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
In this activity, students use surface area as a context to extend the order of operations to expressions with exponents. The context provides a reason to evaluate the exponent before performing the multiplication.
Launch
Give students 10 minutes of quiet work time, followed by a class discussion.
Reading: MLR6 Three Reads.
Use this routine to support students’ comprehension of the situation. In the first read, students read the text with the goal of comprehending the situation (e.g., Jada and Noah have different solutions for the surface area of the same cube). In the second read, ask students to identify important quantities that can be counted or measured (e.g., the side length of the cube; the number of faces of a cube; the area of each face of the cube). In the third read, reveal the question, “Do you agree with either of them? Explain your reasoning.” Ask students to brainstorm possible strategies to answer the question (e.g., Find the area of each face of the cube. Multiply the area by 6 to find the surface area of the cube). This will help students concentrate on making sense of the situation before rushing to a solution or method.
Design Principle(s): Support sense-making
Student Facing
A cube has side length 10 inches. Jada says the surface area of the cube is 600 in
2
, and Noah says the surface area of the cube is 3,600 in
2
. Here is how each of them reasoned:
Jada’s Method:
\(6 \boldcdot 10^2\)
\(6 \boldcdot 100\)
\(600\)
Noah’s Method:
\(6 \boldcdot 10^2\)
\(60^2\)
\(3,\!600\)
Do you agree with either of them? Explain your reasoning.
Student Response
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Activity Synthesis
In finding the surface area, there is a clear reason to find
\(10^2\)
and then multiply by 6. Tell students that sometimes it is not so clear in which order to evaluate operations. There is an order that we all generally agree on, and when we want something done in a different order, brackets are used to communicate what to do first. When an exponent occurs in the same expression as multiplication or division, we evaluate the exponent first, unless brackets say otherwise. Examples:
\((3\boldcdot 4)^2 = 12^2 = 144\)
, since the brackets tell us to multiply
\(3\boldcdot 4\)
first. But
\(3\boldcdot  4^2 = 3\boldcdot  16 = 48\)
, because since there are no brackets, we evaluate the exponent before multiplying.
If students bring up PEMDAS or another mnemonic for remembering the order of operations, point out that PEMDAS can be misleading in indicating multiplication before division, and addition before subtraction. Discuss the convention that brackets or parentheses indicate that something should be evaluated first, followed by exponents, multiplication, or division (evaluated left to right), and last, addition or subtraction (evaluated left to right).


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