This Math Talk focuses on the meaning of numbers and symbols in expressions. It encourages students to relate repeated addition to multiplication and to relate repeated multiplication to exponents. The numbers are selected to discourage students from computing the values of the expressions. Instead, they prompt students to rely on what they know about operations and exponents to mentally make comparisons. The understanding elicited here will be helpful later in the lesson when students write expressions to represent the surface area and volume of cubes.
To decide, without calculations, which of two expressions has the greater value, students need to look for and make use of structure (MP7). In explaining their reasoning, students need to be precise in their word choice and use of language (MP6).
Student Lesson in Spanish
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies and record and display their responses for all to see.
Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Decide mentally which expression has a greater value.
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Student Response
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Building on Student Thinking
When given an expression with an exponent, students may misinterpret the base and the exponent as factors and multiply the two numbers. Remind them about the meaning of the exponent notation. For example, show that = 15, which is much smaller than , which equals 125.
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
To support students in upcoming work, highlight the following ideas if they are not already mentioned by students:
We can express repeated addition with multiplication. is a more concise way to write .
We can express repeated multiplication with an exponent. is a more concise way to write .
The parentheses in the last expression tells us that it is the value of , not just one of the numbers, that is being squared.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
Math Community
Display the Math Community Chart and a list of 2–5 revisions suggested by the class in the previous exercise for all to see. Remind students that norms are agreements that everyone in the class shares responsibility for, so everyone needs to understand and agree to work on upholding the norms. Briefly discuss any revisions and make changes to the “Norms” sections of the chart as the class agrees. Depending on the level of agreement or disagreement, it may not be possible to discuss all suggested revisions at this time. If that happens, plan to discuss the remaining suggestions over the next few lessons.
Tell students that the class now has an initial list of norms or “hopes” for how the classroom math community will work together throughout the school year. This list is just a start, and over the year it will be revised and improved as students in the class learn more about each other and about themselves and math learners.
18.2
Activity
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression as a product of two factors; view as both a single entity and a sum of two terms.
This activity serves two goals. One goal is to allow students to practice drawing a net and finding the surface area and volume of a cube. The other goal is to encourage students to write expressions to represent the surface area and volume of a cube.
In the first set of questions, the edge length of the cube is a small number (5), enabling students to compute the surface area and volume numerically. Use students’ work here to check that they are drawing a net correctly. In the next set of questions, the edge length of the cube is a larger number (17), making computation more cumbersome and prompting students to build expressions rather than evaluating them.
As students work on the questions about a cube with a 17-inch edge length, monitor for students who write different expressions for the second question. Here are some expressions they may write, from longer (more expanded) to shorter (more succinct):
Products, such as , or
Sums of products, such as
Combination of like terms, such as
Exponents, such as , or
Completed calculations, such as 1,734 or 4,913
A note about notation:
In a later unit, students will learn that means the same as . At this point, expect them to write instead of . It is not critical that they understand that a number placed next to a variable (or a number placed next to an expression in parentheses) are being multiplied.
Launch
Arrange students in groups of 2. Give students access to their geometry toolkits. Tell students to use graph paper to draw a net of the first cube and to try to answer the questions without using a calculator. Give students 8–10 minutes of quiet work time followed by 1–2 minutes to share their responses with their partner.
Select students with different expressions, such as those described in the activity narrative, to share later.
Representation: Develop Language and Symbols. Activate or supply background knowledge. To help students recall the meanings of the terms “net,” “surface area,” and “volume,” ask, “How is a net related to a three-dimensional shape?”, “What parts of a shape contribute to its surface area?”, or “How are volume and surface area different?” Supports accessibility for: Memory, Language
Activity
None
A cube has an edge length of 5 units.
Draw a net for this cube on graph paper. Label its sides with measurements.
What is the shape of each face?
What is the area of each face?
What is the surface area of this cube?
What is the volume of this cube?
A second cube has an edge length of 17 units.
Sketch a net for this cube. Label its sides with measurements.
Explain why the area of each face of this cube is square units.
Write an expression for the surface area, in square units.
Write an expression for the volume, in cubic units.
Activity Synthesis
The purpose of this discussion is to emphasize that exponents can be used to concisely express calculations for surface area and volume. Briefly discuss the answers to the questions about the cube with a 5-inch edge length. Then, focus the discussion on the larger cube with a 17-inch edge length.
Invite previously selected students to share their expressions for the last two questions. Sequence the responses in the following order to help students see how the expressions and come about. If any expressions are missing but are needed to illustrate the idea of writing succinct expressions, add them to the lists. Refer to parts of expressions using terms such as “sum,” “product,” and “factor” to support students in using mathematical terms when working with expressions.
