What Are Scaled Copies?
10 min
Teacher Prep
Setup
Required Preparation
Make a space for students to place their sticky notes at the end of the Warm-up. For example, hang a sheet of chart paper on a wall near the door.
For the digital version of the activity, acquire devices that can run the applet.
Narrative
This opening task introduces the term scaled copy. It prompts students to observe several copies of a picture, visually distinguish scaled and unscaled copies, and articulate the differences in their own words. Besides allowing students to have a mathematical conversation about properties of figures, it provides an accessible entry into the concept and gives an opportunity to hear the language and ideas that students associate with scaled figures.
Students are likely to have some intuition about the term “to scale,” either from previous work in grade 6 (e.g., scaling a recipe, or scaling a quantity up or down on a double number line) or from outside the classroom. This intuition can help them identify scaled copies. As students apply their previous experience with scaling to analyze the images, they are making sense of problems (MP1).
Expect them to use adjectives such as “stretched,” “squished,” “skewed,” “reduced,” and so on, in imprecise ways. This is fine because students’ intuitive definitions of scaled copies will be refined over the course of the lesson. As students discuss the pictures, note the range of descriptions used. Monitor for students whose descriptions are particularly supportive of the idea that lengths in a scaled copy are found by multiplying the original lengths by the same value. Invite them to share their responses later.
In the digital version of the activity, students use an applet to manipulate copies of the original image. The applet allows students to explore different transformations of the original portrait. This activity works best when each student has access to the applet because students will benefit from seeing the transformations in a dynamic way. If students don't have individual access, displaying the applet for all to see would be helpful during the launch and synthesis.
Launch
Arrange students in groups of 2. Give students 2–3 minutes of quiet think time and a minute to share their response with their partner.
Student Task
Here is a portrait of a student.
-
Look at Portraits A–E. How is each one the same as or different from the original portrait of the student?
- Some of the Portraits A–E are scaled copies of the original portrait. Which ones do you think are scaled copies? Explain your reasoning.
- What do you think “scaled copy” means?
Sample Response
- Sample response:
- Similarities: Pictures A–E are all based on the same original portrait. They all show the same brown-haired boy wearing a blue shirt. They all have the same white background.
- Differences: They each are a different size; some have different shapes. Pictures A, B, and E have been stretched or somehow distorted. Pictures C and D are not stretched or distorted but are each of a different size than the original.
- C and D are scaled copies. Sample reasoning:
- Pictures A, B, and E are not scaled copies because they have changed in shape compared to the original portrait. Portrait A is stretched vertically, so the vertical side is now much longer than the horizontal side. Picture B is stretched out sideways, so the horizontal sides are now longer than the vertical side. Picture E seems to have its upper-left and lower-right corners stretched out in opposite directions. The portrait is no longer a rectangle.
- Picture C is a smaller copy and Picture D is a larger copy of the original, but their shapes remain the same as the shape of the original.
- Sample responses:
- A scaled copy is a copy of a picture that changes in size but does not change in shape.
- A scaled copy is a duplicate of a picture with no parts of it distorted, though it could be larger, smaller, or the same size.
- A scaled copy is a copy of a picture that has been enlarged or reduced in size but nothing else changes.
Activity Synthesis (Teacher Notes)
Select a few students to share their observations. Record and display students’ explanations for the second question. Consider organizing the observations in terms of how certain pictures are or are not distorted. For example, students may say that C and D are scaled copies because each is a larger or smaller version of the picture, but the face (or the sleeve, or the outline of the picture) has not changed in shape. They may say that A, B, and E are not scaled copies because something other than size has changed. If not already mentioned in the discussion, guide students in seeing features of C and D that distinguish them from A, B, and E.
Invite a couple of students to share their working definition of scaled copies. Some of the students’ descriptions may not be completely accurate. That is appropriate for this lesson because the goal is to build on and refine this language over the course of the next few lessons until students have a more precise notion of what it means for a picture or figure to be a scaled copy.
Math Community
After the Warm-up, tell students that today is the start of planning the type of mathematical community they want to be a part of for this school year. The start of this work will take several weeks as the class gets to know one another, reflects on past classroom experiences, and shares their hopes for the year.
Display and read aloud the question “What do you think it should look like and sound like to do math together as a mathematical community?” Give students 2 minutes of quiet think time and then 1–2 minutes to share with a partner. Ask students to record their thoughts on sticky notes and then place the notes on the sheet of chart paper. Thank students for sharing their thoughts and tell them that the sticky notes will be collected into a class chart and used at the start of the next discussion.
After the lesson is complete, review the sticky notes to identify themes. Make a Math Community Chart to display in the classroom. See the blackline master Blank Math Community Chart for one way to set up this chart. Depending on resources and wall space, this may look like a chart paper hung on the wall, a regular sheet of paper to display using a document camera, or a digital version that can be projected. Add the identified themes from the students’ sticky notes to the student section of the “Doing Math” column of the chart.
Standards
- 7.G.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
- 7.G.A.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
10 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
What Are Scaled Copies?
