Lesson 2

This number talk allows students to review multiplication strategies, refreshing the idea that multiplying by a unit fraction is the same as dividing by its whole number reciprocal. It encourages students to use the structure of base ten numbers and the properties of operations to find the product of two whole numbers (MP7). For example, a student might find \(72 \boldcdot \frac{1}{9}\) (or \(72 \div 9\)) and then shift the decimal one place to the right in order to evaluate \((7.2) \boldcdot \frac{1}{9}\). Each problem was chosen to elicit different approaches, so as students share theirs, ask how the factors in each problem impacted their strategies.

Before students begin, consider establishing a small, discreet hand signal (such as a thumbs-up) students can display to indicate they have an answer that they can support by reasoning. Discreet signaling is a quick way for teachers to gather feedback about timing. It also keeps students from being distracted or rushed by raised hands around the class.

Display one problem at a time. Give students up to 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Follow with a brief whole-class discussion.

Ask students to share their strategies for each problem. Record and display their explanations for all to see. If students express strategies in terms of division, ask if that strategy would work for any multiplication problem involving fractions. Highlight that these problems only involve unit fractions and division by the denominator is a strategy that works when multiplying by a unit fraction.

To involve more students in the conversation, consider asking:

• Who can restate ___’s reasoning in a different way?
• Did anyone solve the problem the same way but would explain it differently?
• Did anyone solve the problem in a different way?
• Does anyone want to add on to _____’s strategy?
• Do you agree or disagree? Why?

In this activity, students continue to practice identifying corresponding parts of scaled copies. By organizing corresponding lengths in a table, students see that there is a single factor that relates each length in the original triangle to its corresponding length in a copy (MP8). They learn that this number is called a scale factor.

As students work on the first question, listen to how they reason about which triangles are scaled copies. Identify groups who use side lengths and angles as the basis for deciding. (Students are not expected to reason formally yet, but should begin to look to lengths and angles for clues.)

As students identify corresponding sides and their measures in the second and third questions, look out for confusion about corresponding parts. Notice how students decide which sides of the right triangles correspond.

If students still have access to tracing paper, monitor for students who use this tool strategically (MP5).

Due to students’ prior work with percent increase and decrease, they may want to describe the scale factor as a percent change. For example, the side lengths of Triangle D are \(50\%\) the side lengths of Triangle O. Make sure that these students understand that a \(50\%\) decrease in length corresponds to a scale factor of \(\frac{1}{2}\). In a later lesson, students will identify the effect a scale factor has on the area of a figure.

Arrange students into groups of 4. Assign each student one of the following pairs of triangles in the first question.

• A and E
• B and F
• C and G
• D and H

Give students 2 minutes of quiet think time to determine if their assigned triangles are scaled copies of the original triangle. Give another 2–3 minutes to discuss their responses and complete the first question in groups.

Discuss briefly as a class which triangles are scaled copies and select a couple of groups who reasoned in terms of lengths and angles to explain their reasoning. Some guiding questions:

• What information did you use to tell scaled copies from those that are not?
• How were you able to tell right away that some figures are not scaled copies?

Give students quiet work time to complete the rest of the task after the class recognizes that A, C, F, and H are not scaled copies.

Students may think that Triangle F is a scaled copy because just like the 3-4-5 triangle, the sides are also three consecutive whole numbers. Point out that corresponding angles are not equal.

Display the image of all triangles and invite a couple of students to share how they knew which sides of the triangles correspond. Then, display a completed table in the third question for all to see. Ask each group to present its observations about one triangle and how the triangle has been scaled from the original. Encourage the use of “corresponding” in their explanations. As students present, record or illustrate their reasoning on the table, e.g., by drawing arrows between rows and annotating with the operation students are describing, as shown here.

Use the language that students use to describe the side lengths and the numerical relationships in the table to guide students toward scale factor. For example: “You explained that the lengths in Triangle F are all twice those in the original triangle, so we can write those as “2 times” the original numbers. Lengths in Triangle A are half of those in the original; we can write “\(\frac12\) times” the original numbers. We call those multipliers—the 2 and the \(\frac12\)-scale factors. We say that scaling Triangle O by a scale factor of 2 produces Triangle F, and that scaling Triangle O by \(\frac12\) produces Triangle A.”

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. 

This activity is optional. Use this activity if your students need more practice with the ideas from Which Operations? (Part 1).

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?