The purpose of this activity is to give students more practice drawing scaled copies of simple geometric figures on a grid. When trying to scale non-horizontal and non-vertical segments, students may think of using tracing paper or a ruler to measure lengths and a protractor to measure angles. Make sure they have a chance to see how the structure of the grid can be useful for scaling the lengths of non-vertical and non-horizontal segments (MP7).

To create scaled copies, students need to attend to all parts of the original figure, or else the copy will not be scaled correctly. Use of the grid for scaling non-horizontal and non-vertical segments is a good example of using tools strategically (MP5).

As students work, monitor for students who find a way to scale segment lengths properly but neglect to consider the size of corresponding angles (especially in making a copy of Figure B or Figure D). 

In the digital version of the activity, students use an applet to draw scaled copies of figures on a grid. The applet allows students to add, remove, adjust, and label points and line segments. The digital version may help students draw quickly and accurately so they can focus more on the mathematical analysis of scale factor.

Invite students to share their strategies of how they used the grid (or other tools) to make sure their drawings were scaled copies. Consider asking questions like:

• How did you know how long to make each side in your scaled copy?
• How did you know how big to make each angle in your scaled copy?
• If you made a mistake while drawing your scaled copy, how could you tell?

Model, prompt, and listen for the language students are using to distinguish between scaled and not scaled figures. Emphasize the usefulness of the grid in drawing and checking right angles, and for drawing and checking lengths of segments. All correct answers will be the same size and shape, but they could be drawn in different positions on the grid.

The purpose of this activity is to contrast the effects of multiplying side lengths versus adding to side lengths when creating copies of a polygon. To find the corresponding side lengths on a scaled copy, the side lengths of a figure are all multiplied (or divided) by the same number. However, students often mistakenly think that adding or subtracting the same number to all the side lengths will also create a scaled copy. When students recognize that there is a multiplicative relationship between the side lengths rather than an additive one, they are looking for and making use of structure (MP7).

Monitor for students who:

• Notice that Diego's copy is no longer a polygon while Jada's still is.
• Notice that all the corresponding angles have equal measures (either 90 or 270 degrees).
• Notice that the relationships between side lengths in Diego's copy have changed (for example, Side 1 is no longer twice as long as Side 2) while in Jada's copy these relationships stayed the same as in the original.
• Describe Jada's copy as having all side lengths divided by 3.
• Describe Jada's copy as having all side lengths a third as long as the original lengths.
• Describe Jada's copy as having a scale factor of .

Invite previously-selected students to share their answers and reasoning. Sequence their explanations from most general to most technical. Before moving to the next activity, consider asking questions like these:

• “What is the scale factor used to create Jada’s drawing? What about for Diego’s drawing?” ( for Jada’s. There isn't one for Diego's, because it is not a scaled copy.)
• “What can you say about the corresponding angles in Jada and Diego’s drawings?” (They are all equal, even though one is a scaled copy and one is not.)
• “Subtraction of side lengths does not (usually) produce scaled copies. Do you think addition would work?” (Answers vary.)

Note: There are rare cases when adding or subtracting the same length from each side of a polygon (and keeping the angles the same) will produce a scaled copy, namely if all side lengths are the same. If not mentioned by students, it is not important to discuss this at this point.

This activity reinforces the idea that scaling involves multiplication. In the previous activity, students saw that subtracting the same value from all side lengths of a polygon did not produce a (smaller) scaled copy. This activity makes the case that adding the same value to all lengths also does not produce a (larger) scaled copy. As students explain why Andre’s method would not create a scaled copy, they are constructing arguments (MP3).

This activity gives students a chance to draw a scaled copy without a grid and to use paper as a measuring tool. To create a copy using a scale factor of 2, students need to mark the length of each original segment and transfer it twice onto their drawing surface, reinforcing—in a tactile way—the meaning of scale factor. The angles in the polygon are right angles (and a 270 degree angle in one case) and can be made using the corner of an index card.

Monitor for students who use these strategies to create the new drawing:

• Add 4 units to all the sides and draw a shape that is not a scaled copy (either by changing the angles or creating a figure that isn’t closed).
• Sketch a figure that looks like a scaled copy and label it with side lengths that are 4 units longer, without attending to whether the numbers match the actual lengths.
• Show that adding 4 units would not result in a consistent scale factor between pairs of corresponding side lengths.
• Use a ruler to measure the side lengths of the original shape and multiply these measurements by 2.
• Use an index card to measure the side lengths of the original shape and draw lengths that are twice as long.
• Use an index card to measure and recreate the right angles in the figure.

Plan to have students present in this order to support moving them from additive to multiplicative reasoning about the side lengths.

The routine of Anticipate, Monitor, Select, Sequence, Connect (5 Practices) requires a balance of planning and flexibility. The anticipated approaches might not surface in every class, and there may be reason to change the order in which strategies are presented. While monitoring, keep in mind the learning goal and adjust the order to ensure all students have access to the first idea presented (whether that be a common misconception or a different approach).
 

The purpose of the activity is to explicitly call out a potential misunderstanding about how scale factors work, emphasizing that scale factors work by multiplying existing side lengths by a common factor, rather than by adding a common length to each.  

Invite previously selected students to share their explanations or illustrations that adding 4 units to the length of each segment would not work (for example, the copy is no longer a polygon, or the copy has angles that are different from those in the original figure). Then, select a couple of other students to show their scaled copies and share how they created the copies. Sequence the discussion of the approaches in the order listed in the Activity Narrative. If possible, record and display the students' work for all to see.

As students share their approaches, consider asking:

• “What scale factor did you use to create your copy? Why?”
• “How did you measure the side lengths for the copy?”
• “How did you measure the angles for the copy?”

After the pre-selected students have finished sharing with the whole class, connect the different responses to the learning goals by asking questions such as:

• “Did this approach create a scaled copy? Why or why not?”
• “How did the scale factor show up in each method?”
• “What worked well in _____’s approach? What did not work well?”
• “What role does multiplication play in each approach?”