Lesson 3

This warm-up prompts students to use what they know about numbers and multiplication to reason about decimal computations. The problems are designed to result in an answer very close to the given choices, so students must be more precise in their reasoning than simply rounding and calculating. 

Whereas a number talk typically presents a numerical expression and asks students to explain strategies for evaluating it, this activity asks a slightly different question because students don't necessarily need to evaluate the expression. Rather, they are asked to judge whether the expression is greater than or less than a given value. Although this activity is not quite the same thing as a number talk, the discussion might sound quite similar.

Display the problems for all to see. Give students 2 minutes of quiet think time. Tell students they may not have to calculate, but could instead reason using what they know about the numbers and operation in each problem. Ask students to give a signal when they have an answer and a strategy for every problem.

Students may attempt to solve each problem instead of reasoning about the numbers and operations. If a student is calculating an exact solution to each problem, ask them to look closely at the characteristics of the numbers and how an operation would affect those numbers.

Discuss each problem one at a time with this structure:

• Ask students to indicate which option they agree with.
• If everyone agrees on one answer, ask a few students to share their reasoning, recording it for all to see.
• If there is disagreement on an answer, ask students with differing answers to explain their reasoning and come to an agreement on an answer.

Students continue to work with scaled copies of simple geometric figures, this time 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-horizonal segments.

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 and D). 

Some students may think that Figure C cannot be scaled by a factor of \(\frac12\) because some vertices will not land on intersections of grid lines. Clarify that the grid helps us see lengths in whole units but segments we draw on them are not limited to whole units in length.

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 the relationships between side lengths in Diego's copy have changed (e.g., Side 1 is twice as long as Side 2 in the original but is not twice as long as Side 2 in the copy.) while in Jada's copy they have not
• notice that all the corresponding angles have equal measures (i.e., 90 or 270 degrees)
• 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 their original lengths
• describe Jada's copy as having a scale factor of \(\frac13\)

Give students 2–3 minutes of quiet think time, and then 2 minutes to share their thinking with a partner.  See MLR 3 (Clarify, Critique, Correct) and use the strategy "Critique a Partial or Flawed Explanation".

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? (\(\frac13\) 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.

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, reinforcing the idea that scaling involves multiplication.

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.

Some students may struggle to figure out how to use an index card or a sheet of paper to measure lengths. Before demonstrating, encourage them to think about how a length in the given polygon could be copied onto an index card and used as an increment for measuring. If needed, show how to mark the 4-unit length along the edge of a card and to use the mark to determine the needed lengths for the copy. 

Have students read the task statement and check that they understand which side of the polygon Andre would like to be 8 units long on his drawing. Provide access to index cards, so that students can use it as a measuring tool. Consider not explicitly directing students as to its use to give them a chance to use tools strategically (MP5). Give students 5–6 minutes of quiet work time, and then 2 minutes to share their work with a partner.

Some students might not be convinced that making each segment 4 units longer will not work. To show that adding 4 units would work, they might simply redraw the polygon and write side lengths that are 4 units longer, regardless of whether the numbers match the actual lengths. Urge them to check the side lengths by measuring. Tell them (or show, if needed) how the 4-unit length in Jada’s drawing could be used as a measuring unit and added to all sides.

Other students might add 4 units to all sides and manage to make a polygon but changing the angles along the way. If students do so to make the case that the copy will not be scaled, consider sharing their illustrations with the class, as these can help to counter the idea that “scaling involves adding.” If, however, students do this to show that adding 4 units all around does work, address the misconception. Ask them to recall the size of corresponding angles in scaled copies, or remind them that angles in a scaled copy are the same size as their counterparts in the original figure.

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

Invite a couple of students to share their explanations or illustrations that adding 4 units to the length of each segment would not work (e.g. the copy is no longer a polygon, or the copy has angles that are different than in the original figure). Then, select a couple of other students to show their scaled copies and share how they created the copies. Consider asking:

• What scale factor did you use to create your copy? Why?
• How did you use an index card (or a sheet of paper) to measure the lengths for the copy?
• How did you measure the angles for the copy?