Stained-Glass Windows
25 min
Teacher Prep
Setup
Narrative
In this activity students apply what they have learned about circles to solve a multi-step problem (MP1). Students find the area and perimeter of geometric figures whose boundaries are line segments and fractions of circles and use that information to calculate the cost of a project. The shape of the regions in the stained-glass window are left unspecified on purpose to give students an opportunity to engage in an important step of the mathematical modeling cycle—making simplifying assumptions (MP4). Assuming that the curves in the design are arcs of a circle is reasonable and expedient.
Another opportunity for mathematical modeling in this activity is to discuss if it is reasonable that a person has to pay only for the glass used in the final window and not for possible scraps of glass left over from cutting out the shapes. In reality, if they had to buy the glass at a store, the glass would likely come in square or rectangular sheets, and they would need to buy more than they were going to use. If these issues come up, encourage students to keep note of the decisions they are making and to recognize that different choices would lead to different results.
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.
Supports accessibility for: Organization, Attention
Student Task
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.
- 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?
Sample Response
Yes, they need about $93. Sample reasoning:
Assume that the students have to pay only for the glass they use and not for the scraps they cut away.
First, we need to find the area of the clear glass and the area of the colored glass. The entire window is 3 ft by 4 ft and has an area of 12 ft2. There are 6 smaller rectangles. Each of these rectangles has a total of 2 full circles of clear glass, because 24=2, and 22+44=2. In the entire window, there are 12 complete circles of clear glass. Each circle has a diameter of 1 ft, a radius of 21 ft, and an area of 41π ft2. The area of the clear glass is 12⋅41π, or approximately 9.42 ft2. That means the area of colored glass is approximately 12−9.42, or 2.58 ft2.
Next, we need to find the total length of the seams between the pieces of glass and the total length of the frame around the window. The 12 circles each have a circumference of 1π ft, which makes 12π ft or about 37.68 ft of curved seams. There are also 11 ft of straight seams, because 4+4+3=11. All together there are about 48.68 ft of seams. Finally, there is 14 ft of frame all the way around the window.
Next, we can calculate how much each material will cost. The clear glass will cost 9.42⋅2, or $18.84. The colored glass will cost 2.58⋅5, or $12.90. The seams will cost 48.68⋅0.10, or $4.87. The frame will cost 14⋅4, or $56. The total cost of all the materials is about $93, because 18.84+12.90+4.87+56.00=92.61. If these assumptions are accurate, they have just enough money to buy the materials, but if they need to pay for the scraps they cut off or if accidentally they break pieces as they go, they don’t have a lot of extra money.
Activity Synthesis (Teacher Notes)
The purpose of this discussion is to highlight assumptions that students made while solving the problem.
As groups complete the activity, combine groups of 2 to make groups of 4. If possible, combine groups who solved the problem in different ways. Display the following questions for all to see and ask the groups to discuss:
- "Did you get the same answer? Why or why not?"
- "Did you use the same strategy? What was the same or different in your work?"
- "Did you make any assumptions as you worked on the problem?"
For the whole-class discussion, invite groups to share the similarities and differences they noticed between their small groups. As students share, revoice comparison statements and assumptions that students state. Ask for details and examples as needed to help clarify students’ reasoning.
After each group shares, consider asking the class to indicate if they had any of the same conversations in their own group so as to not have repetitive explanations. Every group does not need to share if the same conversation was had.
If not mentioned in students’ explanations, highlight the assumptions that the shapes are parts of circles and that the total cost takes into account only the exact area and lengths shown in the figure.
Anticipated Misconceptions
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.
Standards
- 7.EE.3·Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. <em>For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.</em>
- 7.EE.B.3·Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
- 7.G.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
- 7.G.B.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Knowledge Components
All skills for this lesson
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Stained-Glass Windows
25 min
Teacher Prep
Setup
Narrative
In this activity students apply what they have learned about circles to solve a multi-step problem (MP1). Students find the area and perimeter of geometric figures whose boundaries are line segments and fractions of circles and use that information to calculate the cost of a project. The shape of the regions in the stained-glass window are left unspecified on purpose to give students an opportunity to engage in an important step of the mathematical modeling cycle—making simplifying assumptions (MP4). Assuming that the curves in the design are arcs of a circle is reasonable and expedient.
Another opportunity for mathematical modeling in this activity is to discuss if it is reasonable that a person has to pay only for the glass used in the final window and not for possible scraps of glass left over from cutting out the shapes. In reality, if they had to buy the glass at a store, the glass would likely come in square or rectangular sheets, and they would need to buy more than they were going to use. If these issues come up, encourage students to keep note of the decisions they are making and to recognize that different choices would lead to different results.
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.
Supports accessibility for: Organization, Attention
Student Task
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.
- 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?
Sample Response
Yes, they need about $93. Sample reasoning:
Assume that the students have to pay only for the glass they use and not for the scraps they cut away.
First, we need to find the area of the clear glass and the area of the colored glass. The entire window is 3 ft by 4 ft and has an area of 12 ft2. There are 6 smaller rectangles. Each of these rectangles has a total of 2 full circles of clear glass, because 24=2, and 22+44=2. In the entire window, there are 12 complete circles of clear glass. Each circle has a diameter of 1 ft, a radius of 21 ft, and an area of 41π ft2. The area of the clear glass is 12⋅41π, or approximately 9.42 ft2. That means the area of colored glass is approximately 12−9.42, or 2.58 ft2.
Next, we need to find the total length of the seams between the pieces of glass and the total length of the frame around the window. The 12 circles each have a circumference of 1π ft, which makes 12π ft or about 37.68 ft of curved seams. There are also 11 ft of straight seams, because 4+4+3=11. All together there are about 48.68 ft of seams. Finally, there is 14 ft of frame all the way around the window.
Next, we can calculate how much each material will cost. The clear glass will cost 9.42⋅2, or $18.84. The colored glass will cost 2.58⋅5, or $12.90. The seams will cost 48.68⋅0.10, or $4.87. The frame will cost 14⋅4, or $56. The total cost of all the materials is about $93, because 18.84+12.90+4.87+56.00=92.61. If these assumptions are accurate, they have just enough money to buy the materials, but if they need to pay for the scraps they cut off or if accidentally they break pieces as they go, they don’t have a lot of extra money.
Activity Synthesis (Teacher Notes)
The purpose of this discussion is to highlight assumptions that students made while solving the problem.
As groups complete the activity, combine groups of 2 to make groups of 4. If possible, combine groups who solved the problem in different ways. Display the following questions for all to see and ask the groups to discuss:
- "Did you get the same answer? Why or why not?"
- "Did you use the same strategy? What was the same or different in your work?"
- "Did you make any assumptions as you worked on the problem?"
For the whole-class discussion, invite groups to share the similarities and differences they noticed between their small groups. As students share, revoice comparison statements and assumptions that students state. Ask for details and examples as needed to help clarify students’ reasoning.
After each group shares, consider asking the class to indicate if they had any of the same conversations in their own group so as to not have repetitive explanations. Every group does not need to share if the same conversation was had.
If not mentioned in students’ explanations, highlight the assumptions that the shapes are parts of circles and that the total cost takes into account only the exact area and lengths shown in the figure.
Anticipated Misconceptions
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.
Standards
- 7.EE.3·Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. <em>For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.</em>
- 7.EE.B.3·Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
- 7.G.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
- 7.G.B.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.