standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
This task enables students to describe more precisely the characteristics of scaled copies and to refine the meaning of the term. Students observe copies of a line drawing on a grid and notice how the lengths of line segments and the angles formed by them compare to those in the original drawing.
Students engage in MP7 in multiple ways in this task. Identifying distinguishing features of the scaled copies means finding similarities and differences in the shapes. In addition, the fact that corresponding parts increase by the
same
scale factor is a vital structural property of scaled copies.
For the first question, expect students to explain their choices of scaled copies in intuitive, qualitative terms. For the second question, students should begin to distinguish scaled and unscaled copies in more specific and quantifiable ways. If it does not occur to students to look at lengths of segments, suggest they do so.
As students work, monitor for students who notice the following aspects of the figures. Students are not expected to use these mathematical terms at this point, however.
The original drawing of the letter F and its scaled copies have equivalent width-to-height ratios.
We can use a scale factor (or a multiplier) to compare the lengths of different figures and see if they are scaled copies of the original.
The original figure and scaled copies have corresponding angles that have the same measure.
Digital
Launch
Keep students in the same groups. Give them 3–4 minutes of quiet work time, and then 1–2 minutes to share their responses with their partner. Tell students that how they decide whether each of the seven drawings is a scaled copy may be very different than how their partner decides. Encourage students to listen carefully to each other’s approach and to be prepared to share their strategies. Use gestures to elicit from students the words “horizontal” and “vertical” and ask groups to agree internally on common terms to refer to the parts of the F (e.g., “horizontal stems”).
Engagement: Internalize Self Regulation.
Display sentence frames to support small group discussion. For example, “That could/couldn’t be true because…,” “We can agree that…,” and “Is there another way to say/do...?”
Supports accessibility for: Social-emotional skills; Organization; Language
Speaking: Math Language Routine 1 Stronger and Clearer Each Time.
This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners, and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means.
Design Principle(s): Optimize output (for explanation)
How It Happens:
Use this routine to provide students a structured opportunity to refine their explanations for the first question: “Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.” Allow students 2–3 minutes to individually create first draft responses in writing.
Invite students to meet with 2–3 other partners for feedback.
Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Provide the listener with these prompts for feedback that will help their partner strengthen their ideas and clarify their language: “What do you mean when you say….?”, “Can you describe that another way?”, “How do you know that _ is a scaled copy?”, “Could you justify that differently?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss.
Signal for students to move on to their next partner and repeat this structured meeting.
Close the partner conversations and invite students to revise and refine their writing in a second draft.
Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Drawing _ is a scaled copy of the original, and I know this because.…”, “When I look at the lengths, I notice that.…”, and “When I look at the angles, I notice that.…”
Here is an example of a second draft:
“Drawing 7 is a scaled copy of the original, and I know this because it is enlarged evenly in both the horizontal and vertical directions. It does not seem lopsided or stretched differently in one direction. When I look at the length of the top segment, it is 3 times as large as the original one, and the other segments do the same thing. Also, when I look at the angles, I notice that they are all right angles in both the original and scaled copy.”
If time allows, have students compare their first and second drafts. If not, have the students move on by working on the following problems.
Student Facing
Here is an original drawing of the letter F and some other drawings.
[IMG]
Identify
all
the drawings that are scaled copies of the original letter F drawing. Explain how you know.
Examine all the scaled copies more closely, specifically, the lengths of each part of the letter F. How do they compare to the original? What do you notice?
On the grid, draw a different scaled copy of the original letter F.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Keep students in the same groups. Give them 3–4 minutes of quiet work time, and then 1–2 minutes to share their responses with their partner. Tell students that how they decide whether each of the seven drawings is a scaled copy may be very different than how their partner decides. Encourage students to listen carefully to each other’s approach and to be prepared to share their strategies. Use gestures to elicit from students the words “horizontal” and “vertical” and ask groups to agree internally on common terms to refer to the parts of the F (e.g., “horizontal stems”).
Engagement: Internalize Self Regulation.
Display sentence frames to support small group discussion. For example, “That could/couldn’t be true because…,” “We can agree that…,” and “Is there another way to say/do...?”
Supports accessibility for: Social-emotional skills; Organization; Language
Speaking: Math Language Routine 1 Stronger and Clearer Each Time.
This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners, and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means.
Design Principle(s): Optimize output (for explanation)
How It Happens:
Use this routine to provide students a structured opportunity to refine their explanations for the first question: “Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.” Allow students 2–3 minutes to individually create first draft responses in writing.
Invite students to meet with 2–3 other partners for feedback.
Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Provide the listener with these prompts for feedback that will help their partner strengthen their ideas and clarify their language: “What do you mean when you say….?”, “Can you describe that another way?”, “How do you know that _ is a scaled copy?”, “Could you justify that differently?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss.
Signal for students to move on to their next partner and repeat this structured meeting.
Close the partner conversations and invite students to revise and refine their writing in a second draft.
Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Drawing _ is a scaled copy of the original, and I know this because.…”, “When I look at the lengths, I notice that.…”, and “When I look at the angles, I notice that.…”
Here is an example of a second draft:
“Drawing 7 is a scaled copy of the original, and I know this because it is enlarged evenly in both the horizontal and vertical directions. It does not seem lopsided or stretched differently in one direction. When I look at the length of the top segment, it is 3 times as large as the original one, and the other segments do the same thing. Also, when I look at the angles, I notice that they are all right angles in both the original and scaled copy.”
If time allows, have students compare their first and second drafts. If not, have the students move on by working on the following problems.
Student Facing
Here is an original drawing of the letter F and some other drawings.
[IMG]
Identify
all
the drawings that are scaled copies of the original letter F. Explain how you know.
Examine all the scaled copies more closely, specifically the lengths of each part of the letter F. How do they compare to the original? What do you notice?
On the grid, draw a different scaled copy of the original letter F.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may make decisions by “eyeballing” rather than observing side lengths and angles. Encourage them to look for quantifiable evidence and notice lengths and angles.
Some may think vertices must land at intersections of grid lines (e.g., they may say Drawing 4 is not a scaled copy because the endpoints of the shorter horizontal segment are not on grid crossings). Address this during the whole-class discussion, after students have a chance to share their observations about segment lengths.
Activity Synthesis
Display the seven copies of the letter F for all to see. For each copy, ask students to indicate whether they think each one is a scaled copy of the original F. Record and display the results for all to see. For contested drawings, ask 1–2 students to briefly say why they ruled these out.
Discuss the identified scaled and unscaled copies.
What features do the scaled copies have in common? (Be sure to invite students who were thinking along the lines of scale factors and angle measures to share.)
How do the other copies fail to show these features? (Sometimes lengths of sides in the copy use different multipliers for different sides. Sometimes the angles in the copy do not match the angles in the original.)
If there is a misconception that scaled copies must have vertices on intersections of grid lines, use Drawing 1 (or a relevant drawing by a student) to discuss how that is not the case.
Some students may not be familiar with words such as “twice,” “double,” or “triple.” Clarify the meanings by saying “two times as long” or “three times as long.”


aligns: no