This Warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three expressions that go together and can explain why. Next, tell students to share their response with their group and then together find as many sets of three as they can.
Student Lesson in Spanish
Activity
None
Which three go together? Why do they go together?
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “numbers,” “digits,” or “exponents,” and to clarify their reasoning as needed. Consider asking:
“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”
13.2
Activity
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
The purpose of this activity is to give students opportunities to look for and make use of structure to compare exponential expressions and determine if statements about them are true (MP7). These problems can also be reasoned by evaluating the expressions, which is fine at this point, students should also look for other ways to reason about each statement.
Monitor for students who use these different strategies:
Find the value of each expression (for instance, calculate both and , and see that they both have a value of 125).
Rewrite an expression in an equivalent form to compare structure on both sides of an equation (for instance, rewrite as .
Reason about the meaning of the operations on both sides of an equation (for instance, reason that means .)
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
Arrange students in groups of 2. Give students 2–3 minutes of quiet work time, followed by time to discuss their thinking with a partner before completing the remainder of the activity.
Select students with different strategies, such as those described in the Activity Narrative, to share later. If most students are evaluating every given expression, consider asking: “Are there ways to compare the expressions on the two sides of the equal sign without calculating their values?
Representation: Internalize Comprehension. Begin with the demonstration as described in the Launch to support connections between new situations that involve evaluating exponential expressions and prior understandings. Use color or annotations to highlight what changes and what stays the same at each step. Supports accessibility for: Conceptual processing, Visual-Spatial Processing
Activity
None
Decide whether each equation is true or false. Explain or show how you know.
Activity Synthesis
The goal of this discussion is to draw students' attention to the structure of expressions and the meaning of exponent notation. Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach, then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
“What do these approaches have in common? How are they different?”
“How does repeated multiplication show up in each method?”
“Are there any benefits or drawbacks to one approach compared to another?”
13.3
Activity
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
In this partner activity, students take turns explaining why they think two expressions are equivalent. The reasoning here allows students to practice looking for and making use of structure in analyzing exponential expressions (MP7). As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
Launch
Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms, or invite a student to read them out loud. Tell students that during this activity, they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how the norms supported the mathematical community during the activity.
Arrange students in groups of 2. Display the task for all to see. Tell students that for each group of 4 expressions, there are two equivalent expressions, and maybe more. If time allows, choose a student to be your partner and demonstrate how to set up and do the activity, otherwise share these steps:
One partner identifies two equivalent expressions from a group of four expressions and explains why they think those expressions are equivalent.
The other partner listens to the speaker’s reasoning and makes sure it makes sense.
If there is a disagreement, the partners discuss until coming to an agreement.
Both partners look for any other expressions in the group that are also equivalent and explain their reasoning.
For the next group of expressions, the students swap roles.
Remind students that we can often tell if two expressions are equivalent by looking for structure in the expressions and applying the meaning of exponents. It is not always necessary to calculate the values of the expressions.
Action and Expression: Develop Expression and Communication. T To help get students started, display sentence frames, such as “ and are equivalent because . . . .” and “I agree/disagree because . . . .” Supports accessibility for: Language, Organization
Activity
None
Take turns with your partner to find two equivalent expressions in each list.
For each two that you find, explain to your partner how you know they are equivalent.
For each two that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
Invite students to share their responses and strategies for finding equivalent expressions. Highlight strategies that rely on the structure of the expressions. As students respond, record the equivalent expressions using an equal sign.
Consider asking students:
“Were there any expressions that you and your partner disagreed about and needed to discuss to come to an agreement? If so, which ones?”
“Is it possible to figure out the expressions that are equivalent in each set without calculating the value of any of the expressions?”
Math Community
Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:
“I noticed our norm ‘’ in action today, and it really helped me/my group because .”
Lesson Synthesis
The purpose of this discussion is to analyze some of the typical errors and misconceptions that often come up when working with exponent notation. Based on student responses and discussions in the activities, consider asking questions, such as:
“Are and equivalent? How do you know?” (No, we can try different values of and or use the meaning of exponents to see that multiplied times is not always the same as multiplied times.)
“What change can we make to the equation to make it true?” (Change addition to multiplication on the left, or change the exponent to multiply by 5 on the right.)
“Your friend claims the equation is true. What do you think they are misunderstanding? How can you convince them it is false?” (They likely are thinking that multiplying four of a number is the same as multiplying that number by 4. I would explain that means , rather than . One expression has a value of 2, while the other is . I could show similar examples with other numbers, but the best way to convince them is to talk about what exponents and multiplication mean.)
“Can we show that is true without calculating the value of or ? How?” (Yes. means or , which also equals or . Another way to write is . We know we can sometimes write a number as its factors, rearrange the factors, and multiply them in any order.)
Student Lesson Summary
When adding or multiplying numbers, the order of the numbers doesn’t affect the result of addition or multiplication. When working with exponents, each number means something specific, so its placement does matter.
equals .
always equals .
equals .
always equals .
does not equal .
does not always equal .
means , while means .
It is also important to remember that we use multiplication as a quicker way to express repeated addition and we use exponent as a quicker way to express repeated multiplication.
When working with exponents, the numbers being multiplied don’t have to always be whole numbers. They can also be other kinds of numbers, like fractions, decimals, and even variables. For example, we can use exponents in each of the following ways:
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
