Lesson 5: Reasoning about Equations and Tape Diagrams (Part 2)

Reasoning about Equations and Tape Diagrams (Part 2)

10 min

Teacher Prep

Setup

Display one problem at a time. Allow 30 seconds of quiet think time, followed by a whole-class discussion.

Narrative

This Math Talk focuses on analyzing a specific equation structure, $p(x+q)=r$ , with an eye toward options for solving such equations. It encourages students to notice when $r$ is divisible by $p$ , and to rely on the structure of expressions in this form to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students solve equations of this form.

To notice that each equation is a template where just two numbers vary, and therefore all the equations have the same solution, students need to look for and make use of structure (MP7).

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 access to sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Solve each equation mentally.

$x +1 = 5$
$2(x+1) = 10$
$3(x+1) = 15$
$500 = 100(x+1)$

Sample Response

$x=4$ . Sample reasoning: $5-1=4$ .
$x=4$ . Sample reasoning: When $x$ is 1, $2(x+1)=4$ . When $x$ is 2, $2(x+1)=6$ . When $x$ is 3, $2(x+1)=8$ . When $x$ is 4, $2(x+1)=10$ .
$x=4$ . Sample reasoning: 15 equals 3 times 5. So, $(x+1)$ must equal 5.
$x=4$ . Sample reasoning: 500 equals 100 times 5. So, $(x+1)$ must equal 5.

Activity Synthesis (Teacher Notes)

Ask students to identify what all the equations have in common. If it doesn’t come up, point out that $x+1$ must equal 5 in every equation. Therefore, the solution to each equation is $x=4$ .

To involve more students in the conversation, consider asking:

“Who can restate $\underline{\hspace{.5in}}$ ’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 $\underline{\hspace{.5in}}$ ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I

\underline{\hspace{.5in}}

because . . . .” or “I noticed

\underline{\hspace{.5in}}

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

Standards

Addressing

7.EE.4·Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.4.a·Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.B.4·Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.B.4.a·Solve word problems leading to equations of the form $px + q = r$ and $p(x + q) = r$, where $p$, $q$, and $r$ are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is $54$ cm. Its length is $6$ cm. What is its width?

15 min

10 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Reasoning about Equations and Tape Diagrams (Part 2)

10 min

Teacher Prep

Setup

Display one problem at a time. Allow 30 seconds of quiet think time, followed by a whole-class discussion.

Narrative

To notice that each equation is a template where just two numbers vary, and therefore all the equations have the same solution, students need to look for and make use of structure (MP7).

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.

Student Task

Solve each equation mentally.

$x +1 = 5$
$2(x+1) = 10$
$3(x+1) = 15$
$500 = 100(x+1)$

Sample Response

$x=4$ . Sample reasoning: $5-1=4$ .
$x=4$ . Sample reasoning: When $x$ is 1, $2(x+1)=4$ . When $x$ is 2, $2(x+1)=6$ . When $x$ is 3, $2(x+1)=8$ . When $x$ is 4, $2(x+1)=10$ .
$x=4$ . Sample reasoning: 15 equals 3 times 5. So, $(x+1)$ must equal 5.
$x=4$ . Sample reasoning: 500 equals 100 times 5. So, $(x+1)$ must equal 5.

Activity Synthesis (Teacher Notes)

Ask students to identify what all the equations have in common. If it doesn’t come up, point out that $x+1$ must equal 5 in every equation. Therefore, the solution to each equation is $x=4$ .

To involve more students in the conversation, consider asking:

“Who can restate $\underline{\hspace{.5in}}$ ’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 $\underline{\hspace{.5in}}$ ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I

\underline{\hspace{.5in}}

because . . . .” or “I noticed

\underline{\hspace{.5in}}

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

Standards

Addressing

7.EE.4·Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.4.a·Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.B.4·Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
7.EE.B.4.a·Solve word problems leading to equations of the form $px + q = r$ and $p(x + q) = r$, where $p$, $q$, and $r$ are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is $54$ cm. Its length is $6$ cm. What is its width?

Reasoning about Equations and Tape Diagrams (Part 2)

Warm-upMath Talk: Seeing Structure10 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityMore Situations and Diagrams15 minKCs

ActivityMore Situations, Diagrams, and Equations10 minKCs

Knowledge Components

Reasoning about Equations and Tape Diagrams (Part 2)

Warm-upMath Talk: Seeing Structure10 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityMore Situations and Diagrams15 minKCs

ActivityMore Situations, Diagrams, and Equations10 minKCs

10 min

15 min

10 min

10 min

15 min

10 min