Evaluating Expressions with Exponents

10 min

Narrative

The purpose of this Warm-up is for students to recall previous understandings of area, volume, and surface area of cubes, and how to record these measurements as expressions with exponents. Given the side length of a square and the edge length of a cube, students are prompted to describe other measurements that could be determined. Students might respond with either verbal or numerical descriptions, saying, for example, “We can find the area of the square,” or “The area of the square is 9 square units.”

Launch

Arrange students in groups of 2. Give students 2 minutes to read the problem and discuss it with their partner.  After students share their responses, display the following table for all to see and give students time to discuss the information with a partner.

side length of the square area of the square volume of the cube surface area of the cube
as a number 3
as an expression using an exponent 313^1

Give students 1 minute of quiet work time to complete as much of the table as they can independently. Then ask them to discuss their responses with their partner and complete the rest of the table.

Student Task

Based on the given information, what other measurements of the square and cube could we find?

A square with a side length of three units. A cube with a side length of three units.

Sample Response

Sample responses:

  • We can find the area of the square.
  • The area of the square is 9 square units.
  • The perimeter is 12 units.
  • We can find the volume of the cube.
  • The volume of the cube is 27 cubic units.
  • The surface area is 54 square units.
side length of the square area of the square volume of the cube surface area of the cube
as a number 3 9 27 54
as an expression using an exponent 313^1 323^2 333^3 6(32)6(3^2)

Synthesis

Ask students to share their responses for the first row in the table and their reasoning. Record and display the responses for all to see. Clarify their answers with questions such as:

  • “What calculation did you do to arrive at that answer?"
  • "Where are those measurements in the image?”

Then invite students to share their responses for the second row in the table. Ask questions such as:

  • “How did you decide on the exponent for your answer?"
  • "Where are those measurements in the image?”

In the next activity, students will analyze calculations of the surface area of a cube. If the reason that 6(32)6(3^2) expresses the surface area of the cube is not yet discussed, ask students:

  • “Where did the 323^2 come from?” (It's the area of one face of the cube.) 
  • “Why are we multiplying by 6?” (The cube has 6 faces, and the surface area of the cube is the total of those 6 areas. Multiplying 323^2 by 6 is the same as adding six 323^2s.) 
Standards
Addressing
  • 6.EE.1·Write and evaluate numerical expressions involving whole-number exponents.
  • 6.EE.2.c·Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). <em>For example, use the formulas V = s³ and A = 6 s² to find the volume and surface area of a cube with sides of length s = 1/2.</em>
  • 6.EE.A.1·Write and evaluate numerical expressions involving whole-number exponents.
  • 6.EE.A.2.c·Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). <span>For example, use the formulas <span class="math">\(V = s^3\)</span> and <span class="math">\(A = 6 s^2\)</span> to find the volume and surface area of a cube with sides of length <span class="math">\(s = 1/2\)</span>.</span>

10 min

15 min