10 min
Teacher Prep
Setup
Required Preparation
Make a space for students to place their sticky notes at the end of the Warm-up. For example, hang a sheet of chart paper on a wall near the door.
For the digital version of the activity, acquire devices that can run the applet.
Narrative
This opening task introduces the term scaled copy. It prompts students to observe several copies of a picture, visually distinguish scaled and unscaled copies, and articulate the differences in their own words. Besides allowing students to have a mathematical conversation about properties of figures, it provides an accessible entry into the concept and gives an opportunity to hear the language and ideas that students associate with scaled figures.
Students are likely to have some intuition about the term “to scale,” either from previous work in grade 6 (e.g., scaling a recipe, or scaling a quantity up or down on a double number line) or from outside the classroom. This intuition can help them identify scaled copies. As students apply their previous experience with scaling to analyze the images, they are making sense of problems (MP1).
Expect them to use adjectives such as “stretched,” “squished,” “skewed,” “reduced,” and so on, in imprecise ways. This is fine because students’ intuitive definitions of scaled copies will be refined over the course of the lesson. As students discuss the pictures, note the range of descriptions used. Monitor for students whose descriptions are particularly supportive of the idea that lengths in a scaled copy are found by multiplying the original lengths by the same value. Invite them to share their responses later.
In the digital version of the activity, students use an applet to manipulate copies of the original image. The applet allows students to explore different transformations of the original portrait. This activity works best when each student has access to the applet because students will benefit from seeing the transformations in a dynamic way. If students don't have individual access, displaying the applet for all to see would be helpful during the launch and synthesis.
Launch
Arrange students in groups of 2. Give students 2–3 minutes of quiet think time and a minute to share their response with their partner.
Student Task
Here is a portrait of a student.
-
Look at Portraits A–E. How is each one the same as or different from the original portrait of the student?
- Some of the Portraits A–E are scaled copies of the original portrait. Which ones do you think are scaled copies? Explain your reasoning.
- What do you think “scaled copy” means?
Sample Response
- Sample response:
- Similarities: Pictures A–E are all based on the same original portrait. They all show the same brown-haired boy wearing a blue shirt. They all have the same white background.
- Differences: They each are a different size; some have different shapes. Pictures A, B, and E have been stretched or somehow distorted. Pictures C and D are not stretched or distorted but are each of a different size than the original.
- C and D are scaled copies. Sample reasoning:
- Pictures A, B, and E are not scaled copies because they have changed in shape compared to the original portrait. Portrait A is stretched vertically, so the vertical side is now much longer than the horizontal side. Picture B is stretched out sideways, so the horizontal sides are now longer than the vertical side. Picture E seems to have its upper-left and lower-right corners stretched out in opposite directions. The portrait is no longer a rectangle.
- Picture C is a smaller copy and Picture D is a larger copy of the original, but their shapes remain the same as the shape of the original.
- Sample responses:
- A scaled copy is a copy of a picture that changes in size but does not change in shape.
- A scaled copy is a duplicate of a picture with no parts of it distorted, though it could be larger, smaller, or the same size.
- A scaled copy is a copy of a picture that has been enlarged or reduced in size but nothing else changes.
Activity Synthesis (Teacher Notes)
Select a few students to share their observations. Record and display students’ explanations for the second question. Consider organizing the observations in terms of how certain pictures are or are not distorted. For example, students may say that C and D are scaled copies because each is a larger or smaller version of the picture, but the face (or the sleeve, or the outline of the picture) has not changed in shape. They may say that A, B, and E are not scaled copies because something other than size has changed. If not already mentioned in the discussion, guide students in seeing features of C and D that distinguish them from A, B, and E.
Invite a couple of students to share their working definition of scaled copies. Some of the students’ descriptions may not be completely accurate. That is appropriate for this lesson because the goal is to build on and refine this language over the course of the next few lessons until students have a more precise notion of what it means for a picture or figure to be a scaled copy.
Math Community
After the Warm-up, tell students that today is the start of planning the type of mathematical community they want to be a part of for this school year. The start of this work will take several weeks as the class gets to know one another, reflects on past classroom experiences, and shares their hopes for the year.
Display and read aloud the question “What do you think it should look like and sound like to do math together as a mathematical community?” Give students 2 minutes of quiet think time and then 1–2 minutes to share with a partner. Ask students to record their thoughts on sticky notes and then place the notes on the sheet of chart paper. Thank students for sharing their thoughts and tell them that the sticky notes will be collected into a class chart and used at the start of the next discussion.
After the lesson is complete, review the sticky notes to identify themes. Make a Math Community Chart to display in the classroom. See the blackline master Blank Math Community Chart for one way to set up this chart. Depending on resources and wall space, this may look like a chart paper hung on the wall, a regular sheet of paper to display using a document camera, or a digital version that can be projected. Add the identified themes from the students’ sticky notes to the student section of the “Doing Math” column of the chart.
Standards
- 7.G.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
- 7.G.A.1·Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.