Lesson 4: Rational and Irrational Numbers

Rational and Irrational Numbers

5 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 reviewing multiplication of fractions. It encourages students to think about squaring integers and to rely on the structure of fractions to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students estimate solutions to the equation $x^2=2$ .

For this activity, it is best if students work with fractions and do not convert numbers to their decimal forms. Answers expressed in decimal form aren’t wrong, but working with decimal forms will miss out on the purpose of this Warm-up.

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.

$x^2=36$
$x^2=\frac94$
$x^2=\frac14$
$x^2=\frac{49}{25}$

Sample Response

6. Sample reasoning: $6 \boldcdot 6=36$
$\frac32$ . Sample reasoning: $\frac32 \boldcdot \frac32=\frac94$
$\frac12$ . Sample reasoning: $\frac12 \boldcdot \frac12=\frac14$
$\frac75$ . Sample reasoning: Since $7^2=49$ and $5^2=25$ , we know $(\frac75)^2=\frac{49}{25}$ .

Activity Synthesis (Teacher Notes)

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

Building On

5.NF.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
6.EE.1·Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.1·Write and evaluate numerical expressions involving whole-number exponents.

Building Toward

8.NS.2·Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
8.NS.A.2·Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., $\pi^2$). For example, by truncating the decimal expansion of $\sqrt{2}$, show that $\sqrt{2}$ is between $1$ and $2$, then between $1.4$ and $1.5$, and explain how to continue on to get better approximations.

10 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Rational and Irrational Numbers

5 min

Teacher Prep

Setup

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

Narrative

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.

$x^2=36$
$x^2=\frac94$
$x^2=\frac14$
$x^2=\frac{49}{25}$

Sample Response

6. Sample reasoning: $6 \boldcdot 6=36$
$\frac32$ . Sample reasoning: $\frac32 \boldcdot \frac32=\frac94$
$\frac12$ . Sample reasoning: $\frac12 \boldcdot \frac12=\frac14$
$\frac75$ . Sample reasoning: Since $7^2=49$ and $5^2=25$ , we know $(\frac75)^2=\frac{49}{25}$ .

Activity Synthesis (Teacher Notes)

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

Building On

5.NF.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.4·Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
6.EE.1·Write and evaluate numerical expressions involving whole-number exponents.
6.EE.A.1·Write and evaluate numerical expressions involving whole-number exponents.

Building Toward

8.NS.2·Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
8.NS.A.2·Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., $\pi^2$). For example, by truncating the decimal expansion of $\sqrt{2}$, show that $\sqrt{2}$ is between $1$ and $2$, then between $1.4$ and $1.5$, and explain how to continue on to get better approximations.

Rational and Irrational Numbers

Warm-upMath Talk: Positive Solutions5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityThree Squares10 minKCs

ActivityLooking for a Solution10 minKCs

ActivityLooking for $\sqrt2$10 minKCs

Knowledge Components

Rational and Irrational Numbers

Warm-upMath Talk: Positive Solutions5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityThree Squares10 minKCs

ActivityLooking for a Solution10 minKCs

ActivityLooking for $\sqrt2$10 minKCs

5 min

10 min

10 min

10 min

5 min

10 min

10 min

10 min