Surface area:
1,734
Volume:
4,913
Connect the expressions to the learning goals by asking questions such as:
“In each expression, where do you see the area of one face of the cube?” (, , 289)
“In one expression, the number 17 is written twelve times. In another expression, it is written six times. How do they both represent the same surface area?” (They both show the sum of the areas of the six square faces. The second expression uses the exponent 2 to represent the product of two 17s.)
“Some expressions show addition but others don’t. How do they all represent the surface area?” (Multiplication is used for repeated addition. Adding six of the same expression is the same as multiplying it by 6.)
“Which expression doesn't require a lot of computation and is the most efficient way to represent the surface area of the cube?” ()
“Which expression doesn't require a lot of computation and is the most efficient way to represent the volume of the cube?” ()
To further encourage students to see regularity in the expressions, consider offering another example, for instance: “Suppose the edge length of a cube is 38 cm. How can we express its surface area and volume?” ( cm2 and cm3, respectively) Using a large number for the edge length will discourage calculation and prompt students to focus on building an expression and using exponents.
If time permits, consider directing students’ attention to the units of measurements. Remind students that, rather than writing square units, we can write units2, and instead of cubic units, we can write units3. Unit notations will appear again later in the course, so it can also be reinforced later.
18.3
Activity
Standards Alignment
Building On
Addressing
6.EE.A.2.a
Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract from 5” as .
In this activity, students develop the formulas for the surface area and the volume of a cube in terms of a variable edge length .
Students should be encouraged to refer to their work in the preceding activity as much as possible and to generalize from it. As before, monitor for different ways of writing expressions for surface area and volume. The expressions are likely to be in the following forms, listed from longer (more expanded) to shorter (more succinct):
Products, such as or
Sums of products, such as
Combination of like terms, such as
Expressions with exponents, such as , or
As they write and discuss expressions for volume and surface area, students practice looking for and making use of structure (MP7) and seeing regularity through repeated reasoning (MP8).
Launch
Give students access to their geometry toolkits and 7–8 minutes of quiet think time. Tell students that they will be answering the same questions as before, but with a variable for the side length. Encourage them to use the work they did earlier to help them here.
Select students who wrote different expressions to share their work later.
Activity
None
A cube has an edge length of .
Draw a net for the cube.
Write an expression for the area of each face. Label each face with its area.
Write an expression for the surface area.
Write an expression for the volume.
Student Response
Activity Synthesis
To support students in seeing structure and generalizing, discuss the problems in as similar a fashion as was done in the earlier activity involving a cube with an edge length of 17 units.
Ask previously selected students to share their responses with the class. If possible, sequence the responses in the following order (as shown in the activity narrative) to help students see how the expressions and come about. If any expressions are missing but are needed to illustrate the idea of writing succinct expressions, add them to the lists. Refer to parts of expressions using terms such as “sum,” “product,” and “factor” to support students in using mathematical terms when working with expressions.
Surface area:
or
Volume
If students have trouble understanding where the most concise expression of surface area comes from, refer back to the example involving a numerical side length (a cube with an edge length of 17 units).
Present the surface area as .
Lesson Synthesis
In this lesson, students wrote expressions for the volume and surface area of a cube. Consider asking students:
"A cube has an edge length of . Why does the expression describe its volume?" (The volume is , which can be written as
"Why does the expression describe its surface area?" (A cube has 6 faces that are all identical squares. Each face has an area of or , so the total area for all the faces is 6 times .).
A note about materials for an upcoming unit:
For the first lesson on the unit on ratios, students will need to bring in a personal collection of 10–50 small objects. Examples include rocks, seashells, trading cards, or coins. Inform or remind students about this.
Student Lesson Summary
The volume of a cube with an edge length of is .
A cube has 6 faces that are all identical squares. For a cube with an edge length of , the area of each square face is . This means that the surface area of the cube is .
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
Students might think the surface area is . Prompt students to write down how they would compute surface area step by step, before trying to encapsulate their steps in an expression. Dissuade students from using calculators in the last two problems and assure them that building an expression does not require extensive computation.
Students may think that refraining from using a calculator means performing all calculations—including those of larger numbers—on paper or mentally, especially if they are unclear about the meaning of the term “expression.” Ask them to refer to the expressions in the Warm-up, or share examples of expressions in a few different forms, to help them see how surface area and volume can be expressed without computation.
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Building on Student Thinking
If students are unclear or unsure about using the variable , explain that we are looking for an expression that would work for any edge length, and that a variable, such as , can represent any number. The could be replaced with any edge length in finding surface area and volume.
To connect students’ work to earlier examples, point to the cube with edge length 17 units from the previous activity. Ask: “If you wrote the surface area as before, what should it be now?”
As students work, encourage those who may be more comfortable using multiplication symbols to instead use exponents whenever possible.
6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression as a product of two factors; view as both a single entity and a sum of two terms.
