In this Warm-up, students take two numbers to different powers and look for patterns. The first number is a whole number, 3, and the second is its reciprocal, . The goal is for students to notice that when a fraction is raised to a positive exponent, its value decreases as the exponent increases. Aside from the presence of exponents, these observations are largely a review of work from grade 5.
As students complete the table, monitor for those who can describe some of the following patterns:
The values in the column increase as the exponent increases.
The values in the column decrease as the exponent increases.
The values in the column are reciprocals of the values in the corresponding row of the column.
Student Lesson in Spanish
Launch
Give students 2 minutes of quiet work time, followed by a whole-class discussion.
Activity
None
Student Task Statement
Find the values of and for each value of . What patterns do you notice?
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2
3
4
Student Response
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Building on Student Thinking
Activity Synthesis
Display the table for all to see. Ask students to share their responses and record them in the table. Ask selected students to share the patterns they noticed in the table and ask others to explain why they think these patterns happen. If the ideas described in the Student Response do not arise from students during this discussion, bring those ideas to students’ attention.
15.2
Activity
10 mins
What's the Value?
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
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). For example, use the formulas and to find the volume and surface area of a cube with sides of length .
In this activity, students continue working with exponential expressions with variables. They evaluate expressions when given a value for the variable. As students discuss and compare their thinking with others, they also deepen their understanding of the order of operations, such as when considering the difference between expressions like and . During class discussion, students also have opportunities to practice using mathematical vocabulary related to expressions, such as “coefficient,” “variable,” and “exponent.”
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Reiterate that the expression means to multiply 6 by the result of . Remind students that the number part of such a product is called the coefficient of the expression, so in this example, 6 is the coefficient of .
Give students 5 minutes of quiet work time to evaluate the expressions. Follow with a whole-class discussion.
Use Collect and Display to direct attention to words collected and displayed from earlier lessons. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. After students have evaluated the first 2–3 expressions, check in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing. Supports accessibility for: Organization, Attention
Activity Synthesis
The purpose of the discussion is to ensure that students understand how to evaluate exponential expressions with variables for a given value of the variable. It is also an opportunity for students to practice interpreting and using vocabulary like “coefficient,” “variable,” “power,” and “exponent.”
Direct students’ attention to the reference created using Collect and Display. Ask students to share the steps they used to find the value of an expression. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional words or phrases.
If more than a few students multiply the coefficient and the variable in and then square the product, which results in a value of 900 when is 10 and when is , discuss the difference between and and how each expression should be evaluated based on the order of operations.
If time permits, consider asking questions such as:
"In each expression, what is the coefficient?" (3, 3, 3, , 1, 1.)
"How is evaluating the expressions when is a fraction similar to when is a whole number? How is it different?" (It's similar because we're still just multiplying by itself a certain number of times. It's different because multiplying a fraction by a fraction is a bit more complicated than multiplying a whole number by itself.)
15.3
Activity
15 mins
Exponent Experimentation
Standards Alignment
Building On
Addressing
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). For example, use the formulas and to find the volume and surface area of a cube with sides of length .
In this activity, students recall what is meant by a solution to an equation as they look to replace a variable with a number that makes two expressions equal. Here, one or both expressions in each equation contains an exponent.
Students are not expected to use algebraic methods to solve equations such as or in grade 6. Instead, they are to look for and make use of the structure to strategically choose and test values that could be solutions from a list, or by reasoning about the meaning of each expression. For example, to find the solution for , they may think: “What number, when squared, will give 64?”
Note that some of the equations also have solutions that are negative. Because operations on negative numbers are not part of grade 6 standards, students are only expected to consider positive values in this activity.
Launch
Ask students to close their books or devices. Display the equation and the values 2, 10, and 50. Discuss what it would mean for 2, 10, or 50 to be a solution to the equation. As needed, remind students that a solution is a value for that makes the equation true. Discuss why 10 is a solution, and why 2 or 50 are not solutions.
Arrange students in groups of 2. Give students 5 minutes of quiet work time and then time to discuss their responses with a partner, followed by a whole-class discussion.
Representation: Internalize Comprehension. Activate or supply background knowledge about exponential expressions and finding a solution to an equation. Allow students to use calculators to ensure inclusive participation in the activity. Supports accessibility for: Memory, Conceptual Processing
Activity Synthesis
The goal of the discussion should focus on how the meaning of the equal sign, exponent, and solution to an equation can help us find a value that makes each equation true.
Invite students to share how they found the solution to each equation. Discuss questions such as:
“What was your strategy when is by itself on one side of the equation?”
“What was your strategy when is the number being multiplied or raised to a power?”
“What was your strategy when x was the exponent?”
“Did you reason about the last equation the same way you did with the first equation? What did you do differently, if anything?”
Lesson Synthesis
Summarize the key ideas from the lesson by asking questions such as:
“In this lesson, we saw expressions such as , , and . How are these different from those in earlier lessons?” (They involve both a variable and an exponent.)
“If the value of is 2, what is the value of and ? (9 and )
“If we use a greater value of , say 4, 5, or 10, how would the value of each expression change?” (The value of would get larger. The value of would get smaller.)
“If is 2, to find the value of , do we multiply 3 and 2 first and then raise the product to the power of 4, or do we find first, and then multiply the result by 3?” (The latter.) “Why?” (Following the order of operations, we compute the expression with an exponent first before multiplying.)
“Which number is a solution to ? How do you know?” (, because it is the value that would make the equation true.)
Student Lesson Summary
We can find the value of expressions with an exponent and a variable for different values of the variable. For example:
To find the value of the expression when is 5, we replace the variable with 5 to get .
This is equal to , or just 250.
So, the value of is 250 when is 5.
To find the value of when is 4, we replace the variable with 4 to get , which equals 2.
So, has a value of 2 when is 4.
Equations may also have an exponent and a variable. We can find out what value of the variable would make such an equation true.
Suppose we have an equation and a list of possible solutions: .
The only value of that makes the equation true is 2 because , which equals 90.
So, 2 is the solution to the equation, which we can express as .
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
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). For example, use the formulas and to find the volume and surface area of a cube with sides of length .
Find the value of each expression for the given value of .
when is 10
when is
when is 4
when is
when is 1
when is
Student Response
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Building on Student Thinking
Students may use the wrong order when evaluating expressions, such as in multiplying the given value of by 3 first in and then squaring. Ask students to explain their calculations. Remind them of the conventional order of operations in numeric expressions. Explain that the same order applies to expressions with variables.
Activity
None
Student Task Statement
Find a solution to each equation in the list. (Numbers in the list may be a solution to more than one equation, and not all numbers in the list will be used.)
List:
1
2
3
4
5
6
8
Student Response
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Building on Student Thinking
Building Toward
6.EE.B.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
