Explain why each of these expressions represents the area, in cm
2
, of the shaded region.
\(6w-24\)
\(6(w-4)\)
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Student Response
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Activity Synthesis
Ask students to share their reasoning.
\(6w-24\)
should be straighforward: the area of the entire rectangle is
\(6w\)
, and the area of the unshaded portion is
\(6 \boldcdot 4\)
or 24, so the area of the shaded portion is
\(6w-24\)
.
Students may have more trouble with
\(6(w-4)\)
. The key is to understand that the longer side of the shaded portion can be represented by
\(w-4\)
.
[IMG]


aligns: no