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Launch

Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the diagram, without revealing the question.

For the first read, read the problem aloud and then ask, “What is this situation about?” (designing a stained-glass window). Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured (the length and width of the window, the cost per square foot of colored glass, the cost per square foot of clear glass, the cost of the frame).
After the third read, reveal the question: “Do they have enough money to cover the cost of making the window?” and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read (decompose the design into parts of circles, determine the radius of the circular pieces).

Arrange students in groups of 2. Give students 8–10 minutes of partner work time followed by small-group and whole-class discussions.

As students work, prompt them to recognize any assumptions that they are making and to make them explicit.

Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, begin by providing students with just one of the six rectangular pieces of the window to analyze and make sense of the shapes.
Supports accessibility for: Organization, Attention

Activity

None

The students in art class are designing a stained-glass window to hang in the school entryway. The window will be 3 feet tall and 4 feet wide. Here is their design.

A stained glass window composed of six rectangular panels of equal size.

They have raised \$100 for the project.
The colored glass costs \$5 per square foot.
The clear glass costs \$2 per square foot.
The material they need to join the pieces of glass together costs 10 cents per foot.
The frame around the window costs \$4 per foot.

Do they have enough money to cover the cost of making the window?

Student Response

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Building on Student Thinking

Because there are multiple steps in solving this problem, some students may benefit from having their calculations checked along the way so that one early error does not impact the final result.

Some students may struggle to find the diameter or radius lengths. Encourage these students to cut one individual panel, separate the clear glass from the colored glass, and rearrange the figures to see how to determine the length of the diameter and radius.

11.2

Activity

Optional

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR3: Critique, Correct, Clarify

Materials

None

Activity Narrative

In this activity, students calculate the cost of a larger version of the stained-glass window from the previous activity. The window is scaled by a factor of 3. Students recognize that the lengths of the frame and seams will increase by a factor of 3, while the area of the glass will increase by a factor of .

If students observe that the material for the seams and the frame has width and that the scale factor would need to be applied to this measurement, ask them if they can make a simplifying assumption. The width of the seams is never specified or taken into account in the calculations in the previous activity so it is appropriate to continue to put this to the side, as part of the modeling process.

This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.

Launch

Keep students in the same groups. Give students 2–3 minutes of quiet work time followed by partner work time.

As students work, monitor for how students make use of the given scale factor. Select students who solve the problem in different ways to share during the whole-class discussion. If there is a student who quickly assumes they could just multiply their cost from the previous activity by 3, but later realizes why they could not do that, select them to share their reasoning.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “Let’s try . . . .”, “We are trying to . . . .”, “We already know . . . .”, and “We will need to know . . . .”
Supports accessibility for: Language, Social-Emotional Functioning

Activity

None

A local community member sees the school’s stained-glass window and really likes the design. They ask the students to create a larger copy of the window using a scale factor of 3.

Would \$450 be enough to buy the materials for the larger window? Explain or show your reasoning.

Student Response

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Activity Synthesis

The purpose of this discussion is for students to reiterate that when all the lengths in a figure are scaled by a scale factor, the area changes by .

Ask selected students to share their reasoning. If there are students who still think that \$450 is enough money, ask them to share their reasoning. Discuss why you cannot just multiply the price of the original design by 3 to find the price of the scaled stained-glass window.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response by correcting errors, clarifying meaning, and adding details.

Display this first draft:
“I learned that to scale something, you multiply it by the scale factor. Since they want a window that is 3 times as big, I multiplied the cost of the small window by 3. Because , then \$450 dollars is more than enough.”

Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

11.3

Activity

Optional

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Blank paper Compasses Geometry toolkits

Activity Narrative

In this activity students create their own stained-glass design for a given amount of money. The activity is purposefully left open to allow students to either tweak the previous design or create something completely new. As students apply what they have learned about perimeter and area, choose tools and strategies, and come up with a design that fits the given constraints, they are modeling with mathematics (MP4).

Launch

Keep students in the same groups. Encourage students to include whole or partial circles in their designs.

Activity

None

Draw a stained-glass window design that could be made for less than \$450. Show your thinking. Organize your work so it can be followed by others.

Student Response

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Building on Student Thinking

Some students may think that they need to create a new design and, therefore, struggle to get started. Point these students to the designs in previous activities and ask how they could modify these designs to meet the cost requirement.

Activity Synthesis

Display students’ designs for all to see and ask students to explain how they knew their design met the cost requirement. Allow other students to ask questions of the student who is sharing their design. Consider displaying sentence frames to support students in asking questions of the presenters, such as:

“Can you say more about . . . ?”
“Why did you . . . ?”
“How do you know . . . ?”

MLR8 Discussion Supports. At the appropriate time, give groups 2–3 minutes to plan what they will say if they are chosen to present to the class. “Practice what you will say when you share your design with the class. Talk about what is important to say, and decide who will share each part.” During group presentations, invite the student who is not speaking to follow along and point to the corresponding parts of the display.
Advances: Speaking, Conversing, Representing