Students extend their understanding from the previous lessons to recognize the structure of a linear equation for all possible types of solutions: one solution, no solution, or infinitely many solutions. Students are still using language such as “true for one value of ,” “always true” or “true for any value of ,” and “never true.” Students should be able to articulate that this depends both on the coefficient of the variable and on the constant term on each side of the equation.
Student Lesson in Spanish
Launch
Give students 2–3 minutes of quiet think time followed by a whole-class discussion.
Activity
None
Student Task Statement
Match each equation with the number of values that solve the equation.
true for only 1 value
true for no values
true for any value
Student Response
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Building on Student Thinking
Activity Synthesis
In order to highlight the structure of these equations, ask students:
“What do you notice about equations that are true for no values?” (These equations have equal or equivalent coefficients for the variable, but unequal values for the constants on each side of the equation.)
“What do you notice about equations that are true for all values?” (These equations have equivalent expressions on each side of the equation, so the coefficients are equal and the constants are equal or equivalent on each side.)
“What do you notice about equations that have exactly one value that makes them true?” (These equations have different values for the coefficients on each side of the equation and it doesn’t matter what the constant term says.)
Display the equation for all to see. Ask students how this fits with their explanations. (We can see that there is one solution. Another way to think of this is that the coefficient of is 1 on the left side of the equation, and the coefficient of is 0 on the right side of the equation. So the coefficients of are different, just as the explanation states.)
11.2
Activity
15 mins
Card Sort: Solutions
Standards Alignment
Building On
Addressing
8.EE.7.a
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
Students sort different equations during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).
Monitor for the different ways in which groups choose to categorize the equations, but especially for categories that distinguish between the number of solutions.
Students should look for ways to sort the equations without solving them. In particular, they should pay attention to the coefficients of the variable term as well as any constant terms.
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2 and distribute pre-cut cards. Allow students to familiarize themselves with the representations on the cards:
Give students 1 minute to place all the cards face up and to start thinking about possible ways to sort the cards into categories.
Pause the class, and select 1–3 students to share the categories that they identified.
Discuss as many different categories as time allows.
Attend to the language that students use to describe their categories and equations, giving them opportunities to describe their equations more precisely. Highlight the use of terms like “solution,” “coefficient,” and “term.” After a brief discussion, invite students to continue with the activity.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with, and introduce the remaining cards after students have completed their initial set of matches. Supports accessibility for: Conceptual Processing, Organization, Memory
Activity
None
Student Task Statement
Your teacher will give you a set of cards containing equations.
Sort the cards into categories of your choosing.
Describe the defining characteristics of the categories, and be prepared to share your reasoning with the class.
Student Response
Activity Synthesis
Select 2–3 groups to share one of their categories’ defining characteristics and which equations they sorted into that category. Given the Warm-up, categories based on the number of values that make the equations true labeled “one value,” “no values,” or “any values” are likely. Introduce students to the language “infinite number of solutions” if it has not already come up in discussion.
Ask students if anyone noticed a way to categorize the cards without completely solving them. If not, ask students to look for what the equations have in common when the equations got to the point where they looked like .
During the discussion, it is likely that students will want to refer to specific parts of an expression. Encourage students to use the words “coefficient” and “variable.” Define the word constant term as the term in an expression that doesn't change, or the term that does not have a variable part. For example, in the expression the 2 is the coefficient of , the 3 is a constant, and is the variable.
MLR8 Discussion Supports. Display sentence frames to support whole-class discussion: “We grouped these equations into the category because .” and “Some characteristics of this category are .” Invite students to press for details by asking clarifying questions such as, “What other features do those equations have in common?” and “Can you explain how to create a different equation that would fall into that category?” Advances: Conversing, Representing
11.3
Activity
20 mins
Make Use of Structure
Standards Alignment
Building On
Addressing
8.EE.7.a
Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
The purpose of this activity is to give students an opportunity to compare the structure of equations that have no solution, one solution, and infinitely many solutions. This may be particularly useful for students who need more practice identifying which equations will have each of these types of solutions before they attempt to solve the equations.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet think time to work on the problems, followed by 2–3 minutes of partner discussion to work together. Follow with a whole-class discussion.
Tell partners to divide the sets of questions so that they each solve 2 sets on their own and 1 set together, and then to discuss the last question together.
Activity
None
Student Task Statement
For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of (and has infinitely many solutions). If an equation has one solution, solve the equation to find the value of that makes the statement true.
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What do you notice about equations with one solution? How is this different from equations with no solutions and equations that are true for every ?
Activity Synthesis
Briefly review each of the equations and how many solutions it has. If students disagree, ask them to explain their thinking about the equation and work to reach agreement.
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to what they notice about equations with 1, 0, or infinitely many solutions. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
If time allows, display these prompts for feedback:
“ makes sense, but what do you mean when you say. . . ?”
“Can you describe that another way?”
“How do you know . . . ? What else do you know is true?”
Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words that they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
After Stronger and Clearer Each Time, make clear for students how the form of the expressions on each side of the equation can help make it clear to identify the number of solutions. In particular, it is helpful to rewrite the original equation into an equivalent equation of the form . For example, it is not as easy to see how many solutions there are for the equation . Rewriting the equation using the distributive property and combining like terms on each side, we can get and can more easily see that there are no solutions to this equation.
Students should notice that, when the original equation is rewritten into an equivalent equation of the form ,
There is 1 solution if the coefficients and are not equal.
There is no solution when the coefficients of are equal () and the constant terms are not equal ().
There are infinitely many solutions when the coefficients of are equal () and the constant terms are also equal ().
Lesson Synthesis
Instruct students to write three equations with a variable term and a constant term on each side of the equation. Their equations should be one with no solution, one with infinitely many solutions, and one with exactly one solution. When they think they have three equations that meet these requirements, tell students to trade with a partner, and then identify which equation is each type. Give partners 2–3 minutes to check their solutions and to discuss how they came up with their equations.
Ask students, “How did you know how to make each type of equation?” (I knew that the single-solution equation should have different coefficients for the variable terms, I knew that the many-solution equation should have equivalent expressions on each side, and I knew that the no-solution equation should differ only by a constant term on each side.)
If time allows, consider making a poster for permanent display that shows an equation with coefficients, variables, and constant terms labeled.
Student Lesson Summary
Sometimes it's possible to look at the structure of an equation and tell if it has infinitely many solutions or no solutions. For example, look at
Using the distributive property on the left and right sides, we get
From here, collecting like terms gives us
Without doing any more moves, we know that this equation is true for any value of because the left and right sides of the equation are the same.
Similarly, we can sometimes use structure to tell if an equation has no solutions. For example, look at
If we think about each move as we go, we can stop when we realize there is no solution:
Because the coefficient of is 6 on each side, we know that there is either no solution or infinitely many solutions. The last move makes it clear that the constant terms on each side, 5 and , are not the same. Because adding 5 to an amount is always less than adding to that same amount, we know that there are no solutions.
Doing moves to keep an equation balanced is a powerful part of solving equations, but thinking about what the structure of an equation tells us about the solutions is just as important.
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Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
On the second question, students may think that have the same coefficients of . Ask students to rewrite each side in the form where the term with is first and it is added to the constant term.
