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: Activity
One purpose of this activity is to practice seeing the total length of a segment as the sum of its pieces, and using the whole side length of a rectangle to express its area. Another is to generate examples of equivalent expressions, and understand why they are equivalent based on an understanding of area and the distributive property.
Monitor for students who write different but equivalent expressions to represent the area of a rectangle. For example,
\(2(t+5+t)\)
,
\(2t+10+2t\)
, and
\(4t+10\)
for figure B.
Student Facing
Complete the table with the length, width, and area of each rectangle.
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "rectangle"}, {"_attributes": {"scope": "col"}, "_value": "length (units)"}, {"_attributes": {"scope": "col"}, "_value": "width (units)"}, {"_attributes": {"scope": "col"}, "_value": "area (square units)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "A"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(a+4\\)"}]}, {}, {}]}, {"td": [{"_value": "B"}, {}, {"_value": "2"}, {}]}, {"td": [{"_value": "C"}, {}, {}, {}]}, {"td": [{"_value": "D"}, {}, {}, {}]}, {"td": [{"_value": "E"}, {}, {}, {}]}]}]}]}
Student Response
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Activity Synthesis
Invite previously-selected students to share different but equivalent expressions for length and area. (Reassure students that it doesn’t matter which side is considered the length or the width.) If any students express confusion over a response, encourage the student who came up with the response to explain. For example, a student might wonder why
\(a+a+a\)
can be expressed as
\(3a\)
.
Display the table from the task statement for all to see, and write any equivalent expressions that are mentioned next to each other. For example, for the area of figure A, you might write
\(5(a+4)\)
,
\((4+a) \boldcdot 5\)
, and
\(5a+20\)
. Show in the figure how you can see
\(5(a+4)\)
(the product of one side and the other whole side) and
\(5a+20\)
(the area of each smaller rectangle, added up). Emphasize that all of these expressions correctly represent the area of the whole figure, and highlight use of the distributive property where it occurs.


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