Lesson 18: Scaling Two Dimensions

Scaling Two Dimensions

5 min

Teacher Prep

Setup

Groups of 2. 1–2 minutes of quiet work time followed by a partner discussion, then a whole-class discussion.

Narrative

The purpose of this Warm-up is for students to explore how scaling the addends or factors in an expression affects their sum or product. Students determine which statements are true and then create one statement of their own that is true. This will prepare students to see structure in the equations they will encounter in the lesson. Identify students who:

Choose the correct statements (b, c).
Pick numbers to test the validity of statements.
Use algebraic structure to show that the statements are true.

Ask these students to share during the discussion.

Launch

Arrange students in groups of 2. Give students 1–2 minutes of quiet work time followed by time to discuss their chosen statements with their partner. Follow with a whole-class discussion.

Student Task

$m$ , $n$ , $a$ , $b$ , and $c$ all represent positive integers. Consider these two equations:

$\displaystyle m=a+b+c$

$\displaystyle n=abc$

Which of these statements are true? Select all that apply.
1. If $a$ is tripled, $m$ is tripled.
2. If $a$ , $b$ , and $c$ are all tripled, then $m$ is tripled.
3. If $a$ is tripled, $n$ is tripled.
4. If $a$ , $b$ , and $c$ are all tripled, then $n$ is tripled.
Create a true statement of your own about one of the equations.

Sample Response

b, c
Sample response: If $a$ , $b$ , and $c$ are all tripled, then $n$ is 27 times as large. When $a$ , $b$ , and $c$ are tripled, the result is $3a\boldcdot 3b\boldcdot 3c$ , which can be written as $(3\boldcdot 3\boldcdot 3)abc$ or $27abc$ .

Activity Synthesis (Teacher Notes)

Ask previously identified students to share their reasoning about which statements are true (or not true). Display any examples (or counterexamples) for all to see, and ask students to refer to them while sharing. If using the algebraic structure is not brought up in students’ explanations, display for all to see:

If $a$ , $b$ , and $c$ are all tripled, the expression becomes $3a +3b+3c$ , which can be written as $3(a+b+c)$ by using the distributive property to factor out the 3. So if all the addends are tripled, their sum, $m$ , is also tripled.
Looking at the third statement, if $a$ is tripled, the expression becomes $(3a)bc$ , which, by using the associative property, can be written as $3(abc)$ . So if just $a$ is tripled, then $n$ , the product of $a$ , $b$ , and $c$ is also tripled.

Standards

Building On

6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.

15 min

Knowledge Components

All skills for this lesson

No KCs tagged for this lesson

Scaling Two Dimensions

5 min

Teacher Prep

Setup

Groups of 2. 1–2 minutes of quiet work time followed by a partner discussion, then a whole-class discussion.

Narrative

Choose the correct statements (b, c).
Pick numbers to test the validity of statements.
Use algebraic structure to show that the statements are true.

Ask these students to share during the discussion.

Launch

Arrange students in groups of 2. Give students 1–2 minutes of quiet work time followed by time to discuss their chosen statements with their partner. Follow with a whole-class discussion.

Student Task

$m$ , $n$ , $a$ , $b$ , and $c$ all represent positive integers. Consider these two equations:

$\displaystyle m=a+b+c$

$\displaystyle n=abc$

Which of these statements are true? Select all that apply.
1. If $a$ is tripled, $m$ is tripled.
2. If $a$ , $b$ , and $c$ are all tripled, then $m$ is tripled.
3. If $a$ is tripled, $n$ is tripled.
4. If $a$ , $b$ , and $c$ are all tripled, then $n$ is tripled.
Create a true statement of your own about one of the equations.

Sample Response

b, c
Sample response: If $a$ , $b$ , and $c$ are all tripled, then $n$ is 27 times as large. When $a$ , $b$ , and $c$ are tripled, the result is $3a\boldcdot 3b\boldcdot 3c$ , which can be written as $(3\boldcdot 3\boldcdot 3)abc$ or $27abc$ .

Activity Synthesis (Teacher Notes)

If $a$ , $b$ , and $c$ are all tripled, the expression becomes $3a +3b+3c$ , which can be written as $3(a+b+c)$ by using the distributive property to factor out the 3. So if all the addends are tripled, their sum, $m$ , is also tripled.
Looking at the third statement, if $a$ is tripled, the expression becomes $(3a)bc$ , which, by using the associative property, can be written as $3(abc)$ . So if just $a$ is tripled, then $n$ , the product of $a$ , $b$ , and $c$ is also tripled.

Standards

Building On

6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.A·Apply and extend previous understandings of arithmetic to algebraic expressions.

Scaling Two Dimensions

Warm-upTripling Statements5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityA Square Base15 minKCs

ActivityPlaying with Cones15 minKCs

Knowledge Components

Scaling Two Dimensions

Warm-upTripling Statements5 minKCs

Setup

Narrative

Launch

Student Task

Sample Response

ActivityA Square Base15 minKCs

ActivityPlaying with Cones15 minKCs

5 min

15 min

15 min

5 min

15 min

15 min